Seminars
By Year
Title: Eigenfunctions Seminar: TBA
Speaker: Fred Diamond (King's College London, UK)
Date: Fri, 24 Jan 2025
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Title: Eigenfunctions Seminar: Adjoint L-value and the Tate conjecture
Speaker: Haruzo Hida (University of California at Los Angeles, USA)
Date: Mon, 23 Dec 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
We sketch a strategy to prove the Tate conjecture on algebraic cycles for a good amount of quaternionic Shimura varieties. A key point is a twisted adjoint L-value formula relative to each quaternion algebra $D/F$ for a totally real field $F$ and its scalar extension $B=D\otimes_F E$ for a totally real quadratic extension $E_{/F}$. The theta base-change lift $\mathcal{F}$ of a Hilbert modular form $f$ to $B^\times$ has period integral over the Shimura subvariety $Sh_D\subset Sh_B$ given by $L(1,Ad(f)\otimes(\frac{E/F}{}))\ne0$; so, $Sh_D$ gives rise to a non-trivial Tate cycle in $H^{2r}(Sh_B,\mathbb{Q}_l(r))$ for $r=\dim Sh_D=\dim Sh_B/2$.
Title: APRG Seminar: The Geometry of Distances: A Tale of Points, Patterns, and Problems: II
Speaker: Malabika Pramanik (University of British Columbia, Vancouver, Canada)
Date: Wed, 18 Dec 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
For millennia, the Pythagorean theorem has provided a simple yet profound formula for computing distances between points. Remarkably, this timeless concept continues to captivate mathematicians, inspiring new discoveries and posing challenging open questions.
A central theme in this journey involves distance sets: collections of distances determined by points in a set. How large can a distance set be if the set of points it comes from is “large”? How does the structure of a set shape the distances it contains? These questions bridge classical geometry and modern mathematics, with applications reaching far beyond disciplinary boundaries.
This two-part series explores the surprising depth of these problems. The first talk introduces key ideas and landmark results in a way accessible to a general audience, requiring no advanced mathematical background. The second talk delves into analytical perspectives, uncovering deeper mathematical mysteries for those eager to explore the intricate beauty of this field.
Title: APRG Seminar: The Geometry of Distances: A Tale of Points, Patterns, and Problems: I
Speaker: Malabika Pramanik (University of British Columbia, Vancouver, Canada)
Date: Tue, 17 Dec 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
For millennia, the Pythagorean theorem has provided a simple yet profound formula for computing distances between points. Remarkably, this timeless concept continues to captivate mathematicians, inspiring new discoveries and posing challenging open questions.
A central theme in this journey involves distance sets: collections of distances determined by points in a set. How large can a distance set be if the set of points it comes from is “large”? How does the structure of a set shape the distances it contains? These questions bridge classical geometry and modern mathematics, with applications reaching far beyond disciplinary boundaries.
This two-part series explores the surprising depth of these problems. The first talk introduces key ideas and landmark results in a way accessible to a general audience, requiring no advanced mathematical background. The second talk delves into analytical perspectives, uncovering deeper mathematical mysteries for those eager to explore the intricate beauty of this field.
Title: Number Theory Seminar: Integral period relations for $\mathrm{GL}_n$, $n$ odd
Speaker: Jacques Tilouine (LAGA, Universite Paris 13, Paris, France)
Date: Thu, 05 Dec 2024
Time: 2 pm
Venue: LH-1
In a joint work in progress with B. Balasubramanyam (IISER Pune),
we generalize in two different directions, for higher odd dimensions,
an earlier work of T.-Urban on integral period relations for $\mathrm{GL}_2$.
Title: Geometry & Topology Seminar: Maps between Boundaries of Relatively Hyperbolic Groups
Speaker: Abhijit Pal (IIT Kanpur)
Date: Mon, 02 Dec 2024
Time: 04:00 pm
Venue: LH-1, Department of Mathematics
Quasi-isometric classification of groups is one of the central problems in Geometric group theory. In this talk, we will be focusing on quasi-isometries obtained from a homeomorphism between boundaries of hyperbolic groups and relatively hyperbolic groups. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalence, then those two hyperbolic groups are quasi-isometric. In this talk, we will extend Paulin’s results to relatively hyperbolic groups by introducing the notion of ‘relative quasi-Mobius maps’ between their Bowditch boundaries.
Title: Algebra & Combinatorics Seminar: A bijection between two branching models
Speaker: V. Sathish Kumar (Harish-Chandra Research Institute, Prayagraj)
Date: Fri, 29 Nov 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We prove a bijection between the branching models of Kwon and Sundaram, conjectured by Lenart–Lecouvey. To do so, we use a symmetry of the Littlewood–Richardson coefficients in terms of the hive model. Along the way, we introduce a new branching rule with flagged hives. This talk is based on a joint work with Dr. Jacinta Torres.
Title: Geometry & Topology Seminar: Tangent cones of Kahler-Ricci flow singularity models
Speaker: Maximilien Hallgren (Rutgers University)
Date: Mon, 25 Nov 2024
Time: 10:00 am
Venue: online, MS Teams
By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.
Title: Number Theory Seminar: p-adic L-functions for U(2,1) X U(1,1)
Speaker: Ming-Lun Hsieh (National Taiwan University, Taiwan)
Date: Fri, 22 Nov 2024
Time: 11 am
Venue: LH-1
In this talk, I plan to explain a construction of a five-variable $p$-adic $L$-function associated with Hida families on $U(2, 1) \times U(1, 1)$ and explain a possible connection with the p-adic Ichino-Ikeda formula.
This construction consists of two steps:
(i) the p-adic interpolation of the GGP period integrals for Hida families on $U(2, 1) \times U(1, 1)$;
(ii) the explicit evaluation of GGP period integrals via the the Ichino-Ikeda formula.
I will primarily focus on some key ingredients in the step (i), in particular, the use of $p$-integrality of the Taylor expansion of CM theta functions due to K. Bannai and S. Kobayashi. This is a joint work with M. Harris and S. Yamana.
Title: APRG Seminar: Hardy–Littlewood maximal operators on trees and Riemannian manifolds
Speaker: Stefano Meda (Università degli Studi di Milano Bicocca, Italy)
Date: Wed, 20 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
We consider the centred and the uncentred Hardy–Littlewood maximal operators on metric measure spaces with exponential volume growth and “locally bounded geometry”. We discuss the problem of finding the optimal range of $p$’s such that either the centred or the uncentred HL maximal operator is bounded on $L^p$.
The prototypes of the metric measure spaces we consider are the symmetric spaces of the noncompact type and the homogeneous trees, where sharp $L^p$ estimates on the HL operators are available in the literature.
We shall show that interesting phenomena arise when homogeneous trees are replaced by non-homegeoeus ones, and noncompact symmetric spaces by Riemannian manifolds with pinched negative sectional curvature.
Title: MS Thesis colloquium: λ-Decomposition of Hochschild Homology and Cyclic Homology
Speaker: Sahil Chavda Ramjibhai
Date: Wed, 20 Nov 2024
Time: 2 pm
Venue: LH-3, Department of Mathematics
Alain Connes developed Cyclic homology as a non-commutative analogue of de Rham cohomology, which can be viewed as a Lie analogue of algebraic $K$-theory. In this talk, we will begin by defining chain complexes, followed by an introduction to Hochschild homology and various perspectives on Cyclic homology. A key focus will be on the theorem stating that if $k$ contains $\mathbb{Q}$, the $k$-algebra $A$ is commutative, and the $A$-bimodule $M$ is symmetric, then the Hochschild complex $C_*(A, M)$ splits into a sum of sub-complexes $C_*^{(i)}$, $i \geq 0$, and the bicomplex $\mathcal{B}(A)$ similarly decomposes into sub-complexes $\mathcal{B}(A)^{(i)}$, $i \geq 0$. We will also discuss the Eulerian idempotents, the Eulerian decomposition of $S_n$, and Cyclic descents, which facilitate a $\lambda$-decomposition of Hochschild and Cyclic homology.
Title: APRG Seminar: Slides Quantum resonances and scattering poles
Speaker: Joachim Hilgert (Universität Paderborn, Germany)
Date: Wed, 13 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
For negatively curved symmetric spaces such as real hyperbolic spaces scattering matrices are defined via the standard intertwining operators for the spherical principal representations of the isometry group. They depend meromorphically on spectral parameters and it is known that their poles are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. In this talk I will explain these facts and discuss extensions to locally symmetric spaces of negative curvature.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Slides Exceptional directions in hyperbolic FPP
Speaker: Mahan Mj (TIFR Mumbai)
Date: Fri, 08 Nov 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
First passage percolation (FPP) gives a well-known model of random geometry on a fixed background infinite graph. When we specialize to Cayley graphs of Gromov-hyperbolic groups $G$, random trees $T$ emerge naturally. The first part of the talk will dwell on setting up hyperbolic FPP and outlining its basic properties. This will have a probabilistic emphasis.
In the second part, we will specialize to the study of exceptional directions, i.e. distinct random geodesics in $T$ that converge asymptotically to the same point in the boundary $\partial G$ of $G$. This will have a geometric group theoretic emphasis (joint work with Riddhipratim Basu).
If time permits, we will describe how to reconstruct the bulk random hyperbolic geometry from the boundary. Random trees leave a trace on the boundary $\partial G$ of $G$ in the form of an evolving random partition. It turns out that there is an inverse construction, where the random metric can be reconstructed (up to bounded errors) from the data of an evolving random partition on the boundary.
Title: APRG Seminar: Slides How to represent a function in a quantum computer?
Speaker: Christoph Thiele (Universität Bonn, Germany)
Date: Wed, 06 Nov 2024
Time: 3 pm
Venue: Microsoft Teams (online)
Recently it was observed that an important algorithm in quantum signal processing (QSP) is the same as what is called elsewhere the nonlinear Fourier transform (NLFT). This has led to a new provably stable algorithm to compute the tuning parameters for this quantum algorithm. We will discuss QSP, NLFT, and the algorithm. This is joint work with Michel Alexis and Gevorg Mnatsakanyan and in part also with Lin Lin and Jiasu Wang.
The video of this talk is available on the IISc Math Department channel.
Title: Bangalore Probability Seminar: Probabilistic Solution of a Conjecture on Graphical Partitions
Speaker: Somabha Mukherjee (National University of Singapore)
Date: Mon, 04 Nov 2024
Time: 2 pm
Venue: LH-3, Department of Mathematics, Indian Institute of Science
An integer partition is called graphical if it is the degree sequence of a simple graph. In this talk, we will show that the probability that a uniformly chosen partition of size $n$ is graphical, decreases to 0 faster than $n^{-0.003}$, thereby addressing a conjecture of Boris Pittel. A lower bound of $n^{-0.5}$ for this rate was already proved by Paul Erdős and Bruce Richmond in 1993, and so this demonstrates that the probability actually decreases polynomially. Key to our argument is an asymptotic characterization of the joint distribution of the leading rows and columns of the Young diagram of a uniformly random partition, combined with a celebrated characterization of graphical partitions due to Erdős and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.
Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480
Title: Geometry & Topology Seminar: Visibility domains in complex manifolds
Speaker: Rumpa Masanta (IISc)
Date: Mon, 04 Nov 2024
Time: 04:00 pm
Venue: LH-1, Department of Mathematics
In this talk, we extend the notion of visibility with respect to the Kobayashi distance to domains in arbitrary complex manifolds. Visibility here is a weak notion of negative curvature and refers to a property resembling visibility in the sense of Eberlein–O’Neill for Riemannian manifolds. However, we do not assume Cauchy-completeness, with respect to the Kobayashi distance, of the domains in question. The visibility property of a domain $D$ can be used to deduce many properties of certain holomorphic mappings into $D$, ranging from their continuous extendibility to the iterative dynamics of such self-maps of $D$. Here, we present a few sufficient conditions for visibility in the above setting, and with these conditions, we see that the class of domains with the visibility property is very large. Time permitting, we will discuss an application of visibility and establish a far-reaching generalisation of the classical Wolff–Denjoy theorem.
Title: Bangalore Probability Seminar: Condensation in scale free geometric random graphs
Speaker: Neeladri Maitra (T.U. Eindhoven)
Date: Mon, 04 Nov 2024
Time: 3:15 pm
Venue: LH-3, Department of Mathematics, Indian Institute of Science
Geometric random graphs with a scale-free degree distribution has been the go-to model of random graphs in the network science community to model a large class of real-life networks. In this talk, we will focus on how condensation effects arise in such models. More specifically, we look at upper tail large deviations of the total number of edges in such graphs and show that the excess number of edges leading to the large deviation event, come from a condensation effect in the underlying degree distribution. This is in sharp contrast with the condensation effect in the `classical’ random geometric graph observed by Chatterjee and Harel (https://arxiv.org/abs/1401.7577), where the condensation instead takes place in the underlying space. This difference is due to the scale-free nature of our model. Here, the randomness coming from the highly variable degrees overpower the randomness of the underlying vertex locations, which gives rise to degree condensates - vertices with high degrees responsible for the excess number of edges in the large deviation event. We will also review the local structure and the degree distribution of the conditional graph. Based on joint work with Remco van der Hofstad, Pim van der Hoorn, Céline Kerriou and Peter Mörters.
Zoom link: https://us02web.zoom.us/j/88670406480 Meeting ID: 886 7040 6480
Title: APRG Seminar: Slides Norms on Harish-Chandra modules
Speaker: Pritam Ganguly (Universität Paderborn, Germany)
Date: Wed, 30 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, we plan to explore the natural question of how to compare different $G$-continuous norms on a Harish-Chandra module with a fixed growth rate. We demonstrate that the space of equivalence classes of these norms features both maximal and minimal elements, as well as a natural Sobolev distance. This leads to a new invariant, which we term the “Sobolev gap.” We will also discuss the natural question of estimating the Sobolev gap uniformly over all irreducible Harish-Chandra modules, offering a quantitative version of the Casselman-Wallach globalization theorem. Additionally, we provide explicit values for the Sobolev gap in the case of Harish-Chandra modules for $SL(2,\mathbb{R})$.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Geometry of Hyperconvex representations
Speaker: James Farre (Max Plank Institute, Leipzig)
Date: Mon, 28 Oct 2024
Time: 04:00 pm
Venue: online on MS Teams
A discrete group of conformal automorphisms of the Riemann sphere is called a Kleinian group. Anosov Kleinian surface groups are quasi-conformal deformations of Fuchsian surface groups; they preserve a Jordan curve cutting the sphere into two disks on which the action of the group is nice. Bers proved that quasi-Fuchsian surface group representations are determined uniquely by the pair of conformal structures at infinity, giving a natural parameterization by a product of Teichmüller spaces. We study the higher rank analogue: hyperconvex Anosov surface group representations into $\mathrm{PSL}(d,C)$, introduced by Pozzetti—Sambarino—Wienhard. We define a natural map into a product of Teichmüller spaces of Riemann surface foliations and prove the following analogue of a famous theorem of Bowen from the 70’s: The Hausdorff dimension of the limit set of a fully hyperconvex surface subgroup into $\mathrm{PSL}(d,C)$ is equal to 1 iff it is conjugated into $\mathrm{PSL}(d,R)$. This is joint work with Beatrice Pozzetti and Gabriele Viaggi.
Title: APRG Seminar: Symmetry for solutions of a Quasilinear equation on Hyperbolic Space
Speaker: Ramya Dutta (Institut Camille Jordan, Universite Claude Bernard Lyon 1, Villeurbanne, France)
Date: Fri, 25 Oct 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We will use the Alexandrov moving plane method to prove radial symmetry of positive finite energy solutions of a quasilinear elliptic equation arising from the Poincare Sobolev inequality in the Hyperbolic space. We will also discuss the existence and non-existence results in some particular cases.
Title: APRG Seminar: Slides Estimates for maximal Riesz transforms in terms of the Riesz transforms: dimension-free bounds on L^p
Speaker: Blazej Wrobel (Institute of Mathematics of the Polish Academy of Science and University of Wroclaw, Poland)
Date: Wed, 23 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In 1983 E. M. Stein proved that the vector of classical Riesz transforms has $L^p$ bounds on $\mathbb R^d$ which are independent of the dimension $d$. I will discuss an analogous result for the vector of maximal Riesz transforms. I will also mention generalizations to higher order Riesz transforms. A main principle of our approach is a comparison between a maximal Riesz transform and the corresponding Riesz transform. The talk is based on joint work with Maciej Kucharski and Jacek Zienkiewicz (Wrocław).
The video of this talk is available on the IISc Math Department channel.
Title: MS Thesis defence: On the study of Toeplitz operators on Hardy Space over a multiply connected domain
Speaker: Sushoban Das (IISc Mathematics)
Date: Tue, 22 Oct 2024
Time: 11 am
Venue: Microsoft Teams (online)
The classical theory of Toeplitz Operators on Hardy Space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the Theory for Toeplitz Operators on Hardy Space over multiply connected domains $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^{\infty+C}(\partial D)$ symbols, and the characterization of Fredholm operators with $H^{\infty+C}$ symbols to those for the multiply connected domain $D$. Also, he came up with a striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.
In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.
Title: Geometry & Topology Seminar: Positivity properties of the vector bundle Monge-Ampere equation
Speaker: Aashirwad Ballal (IISc)
Date: Mon, 21 Oct 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department
Monge-Ampere-positivity (MA-positivity) is a notion of positivity which arises naturally in the study of a generalization the complex Monge-Ampere equation to vector bundles. In particular, preservation of MA-positivity along a continuity path turns out to be crucial in proving the existence of solutions to the vector bundle Monge-Ampere (vbMA) equation. In this talk, we briefly introduce the vbMA equation and discuss some recent results such as the preservation of MA-positivity for rank-two holomorphic bundles over complex surfaces and the existence of counterexamples to an algebraic version of MA-positivity preservation for vector bundles of rank-three and higher over complex manifolds of dimension greater than one.
Title: Eigenfunctions Seminar: Phase transition for Elephant Random Walks on lattices
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Fri, 18 Oct 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Random processes with strong memory and/or self-excitation properties arise naturally in various disciplines including physics, economics, biology, engineering, geology, etc. Many of these processes exhibit superdiffusive growth due to the effect of self-excitation. In order to model such situations, two statistical physicists Schütz and Trimper (2004) introduced a class of processes called Elephant Random Walks, in which the random walker (an elephant 🐘 with strong memory!) remembers the past steps and repeats them with some probability. In the first half of this talk, we shall discuss the basics of random walks and the elephant random walk in a lucid language. The second half will focus on a couple of models investigated by the speaker and his collaborators leading to phase transition results that verify a bunch of conjectures of Saha (2022).
Special care will be taken so that students can follow both halves of this talk. Most of the notions will be introduced and nothing more than basic probability will be prerequisite for this talk. (Based on collaborations with A. Haldar, K. Maulik, S. S. Manna and T. Sadhukhan.)
Title: Algebra & Combinatorics Seminar: Expansion, Group Homomorphism Testing, and Cohomology
Speaker: Bharatram Rangarajan (Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel)
Date: Thu, 17 Oct 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Expansion in groups (or their Cayley graphs) is a valuable and well-studied notion in both mathematics and computer science, and describes a robust form of connectivity of graphs (a gap property of fixed points of representations of groups). It can also be interpreted as a graph on which connectivity is efficiently locally testable.
Group stability, on the other hand, is concerned with another robustness property – but of homomorphisms (or representations). Namely, is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This too can be interpreted as a local testability property of group homomorphisms in the right settings.
Expansion in groups (or property (T)) had been classically reformulated in the language of algebraic topology – in terms of the vanishing of the first cohomology of the group. In this talk we will see approaches in capturing group stability in terms of the vanishing of a second cohomology of the group, motivating higher-dimensional generalizations of expansion.
Based on joint (previous and ongoing) works with Monod, Glebsky, Lubotzky, Fournier-Facio, Dogon.
Title: APRG Seminar: Slides Quantitative characterizations of weights and parabolic boundary value problems
Speaker: Olli Saari (Universitat Politècnica de Catalunya, Barcelona, Spain)
Date: Wed, 16 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
Weights (non-negative locally integrable functions) satisfying a reverse H"older condition are important in the study of harmonic measure and boundary value problems for elliptic and parabolic partial differential equations. In this talk, I will discuss a quantitative version of a Carleson measure characterization of reverse H"older weights (originally found by Fefferman, Kenig and Pipher in the 90s) and its application to elliptic measure. In addition, I will explain extensions and modifications of the results needed in the analogous theory for parabolic measures. This is based on joint work (and work in progress) with Simon Bortz and Moritz Egert.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: The monopole-dimer model and eccentricities for Cartesian product of graphs
Speaker: Anita Arora (IISc Mathematics)
Date: Tue, 15 Oct 2024
Time: 4 PM
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This thesis comprises two main parts. The details of the two parts are as follows:
The first part of the thesis deals with the monopole-dimer model. The dimer (resp. monomer-dimer) model deals with weighted enumeration of perfect matchings (resp. matchings). The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. A more general model called the loop-vertex model has also been defined for an oriented graph and the partition function in this case can also be written as a determinant. However, this model depends on the orientation of the graph. The monopole-dimer model interprets the loop-vertex model independent of the orientation for planar graphs with Pfaffian orientation.
The first part of the thesis focuses on the extension of the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as they are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley–Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Further, we generalise this product formula to higher dimensions, again obtaining an explicit product formula. We also discuss about the asymptotic formulas for the free energy and monopole densities.
Lu and Wu (Physics Letters A, 1999) evaluated the partition function of the dimer model on two-dimensional grids embedded on a M"obius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional M"obius and Klein grids, and evaluate the partition function for three-dimensional M"obius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and M"obius boundary conditions, generalising a result of Lu and Wu.
Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max{d(v,w): w\in V(G)}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of Cartesian product of trees becomes invertible.
Title: Bangalore Probability Seminar: Optimization and dynamics on large graphs
Speaker: Raghavendra Tripathi (UW Seattle/NYU Abu Dhabi)
Date: Mon, 14 Oct 2024
Time: 2 pm to 3:45 pm
Venue: LH-1, Mathematics Department
The problem of optimizing the edge weights of large networks arises in many contexts, for example, machine learning, extremal graph theory, and large deviations of exponential random graphs. Finding the exact minimizers in these problems is often very difficult. In practice, various dynamic optimization schemes such as gradient descent or stochastic gradient descent are used to find the approximate minimizers. From a practical viewpoint, it is often important to study these algorithms or optimization problems in large dimensions. These algorithms yield a process of graphs/matrices and it is natural to investigate the limits of these processes as the dimension goes to infinity. This is done via taking the graphon limit of these processes.
In the first part of this talk, we give several motivating examples where such optimization problems arise and the algorithms that are used in practice. We give a brief introduction to the theory of graphons and recast these problems in the framework of graphons. In the second part of the talk, we introduce the notion of gradient flow on graphons and argue the existence and uniqueness of gradient flow for a rich class of functions. We then show that a large class of algorithms on large graphs does have a graphon limit and the limit can be described as the gradient flow or regularization. Time permitting, we will discuss some applications and limitations of this approach and end with several problems that remain to be investigated.
Title: Algebra & Combinatorics Seminar: Plücker inequalities for weakly separated coordinates in the TNN Grassmannian
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Wed, 09 Oct 2024
Time: 4:15 pm
Venue: LH-1, Mathematics Department
We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2024). In fact we obtain a characterization of weak separability, by showing that no other pair of Plücker coordinates satisfies this property.
Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Daniel Soskin.
Title: APRG Seminar: Slides $L^p$ bounds of maximal operators given by Fourier multipliers with some dilation sets
Speaker: Jin Bong Lee (Seoul National University, South Korea)
Date: Wed, 09 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, the speaker considers two types of maximal operators given by Fourier multipliers, and suggests criteria for Fourier multipliers so that the associated maximal operators are bounded on $L^p$ for each $p$. In other words, we first consider maximal operators given by taking supremum over $t > 0$ where $t$ is a dilation factor of Fourier multipliers, $m(t\xi)$. The condition for $m$ may be understood as a vector-valued analogue of the Hörmander–Mikhlin multiplier theorem. For the second type of maximal operators, we take the supremum over $t \in E$ with $0 \leq \dim(E) < 1$. Together with the dimension of $E$, the condition for $m$ associated with the first maximal operators is still valid for the second maximal operators.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: A Shannon–Kneser–Poulsen Theorem
Speaker: Gautam Aishwarya (Technion, Haifa, Israel)
Date: Tue, 08 Oct 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider the following questions.
Question 1: Does the volume of a union of balls decrease if their centres are brought pairwise closer?
Question 2: Does communication over an additive white Gaussian noise channel worsen if the transmitters are brought pairwise closer?
These questions appeal to our basic intuition about geometry and information transmission, which seems to suggest the answer to both of them is yes. The first question is open; the Kneser–Poulsen conjecture asserts that it has an affirmative answer. In this talk, based on well-known analogies between convex geometry and information theory, we will frame (and prove) the natural entropic formulation of the Kneser–Poulsen conjecture. As a corollary, an affirmative answer to the second question is obtained. This talk is based on joint work with Dongbin Li.
Title: Geometry & Topology Seminar: Dynamics of Fuchsian meromorphic connections with real period
Speaker: Marco Abate (Università di Pisa and Università degli Studi Guglielmo Marconi, Italy)
Date: Mon, 07 Oct 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department (Joint with the APRG Seminar)
An interesting class of examples of holomorphic maps tangent to the identity at a fixed point in several complex variables is given by the time-1 maps of homogeneous vector fields. It is known that the study of the dynamics of these maps can be reduced to the study of the dynamics of the geodesic field of meromorphic connections on Riemann surfaces. In this talk we shall describe some recent results, obtained in collaboration with Karim Rakhimov, on the dynamics of the geodesic field for Fuchsian meromorphic connections having real periods. The main tools used are: a generalization to general Fuchsian meromorphic connections of a classical formula proved by Teichmüller for quadratic differentials, and the relationship between Fuchsian meromorphic connections with real periods and singular flat Hermitian metrics. In particular, we obtain a description of the possible $\omega$-limit sets of simple geodesics that extends and makes more precise results known for the particular case of Riemann surfaces endowed with a meromorphic $k$-differential.
Title: PhD Thesis defence: The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK Metrics
Speaker: Rajas Sandeep Sompurkar (IISc Mathematics)
Date: Fri, 04 Oct 2024
Time: 3:30 pm
Venue: Hybrid - Google Meet (online) and LH-3, Mathematics Department
This Ph.D. thesis consists of two parts. In both the parts we study two new notions of canonical Kahler metrics introduced by Pingali viz. ‘higher extremal Kähler metric’ and ‘higher constant scalar curvature Kähler (higher cscK) metric’ both of whose definitions are analogous to the definitions of extremal Kähler metric and constant scalar curvature Kähler (cscK) metric respectively. On a compact Kähler manifold a higher extremal Kähler metric is a Kähler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is a Kahler metric whose top Chern form is a real constant multiple of its volume form or equivalently whose top Chern form is harmonic. In both the parts we consider a special family of minimal ruled complex surfaces called as ‘pseudo-Hirzebruch surfaces’ which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and the zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical metrics on them.
In the first part of this Ph.D. thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every Kahler class admits a higher extremal Kahler metric which is not a higher cscK metric. The construction of the required metric boils down to solving an ODE depending on a parameter on a closed and bounded interval with some boundary conditions, but the ODE is not directly integrable and requires a very delicate analysis for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude from this that higher cscK metrics (momentum-constructed or otherwise) do not exist in any Kahler class on this Kahler surface. We will briefly see the analogy of this problem with the related problem of constructing extremal Kähler metrics which are not cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by Tønnesen-Friedman and Apostolov et al..
In the second part of this Ph.D. thesis we will see that if we relax the smoothness condition on our metrics a bit and allow our metrics to develop ‘conical singularities’ along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface then we do get ‘conical higher cscK metrics’ in each Kahler class of the Kahler surface by the momentum construction method. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions. We will show that our constructed metrics are conical Kahler metrics satisfying the strongest condition for conical metrics viz. the ‘polyhomogeneous condition’ of Jeffres-Mazzeo-Rubinstein, and we will interpret the conical higher cscK equation globally on the surface in terms of currents by using Bedford-Taylor theory. We will then employ the top ‘log Bando-Futaki invariant’ to obtain the linear relationship between the cone angles of the conical singularities of the metrics at the zero and infinity divisors of the surface.
Title: PhD Thesis defence: Holomorphic mappings and Kobayashi geometry of domains
Speaker: Annapurna Banik (IISc Mathematics)
Date: Fri, 04 Oct 2024
Time: 10 am
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
In this presentation, we shall study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We will focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.
In the initial part of this talk, we shall present a couple of results on local continuous extension of proper holomorphic mappings $F:D \to \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $bD$ and $b\Omega$. These results are motivated by a well-known work by Forstneric–Rosay. However, our results allow us to have much lower regularity, for the patches of $bD, b\Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $b\Omega$ to contain boundary points of infinite type.
We will also discuss another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph–Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds – viewed as complex submanifolds embedded in some ambient complex manifold – with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $b\Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge–Ampere equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called “visibility”.
In the concluding part of this presentation, we will explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincare disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali–Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
Title: APRG Seminar: Slides Bilinear Bochner–Riesz operator for convex domains in the plane
Speaker: Surjeet Singh Choudhary (IISER Bhopal)
Date: Thu, 03 Oct 2024
Time: 3 pm
Venue: Microsoft Teams (online)
For $\alpha\geq0$, the bilinear Bochner–Riesz operator of order $\alpha$ in $\mathbb{R}^n$ is defined as
\begin{equation} \mathcal{B}^\alpha_R(f,g)(x)=\int_{\mathbb{R}^n\times \mathbb{R}^n} \left(1-\frac{|\xi|^2+|\eta|^2}{R^2}\right)^\alpha_+ \hat{f}(\xi) \hat{g}(\eta) e^{2\pi ix\cdot(\xi+\eta)} d\xi d\eta. \end{equation}
For $\alpha>n-\frac{1}{2}$, $\mathcal{B}^{\alpha}_R$ maps $L^{p_1}(\mathbb{R}^n)\times L^{p_2}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ whenever $p_1,p_2\geq1$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. Thus, $\alpha=n-\frac{1}{2}$ is commonly referred as the critical index for the bilinear Bochner–Riesz operator. Recently, there have been some results on $L^p$-boundedness of the bilinear Bochner–Riesz operator $\mathcal{B}^{\alpha}_R$ when $\alpha\leq n-\frac{1}{2}$.
In this talk, we extend the bilinear Bochner–Riesz operator to convex domains in the plane and discuss some $L^p$-boundedness results.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Invariants Associated with Complete Nevanlinna-Pick Spaces
Speaker: Abhay Jindal (IISc Mathematics)
Date: Tue, 01 Oct 2024
Time: 11:15 am
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.
We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.
We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.
Title: Geometry & Topology Seminar: Stability, moment maps and holomorphic submersions
Speaker: Annamaria Ortu (University of Gothenburg)
Date: Mon, 30 Sep 2024
Time: 04:00 pm
Venue: online, MS Teams
A guiding problem in Kähler geometry is to find an equivalence between the existence of a given special metric and an algebro-geometric stability condition. In this context, a natural step towards proving the equivalence is to show that, given a manifold that admits a special metric, stable deformations of the manifold still admit a special metric. We propose a new technique to address this type of problem that relies on restricting to a finite-dimensional problem and applying the theory of moment maps and the moment map flow. We will show how to apply it to constant scalar curvature metrics and, if time permits, to holomorphic submersions.
Title: APRG Seminar: Slides Directional singular integrals in codimension one
Speaker: Ioannis Parissis (University of the Basque Country, Leioa, Spain)
Date: Wed, 25 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk I will give a brief overview of the theory of maximal directional averages and singular integrals in 2 and higher dimensions. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for maximal directional singular integrals in codimension one and general ambient dimension. This reports on joint work with O. Bakas (U. of Patras), F. Di Plinio (Napoli, Federico II) and L. Roncal (BCAM).
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Slides Algebraic positivity versus analytic positivity
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Fri, 20 Sep 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Elementary root separation arguments imply that a non-negative polynomial on the real line is a sum of two squares of polynomials. A non-constructive and ingenious enumerative geometry observation of Hilbert shows that in two or more variables, non-negative polynomials are not always sums of squares. This led to Hilbert’s 17-th problem, asking whether such a decomposition is possible in the field of rational functions. The answer is yes, due to Emil Artin. The gap between polynomials which are positive on semi-algebraic sets and corresponding weighted sums of squares was elucidated by Tarski’s elimination of quantifiers principle. The first part of the lecture will contain accessible details and historical notes on these topics, now part of Real Algebra and Real Algebraic Geometry.
In the second part of the lecture, I will show how sums of squares decompositions led F. Riesz to the definitive form of the spectral theorem for self-adjoint transforms of a Hilbert space. Implying for instance novel positivity results of harmonic analysis. Reversing the historical arrow, I will show how operator theory has put some essential marks on purely Real Algebra chapters. We will also touch positivity in non-commutative *-algebras and Lie-algebras.
The third part of the lecture will contain applications of relatively recent positivity certificates to global, non-convex optimization, stability of dynamical systems and construction of wavelet frames.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis defence: Maps between non-compact surfaces
Speaker: Sumanta Das (IISc Mathematics)
Date: Thu, 19 Sep 2024
Time: 11:30 am
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.
Topological rigidity is the property that every homotopy equivalence between two closed $n$-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces.
We prove that all non-compact, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any $\pi_1$-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane.
An oriented manifold $M$ is said to be Hopfian if every self-map $f : M \to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface $S$ is of finite type if and only if every proper self-map of $S$ of degree one is homotopic to a homeomorphism.
Title: Algebra & Combinatorics Seminar: Positivity preservers over finite fields
Speaker: Prateek Kumar Vishwakarma (IISc Mathematics)
Date: Wed, 18 Sep 2024
Time: 4:15 pm
Venue: LH-1, Mathematics Department
We discuss an algebraic version of Schoenberg’s celebrated theorem [Duke Math. J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider matrices with entries in a finite field and obtain a complete characterization of such preservers for matrices of a fixed dimension. When the dimension of the matrices is at least 3, we prove that, surprisingly, the positivity preservers are precisely the positive multiples of the field’s automorphisms. Our work makes crucial use of the well-known character-sum bound due to Weil, and of a result of Carlitz [Proc. Amer. Math. Soc., 1960] that leads to characterizing the automorphisms of Paley graphs. This is joint with Dominique Guillot and Himanshu Gupta.
Title: APRG Seminar: Slides Hyponormal quantization
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Wed, 18 Sep 2024
Time: 10 am – 12:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
After a review of the Markov–Krein transform in the case of a one variable, and the Hilbert space interpretation of it (the phase shift), we will specialize the Markov–Krein transform to 2D. This will bring us to a relaxation of the Heisenberg commutation relation, this time filled by bounded linear transforms. The spectral invariant of this class of so called hyponormal operators is called the principal function. It is a measurable function of compact support, carrying a degree of shade. We will sketch the main specific results pertaining to hyponormal operators.
For the rest of the lecture we will link the resulting inverse spectral problem to image processing, potential theory, Hele–Shaw flows, integrable systems, and the regularity of free boundaries. Current advances with precise open questions will be detailed.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides Fourier uncertainty and bandlimited functions
Speaker: Emanuel Carneiro (ICTP Trieste, Italy)
Date: Wed, 18 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
This will be a broad discussion on a few different types of Fourier uncertainty principles for bandlimited functions (i.e. functions that have Fourier transforms compactly supported), and their connections to multiplication operators in certain Hilbert spaces of entire functions. Some of these problems are related to applications in analysis, PDEs and number theory, and I might describe a few of these if time permits. The talk should be accessible to a broad audience.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides The Fantappie transform
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Fri, 13 Sep 2024
Time: 3 – 5:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
There are several analogs of Cauchy’s integral transform in the theory of functions of several complex variables. Fantappie’s transform is one of them, possibly the simplest and oldest, with some remarkable properties we will discuss in detail. First, it is immediately connected to Radon transform, much exploited today in inverse problems. The characterization of Fantappie transforms of positive measures resonates with Bernstein’s theorem linking Laplace transforms and completely monotonic functions. One of the spectacular applications of Fantappie’s transform is the multivariate analog of Koethe–Grothendieck duality of spaces of analytic functions. The multiplicative structure of Fantappie’s transform brings us to the classical Markov–Krein correspondence, much investigated these days by probabilists.
An application to mathematical economics will be sketched.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Computing the zeta function of varieties over finite fields
Speaker: Madhavan Venkatesh (IIT Kanpur)
Date: Fri, 13 Sep 2024
Time: 4 pm
Venue: LH-2
The focus of this talk will be on computing the point counts
of algebraic varieties, i.e., number of solutions of a system of
polynomial equations over finite fields. The zeta function encodes the
point counts over an infinite tower of finite field extensions and enjoys
the property of being a rational function over $\mathbb{Q}$. Further, the zeta
function can be recovered from certain invariants of the variety in
question, using an appropriate cohomology theory. I will review the state of the art on efficient algorithms to compute the
zeta function of varieties, including the dimension one case of curves
(covering the works of Schoof, and Pila) and report on our generalisations
for the first cohomology (joint work with Diptajit Roy and Nitin
Saxena) and ongoing work on the second cohomology, which addresses a question
of Edixhoven.
Title: APRG Seminar: Slides Strichartz estimates in the Heisenberg group and generalizations
Speaker: Davide Barilari (Università degli Studi di Padova, Italy)
Date: Wed, 11 Sep 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk I will discuss Strichartz estimates on the Heisenberg group for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available.
Our approach, inspired by the Fourier transform restriction method initiated by Tomas and Stein, is based on Fourier restriction theorems, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.
Title: Bangalore Probability Seminar: Classical random matrix ensembles and planar random growth models
Speaker: Riddhipratim Basu (ICTS Bangalore)
Date: Mon, 09 Sep 2024
Time: 3:15 pm
Venue: LH-1, Mathematics Department
There are some remarkable bijections that connect the spectrum of classical random matrix ensembles (such as Gaussian and Laguerre unitary ensembles) to certain planar random growth models. I shall review some of these connections and discuss a few results on either side that uses the interplay.
Based on joint works with Jnaneshwar Baslingker, Sudeshna Bhattacharjee and Manjunath Krishnapur.
Title: APRG Seminar: Slides Moment indeterminateness
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Mon, 09 Sep 2024
Time: 10 am – 12:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Contrary to Fourier, Laplace, Cauchy, or Poisson transforms, the power moments of a positive measure, fast decaying at infinity on the real line, do not determine the original measure. The non-uniqueness phenomenon was analyzed in depth by Stieltjes, via continued fraction expansions of the formal generating series of moments. I will sketch the main ideas of Stieltjes celebrated memoir, to continue with an account of a not less foundational contribution put by Carleman in book format. Then we will touch on Marcel Riesz’s pioneering work on extensions of positive linear functionals, to return from another perspective to the Christoffel-Darboux kernel. All in 1D.
Some unfinished parallel studies in $n$D, marred by pitfalls and open problems will be discussed.
The video of this talk is available on the IISc Math Department channel.
Title: Geometry & Topology Seminar: Ray structures on Teichmuller spaces
Speaker: Huiping Pan (South China University of Technology)
Date: Mon, 09 Sep 2024
Time: 06:30 pm
Venue: online, MS Teams
Teichmuller space admits several ray structures, such as the Teichmuller geodesic ray, Thurston stretch ray, harmonic map (dual) ray, grafting ray, etc. In the first part of this talk, we will depict harmonic map ray structures on Teichmuller space as a geometric transition between Teichmuller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces, the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to a Teichmuller geodesic. In the second part, we will discuss applications to the Thurston metric. This is a joint work with Michael Wolf.
Title: Bangalore Probability Seminar: Interchangeability theorems for interacting particle systems
Speaker: Arvind Ayyer (IISc Mathematics)
Date: Mon, 09 Sep 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
I will focus on two interacting particle systems with inhomogeneous rates on the finite ring with site-dependent rates. The first is the Totally Asymmetric Zero Range Process (TAZRP) and the second is the PushTASEP, where TASEP stands for the Totally Asymmetric Simple Exclusion Process. In both cases, I will present results which say that the distribution of the path of the process restricted to certain intervals is unchanged under permutations of the rates. Our proofs build on Weber’s theorem for exponential queues. I will aim to keep the talk self-contained.
This is joint work with O. Mandelshtam and J. Martin (arXiv:2209.09859, Math. Z., to appear) and with J. Martin (arXiv:2310.09740).
Title: Eigenfunctions Seminar: Slides Two perspectives in number theory: explicit and probabilistic
Speaker: Kaneenika Sinha (IISER Pune)
Date: Fri, 06 Sep 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
“Explicit” number theory is the name given to the study of what are called zero-free regions of the Riemann zeta function and other L-functions. An explicit determination of such regions often reveals deep arithmetic properties of the underlying object attached to the concerned L-function. More generally, it could refer to the use of “explicit”, often technical methods to understand an arithmetic object. On the other hand, “probabilistic” number theory attempts to investigate arithmetic properties of an object by treating the object as one in a family of many, and exploring these families of varying sizes through the viewpoint of probability. In this talk, we will explore both perspectives and compare the wealth of information each perspective presents to us. In particular, we will address the study of Fourier coefficients of certain modular forms (called the Hecke eigenforms) through both the above viewpoints.
Title: Algebra & Combinatorics Seminar: Graphical models, causality and algebraic perspectives
Speaker: Prateek Misra (Technical University of Munich, Germany)
Date: Thu, 05 Sep 2024
Time: 4 pm
Venue: LH-3, Mathematics Department
Algebraic Statistics is a relatively new field of research where tools from Algebraic Geometry, Combinatorics and Commutative Algebra are used to solve statistical problems. A key area of research in this field is the Gaussian graphical models, where the dependence structure between jointly normal random variables is determined by a graph. In this talk, I will explain the algebraic perspectives on Gaussian graphical models and present some of my key results on understanding the defining equations of these models. In the end, I will talk about the problem of structural identifiability and causal discovery and how algebraic techniques can be implemented to tackle them.
Title: APRG Seminar: Slides The Christoffel–Darboux Kernel
Speaker: Mihai Putinar (University of California at Santa Barbara, USA and Newcastle University, UK)
Date: Thu, 05 Sep 2024
Time: 2 – 4:15 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Given a positive measure in Euclidean space, rapidly decaying at infinity, a point $a$, and a filtration of the polynomial ring by the degree, the optimal bounds for point evaluations at $a$ in Lebesgue space $L^2$ are provided by a reproducing kernel known as the Christoffel–Darboux kernel. For a century and a half, this object continues to intrigue by surprising new turns.
We will touch the asymptotics of orthogonal polynomials in the complex plane, with emphasis on Szego’s Limit Theorem. Then turn to spectral analysis on the line, culminating in Weyl’s circle phenomenon.
Recent applications to dynamical systems (via Koopman’s operator formalism) and the statistics of geometric data will be presented together with some numerical experiments.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Algebras of functions and heat kernel estimates
Speaker: Tommaso Bruno (University of Genova, Italy)
Date: Wed, 04 Sep 2024
Time: 4 pm
Venue: Microsoft Teams (online)
I will discuss a theory of function spaces defined in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. This includes inhomogeneous and homogeneous Sobolev, Besov and Triebel–Lizorkin spaces on Lie groups and Grushin settings.
Title: Number Theory Seminar: A finer Sato-Tate conjecture and Bias conjecture for certain families of elliptic curves
Speaker: Sudhir Pujahari (NISER Bhubaneshwar)
Date: Fri, 30 Aug 2024
Time: 2 pm
Venue: LH-1
In this talk, we will study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In conclusion, we will obtain the Sato-Tate distribution for the trace of certain families of Elliptic curves in arithmetic progressions. As a special case we will recover a result of Birch proving Sato-Tate distribution for certain families of elliptic curves. Moreover, we will see that these results follow from asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. Finally, if time permits, we will discuss the bias conjecture in the finite field setting. These are joint works with Kathrin Bringmann, Ben Kane, and Zichen Yang.
Title: APRG Seminar: Slides Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on hyperbolic spaces
Speaker: Effie Papageorgiou (Universität Paderborn, Germany)
Date: Wed, 28 Aug 2024
Time: 3 pm
Venue: Microsoft Teams (online)
On a Riemannian manifold, consider the Laplace-Beltrami operator $-\Delta$, and the associated extension problem
\begin{equation} \Delta v+\frac{(1-2\sigma)}{t}\frac{\partial v}{\partial t}+\frac{\partial^2 v}{\partial t^2}=0, \quad 0 <\sigma < 1,\quad t>0, \end{equation}
introduced by Caffarelli and Silvestre on Euclidean space to recover the fractional Laplacian $(-\Delta)^{\sigma}$, as $t$ approaches zero.
On hyperbolic spaces this gives rise to a family of convolution operators, including the Poisson operator $e^{-t\sqrt{-\Delta}}$, $t>0$; moreover,
the kernels of these operators are subordinated to the heat kernel.
Motivated by Euclidean results for the Poisson semigroup, but also by results on the heat semigroup on Riemannian manifolds and the influence of underlying geometry, we study the long-time asymptotic behavior of solutions to the extension problem for $L^1$ initial data. If the initial datum is radial, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence can break down in the non-radial case. The results extend to all noncompact symmetric spaces of arbitrary rank.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Old and new results on bounds for codes and sphere packings: An overview
Speaker: Alexander Barg (University of Maryland, College Park, USA)
Date: Fri, 23 Aug 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The problem of maximum packing density of the $n$-dimensional real space with spheres is a classic one. Exact values of the density are known only in a few cases ($n=1,2,3,8,24$), and there have been several recent improvements of the bounds for other small dimensions. In a parallel development, researchers have studied the maximum size of packings of the $n$-dimensional Hamming space, known as error-correcting codes. While existence bounds in both cases are found by random choice, the best known impossibility results are obtained by an application of a general method commonly known as Delsarte’s linear programming. The best known upper bound on the maximum size of a code with a given minimum distance for large $n$ was obtained in 1977, and it has proved surprisingly resistant to various improvement attempts, including the semidefinite programming extension of LP.
In the first part of the talk we will introduce the general problem and give an overview of the known results on upper bounds on codes and related problems such as equiangular lines and families of finite sets with restricted intersections. In the second part, we will delve into the details of the proofs for the case of codes and highlight some obstacles for further improvements.
Title: PhD Thesis defence: On two complex Hessian equations and convergence of corresponding flows
Speaker: Ramesh Mete (IISc Mathematics)
Date: Thu, 22 Aug 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Google meet (online)
This thesis consists of three parts. Two important complex Hessian equations are studied on certain compact Kahler manifolds from different perspectives. The first one is the J-equation introduced independently by S.K. Donaldson and X.X. Chen from different point of view. The second one is the deformed Hermitian Yang Mills (dHYM) equation which has connection to the mirror symmetry in string theory.
There is a notion of (global) slopes for both equations. It is known that they admit smooth solutions with the global slopes if and only if certain Nakai-Moishezon (NM) type criterion holds. In the first part, our aim is to find some appropriate singular solutions of the equations when the NM-type criterion fails–this is the so-called unstable case. An algebro-geometric characterization of the slopes is formulated – which we call the minimal J-slope for the J-equation and the maximal dHYM-slope for the dHYM equation. There is a natural weak (i.e. singular) version of the equations replacing the standard wedge product with a more generalized product, called the non-pluripolar product. We settle the existence and uniqueness problem for the singular J and dHYM equation on compact Kahler surfaces. More precisely, for the J-equation we show that there exists a unique closed $(1,1)$-Kahler current solving the singular J-equation on a compact K"ahler surface with the minimal J-slope. Analogous result is established for the singular dHYM equation on compact Kahler surfaces with the maximal dHYM-slope. Furthermore, we conjecture analogous existence and uniqueness result for higher dimensions.
In the second part, we study the convergence behaviour of the J-flow, which is the parabolic version of the J-equation, on certain generalized projective bundles using the Calabi Symmetry in the J-unstable case. An invariant version of the minimal J-slope is introduced for these bundles. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the singular J-equation with the invariant minimal J-slope. This result resolves the invariant version of our conjecture for the J-equation on these examples with symmetry.
In the third part, we study the convergence behaviour of a flow, called the cotangent flow, for the dHYM equation in the dHYM-unstable case on the blowup of $\mathbb{C}\mathbb{P}^2$ or $\mathbb{C}\mathbb{P}^3$. Analogous to our results in the second part, we show that the flow converges to some unique limit, and the limiting current solves the singular dHYM equation with the (invariant) maximal dHYM-slope.
Title: APRG Seminar: Slides Smooth barycentric decompositions of Weyl chambers - a helpful tool to handle inverse spherical Fourier transforms on noncompact symmetric spaces of higher rank Fourier and Schur idempotents
Speaker: Jean-Philippe Anker (Institut Denis Poisson, Université d'Orléans, France)
Date: Wed, 21 Aug 2024
Time: 4 pm
Venue: Microsoft Teams (online)
The study of invariant dispersive PDE on noncompact symmetric spaces, such as the wave equation or the Schrödinger equation, requires to analyze oscillating integrals arising from the inverse spherical Fourier transform. While this can be achieved by classical though nontrivial tools in rank one, a major problem in higher rank lies in the fact that the Plancherel density is not a differentiable symbol in general, and thus integration by parts produces no additional global decay at infinity. In this talk, we will explain a way to overcome this problem by introducing a smooth barycentric decomposition of Weyl chambers, which leads eventually to the same dispersive and Strichartz estimates as in rank one. This work started 15 years ago as a joint project with S. Meda, V. Pierfelice, M. Vallarino and was finally achieved in collaboration with H.-W. Zhang.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: Slides Hypoelliptic damped wave equations on graded lie groups with initial data from negative order Sobolev spaces
Speaker: Vishvesh Kumar (Ghent University, Ghent, Belgium)
Date: Wed, 14 Aug 2024
Time: 3 pm
Venue: Microsoft Teams (online)
In this talk, I will discuss the semilinear hypoelliptic damped wave equation with power-type nonlinearity associated with a Rockland operator on graded Lie groups. Specifically, we will concentrate on the case when the initial data belongs to Sobolev spaces of negative order. We show the global-in-time existence of small data Sobolev solutions of lower regularity for the supercritical range and a finite-time blow-up of weak solutions for the subcritical range. For the particular settings of the Heisenberg group and Euclidean space, we will prove that the critical exponent belongs to the blow-up case. Furthermore, to precisely characterize the blow-up time, we derive sharp upper and lower bound estimates for the lifespan in the subcritical cases.
This talk is based on my joint research with Aparajita Dasgupta (IIT Delhi), Shyam Swarup Mondal (ISI Kolkata), Michael Ruzhansky (Ghent University), and Berikbol Torebek (Ghent University).
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: p and hp Spectral Element Methods for Elliptic Boundary Layer Problems
Speaker: Akhlaq Husain (Jamia Millia Islamia, Delhi)
Date: Tue, 13 Aug 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
In this talk, we consider p and h-p least-squares spectral element methods for elliptic boundary layer problems in one dimension. We derive stability estimates and design a numerical scheme based on minimizing the residuals in the sense of least-squares in appropriate Sobolev norms. We prove parameter robust uniform error estimates i.e. error in the approximation is independent of the boundary layer parameter for the p and hp-version. Numerical results are presented for a number of model elliptic boundary layer problems confirming the theoretical estimates and uniform convergence results.
Title: Geometry & Topology Seminar: Ribbon concordance is a partial order (after Ian Agol)
Speaker: Siddhartha Gadgil (IISc)
Date: Mon, 12 Aug 2024
Time: 11:00 am
Venue: LH-1, Mathematics Department
A couple of years ago Ian Agol proved a long-standing conjecture showing that a relation called “Ribbon concordance” on knots is a partial order. The proof was a six page paper involving a blend of topology, combinatorial group theory and real algebraic geometry, with ribbon concordance implying a relation between representation varieties. In this talk I describe Agol’s proof and some background results.
Title: Geometry & Topology Seminar: A Moduli Space of Marked Hyperbolic Structures for Big Surfaces.
Speaker: Chaitanya Tappu (Cornell University)
Date: Mon, 12 Aug 2024
Time: 04:00 pm
Venue: LH-1, Mathematics Department
We introduce the moduli space of marked, complete, Nielsen-convex hyperbolic structures on a surface of negative, but not necessarily finite, Euler characteristic. The emphasis is on infinite type surfaces, the aim being to study mapping class groups of infinite type surfaces via their action on this marked moduli space. We define a topology on the marked moduli space. This marked moduli space reduces to the usual Teichm"uller space for finite type surfaces. Since a big mapping class group is a topological group, a basic question is whether its action on the marked moduli space is continuous. We answer this question in the affirmative.
Title: APRG Seminar: Dentability and related concepts in the unit ball of Banach spaces
Speaker: Sudeshna Basu (Loyola University, Baltimore, USA)
Date: Fri, 09 Aug 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
Let $X$ be a Banach space. Let $C$ be a subset of $X$. Let $x^*$ be a functional on $X$. Then $S(C, x^*, \alpha) := \{ x \in C : x^*(x) > \sup x^*(C) - \alpha \}$, $\alpha > 0$, is called the open slice of $C$ determined by $x^*$ and $\alpha$. $X$ has Radon Nikodym Property if and only all closed bounded convex sets admit slices of arbitrarily small diameter i.e. these sets are dentable. The geometry of Banach space is an area of research which characterizes the topological and measure theoretic concepts in Banach spaces in terms of geometric structure of the space. The related concepts were initiated developed and extensively studied in the context of Radon Nikodym Property and Krein Milman Property by Ghoussoub, Godefroy, Maurey, and Scachermayer [Memoirs AMS 1987]. In this work, we look at Banach spaces where the unit ball admits slices of arbitrarily small diameter. We look at some related properties as well. We prove that all these properties are stable under $l_p$ sum for $1 \leq p \leq ∞$, sum and Lebesgue Bochner spaces. We show that these are three space properties under certain conditions on the quotient space. We also study these properties in ideals of Banach spaces. This is based on two papers jointly written with my graduate student, Susmita Seal in [J. Math. Anal. Appl. 2022] and [J. Convex Anal. 2023]. The only prerequisite for this talk is the statement of the Hahn Banach Theorem.
Title: APRG Seminar: On the phase diagram of the polymer model
Speaker: Arjun Krishnan (University of Rochester, USA)
Date: Thu, 08 Aug 2024
Time: 3:40 pm
Venue: LH-1, Mathematics Department
In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.
Title: APRG Seminar: Ruin in double halve gambling
Speaker: Aditya Guha Roy (Indiana University, Bloomington, USA)
Date: Thu, 08 Aug 2024
Time: 2:30 pm
Venue: Hybrid - LH-1, Mathematics Department and Microsoft Teams (online)
We study the ruin probability of a gambler in a scheme where the bet size is doubled after every win in a round and halved after every loss; we show some paradoxical results such as that the ruin probability is one if and only if the probability of winning in any round is at least half. We also verify some conjectures about the behaviour of the ruin probability as a function of the initial fortune and the probability of winning.
Title: APRG Seminar: Slides Fourier and Schur idempotents
Speaker: Javier Parcet (Instituto de Ciencias Matemáticas, Madrid, Spain)
Date: Wed, 07 Aug 2024
Time: 2:30 pm
Venue: Microsoft Teams (online)
What happens to an $L_p$ function when one truncates its Fourier transform to a domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s celebrated theorem for the ball (1971) imposes that, to preserve $L_p$-integrability, the boundary of such domain must be flat. What if we truncate on a curved space like a Lie group? And if we truncate the entries of a given matrix? What happens with the singular numbers of it or with its Schatten $p$-norm? We fully characterize the local geometry of such $L_p$-preserving truncations for these (apparently unrelated) problems, in terms of a surprisingly lax notion of boundary flatness. The matrix ones are all diffeomorphic variations of a fundamental example: the triangular projection. The Lie group ones are all modeled on one of three fundamental examples: the classical Hilbert transform, and two new examples of Hilbert transforms that we call affine and projective. This vastly generalizes Fefferman’s theorem to nontrigonometric and noncommutative scenarios. It confirms the intuition that Schur multipliers share profound similarities with Euclidean Fourier multipliers – even in the lack of a Fourier transform connection – and complete, for Lie groups, a longstanding search of Fourier $L_p$-idempotents. Joint work with M. de la Salle and E. Tablate.
The video of this talk is available on the IISc Math Department channel.
Title: PhD Thesis colloquium: Symmetric real-closed subsets of affine Kac-Moody Lie Algebras
Speaker: Dipnit Biswas (IISc Mathematics)
Date: Mon, 29 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department
In this thesis, we study some important classification problems related to affine Kac–Moody Lie algebras. First, we address the combinatorial problem of classifying symmetric real-closed subsets of affine root systems (which are roots of affine Kac–Moody Lie algebras). Secondly, we understand the correspondence between these symmetric real-closed subsets and the regular subalgebras generated by them, using the aforementioned classification. Motivation for this work comes from the celebrated work of Dynkin (1952), where he classified the semi-simple subalgebras of a given finite-dimensional semisimple Lie algebra 𝔤. He introduced the notion of regular subalgebras in order to achieve this classification. It is not hard to see that the regular subalgebras of 𝔤 correspond to symmetric closed subsets of roots of 𝔤, so the problem of classifying regular subalgebras comes down to the combinatorial problem of classifying these subsets. The analogous problem of studying regular subalgebras of affine Kac–Moody Lie algebras was initiated by Anna Felikson et al. in 2008 and continued by Roy–Venkatesh in 2019, where they addressed the part of the combinatorial problem, namely provided the classification of maximal real closed subroot systems of affine root systems.
In the finite case, it is well known that symmetric closed subsets are in fact closed subroot systems. It is not true in general for affine root systems. So, it is natural to ask the following questions:
- When a given symmetric real-closed subset of affine root system is a closed subroot system?
- Is it possible to classify all symmetric real-closed subsets of affine root systems?
We give affirmative answers to these questions in this thesis. In the untwisted setting, we prove that any symmetric real-closed subset is indeed a closed subroot system. Twisted types need more careful analysis since the finite part of a symmetric real-closed subset has two possibilities in these types, namely closed and semi-closed. For semi-closed cases, the behavior of symmetric real-closed subsets varies for each type. We prove that there are three types of irreducible symmetric real-closed subsets for reduced real affine root systems, one of which did not appear in Roy–Venkatesh’s work. We conclude our classification for the twisted case by determining explicitly when a symmetric closed subset is a closed subroot system, including the case when the ambient Lie algebra is the non-reduced affine Lie algebra $A_{2n}^{(2)}$.
In the second part of the thesis, we explore the correspondence between symmetric real-closed subsets and regular subalgebras generated by them. Roy-Venkatesh proved that the map between closed subroot systems and the regular subalgebras generated by them is injective. We observe that it is not true in general when we extend this map to symmetric real-closed subsets. Let ψ be a real-closed subroot system. We describe the types of symmetric closed subsets that can appear in the fiber of the subalgebra generated by ψ. Moreover, we determine when these fibers are finite. In certain cases, we are able to describe very explicitly the defining parameters of the symmetric closed subsets appearing in the fiber.
Title: Algebra & Combinatorics Seminar: Milnor–Witt cycle modules over excellent DVR
Speaker: Rakesh Pawar (UMPA, École Normale Supériure de Lyon, France)
Date: Wed, 24 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
I will briefly recall Milnor cycle modules over a field as defined by Rost (1996) and their significance and properties. Recently, ‘modules’ over Milnor–Witt K-theory or alternatively Milnor–Witt cycle modules over fields have been formalized by N. Feld (2020).
I will talk about recent joint work with Chetan Balwe and Amit Hogadi, where we considered the Milnor–Witt cycle modules over excellent DVR and studied a subclass of these that satisfy certain lifting conditions on residue maps associated with horizontal valuations. As an important example, Milnor–Witt K-theory of fields belongs to this subclass. Moreover, this condition is sufficient to deduce the local acyclicity property and $A^1$-homotopy invariance of the associated Gersten complex.
Title: Geometry & Topology Seminar: Simplicial cell decompositions of CP^n
Speaker: Basudeb Datta (IAI, TCG Crest, Kolkata)
Date: Thu, 18 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
According to a well-known result in geometric topology, we have $(S^2)^n/Sym(n) = \mathbb{CP}^n$, where $Sym(n)$ acts on $(S^2)^n$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^n$ for each $n > 1$. In more detail, we take the standard two triangle crystallisation $S^2_3$ of the 2-sphere $S^2$, in its $n$-fold Cartesian product. We then simplicially subdivide, and prove that naively taking the $Sym(n)$ quotient yields a simplicial cell decomposition of $\mathbb{CP}^n$. Taking the first derived subdivision of this cell complex produces a triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n > 3$. This is a joint work with Jonathan Spreer, University of Sydney.
Title: MS Thesis defence: Higher categories of n-bordisms
Speaker: Arghan Dutta (IISc Mathematics)
Date: Tue, 16 Jul 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Microsoft Teams (online)
In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.
Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.
One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category $\mathrm{Bord}_n^{(n -1)}$, a variant of the fully extended $\mathrm{Bord}_{n}$. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.
Title: Algebra & Combinatorics Seminar: When do the natural embeddings of classical invariant rings split?
Speaker: Vaibhav Pandey (Purdue University, West Lafayette, USA)
Date: Fri, 12 Jul 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field. Objects of broad interest in commutative algebra, representation theory, and algebraic geometry like generic determinantal rings, Plücker coordinate rings of Grassmannians, symmetric determinantal rings, rings defined by Pfaffians of alternating matrices etc. arise as the invariant rings $S^G$ of such group actions.
In characteristic zero, reductive groups are linearly reductive and therefore the embedding of the invariant ring $S^G$ in the ambient polynomial ring $S$ splits. This explains a number of good algebro-geometric properties of the invariant ring in characteristic zero. In positive characteristic, reductive groups are typically no longer linearly reductive. We determine, for the natural actions of the classical groups, precisely when $S^G$ splits from $S$ in positive characteristic.
This is joint work with Melvin Hochster, Jack Jeffries, and Anurag K. Singh.
Title: PhD Thesis defence: On regular subalgebras of Kac-Moody algebras and related combinatorics
Speaker: Md. Irfan Habib (IISc Mathematics)
Date: Fri, 12 Jul 2024
Time: 11:30 am
Venue: LH-1, Department of Mathematics
In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.
Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.
Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.
In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)
\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}
contains a unique maximal element.
Title: APRG Seminar: An encoding retrieving the edge universality of Wigner matrices
Speaker: Debapratim Banerjee (Ashoka University)
Date: Fri, 12 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In this talk, we give a new combinatorial proof of classical edge universality of Wigner matrices without assuming the entries of the matrices are symmetrically distributed around 0. We complete this proof by giving a new encoding of the Wigner words and a counting strategy which works for traces of very high powers of the matrix. In this talk, we shall introduce the encoding, describe the class of words which capture the randomness and finally give some insight about the proof for general non vanishing odd moments.
Title: Eigenfunctions Seminar: An overview of Fuchs’ problem
Speaker: Sunil Chebolu (Illinois State University, Normal, USA)
Date: Fri, 05 Jul 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. After introducing the problem and known results in the first part of the talk, I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups, and some $p$-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.
Title: APRG Seminar: Isomorphisms between dense random graphs
Speaker: Lutz Warnke (University of California, San Diego, USA)
Date: Thu, 04 Jul 2024
Time: 11:15 am
Venue: LH-1, Mathematics Department
Applied benchmark tests for the famous ‘subgraph isomorphism problem’ empirically discovered interesting phase transitions in random graphs. This motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities $p_1,p_2$. In particular, (i) we prove a sharp threshold result for the appearance of $G_{n,p_1}$ as an induced subgraph of $G_{N,p_2}$, (ii) we show two-point concentration of the size of the maximum common induced subgraph of $G_{N,p_1}$ and $G_{N,p_2}$, and (iii) we show that the number of induced copies of $G_{n,p_1}$ in $G_{N,p_2}$ has an unusual ‘squashed lognormal’ limiting distribution.
These results resolve several open problems of Chatterjee and Diaconis, and confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.
Based on joint work with my PhD students Erlang Surya and Emily Zhu; see arXiv:2305.04850.
Title: APRG Seminar: Algorithmic Discrepancy Theory: A Tale of Combinatorics, Probability, Convex Geometry and Algorithms
Speaker: Abhishek Shetty (University of California, Berkeley, USA)
Date: Thu, 04 Jul 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
Discrepancy theory is a well-studied area in mathematics, concerned with the question of partitioning geometry and combinatorial objects in balanced subsets. Starting from seminal works of Spencer, Banaszczyk, Gluskin amongst others, deep connections between the area and other areas in mathematics such as convex geometry and probability were established. But, for most of its history, the arguments establishing the existence of good partitions were non-constructive, even believed to be fundamentally non-algorithmic.
But, the past decade has seen a flurry of work in algorithmic discrepancy theory, leading to efficient algorithms for several of the most famous settings in discrepancy theory. Perhaps surprisingly, these algorithmic techniques have further strengthened the connection between discrepancy and other areas such as convex geometry and probability.
In this talk, we will survey recent results and techniques in algorithmic discrepancy with the aim to convey connections to various areas. Time permitting, we will end with natural conjectures that would lead to progress on long standing conjectures in discrepancy theory.
No prior exposure to algorithms, computer science or discrepancy will be assumed.
Title: Number Theory Seminar: Towards a refinement of the Bloch-Kato conjecture
Speaker: Sunil Chebolu (Illinois State University, Normal, USA)
Date: Wed, 03 Jul 2024
Time: 11 am
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Let $F$ be a field that has a primitive $p$-th root of unity. According to the Bloch–Kato conjecture, now a theorem by Voevodsky and Rost, the norm-residue map \begin{equation} k_*(F)/pk_*(F) \rightarrow H^*(F, \mathbb{F}_p) \end{equation} from the reduced Milnor $K$-theory to the Galois cohomology of $F$ is an isomorphism of $\mathbb{F}_p$-algebras.
This isomorphism gives a presentation of the rather mysterious Galois cohomology ring through generators and relations. In joint work with Jan Minac, Cihan Okay, Andy Schultz, and Charlotte Ure, we have obtained a second cohomology refinement of the Bloch–Kato conjecture. Using this we can characterize the maximal $p$-extension of $F$, as the “decomposing field” for the cohomology of the absolute Galois group.
Title: Algebra & Combinatorics Seminar: Buchsbaumness and Castelnuovo-Mumford regularity of non-smooth monomial curves
Speaker: Ngo Viet Trung (Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam)
Date: Mon, 01 Jul 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo–Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo–Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.
Title: APRG Seminar: Mated trees and a Peano curve in the directed landscape
Speaker: Manan Bhatia (MIT, Boston, USA)
Date: Fri, 28 Jun 2024
Time: 2:15 pm
Venue: LH-1, Mathematics Department
A recurrent theme encountered in many models of random geometry is that of two trees glued to one another with a space-filling curve snaking in between them. In this talk, we first recall a few examples of this, namely, Brownian geometry, Liouville quantum gravity, and the Brownian web. Subsequently, we discuss the construction of a pair of interlaced trees and the corresponding Peano curve in the directed landscape, the conjectural universal scaling limit of models in the Kardar-Parisi-Zhang universality class. Finally, we look at the question of determining the precise Holder and variation regularity of this space-filling curve and discuss some of the ideas involved in the proof. Based on the works arxiv:2304.03269 (joint with Riddhipratim Basu) and arxiv:2301.07704.
Title: Algebra & Combinatorics Seminar: Local vanishing for toric varieties
Speaker: Sridhar Venkatesh (University of Michigan, Ann Arbor, USA)
Date: Thu, 27 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
Let $f:Y \to X$ be a log resolution of singularities which is an isomorphism over the smooth locus of $X$, and the exceptional locus $E$ is a simple normal crossing divisor on $Y$. We prove vanishing (and non-vanishing) results for the higher direct images of differentials on $Y$ with log poles along $E$ in the case when $X$ is a toric variety. Our consideration of these sheaves is motivated by the notion of $k$-rational singularities introduced by Friedman-Laza. This is joint work with Anh Duc Vo and Wanchun Shen.
Title: PhD Thesis defence: Existence and implications of positively curved metrics on holomorphic vector bundles
Speaker: Arindam Mandal (IISc Mathematics)
Date: Wed, 26 Jun 2024
Time: 3 pm
Venue: LH-3, Mathematics Department
This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann-Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann-Fock spaces on the complex plane. This problem has also been studied for the Bargmann-Fock spaces on the higher dimensional complex Euclidean spaces. Very few results on interpolating and sampling hypersurfaces in higher dimensions are known. We have proven certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerd'{a}, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of them. In dimension two, we provided sufficient conditions for a smooth algebraic hypersurface to be uniformly flat in terms of its projectivization.
The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.
Title: PhD Thesis colloquium: The monopole-dimer model and eccentricities for the Cartesian product of graphs
Speaker: Anita Arora (IISc Mathematics)
Date: Mon, 24 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
This thesis comprises two main parts. The details of the two parts are as follows:
The first part of the thesis deals with the monopole-dimer model. The dimer (resp. monomer-dimer) model deals with weighted enumeration of perfect matchings (resp. matchings). The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. A more general model called the loop-vertex model has also been defined for an oriented graph and the partition function in this case can also be written as a determinant. However, this model depends on the orientation of the graph. The monopole-dimer model interprets the loop-vertex model independent of the orientation for planar graphs with Pfaffian orientation. The first part of the thesis focuses on the extension of the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as they are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley–Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Further, we generalise this product formula to higher dimensions, again obtaining an explicit product formula. We also discuss about the asymptotic formulas for the free energy and monopole densities.
In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional Möbius and Klein grids, and evaluate the partition function for three-dimensional Möbius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and Möbius boundary conditions, generalising a result of Lu and Wu.
Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.
Title: MS Thesis colloquium: On the study of Toeplitz operators on Hardy Space over a multiply connected domain
Speaker: Sushobhan Das (IISc Mathematics)
Date: Fri, 21 Jun 2024
Time: 10 am
Venue: Hybrid - LH-3, Department of Mathematics and Microsoft Teams (online)
The classical theory of Toeplitz operators on Hardy space over the unit disk is a well-developed area in Operator Theory. If we substitute the domain disk $\Delta$ with a bounded multiply connected domain $D$, where $\partial D$ consists of finitely many smooth closed curves, what kinds of difficulties arise? This question motivates us to explore the theory for Toeplitz operators on Hardy space over a multiply connected domain $D$. In 1974, M.B. Abrahamse’s Ph.D. thesis made significant contributions in this topic, extending well-known results for the disk like characterizations of commutator ideals for the Banach Algebra generated by Toeplitz operators with continuous $\mathbb{C}(\partial D)$ or $H^\infty + C(\partial D)$ symbols, and the characterization of Fredholm operators with $H^\infty+C$ symbols to those for the multiply connected domain $D$. Also, he came up with the striking reduction theorem, which roughly says that modulo compact operators, the Toeplitz operator defined on the Hardy space over a multiply connected domain $D$, can be written as the direct sum of Toeplitz operators defined on the Hardy space over the unit disk.
In this talk, we will provide the definition of the Hardy Space $H^p$ over multiple connected domains $D$, where $1 \leq p \leq \infty$, and build some prerequisites to present the aforementioned characterization theorems obtained by Abrahamse for the case of multiple connected domains $D$. We will present the proofs of some of these theorems originally done by Abrahamse. Following that, we will examine the proof of the reduction theorem and explore some of its applications.
Title: APRG Seminar: On Hardy Spaces associated with the Twisted Laplacian and sharp estimates for the corresponding Wave Operator
Speaker: Jotsaroop Kaur (IISER Mohali)
Date: Fri, 21 Jun 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We define atomic Hardy space $H^p_{\mathcal{L}, at}(\mathbb{C}^n), 0<p\leq 1$ for the
twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy
space defined using the maximal function corresponding to the heat
semigroup $e^{-t\mathcal{L}},t>0$. We also prove sharp $L^p, 0<p\leq 1$
estimates for $\left(\mathcal{L}\right)^{-\beta/2}e^{i\sqrt{\mathcal{L}}}$. More
precisely we prove that it is a bounded operator on $H^p_{\mathcal{L},
at}(\mathbb{C}^n)$ when $\beta\geq (2n-1)\left(1/p-1/2\right)$.
Title: PROMYS Guest Lecture: What are Ramsey numbers?
Speaker: Rehana Patel (Wesleyan University, Middletown, USA)
Date: Fri, 14 Jun 2024
Time: 11:45 am
Venue: LH-5, Mathematics Department
Finite Ramsey theory is the study of structure that becomes unavoidable in large finite objects. In this talk, we will provide a brief taste of this rich and beautiful subject. We will start with the following question: In any group of six people, can we always find three who know one another or three who don’t? A far-reaching generalization of this question was first answered in a paper on logic by Frank Ramsey in 1928. Our approach to it will involve graph theory and combinatorics, with a dash of probability. No prerequisites will be needed to understand the talk.
Title: PhD Thesis defence: On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds.
Speaker: P Sivaram (IISc Mathematics)
Date: Fri, 14 Jun 2024
Time: 11 am
Venue: Hybrid - LH-3, Department of Mathematics and Google meet (online)
The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case i.e. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.
In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Goerenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.
Title: Geometry & Topology Seminar: Doing Morse theory without Smale condition
Speaker: Ipsita Datta (ETH Zurich)
Date: Wed, 12 Jun 2024
Time: 04:00 pm
Venue: LH-1 LH-3
We can apply Obstruction bundle gluing introduced by Hutchings and Taubes in Embedded contact homology to the setting of Morse theory. The goal is to understand this gluing technique in a simpler set up and analyse Morse theory with assumptions weaker than Smale condition on surfaces. This is work in progress with Yuan Yao.
Title: PhD Thesis colloquium: The Momentum Construction Method for Higher Extremal Kähler and Conical Higher cscK Metrics
Speaker: Rajas Sandeep Sompurkar (IISc Mathematics)
Date: Fri, 07 Jun 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
This Ph.D. thesis consists of two parts. In both the parts we study two new notions of canonical Kahler metrics introduced by Pingali viz. ‘higher extremal Kahler metric’ and ‘higher constant scalar curvature Kahler (higher cscK) metric’ both of whose definitions are analogous to the definitions of extremal Kahler metric and constant scalar curvature Kähler (cscK) metric respectively. On a compact Kahler manifold a higher extremal Kähler metric is a Kahler metric whose corresponding top Chern form equals its corresponding volume form multiplied by a smooth real-valued function whose gradient is a holomorphic vector field, while a higher cscK metric is a Kahler metric whose top Chern form is a real constant multiple of its volume form or equivalently whose top Chern form is harmonic. In both the parts we consider a special family of minimal ruled complex surfaces called as ‘pseudo-Hirzebruch surfaces’ which are the projective completions of holomorphic line bundles of non-zero degrees over compact Riemann surfaces of genera greater than or equal to two. These surfaces have got some nice symmetries in terms of their fibres and the zero and infinity divisors which enable the use of the momentum construction method of Hwang-Singer (a refinement of the Calabi ansatz procedure) for finding explicit examples of various kinds of canonical metrics on them.
In the first part of this Ph.D. thesis we will prove by using the momentum construction method that on a pseudo-Hirzebruch surface every Kahler class admits a higher extremal Kahler metric which is not a higher cscK metric. The construction of the required metric boils down to solving an ODE depending on a parameter on a closed and bounded interval with some boundary conditions, but the ODE is not directly integrable and requires a very delicate analysis for getting the existence of a solution satisfying all the boundary conditions. Then by doing a certain set of computations involving the top Bando-Futaki invariant we will finally conclude from this that higher cscK metrics (momentum-constructed or otherwise) do not exist in any Kahler class on this Kahler surface. We will briefly see the analogy of this problem with the related problem of constructing extremal Kähler metrics which are not cscK metrics on a pseudo-Hirzebruch surface which has been previously studied by Tønnesen-Friedman and Apostolov et al..
In the second part of this Ph.D. thesis we will see that if we relax the smoothness condition on our metrics a bit and allow our metrics to develop ‘conical singularities’ along at least one of the zero and infinity divisors of a pseudo-Hirzebruch surface then we do get ‘conical higher cscK metrics’ in each Kahler class of the Kahler surface by the momentum construction method. Even in this case the construction of the required metric boils down to solving a very similar ODE on the same interval but with different parameters and slightly different boundary conditions. We will show that our constructed metrics are conical Kahler metrics satisfying the strongest condition for conical metrics viz. the ‘polyhomogeneous condition’ of Jeffres-Mazzeo-Rubinstein, and we will interpret the conical higher cscK equation globally on the surface in terms of currents by using Bedford-Taylor theory. We will then employ the top ‘log Bando-Futaki invariant’ to obtain the linear relationship between the cone angles of the conical singularities of the metrics at the zero and infinity divisors of the surface.
Title: Algebra & Combinatorics Seminar: Ulrich subvarieties and non-existence of low rank Ulrich bundles on complete intersections
Speaker: Debaditya Raychaudhury (University of Arizona, Tucson, USA)
Date: Wed, 05 Jun 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
We characterize the existence of an Ulrich vector bundle on a variety $X\subset{\bf P}^N$ in terms of the existence of a subvariety satisfying certain conditions. Then we use this fact to prove that $(X,\mathcal{O}_X(a))$ where $X$ is a complete intersection of dimension $n\geq 4$, which if n = 4, is either ${\bf P}^4$ with $a\geq 2$, or very general with $a\geq 1$ and not of type (2), (2, 2), does not carry any Ulrich bundles of rank $r\leq 3$. Work in collaboration with A.F. Lopez.
Title: PROMYS Guest Lecture: Sums of n squares
Speaker: Raman Parimala (Emory University, Atlanta, USA)
Date: Mon, 03 Jun 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
We discuss representation of integers as a sum of $n$ squares. We explain the quaternionic composition law for sums of four squares and a proof of a theorem of Lagrange on which positive integers can be expressed as a sum of four squares. We outline general connections to the theory of quadratic forms.
Title: PROMYS Guest Lecture: Counting potato print patterns
Speaker: Tulsi Srinivasan (Azim Premji University, Bengaluru)
Date: Fri, 31 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
You might have carved a piece of potato to create a stamp to print with. How many patterns can we get from a single potato stamp? One approach to answering this question sheds light on the rich connections between objects and their symmetries, and leads us to a more general counting strategy.
Title: APRG Seminar: Rational dilation and constrained algebras
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Fri, 31 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
A set $\Omega$ is a spectral set for an operator $T$ if the spectrum of $T$ is contained in $\Omega$, and von Neumann’s inequality holds for $T$ with respect to the algebra $R(\Omega)$ of rational functions with poles off of the closure of $\Omega$. It is a complete spectral set if for all $n \in \mathbb{N}$, the same is true for $M_n(\mathbb C) \otimes R(\Omega)$. The rational dilation problem asks, if $\Omega$ is a spectral set for $T$, is it a complete spectral set for $T$? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (e.g., over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is an example of a variety naturally associated to a constrained subalgebra of the disk algebra, namely $\mathbb{C} + z^2 A(\mathbb D)$. This talk discusses why rational dilation fails for a large class of such varieties associated to constrained algebras.
Title: APRG Seminar: Products of positive operators
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Thu, 30 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
The study of the class L+2 of Hilbert space operators which are the product of two bounded positive operators first arose in physics in the early ’60s. On finite dimensional Hilbert spaces, it is not hard to see that an operator is in this class if and only if it is similar to a positive operator. We extend the exploration of L+2 to separable infinite dimensional Hilbert spaces, where the structure is much richer, connecting (but not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The (generalized) spectral properties of elements of L+2 are also outlined, as well as membership in L+2 among various special classes of operators, including algebraic and compact operators.
Title: APRG Seminar: The operator Fejer-Riesz theorem in two variables, II
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Wed, 29 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
Here the proof that a positive (but not necessarily strictly positive) trigonometric polynomial with matrix coefficients can be written as a finite sum of hermitian squares of analytic polynomials is sketched. The difficulties in the case of such polynomials with coefficients which are operators on an infinite dimensional Hilbert space is also briefly discussed.
Title: APRG Seminar: The operator Fejer-Riesz theorem in two variables, I
Speaker: Michael Dritschel (Newcastle University, UK)
Date: Tue, 28 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
The Fejer–Riesz thorem states that a positive (i.e., non-negative) trigonometric polynomial of degree $d$ on the unit circle is the hermitian square of an analytic polynomial of the same degree. Rosenblum extended this to polynomials with operator coefficients. The goal of these talks will be to outline a proof of a similar theorem in two variables. Since the techniques used in some proofs of the single variable case play an important role in the two variable proof, this particular talk concentrates primarily on these ideas. An application to strictly positive operator valued multivariable trigonometric polynomials is also considered.
Title: APRG Seminar: The scaling limit of the colored asymmetric simple exclusion process
Speaker: Milind Hegde (Columbia University, New York City, USA)
Date: Fri, 24 May 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
In the colored asymmetric simple exclusion process (ASEP), one places a particle of “color” $-k$ at each integer site $k \in \mathbb{Z}$. Particles attempt to swap places to the left with rate $q \in [0,1)$ and to the right with rate 1; the swap succeeds if the initiating particle has a higher color than the other particle (thus the particles tend to get more ordered over time). We will discuss the space-time scaling limit of this process (as well as a related discrete analog known as the stochastic six-vertex model), captured via a height function given by certain colored particle counts. The limit lies in the Kardar-Parisi-Zhang universality class, and is given by the Airy sheet and directed landscape, which were first constructed in 2018 by Dauvergne-Ortmann-Virág as limits in a very different setting – of fluctuations of a model of a random directed metric. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) with certain Gibbs or spatial Markov properties will play fundamental roles in our discussion. This is based on joint work with Amol Aggarwal and Ivan Corwin.
Title: APRG Seminar: Asymptotic analysis of an optimal control problem for the Stokes system in a perforated domain
Speaker: Bidhan Chandra Sardar (IIT Ropar)
Date: Fri, 24 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
This talk focuses on the asymptotic analysis of an optimal control problem constrained by the stationary Stokes equations within a periodically perforated domain. The problem involves distributive controls applied to the interior region, where the Stokes operator includes oscillating coefficients for the state equations. We aim to demonstrate the convergence of the solutions of the considered optimal control problem to those of the limit OCP governed by the stationary Stokes equations and to establish the convergence of the associated cost functional.
Title: APRG Seminar: Optimal control and homogenization of semi-linear parabolic problems with highly oscillatory coefficients in a pillar-type domain
Speaker: Bidhan Chandra Sardar (IIT Ropar)
Date: Fri, 24 May 2024
Time: 10 am
Venue: LH-1, Mathematics Department
This talk considers an optimal control problem governed by a semi-linear heat equation within a two-dimensional pillar-type domain $\Omega_{\epsilon}$ .The problem features highly oscillatory periodic coefficients in both the state equation and the cost function $A_\epsilon$ and $B_\epsilon$. Our objective is to analyze the convergence of the optimal solutions (as $\epsilon \to 0$ ) and to identify the limit of the optimal control problem in a fixed domain that effectively captures the impact of the oscillatory coefficients.
Title: PhD Thesis colloquium: Holomorphic mappings and Kobayashi geometry of domains
Speaker: Annapurna Banik (IISc Mathematics)
Date: Fri, 24 May 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
In this talk, we shall study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We will focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.
In the initial part of this talk, we shall present a couple of results on local continuous extension of proper holomorphic mappings $F:D \to \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $bD$ and $b\Omega$. These results are motivated by a well-known work by Forstneric–Rosay. However, our results allow us to have much lower regularity, for the patches of $bD, b\Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $b\Omega$ to contain boundary points of infinite type.
We will also discuss another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph–Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds – viewed as complex submanifolds embedded in some ambient complex manifold – with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $b\Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge–Ampere equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called “visibility”.
In the concluding part of this talk, we will explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincare disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali–Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
Title: Algebra & Combinatorics Seminar: Equivariant cohomology in a tensor category
Speaker: Martin Andler (Université de Versailles, France)
Date: Wed, 22 May 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In the 1950s, topologists introduced the notion of equivariant cohomology $H_G(E)$ for a topological space $E$ with an action by a compact group $G$. If the action is free, $H_G(E)$ should be $H(E/G)$, and be computed using de Rham cohomology. In 1950, even before the concept of equivariant cohomology had been formulated, Henri Cartan introduced a complex of equivariant differential forms for a compact Lie group acting on a differential manifold $E$, and proved a result amounting to stating that the cohomology of that complex computes $H_G(E)$. In 1999, Guillemin and Sternberg reformulated Cartan’s work in terms of a supersymmetric extension of the Lie algebra of $G$.
Our aim is to reconsider such considerations, by replacing vector spaces by a $k$-linear symmetric monoidal category, requiring that this category contain an odd unit to account for the supersymmetric dimension plus some further properties, and considering modules of a rigid Lie algebra object in that category. In that context, we obtain a version of Koszul’s homotopy isomorphism theorem, and recover as a consequence some known results as the acyclicity of the Koszul resolution. (Joint work with Siddhartha Sahi.)
Title: The Development of Mathematics in France since the late XIXth Century
Speaker: Martin Andler (Université de Versailles, France)
Date: Wed, 22 May 2024
Time: 11 am
Venue: LH-1, Mathematics Department
The strength of the French mathematical school goes back to the XVIIth century, with major figures like Descartes, Viète, Fermat or Pascal. But history shows that periods when French mathematics thrived alternated with less fruitful times. The factors are diverse, ranging from the role of singular geniuses to social and political causes: structure of higher education and research institutions, political upheavals, wars…
We will focus on the last 150 years: in the period before 1870, Germany had become the dominant scientific power, major advances were being made in England and Italy, to the great dismay of French scientists. In 1870, France was defeated by Prussia, the Emperor was overthrown, and a very favorable period started, allowing the emergence of a remarkable generation of French mathematicians. World War I had a disastrous effect on science (not only on science, of course), abruptly bringing the momentum to a halt. The reconstruction took some years: it is only in the 1950s that French mathematics flourished one again. By that time, however, mathematics had become much more specialized, and applied mathematics were left behind. It took again many years to reach a more balanced landscape where the pure and applied parts can thrive.
Title: PROMYS Guest Lecture: Rounding sequences and counting by congruence classes
Speaker: Caleb Shor (Western New England University, Springfield, USA)
Date: Tue, 21 May 2024
Time: 11:45 am
Venue: LH-1, Mathematics Department
For $n$ a natural number, consider the sequence of $n$ rational numbers $n/1, n/2, n/3, \dots, n/n$. Round each to the nearest integer to obtain sequence of $n$ integers. How many are odd?
In this talk we will see how knowledge of sums of squares and a result of Gauss will help to lead us to a somewhat surprising result. Time permitting, we will discuss similar results.
Title: PROMYS Guest Lecture: Who cut my cheese?
Speaker: Purvi Gupta (IISc Mathematics)
Date: Fri, 17 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Given a box packed with identical cubes of cheese, what is the maximum damage one can cause with a single straight cut through the box? This seemingly simple puzzle represents an old but recurrent mathematical theme that slices through numerous fields such as number theory, functional analysis, probability theory, and computational complexity theory. The cross-sections of convex bodies hold many mysteries, some of which continue to puzzle mathematicians today. We will focus on the deceptively simple case of the cube to demonstrate some of these ideas and open questions. No cheese will be harmed in the making of this talk.
Title: Geometry & Topology Seminar: Constructing smoothings of stable maps
Speaker: Mohan Swaminathan (Stanford University)
Date: Fri, 17 May 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
For positive integers $n$, $g$ and $d$, the moduli space $M(n,g,d)$ of degree d holomorphic maps to $\mathbb{CP}^n$ from non-singular projective curves of genus g is smooth and irreducible for $d > 2g-2.$ It is contained as an open subset within the compact moduli space $K(n,g,d)$ of “stable maps”, i.e., degree d holomorphic maps to $\mathbb{CP}^n$ from at-worst-nodal projective curves of arithmetic genus $g.$ An unfortunate feature of this very natural compactification is that $M(n,g,d)$ is far from being dense in $K(n,g,d)$. Concretely, this means that many stable maps are not “smoothable”, i.e., they don’t arise as limits of non-singular ones. In my talk, I will explain this phenomenon and a new sufficient condition for smoothability of stable maps, obtained in joint work with Fatemeh Rezaee.
Title: PhD Thesis colloquium: On Two complex Hessian equations and convergence of corresponding flows
Speaker: Ramesh Mete (IISc Mathematics)
Date: Fri, 10 May 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
This dissertation consists of three parts, and two important types of complex hessian equations, namely – the J-equation and the deformed Hermitian Yang Mills (dHYM) equation.
In the first part, the main aim is to find out some appropriate “singular” solutions of the equations when they don’t admit smooth solutions (or equivalently, when the so-called “Nakai criteria” fails) - this is the so-called unstable case. An algebro-geometric characterization of the slopes for both the equations is formulated – which we call the “minimal J-slope” in the case of the J-equation and the “maximal dHYM-slope” for the dHYM equation. On compact Kahler surfaces we show that there exists a unique closed $(1,1)$- Kahler current that solves the “weak version” of the equations (i.e. the wedge product is replaced by the “non-pluripolar product”) with the modified slopes. In the higher dimensional case, we conjecture analogous existence and uniqueness results.
In the second part, the convergence behavior of the J-flow is studied on certain generalized projective bundles using Calabi symmetry. For the bundles an “invariant version” of the minimal J-slope is introduced. Furthermore, we show that the flow converges to some unique limit in the weak sense of currents, and the limiting current solves the J-equation with the invariant minimal J-slope. This result resolves our conjecture for J-equation on these examples with symmetry.
In the third part, the convergence behavior of a dHYM flow, called the “cotangent flow”, is studied in the unstable case on the blowup of $\mathbb{CP}^2$ or $\mathbb{CP}^3$. Analogous to our results in the second part, it is shown that this flow converges to some unique limit in the unstable case, and the limiting current solves the dHYM equation with the (invariant) maximal dHYM-slope.
Title: PROMYS Guest Lecture: Are we there yet?
Speaker: Jaclyn Lang (Temple University, Philadelphia, USA)
Date: Fri, 10 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
How far is it from Bangalore to Chennai? Is there a single correct answer to this question? In this talk we will explore different notions of distance as well as why you might choose one over another depending on the context. This will take us on a brief sight seeing tour through geometry, graph theory, and number theory.
Title: Geometry & Topology Seminar: Spectral Theory of the Sub-Riemannian Laplacian
Speaker: Nikhil Savale (University of Cologne)
Date: Mon, 22 Apr 2024
Time: 04:00 pm
Venue: LH-1
Sub-Riemannian (sR) geometry is the geometry of bracket-generating metric distributions on a manifold. Peculiar phenomena in sR geometry include the exotic Hausdorff dimension describing the growth rate of the volumes of geodesic balls. As well as abnormal geodesics that do not satisfy any variational equation. In this talk I will survey my results, to appear in a forthcoming book, which show how both these phenomena are reflected in the spectral theory of the hypoelliptic Laplacian in sR geometry.
Title: MS Thesis colloquium: Higher categories of n-bordisms
Speaker: Arghan Dutta (IISc Mathematics)
Date: Fri, 19 Apr 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.
Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.
One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category $\mathrm{Bord}_n^{(n -1)}$, a variant of the fully extended $\mathrm{Bord}_{n}$. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.
Title: PhD Thesis colloquium: Invariants Associated with Complete Nevanlinna-Pick Spaces
Speaker: Abhay Jindal (IISc Mathematics)
Date: Thu, 18 Apr 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we shall talk about two invariants associated with complete Nevanlinna-Pick (CNP) spaces. One of the invariants is an operator-valued multiplier of a given CNP space, and another invariant is a positive real number. These two invariants are called characteristic function and curvature invariant, respectively. The origin of these concepts can be traced back to the classical theory of contractions by Sz.-Nagy and Foias.
We extend the theory of Sz.-Nagy and Foias about the characteristic function of a contraction to a commuting tuple $(T_{1}, \dots, T_{d})$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. Surprisingly, there is a converse, which roughly says that if a kernel $k$ admits a characteristic function, then it has to be an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characterization explains, among other things, why in the literature an analogue of the characteristic function for a Bergman contraction ($1/k$-contraction where $k$ is the Bergman kernel), when viewed as a multiplier between two vector valued reproducing kernel Hilbert spaces, requires a different (vector valued) reproducing kernel Hilbert space as the domain. So, what can be said if $(T_{1}, \dots,T_{d})$ is $1/k$-contractive when $k$ is an irreducible unitarily invariant kernel, but does not have the complete Nevanlinna-Pick property? We shall see that if $k$ has a complete Nevanlinna-Pick factor $s$, then much can be retrieved.
We associate with a $1/k$-contraction its curvature invariant. The instrument that makes this possible is the characteristic function. We present an asymptotic formula for the curvature invariant. In the special case when the $1/k$-contraction is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of the $1/k$-contraction specifically when its characteristic function is a polynomial.
Title: PhD Thesis colloquium: Maps between Non-compact Surfaces
Speaker: Sumanta Das (IISc Mathematics)
Date: Wed, 17 Apr 2024
Time: 4 pm
Venue: LH-1, Department of Mathematics
In this talk, we discuss proper maps between two non-compact surfaces, with a particular emphasis on facts stemming from two fundamental questions in topology: whether every homotopy equivalence between two $n$-manifolds is homotopic to a homeomorphism, and whether every degree-one self-map of an oriented manifold is a homotopy equivalence.
Topological rigidity is the property that every homotopy equivalence between two closed $n$-manifolds is homotopic to a homeomorphism. This property refines the notion of homotopy equivalence, implying homeomorphism for a particular class of spaces. According to Nielsen’s results from the 1920s, compact surfaces exhibit topological rigidity. However, topological rigidity fails in dimensions three and above, as well as for compact bordered surfaces.
We prove that all non-compact, orientable surfaces are properly rigid. In fact, we prove a stronger result: if a homotopy equivalence between any two noncompact, orientable surfaces is a proper map, then it is properly homotopic to a homeomorphism, provided that the surfaces are neither the plane nor the punctured plane. As an application, we also prove that any $\pi_1$-injective proper map between two non-compact surfaces is properly homotopic to a finite-sheeted covering map, given that the surfaces are neither the plane nor the punctured plane.
An oriented manifold $M$ is said to be Hopfian if every self-map $f\colon M\to M$ of degree one is a homotopy equivalence. This is the natural topological analog of Hopfian groups. H. Hopf posed the question of whether every closed, oriented manifold is Hopfian. We prove that every oriented infinite-type surface is non-Hopfian. Consequently, an oriented surface $S$ is of finite type if and only if every proper self-map of $S$ of degree one is homotopic to a homeomorphism.
Title: Number Theory Seminar: Zig-zag holds for Galois representations
Speaker: Eknath Ghate (TIFR Mumbai)
Date: Fri, 12 Apr 2024
Time: 2 pm
Venue: LH-1
It is an open problem to describe the shape of the reductions of local Galois representations attached to cusp forms at primes away from the level, or more generally, the shape of the reductions of two-dimensional crystalline representations. Partial results go back to Deligne, Fontaine and Edixhoven. One folklore conjecture (attributed to Breuil, Buzzard and Emerton) is that if the weight is even and the slope is fractional, then the reduction is always irreducible.
In this talk we shall state and prove our zig-zag conjecture which deals with large exceptional weights and half-integral slopes. These weights fall squarely outside the scope of the above conjecture. The conjecture states that the reduction in these cases is given by an alternating sequence of irreducible and reducible representations depending on the size of two auxiliary parameters. Special cases of zig-zag have been proved over the years by various authors using Langlands correspondences.
The present general proof uses the reverse of a recent limiting argument due to Chitrao-Ghate-Yasuda in the Colmez-Chenevier rigid analytic blow up space of trianguline representations to reduce the study of the reduction of crystalline representations to results on the reductions of semi-stable representations due to Breuil-Mezard, Guerberoff-Park and most recently Chitrao-Ghate.
Title: APRG Seminar: Pointwise localisation and weighted inequalities for Rubio de Francia square function
Speaker: Luz Roncal (BCAM - Basque Center for Applied Mathematics, Spain)
Date: Wed, 10 Apr 2024
Time: 4 pm
Venue: Microsoft Teams (online)
Let us denote by Rubio de Francia square function the square function formed by frequency projections on a collection of disjoint intervals of the real line. J. L. Rubio de Francia established in 1985 that this operator is bounded on $L^p$ for $p\ge 2$ and on $L^p(w)$, for $p>2$, with weights $w$ in the Muckenhoupt class $A_{p/2}$. What happens in the endpoint $L^1(w)$ for $w \in A_1$ was left open, and Rubio de Francia conjectured the validity of the estimate in this endpoint.
In this talk we will show a new pointwise sparse estimate for the Rubio de Francia square function. Such a bound implies quantitative weighted estimates which, in some cases, improve the available results. We will also confirm that the $L^2(w)$ conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue to the Rubio de Francia square function. In general, the $L^2$ weighted inequality is still an open problem.
Joint work with Francesco Di Plinio, Mikel Flórez-Amatriain, and Ioannis Parissis.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: Epimorphism theorems and allied topics
Speaker: Amartya Kumar Dutta (ISI Kolkata)
Date: Tue, 09 Apr 2024
Time: 11:30 am
Venue: LH-1, Mathematics Department
In the area of Affine Algebraic Geometry, there are several problems on polynomial rings which are easy to state but difficult to investigate. Late Shreeram S. Abhyankar was the pioneer in investigating a class of such problems known as Epimorphism Problems or Embedding Problems. In this non-technical survey talk, we shall highlight some of the contributions of Abhyankar, Moh, Suzuki, Sathaye, Russell, Bhatwadekar and other mathematicians.
Title: In praise of Khahara
Speaker: Amartya Kumar Dutta (ISI Kolkata)
Date: Mon, 08 Apr 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
It was in ancient India that zero received its first acceptance as an integer in its own right. There was an awareness about its perils and yet ancient Indian mathematicians not only embraced zero as an integer but allowed it to participate in all four arithmetic operations, including as a divisor in a division.
But division by zero is strictly forbidden in the present edifice of mathematics. Verses from mathematical giants like Brahmagupta and Bhaskaracharya referring to numbers with “zero in the denominator” shock the modern reader. Certain examples in the Bijaganita of Bhaskaracharya appear as absurd nonsense.
But then there was a time when square roots of negative numbers were considered non-existent and forbidden; even the validity of subtracting a bigger number from a smaller number (i.e., the existence of negative numbers) took a long time to gain universal acceptance. Is it possible that we have simply bound ourselves to a certain safe convention and that there could be other approaches (“mathematical models” in fancy language) in which the ideas of Brahmagupta and Bhaskaracharya, and even the examples of Bhaskaracharya, will appear not only valid but even natural?
Enterprising modern mathematicians have created elaborate legal (or technical) machinery to overcome the limitations imposed by the prohibition against use of zero in the denominator. The most familiar are the methods of calculus with its concept of limit, results like l’Hopital’s rule, and a language which enables one to express intuitive ideas like $\frac{1}{0} = \infty$ through legally permitted euphemisms. Less well-known are the devices of commutative algebra and algebraic geometry like “localisation” which describes a legal structure for directly writing fractions with zero in the denominator without any subterfuge, and the more sophisticated ideas of “valuation theory” which admit multiple levels of infinities and thereby provide higher-dimensional analogues of l’Hopital’s rule.
In this talk we shall present an algebraic model proposed by Prof. Avinash Sathaye for understanding Bhaskaracharya’s treatment of khahara, (numbers with) zero in the denominator. A crucial ingredient of this model is the important concept of “idempotent” in modern abstract algebra (elements $e$ satisfying the relation $e^2=e$). To historians of mathematics who have tried to interpret Bhaskaracharya’s khahara in the light of calculus, the examples in Bijaganiita appear as absurdities. But all difficulties disappear in the light of the algebraic treatment based on idempotents. A verse from the commentary of Kr.s.n.adaivaj˜na indicates that idempotence was indeed envisaged as a natural property of numbers like zero and its reciprocal, the khahara.
Prof. Sathaye’s interpretation of Bhaskaracharya’s khahara also gives a new meaning to certain mysterious utterances of Ramanujan recorded by P.C. Mahalanobis. In the light of valuation theory, Bhaskaracharya’s khahara not only deserves our praise, perhaps they indicate unexplored possibilities!
Title: Eigenfunctions Seminar: Brownian objects
Speaker: Sanchayan Sen (IISc Mathematics)
Date: Fri, 05 Apr 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The Brownian motion is the scaling limit of random walks where the step distribution has finite second moment. Various random objects constructed from the Brownian motion, e.g., the Brownian continuum random tree and the Brownian map, arise naturally in the study of random trees, graphs, and maps. In the first talk, we will give a gentle introduction to these objects. In the second talk, we will discuss some recent advances in establishing certain Brownian objects as the scaling limits of different models of random discrete structures.
Title: PhD Thesis defence: Dominating surface-group representations via Fock-Goncharov coordinates
Speaker: Pabitra Barman (IISc Mathematics)
Date: Thu, 04 Apr 2024
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
Let $S_{g,k}$ be a connected oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S_{g,k})$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher-rank representations.
For a representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, the Hilbert length of a curve $\gamma\in \pi_1(S_{g,k})$ for $n >2$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock-Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$ representation. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.
Let $X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \tau(A)= \sum_{i=1}^{n}\log |\lambda_i(A)|^2, \end{equation} where $\lambda_i(A)$ are the eigenvalues of $A$. We show that the same $j$ dominates $\rho$ with respect to the translation length at the origin as well. Lindström’s Lemma for planar networks and Weyl’s Majorant Theorem are some of the key ingredients of the proof.
In both cases, if $S_{g,k}$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.
Title: PhD Thesis defence: Geometry of Normed Linear Spaces in Light of Birkhoff-James Orthogonality
Speaker: Babhrubahan Bose (IISc Mathematics)
Date: Thu, 04 Apr 2024
Time: 3 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.
In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.
We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.
We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.
Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.
Title: Algebra & Combinatorics Seminar: Approximate packing of independent transversals in locally sparse graphs
Speaker: Debsoumya Chakraborti (Mathematics Institute, University of Warwick, UK)
Date: Wed, 03 Apr 2024
Time: 4 pm
Venue: LH-1, Mathematics Department (Joint with the APRG Seminar)
Consider a multipartite graph $G$ with maximum degree at most $n-o(n)$, parts $V_1,\ldots,V_k$ have size $|V_i|=n$, and every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such $G$ has an independent set, referred to as an ‘independent transversal’, which contains exactly one vertex from each part $V_i$. They further conjectured that the vertex set of $G$ can be decomposed into pairwise disjoint independent transversals. We resolve this conjecture approximately by showing that $G$ contains $n-o(n)$ pairwise disjoint independent transversals. As applications, we give approximate answers to questions on packing list colorings and multipartite Hajnal-Szemerédi theorem. We use probabilistic methods, including a ‘two-layer nibble’ argument. This talk is based on joint work with Tuan Tran.
Title: PhD Thesis defence: Some aspects of weighted kernel functions on planar domains
Speaker: Aakanksha Jain (IISc Mathematics)
Date: Mon, 01 Apr 2024
Time: 4:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.
For a planar domain and an admissible weight function on it, we discuss some aspects of the corresponding weighted Bergman kernel. First, we see a precise relation between the weighted Bergman kernel and the classical Bergman kernel near a smooth boundary point of the domain. Second, the weighted kernel gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the unit disc with respect to the weighted Bergman metrics arising from weights of the form $K(z,z)^{-d}$, where $K$ denotes the classical Bergman kernel and $d$ is a non-negative integer. Specific examples that we discuss in detail include those in which the isometry takes values in polydisk or a cartesian product of a disc and a unit ball, where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.
In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.
Title: PhD Thesis colloquium: On regular subalgebras of Kac-Moody algebras and related combinatorics
Speaker: Md. Irfan Habib (IISc Mathematics)
Date: Thu, 28 Mar 2024
Time: 3 pm
Venue: Hybrid - LH-1, Department of Mathematics and Microsoft Teams (online)
In this thesis, we study two aspects of Kac-Moody algebras. One is to understand the subalgebras that can be embedded inside a Kac-Moody algebra as subalgebras generated by real root vectors. The other one is to explicitly classify the regular subalgebras and the maximal regular subalgebras of an untwisted affine Kac-Moody algebra.
Dynkin classified the semisimple regular subalgebras of a finite-dimensional semisimple Lie algebra back in $1949.$ One of the key tools he used for the classification is $\pi$-systems. For non-finite Kac-Moody algebras, $\pi$-system became an integral part of und erstanding the embedding of different types of algebras in a Kac-Moody algebra. Till now all the articles existing in the literature, which study $\pi$-systems, assume that the $\pi$-systems are either linearly independent or finite. It seems that our work is the first one to address infinite $\pi$ systems in the context of the embedding problem. This paves a way for us to understand the infinite (linearly independent) $\pi$-systems for Borcherds Kac-Moody algebras and understand the embedding problem in that setting. We used Deodhar’s preorder to prove that every closed subroot system in a Kac-Moody root system admits a $\pi$-system and this $\pi$-system need not be finite in general. Moreover, for any closed subroot system $\Psi$ of $\Delta,$ we prove that there exists a unique $\pi$-system $\Pi(\Psi),$ which is contained in the set of positive roots. Since the subroot systems of a root system are not very ‘well behaved’, this is quite surprising and it generalizes the previously well-known fact that they simple systems and positive systems determine each other at the level of the subroot system.
Using this unique $\pi$-system $\Pi(\Psi)$, we prove that for a real closed subroot system $\Psi,$ the real roots of a root generated subalgebra $\mathfrak g(\Psi)$ is equal to $\Psi$. This result was a much-awaited one in the literature because almost after $70$ years of Dynkin’s result, Roy and Venkatesh (Transform. Groups 2019) proved that the same is true for an affine root system. These two results provide a bridge between the algebraic and combinatorial side which shows that the root-generated subalgebras are in bijection with the real closed subroot systems which are in turn in one-to-one correspondence with the $\pi$-systems contained in the positive roots of a Kac-Moody algebra.
In the last part of our analysis of regular subalgebras generated by root vectors, we prove that for any closed subroot system $\Psi,$ the root generated subalgebra is isomorphic to a quotient of the derived subalgebra of the Kac-Moody algebra corresponding to the (infinite) Cartan matrix defined by the unique $\pi$-system of the closed subroot system $\Psi,$ by an ideal contained in the centre of the algebra. This result is a generalization of the existing results when the $\pi$-system is linearly independent and the ideal is zero when the $\pi$-system is linearly independent also follows from our result. In particular, as long as the roots are concerned, to understand the root generated subalgebras, it is enough to consider the derived algebras of Kac-Moody algebras $\mathfrak g’(A)$ corresponding to a(n infinite) GCM $A.$ Classification of regular subalgebras of an affine Kac-Moody Lie algebra is an interesting problem in its own right. Barnea et al. started such classification in $1998.$ Later Felikson et al. used combinatorics of root systems to classify the regular subalgebras in 2008, more precisely the root generated subalgebras of an affine Kac-Moody algebra. We took a completely different approach, namely, using the classification of the closed subroot system of a real affine root system given by Roy and Venkatesh, we classify the regular subalgebras of affine Kac-Moody Lie algebras with a symmetric set of roots and we get the classification of root generated subalgebras as a Corollary. Moreover, we also classify the maximal symmetric regular subalgebras we show a bijective correspondence between the maximal real closed subroot systems of the affine Lie algebra and the maximal symmetric regular subalgebras different from $[\mathfrak g,\mathfrak g].$ Which also shows that in the affine case, given a maximal closed subroot system $\Psi$ of $\Delta,$ the poset (with set inclusion as the partial order)
\begin{equation} A_\Psi:={\mathfrak s:\Delta(\mathfrak s)^{\mathrm{re}}=\Psi} \end{equation}
contains a unique maximal element.
Title: APRG Seminar: Restricted mean value property on Riemannian manifolds
Speaker: Utsav Dewan (ISI Kolkata)
Date: Wed, 27 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
A continuous function in an Euclidean domain is harmonic if and only if it satisfies the spherical mean value property for all spheres contained in that domain. But what happens if a continuous function satisfies instead the following ‘restricted mean value property’: for each point in the domain it satisfies the mean value property precisely on one such sphere (centered at the point). Then is the function still going to be harmonic? This is the classical ‘one-circle problem’ posed by Littlewood. We will see some results dealing with sufficient conditions in terms of the boundary behavior of the function for the above problem to have an affirmative answer in the setting of (1) domains in Riemannian manifolds and (2) Hadamard manifolds of pinched negative sectional curvature, extending classical results of Fenton for the Euclidean unit disc. This is based on a joint work with Prof. Kingshook Biswas.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Poles of Quotients of $L$-functions and primitivity of $L$-functions of $\mathrm{GL}_3$
Speaker: Ravi Raghunathan (IIT Bombay)
Date: Fri, 22 Mar 2024
Time: 2 pm
Venue: LH-1
Given two distinct cuspidal automorphic $L$-functions (of $\mathrm{GL}_n$ and $\mathrm{GL}_m$ over $\mathbb{Q}$) one expects that their quotients will have infinitely poles, but this is surprisingly hard to prove. In this talk, I will discuss my recent work on the case $m=n-2$ and the primitivity of the $L$-functions of cuspidal automorphic $L$-functions of $\mathrm{GL}_3$. These methods also work for Artin $L$-functions and, more generally, for the $L$-functions of Galois representations under further hypotheses.
Title: Number Theory Seminar: On values of the digamma function
Speaker: Siddhi Pathak (CMI)
Date: Fri, 15 Mar 2024
Time: 2 pm
Venue: LH-1
One of the central themes in number theory is the study of special values of $L$-functions, in particular, the investigation of their transcendental nature and algebraic relations among them. A special function governing linear relations among the values $L(1,\chi)$ as $\chi$ varies over Dirichlet characters modulo $q$, is the digamma function, which is the logarithmic derivative of the gamma function. In this talk, we discuss the arithmetic nature and related properties of values of the digamma function at rational arguments, and emphasize their connection with a seemingly unrelated conjecture of Erdos, which is still open.
Title: APRG Seminar: Hardy and Rellich identities and inequalities for Grushin operators via spherical vector fields and Bessel pairs
Speaker: Jotsaroop Kaur (IISER Mohali)
Date: Wed, 13 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
We establish Hardy, Hardy-Rellich and Rellich identities and inequalities with sharp constants for Grushin vector fields. We provide explicit remainder terms which substantially improve those known in the literature. This is based on a joint work with Debdip Ganguly and Prasun Roychowdhury.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: $p$-adic deformations of automorphic forms and Iwasawa Theory
Speaker: Mladen Dimitrov (University of Lille, France)
Date: Thu, 07 Mar 2024
Time: 11.30 am
Venue: LH-1
Families of $p$-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing the correct setup for the study of $p$-adic deformations of automorphic forms. In order to obtain arithmetic applications, such as constructing $p$-adic $L$-functions or proving explicit reciprocity laws for Euler systems, one needs to perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the $p$-stabilization of the automorphic form we are interested in.
While the geometry of an eigenvariety at points of cohomological weight is well understood thanks to classicality results, the study at classical points which are limit of discrete series (such as weight $1$ Hilbert modular forms or weight $(2,2)$ Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch–Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjecture.
Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint $p$-adic $L$-functions.
In this talk we will illustrate this philosophy based on ideas of Joël Bellaïche.
Title: APRG Seminar: Singular integral operators along codimension one subspaces depending on n-1 variables
Speaker: Mikel Florez Amatriain (BCAM – Basque Center for Applied Mathematics, Spain)
Date: Wed, 06 Mar 2024
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, I will report a work in progress in which we show $L^p$ bounds for singular integral operators formed by $(n-1)$-dimensional Hörmander-Mihlin multipliers. In our case, the multipliers act depending on $(n-1)$-dimensional variable subspaces, which depend only on the first $n-1$ variables.
We prove $L^p$ boundedness for these operators for $p>3/2$. Assuming that the frequency support of the function is contained in an annulus, we can show $L^p$ boundedness for $p>1$.
The video of this talk is available on the IISc Math Department channel.
Title: Eigenfunctions Seminar: Waring's problem
Speaker: Ramachandran Balasubramanian (IMSc, Chennai)
Date: Fri, 01 Mar 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Define $g(k) = \min \{ s :$ every positive integer can be written as a sum of $k$th powers of natural numbers with atmost $s$ summands$\}$. Lagrange proved that $g(2) = 4$. Waring conjectured that $g(3) = 9, g(4) = 19$ and so on.
In fact, in this question, there has been a lot of contribution from Indian mathematicians. The method of attacking this problem is called the circle method and it originates from a seminal paper of Hardy and Ramanujan. The final result owes a lot to the contributions of S.S. Pillai. The analogous question over number fields was settled by C.P. Ramanujam. We shall explain their contributions toward this problem.
Title: Number Theory Seminar: Linear congruence relations for exponents of Borcherds products
Speaker: Badri Vishal Pandey (University of Cologne, Germany)
Date: Fri, 01 Mar 2024
Time: 10 am
Venue: LH-1
For all positive powers of primes $p \geq 5$, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic modular form or a suitable vector-valued harmonic Maassform. To this end, we work with the logarithmic derivatives of these twisted Borcherds products, and offer various numerical examples of non-trivial linear congruences between them modulo $p=11$. In the case of positive powers of primes $p = 2, 3$, we obtain similar results by multiplying the logarithmic derivative with a Hilbert class polynomial as well as a power of the modular discriminant function. Both results confirm a speculation by Ono. (joint work with Andreas Mono).
Title: APRG Seminar: Boundedness of pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane SL(2)/SO(2)
Speaker: Tapendu Rana (Ghent University, Belgium)
Date: Wed, 28 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
For a given function $a(x,\xi)$ on $\mathbb{R}^n \times \mathbb{R}^n$, consider the pseudo-differential operator $a(x,D)$ defined by
\begin{equation} a(x,D) (f)(x) =\int_{\mathbb{R}^n} a(x,\xi) \widehat f(\xi) e^{2\pi i x\cdot \xi} d\xi, \end{equation}
where $\widehat{f}$ denotes the Fourier transform of a function $f$. Let $S^0$ be the set of all smooth functions $a: \mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{C}$ satisfying
\begin{equation} \left| \frac{\partial^\beta_x}{\partial_{\xi} ^\alpha} a (x,\xi)\right| \leq {C_{\alpha,\beta} }\, {( 1+ |\xi| )^{-|\alpha|}} \end{equation}
for all $x,\xi \in \mathbb{R}^n $ and for all multi indices $\alpha$ and $\beta$. It is well known that for $a\in S^0$, the associated
pseudo-differential operator $a(x,D)$ extends to a bounded operator on $L^p(\mathbb{R}^n)$ to itself, for $1<p<\infty$.
In this talk, we will discuss an analogue of this result on radial sections of line bundles over the Poincaré upper half plane. More precisely, we will focus on the group $G=\mathrm{SL}(2,\mathbb{R})$, where we will explore the boundedness properties of the pseudo-differential operator defined on functions of fixed $K=\mathrm{SO}(2)$-type in $G$. Additionally, we will explore the case where the symbol exhibits restricted regularity in the spatial variable.
This talk is based on a joint work with Michael Ruzhansky.
The video of this talk is available on the IISc Math Department channel.
Title: Algebra & Combinatorics Seminar: Monomial expansions for q-Whittaker polynomials
Speaker: Ratheesh T V (IMSc Chennai)
Date: Fri, 23 Feb 2024
Time: 3 pm
Venue: LH-1, Mathematics Department
We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$- and $q$-weights, and which are compatible with natural projection and branching maps. We apply this to the limit construction of local Weyl modules and obtain a new character formula for the basic representation of $\widehat{\mathfrak{sl}_n}$.
Title: APRG Seminar: Counting rational points near flat hypersurfaces
Speaker: Rajula Srivastava (Hausdorff Center / Universitaet Bonn / Max Planck Institute, Germany)
Date: Wed, 21 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
How many rational points with denominator of a given size lie within a given distance from a compact hypersurface? In this talk, we shall describe how the geometry of the surface plays a key role in determining this count, and present a heuristic for the same. In a recent breakthrough, J.J. Huang proved that this guess is indeed true for hypersurfaces with non-vanishing Gaussian curvature. What about hypersurfaces with curvature only vanishing up to a finite order, at a single point? We shall offer a new heuristic in this regime which also incorporates the contribution arising from “local flatness”. Further, we will describe how several ideas from Harmonic Analysis can be used to establish the indicated estimates for hypersurfaces of this type immersed by homogeneous functions. Based on joint work with N. Technau.
The video of this talk is available on the IISc Math Department channel.
Title: APRG Seminar: A note on Linear Complementarity via zero sum two person games
Speaker: T.E.S. Raghavan (University of Illinois at Chicago, USA)
Date: Tue, 20 Feb 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
The matrix $M$ of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) The value of $M$ is equal to zero. (ii) For all principal minors of $M^T$ (transpose of $M$) the value is non-negative. (iii) For any optimal mixed strategy $y$ of the maximizer either $y_i>0$ or $(My)_i>0$ for each coordinate $i$.
Title: Eigenfunctions Seminar: Nonlinear Calderon-Zygmund theory and differential forms
Speaker: Swarnendu Sil (IISc Mathematics)
Date: Fri, 16 Feb 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
For any $1 < q <\infty,$ standard representation formulas and the Calderon–Zygmund estimates imply $u \in W^{2,q}_{\text{loc}}\left(\mathbb{R}^{n}\right)$ if $\Delta u \in L^{q}_{\text{loc}}(\mathbb{R}^{n}).$ Combined with the Sobolev–Morrey embeddings for $q>n,$ we deduce that $\nabla u$ is locally Hölder continuous. However, as soon as we pass from the linear case to the quasilinear operator, we no longer have any representation formula for the solution of the following problem \begin{equation} {-}{\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) = f \end{equation} if $p \neq 2$ and CZ estimates for second derivatives of the solution are not yet known. In fact, the solution can fail to be $C^{2}$ even when $f \equiv 0.$
However, one can still establish Hölder continuity of the gradient whenever
${\rm div}\left(\left\lvert \nabla u \right\rvert^{p-2}\nabla u\right) \in L^{q}_{\text{loc}}$ and $q>n.$ These type of results are often called
“Nonlinear Calderon–Zygmund theory”, as the regularity for the gradient is the same, i.e. “as if” Calderon–Zygmund estimates for second derivatives are
valid! This result relies heavily on a fundamental regularity result, commonly known as the DeGiorgi–Nash–Moser estimate, for $p$-harmonic functions.
However, such regularity results are specific to equations and are in general false for elliptic systems. In another groundbreaking work, Uhlenbeck extended
gradient Hölder continuity estimates for solutions to special type of systems, which includes the homogeneous $p$-Laplacian systems.
In this lecture, I would sketch the main ideas involved to establish nonlinear Calderon–Zygmund theory for scalar equations and elliptic systems with Uhlenbeck
structures. In the second half, I would discuss how to extend these estimates to the following $p$-Laplacian type system for vector-valued differential forms
\begin{equation}
d^{\ast}\left(\left\lvert d u \right\rvert^{p-2}d u\right) = f.
\end{equation}
This includes systems which are, strictly speaking, even non-elliptic.
Title: APRG Seminar: The p-Laplace "Signature" for Quasilinear Inverse Problems with Large Boundary Data
Speaker: Ravi Prakash (Universidad de Concepcion, Concepcion, Chile)
Date: Thu, 15 Feb 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
This Talk is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of developments, although breakthroughs are expected in the not-too-distant future.
Nonlinear constitutive relationships which, at a given point in the space, present a behaviour for large arguments that is described by monomials of order $p$ and $q$ is considered in this presentation.
The original contribution this work makes is that the nonlinear problem can be approximated by a weighted $p$-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the $p$-Laplacian in inverse problems with nonlinear materials. Moreover, when $p=2$, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials.
The main result of this work is that for “large” Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted $p$-Laplace problem.
Title: APRG Seminar: Elliptic operators in rough domains
Speaker: José María Martell Berrocal (ICMAT, Madrid, Spain)
Date: Thu, 15 Feb 2024
Time: 2 pm
Venue: Microsoft Teams (online)
Solvability of the Dirichlet problem with data in $L^p$ for some finite $p$ for elliptic operators, such as the Laplacian, amounts to showing that the associated elliptic/harmonic measure satisfies a Reverse Hölder inequality. Under strong connectivity assumptions, it has been proved that such a solvability is equivalent to the fact that that all bounded null-solutions of the operator in question satisfy Carleson measure estimates. In this talk, we will give a historical overview of this theory and present some recent results in collaboration with M. Cao and P. Hidalgo where, without any connectivity, we characterize certain weak Carleson measure estimates for bounded null-solutions in terms of a Corona decomposition for the elliptic measure. This extends the previous theory to non-connected settings where, as a consequence of our method, we establish Fefferman-Kenig-Pipher perturbation results.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Whittaker functions on covers of p-adic groups and quantum groups at roots of unity
Speaker: Manish Patnaik (University of Alberta, Edmonton, Canada)
Date: Fri, 09 Feb 2024
Time: 2 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Since the work of Kubota in the late 1960s, it has been known that certain Gauss sum twisted (multiple) Dirichlet series are closely connected to a theory of automorphic functions on metaplectic covering groups. The representation theory of such covering groups was then initiated by Kazhdan and Patterson in the 1980s, who emphasized the role of a certain non-uniqueness of Whitattaker functionals.
Motivated on the one hand by the recent theory of Weyl group multiple Dirichlet series, and on the other by the so-called “quantum” geometric Langlands correspondence, we explain how to connect the representation theory of metaplectic covers of $p$-adic groups to an object of rather disparate origin, namely a quantum group at a root of unity. This gives us a new point of view on the non-uniqueness of Whittaker functionals and leads, among other things, to a Casselman–Shalika type formula expressed in terms of (Gauss sum) twists of “$q$”-Littlewood–Richardson coefficients, objects of some combinatorial interest.
Joint work with Valentin Buciumas.
Title: PhD Thesis colloquium: On existence and regularity of some complex Hessian equations on Kahler and transverse Kahler manifolds
Speaker: P. Sivaram (IISc Mathematics)
Date: Thu, 08 Feb 2024
Time: 3:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3 LH-1, Mathematics Department
The thesis consists of two parts. In the first part, we study the modified J-flow, introduced by Li-Shi. Analogous to the Lejmi-Szekelyhidi conjecture for the J-equation, Takahashi has conjectured that the solution to the modified J-equation exists if and only if some intersection numbers are positive and has verified the conjecture for toric manifolds. We study the behaviour of the modified J-flow on the blow-up of the projective spaces for rotationally symmetric metrics using the Calabi ansatz and obtain another proof of Takahashi’s conjecture in this special case. Furthermore, we also study the blow-up behaviour of the flow in the unstable case ie. when the positivity conditions fail. We prove that the flow develops singularities along a co-dimension one sub-variety. Moreover, away from this singular set, the flow converges to a solution of the modified J-equation, albeit with a different slope.
In the second part, we will describe a new proof of the regularity of conical Ricci flat metrics on Q-Gorenstein T-varieties. Such metrics arise naturally as singular models for Gromov-Hausdorff limits of Kahler-Einstein manifolds. The regularity result was first proved by Berman for toric manifolds and by Tran-Trung Nghiem in general. Nghiem adapted the pluri-potential theoretic approach of Kolodziej to the transverse Kahler setting. We instead adapt the purely PDE approach to $L^\infty$ estimates due to Guo-Phong et al. to the transverse Kahler setting, and thereby obtain a purely PDE proof of the regularity result.
Title: APRG Seminar: Multilinear maximal averages defined on non-degenerate hypersurfaces
Speaker: Kalachand Shuin (Seoul National University, South Korea)
Date: Wed, 07 Feb 2024
Time: 4 pm
Venue: Microsoft Teams (online)
In this talk, we will explore the $L^{p}$-boundedness of both bilinear and multilinear maximal averages defined on non-degenerate hypersurfaces. Additionally, we will delve into the $L^2(\mathbb{R}^d)\times L^2(\mathbb{R}^d)\times\cdots\times L^2(\mathbb{R}^d) \to L^{2/m}(\mathbb{R}^d)$ estimates for $m$-linear maximal averages, focusing on hypersurfaces with $1\leq \kappa < md-1$ non-zero principal curvatures.
The video of this talk is available on the IISc Math Department channel.
Title: Number Theory Seminar: Euclidean algorithms are Gaussian over imaginary quadratic fields
Speaker: Jungwon Lee (University of Warwick, UK)
Date: Mon, 05 Feb 2024
Time: 2 pm
Venue: LH-1
Baladi and Vallée shows the limit Gaussian distribution of the length of continued fractions as a random variable on the set of rational numbers with bounded denominators based on ergodic methods. We give an analogue of the result for complex continued fractions over imaginary quadratic number fields and discuss applications in value distribution of $L$-functions of $\mathrm{GL}_2$ (joint with Dohyeong Kim and Seonhee Lim).
Title: PhD Thesis colloquium: Existence and implications of positively curved metrics on holomorphic vector bundles
Speaker: Arindam Mandal (IISc Mathematics)
Date: Tue, 30 Jan 2024
Time: 2 pm
Venue: LH-1, Mathematics Department
This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann–Fock spaces on the complex plane started with the work of K. Seip in 1992. In a series of papers, Seip and his collaborators have entirely characterized the interpolating and sampling sequences for the Bargmann–Fock spaces on the complex plane. This problem has also been studied for the Bargmann–Fock spaces on the higher dimensional complex Euclidean spaces. Very few results about the interpolating and sampling hypersurfaces in higher dimensions are known. We have proved certain hypersurfaces are not interpolating in dimensions 2 and 3. Cerda, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied the uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of it. In dimension two, we provided sufficient conditions for a smooth hypersurface to be uniformly flat in terms of its projectivization.
The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, J. P. Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.
Title: APRG Seminar: Geometric spaces at finite resolution
Speaker: Walter van Suijlekom (Radboud Universiteit, Nijmegen, Netherlands and IASc Jubilee Chair)
Date: Mon, 29 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
After a gentle introduction to the spectral approach to geometry, we extend the framework in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by operator algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc.
We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.
Title: Algebra & Combinatorics Seminar: Some combinatorial structures realized by commutative rings
Speaker: Rameez Raja (NIT Srinagar)
Date: Wed, 24 Jan 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
There are many ways to associate a graph (combinatorial structure) to a commutative ring $R$ with unity. One of the ways is to associate a zero-divisor graph $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$ are all elements of $R$ and two vertices $x, y \in R$ are adjacent in $\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a combinatorial structure $\Gamma(R)$ and show that the representatives of some algebraic invariants are eigenvalues of the Laplacian of $\Gamma(R)$. Moreover, we discuss association of another combinatorial structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two positive integers. The Young’s partition lattice $L(m,n)$ is defined to be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq \cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset as Young diagrams ordered by inclusion. We conclude this talk with a discussion on Stanley’s conjecture regarding symmetric saturated chain decompositions (SSCD) of $L(m,n)$.
Title: APRG Seminar: Partial convexity conditions and the d/d(zbar) problem
Speaker: Debraj Chakrabarti (Central Michigan University, Mount Pleasant, USA)
Date: Wed, 24 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
On a so-called Stein manifold the $\overline{\partial}$-problem can be solved in each degree $(p,q)$ where $q\geq 1$, or in other words the Dolbeault cohomology vanishes in these degrees. Sufficient conditions on complex manifolds which ensure that the Dobeault cohomology in degree $(p,q)$ is finite dimensional or vanishes have been studied since Andreotti-Grauert, who introduced the notions of $q$-convex/$q$-complete manifolds, which generalize Steinness. For manifolds with boundary, Hormander and Folland-Kohn introduced the condition now called $Z(q)$ which ensures finite-dimensionality of the cohomology in degree $q$ as well as $\frac{1}{2}$ estimates for the $\overline{\partial}$-Neumann operator. These conditions ($q$-convexity/completeness and $Z(q)$) are biholomorphically invariant characteristics of the underlying complex manifold.
In the context of Hermitian manifolds, a different type of sufficient condition implies that the $L^2$-cohomology in degree $(p,q)$-vanishes. Here one assumes that the sum of any $q$-eigenvalues is positive, and this also leads to the vanishing of the $L^2$-cohomology via the Bochner-Kohn-Morrey formula. These conditions are not biholomorphically invariant (they depend on the choice of the metric).
In this report on ongoing joint work with Andy Raich and Phil Harrington, we discuss the relationship between the two types of the condition. We give new sufficient conditions for the vanishing of the $L^2$-cohomology in degree $(p,q)$ in a domain in a complex manifold and discuss to what extent the conditions are necessary.
Title: APRG Seminar: Quantitative FKG for crossings
Speaker: Ritvik Radhakrishnan (ETH, Zurich, Switzerland)
Date: Wed, 17 Jan 2024
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Consider critical Bernoulli bond percolation on $\mathbb{Z}^2$. We show that the two arm exponent is strictly larger than twice the one arm exponent. This answers a question of Schramm and Steif (2010), and shows that their proof of the existence of exceptional times on the triangular lattice also applies to the square lattice. We use an interpolation formula via noise to obtain asymptotic correlation of crossings and apply this at each scale to obtain the strict inequality of arm exponents. This talk is based on joint work with Vincent Tassion.
Title: PhD Thesis defence: Characters of classical groups twisted by roots of unity
Speaker: Nishu Kumari (IISc Mathematics)
Date: Mon, 15 Jan 2024
Time: 9:30 am
Venue: LH-1, Mathematics Department
This thesis focuses on the study of specialized characters of irreducible polynomial representations of the complex classical Lie groups of types A, B, C and D. We study various specializations where the characters are evaluated at elements twisted by roots of unity. The details of the results are as follows.
Throughout the thesis, we fix an integer $t \geq 2$ and a primitive $t$’th root of unity $\omega$. We first consider the irreducible characters of representations of the general linear group, the symplectic group and the orthogonal group evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$. The case of the general linear group was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In each case, we characterize partitions for which the character value is nonzero in terms of what we call $z$-asymmetric partitions, where $z$ is an integer which depends on the group. This characterization turns out to depend on the $t$-core of the indexed partition. Furthermore, if the character value is nonzero, we prove that it factorizes into characters of smaller classical groups. We also give product formulas for general $z$-asymmetric partitions and $z$-asymmetric $t$-cores, and show that there are infinitely many $z$-asymmetric $t$-cores for $t \geq z+2$.
We extend the above results for the irreducible characters of the classical groups evaluated at similar specializations. For the general linear case, we set the first $tn$ elements to $\omega^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the last $m$ to $y, \omega y, \dots, \omega^{m-1} y$. For the other families, we take the same specializations but with $m=1$. Our motivation for studying these are the conjectures of Wagh–Prasad (Manuscripta Math., 2020) relating the irreducible representations of classical groups.
The hook Schur polynomials are the characters of covariant and contravariant irreducible representations of the general linear Lie superalgebra. These are a supersymmetric analogue of the characters of irreducible polynomial representations of the general linear group and are indexed by two families of variables. We consider similarly specialized skew hook Schur polynomials evaluated at $\omega^p x_i$ and $\omega^q y_j$, for $0 \leq p, q \leq t-1$, $1 \leq i \leq n$, and $1 \leq j \leq m$. We characterize the skew shapes for which the polynomial vanishes and prove that the nonzero polynomial factorizes into smaller skew hook Schur polynomials.
For certain combinatorial objects, the number of fixed points under a cyclic group action turns out to be the evaluation of a nice function at the roots of unity. This is known as the cyclic sieving phenomenon (CSP) and has been the focus of several studies. We use the factorization result for the above hook Schur polynomial to prove the CSP on the set of semistandard supertableaux of skew shapes for odd $t$. Using a similar proof strategy, we give a complete generalization of a result of Lee–Oh (Electron. J. Combin., 2022) for the CSP on the set of skew SSYT conjectured by Alexandersson–Pfannerer–Rubey–Uhlin (Forum Math. Sigma, 2021).
Title: Number Theory Seminar: Fake Projective Planes
Speaker: Gopal Prasad (University of Michigan, Ann Arbor, USA)
Date: Fri, 12 Jan 2024
Time: 12 pm
Venue: LH-1, Mathematics Department (Joint with the Algebra-Combinatorics Seminar)
Title: Geometry & Topology Seminar: On the dense existence of compact invariant sets
Speaker: Rohil Prasad (UC Berkeley)
Date: Wed, 10 Jan 2024
Time: 11:00 am
Venue: LH-1
This is joint work in progress with Dan Cristofaro-Gardiner. We explore the topological dynamics of Reeb flows beyond periodic orbits and find the following rather general phenomenon. For any Reeb flow for a torsion contact structure on a closed 3-manifold, any point is arbitrarily close to a proper compact invariant subset of the flow. Such a statement is false if the invariant subset is required to be a periodic orbit. Stronger results can also be proved that parallel theorems of Le Calvez-Yoccoz, Franks, and Salazar for homeomorphisms of the 2-sphere. In fact, we can also extend their results to Hamiltonian diffeomorphisms of closed surfaces of any genus.
Title: Algebra & Combinatorics Seminar: 'Almost' Representations and Group Stability
Speaker: Bharatram Rangarajan (Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel)
Date: Wed, 10 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.
In this talk, I will survey some of the main results, techniques, and questions in this area.
Title: Geometry & Topology Seminar: A Complete Geodesic Metric on Finite Energy Spaces in Big Cohomology Class.
Speaker: Prakhar Gupta (University of Maryland)
Date: Mon, 08 Jan 2024
Time: 04:00 pm
Venue: LH-1
In this talk, I will describe a complete geodesic metric $d_p$ on the finite energy space $\mathcal{E}^p(X,\theta)$ for $p\geq 1$ where $\theta$ represents a big cohomology class. This work generalizes the complete geodesic metrics in the Kahler setting to the big setting. When p=1, the metric $d_1$ in the Kahler setting has found various applications in the understanding of Kahler-Einstein and Constant Scalar Curvature Kahler metrics. In this talk, I’ll describe how to construct the metric and explain some properties that could have useful applications in the future.
Title: Eigenfunctions Seminar: Macdonald polynomials, interpolation polynomials, and binomial coefficients
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Fri, 05 Jan 2024
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
The Macdonald polynomials are a homogeneous basis for the algebra of symmetric polynomials, which generalize many important families of special functions, such as Schur polynomials, Hall-Littlewood polynomials, and Jack polynomials.
The interpolation polynomials, introduced by F. Knop and the speaker, are an inhomogeneous extension of Macdonald polynomials, which are characterized by very simple vanishing properties.
The binomial coefficients are special values of interpolation polynomials, which play a central role in the higher rank $q$-binomial theorem of A. Okounkov.
We will give an elementary self-contained introduction to all three objects, and discuss some recent results, open problems, and applications.
Title: APRG Seminar: Commuting tuple of operators homogeneous under the unitary group
Speaker: Surjit Kumar (IIT, Madras)
Date: Thu, 04 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
Let $\mathbb B_d$ be the open unit ball in $\mathbb C^d$ and $\boldsymbol T$ be a commuting $d$-tuple of bounded linear operators
on a complex separable Hilbert space $\mathcal H$. Let $\mathcal U(d)$ be the linear group of unitary transformations acting on
$\mathbb C^d$ by the rule: $\boldsymbol z \mapsto u\cdot \boldsymbol z$, $\boldsymbol z \in \mathbb C^d$, where $u\cdot \boldsymbol z$
is the usual matrix product. We say that $\boldsymbol T$ is $\mathcal U(d)$-homogeneous if $u \cdot \boldsymbol T$ is unitarily
equivalent to $\boldsymbol T$ for all $u\in \mathcal U(d)$.
In this talk, we describe $\mathcal U(d)$-homogeneous $d$-tuple $\boldsymbol M$ of multiplication by the coordinate functions acting
on a reproducing kernel Hilbert space $\mathcal H_K(\mathbb B_d, \mathbb C^n) \subseteq {\rm Hol}(\mathbb B_d, \mathbb C^n)$, where
$n$ is the dimension of the joint kernel of $\boldsymbol T^*$. The case $n=1$ is well understood, here, we focus on the case $n=d.$ We
describe this class of $\mathcal U(d)$-homogeneous operators, equivalently, non-negative definite kernels quasi invariant under the
action of the group $\mathcal U(d).$ As a result, we obtain criterion for boundedness, irreducibility and mutual unitary equivalence
among these operators.
This is a joint work with Soumitra Ghara, Gadadhar Misra and Paramita Pramanick.
Title: APRG Seminar: Lyapunov exponents for random matrix products
Speaker: Siddhartha Sahi (Rutgers University, USA)
Date: Wed, 03 Jan 2024
Time: 4 pm
Venue: LH-1, Mathematics Department
We consider probability measures on $GL(n,\mathbb{R})$ that are invariant under the left action of the orthogonal group $O(n,\mathbb{R})$ and satisfy a mild integrability condition. For any such measure we consider the following two quantities: (a) the mean of the log of the absolute value of the eigenvalues of the matrices and (b) the Lyapunov exponents of random products of matrices independently drawn with respect to the measure. Our main result is a lower bound for (a) in terms of (b).
This lower bound was conjectured by Burns-Pugh-Shub-Wilkinson (2001), and special cases were proved by Dedieu-Shub (2002), Avila-Bochi (2003) and Rivin (2005). We give a proof in complete generality by using some results from the theory of spherical functions and Jack polynomials.
This is joint work with Diego Armentano, Gautam Chinta, and Michael Shub. (arXiv:2206.01091), (Ergodic theory and Dynamical systems, to appear).
Title: Algebra & Combinatorics Seminar: The Du Bois complex and some associated singularities
Speaker: Sridhar Venkatesh (University of Michigan, Ann Arbor, USA)
Date: Thu, 28 Dec 2023
Time: 11:30 am
Venue: LH-1, Mathematics Department
For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is a powerful tool to study the geometry of $X$ because of results such as the degeneration of the Hodge-de Rham spectral sequence (when $X$ is proper). For singular varieties, it follows from the work of Deligne and Du Bois that there is a substitute called the Du Bois complex which satisfies many of the nice properties enjoyed by the de Rham complex in the smooth case. In this talk, we will discuss some classical singularities associated with this complex, namely Du Bois and rational singularities, and some recently introduced refinements, namely $k$-Du Bois and $k$-rational singularities. This is based on joint work with Wanchun Shen and Anh Duc Vo.
Title: PhD Thesis defence: Harmonic map heat flow and framed surface-group representations
Speaker: Gobinda Sau (IISc Mathematics)
Date: Thu, 28 Dec 2023
Time: 11 am
Venue: Microsoft Teams (online)
This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.
The focus of the first part is when the domain is the complex plane. In this case, given a finite twisted ideal polygon, there exists a harmonic map heat flow $u_t$ such that the image of $u_t$ is asymptotic to that polygon for all $t\in[0,\infty)$. Moreover, we prove that given any twisted ideal polygon in $\mathbb{H}^3$ with \textit{rotational symmetry}, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.
In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $\text{PSL}_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely \textit{marked and bordered surfaces} (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (\text{PSL}_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$. In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.
Title: Slides Kerala School of Mathematics
Speaker: A. Raghuram (Fordham University, New York City, USA)
Date: Thu, 28 Dec 2023
Time: 2 pm
Venue: LH-1, Mathematics Department
The basic ideas of Calculus started with Archimedes, and reached a highly developed form in the 17th Century with Newton and Leibniz often being credited as its inventors. What was not so well-known until only a few decades ago is that between the 14th and 17th Century there was an unbroken lineage of profound mathematicians working in Kerala who had independently discovered many of the great themes of Calculus. This talk is an introduction to the lives and works of some of the prominent members of the Kerala School of Mathematics. Most of the talk will be accessible to a general audience. Only towards the end of the talk some elementary mathematics will be assumed to explain a few of their contributions.
Title: Algebra & Combinatorics Seminar: Matrix factorisations of polynomials
Speaker: Ravindra Girivaru (University of Missouri, St. Louis, USA)
Date: Wed, 20 Dec 2023
Time: 4 pm
Venue: LH-3, Mathematics Department
A matrix factorisation of a polynomial $f$ is an equation $AB = f \cdot {\rm I}_n$ where $A,B$ are $n \times n$ matrices with polynomial entries and ${\rm I}_n$ is the identity matrix. This question has been of interest for more than a century and has been studied by mathematicians like L.E. Dickson. I will discuss its relation with questions arising in algebraic geometry about the structure of subvarieties in projective hypersurfaces.
Title: Algebra & Combinatorics Seminar: A simple proof for the characterization of chordal graphs using Horn hypergeometric series
Speaker: R. Venkatesh (IISc Mathematics)
Date: Wed, 13 Dec 2023
Time: 11:30 am
Venue: LH-3, Mathematics Department
Let $G$ be a finite simple graph (with no loops and no multiple edges), and let $I_G(x)$ be the multi-variate independence polynomial of $G$. In 2021, Radchenko and Villegas proved the following interesting characterization of chordal graphs, namely $G$ is chordal if and only if the power series $I_G(x)^{-1}$ is Horn hypergeometric. In this talk, I will give a simpler proof of this fact by computing $I_G(x)^{-1}$ explicitly using multi-coloring chromatic polynomials. This is a joint work with Dipnit Biswas and Irfan Habib.
Title: Algebra & Combinatorics Seminar: The structure and representation theory of quantum toroidal algebras
Speaker: Duncan Laurie (University of Oxford, UK)
Date: Fri, 08 Dec 2023
Time: 3 pm
Venue: LH-3, Mathematics Department
Quantum toroidal algebras are the next class of quantum affinizations after quantum affine algebras, and can be thought of as “double affine quantum groups”. However, surprisingly little is known thus far about their structure and representation theory in general.
In this talk we’ll start with a brief recap on quantum groups and the representation theory of quantum affine algebras. We shall then introduce and motivate quantum toroidal algebras, before presenting some of the known results. In particular, we shall sketch our proof of a braid group action, and generalise the so-called Miki automorphism to the simply laced case.
Time permitting, we shall discuss future directions and applications including constructing representations of quantum toroidal algebras combinatorially, written in terms of Young columns and Young walls.
Title: PhD Thesis colloquium: Some aspects of weighted kernel functions on planar domains
Speaker: Aakanksha Jain (IISc Mathematics)
Date: Fri, 08 Dec 2023
Time: 4 pm
Venue: Hybrid - Microsoft Teams (online) and LH-3, Mathematics Department
In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.
For a planar domain $D \subset \mathbb C$ and an admissible weight function $\mu$ on it, we discuss some aspects of the corresponding weighted Bergman kernel $K_{D, \mu}$. First, we see a precise relation between $K_{D, \mu}$ and the classical Bergman kernel $K_D$ near a smooth boundary point of $D$. Second, the weighted kernel $K_{D, \mu}$ gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \geq 0$. Specific examples that we discuss in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.
In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.
Title: Geometry & Topology Seminar: Cubulating hyperbolic mapping tori
Speaker: Suraj Krishna (Technion, Israel)
Date: Wed, 06 Dec 2023
Time: 04:00 pm
Venue: LH-3
A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Some well-known examples are free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds. I will show in the talk that semidirect products of hyperbolic groups with $\mathbb{Z}$ which are again hyperbolic are cubulated, and give some consequences.
Two prominent examples of our setup are
- mapping tori of fundamental groups of closed hyperbolic surfaces over pseudo-Anosov automorphisms, and
- mapping tori of free groups over atoroidal automorphisms.
Both these classes of groups are known to be cubulated by outstanding works. Our proof uses these two noteworthy results as building blocks and places them in a unified framework. Based on joint work with François Dahmani and Jean Pierre Mutanguha.
Title: Algebra & Combinatorics Seminar: A glimpse into the world of Rogers-Ramanujan identities
Speaker: Shashank Kanade (University of Denver, USA)
Date: Fri, 01 Dec 2023
Time: 11 am
Venue: LH-3, Mathematics Department
I will give a gentle introduction to the combinatorial Rogers–Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots.
Title: Number Theory Seminar: Stark-Heegner cycles over arbitrary number fields
Speaker: Guhan Venkat (Ashoka University)
Date: Thu, 30 Nov 2023
Time: 2 pm
Venue: LH-1
In his seminal paper in 2001, Henri Darmon proposed a systematic construction of $p$-adic points, viz. Stark–Heegner points, on elliptic curves over the rational numbers. In this talk, I will report on the construction of local ($p$-adic) cohomology classes/cycles in the $p$-adic Galois representation attached to a cuspidal cohomological automorphic representation of $\mathrm{PGL}_2$ over any number field, building on the ideas of Henri Darmon and Rotger–Seveso. These local cohomology classes are conjectured to be the restriction of global cohomology classes in an appropriate Bloch–Kato Selmer group and have consequences towards the Bloch–Kato conjecture. This work generalises previous constructions of Rotger-Seveso for elliptic cusp forms and earlier joint work with Williams for Bianchi cusp forms. Time permitting, I will also talk about the plectic analogues of these objects.
Title: Algebra & Combinatorics Seminar: Branching multiplicity of symplectic groups as $SL_2$ representations
Speaker: Sagar Shrivastava (TIFR, Mumbai)
Date: Tue, 28 Nov 2023
Time: 11:30 am
Venue: LH-1, Mathematics Department
Branching rules are a systematic way of understanding the multiplicity of irreducible representations in restrictions of representations of Lie groups. In the case of $GL_n$ and orthogonal groups, the branching rules are multiplicity free, but the same is not the case for symplectic groups. The explicit combinatorial description of the multiplicities was given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed that this multiplicity corresponds to the dimension of the multiplicity space, which was a representation of $SL_2$ $(=Sp(2))$. In this talk, we give an alternate proof of the same without invoking any partition function machinery. The only assumption for this talk would be the Weyl character formula.
Title: Number Theory Seminar: Congruences and period relations: a new case
Speaker: Jacques Tilouine (LAGA, Universite Paris 13, Paris, France)
Date: Mon, 20 Nov 2023
Time: 2 pm
Venue: LH-3
In a joint work in progress with K. Prasanna, we
study period relations for the base change to $\mathrm{GL}_4$ of a
cohomological cuspidal representation on $\mathrm{GSp}_4$.
An unexpected period occurs in the period relations.
Title: Geometry & Topology Seminar: Criteria for simplicity
Speaker: Arpan Kabiraj (IIT Palakkad)
Date: Mon, 20 Nov 2023
Time: 04:00 pm
Venue: LH-1
In 80’s Goldman introduced a Lie bracket structure on the free homotopy classes of oriented closed curves on an oriented surface known as the Goldman Lie bracket. In this talk, I will give a brief overview of Goldman Lie algebra and discuss two criteria for a homotopy class of a curve to be simple in terms of the Goldman Lie bracket.
Title: Algebra & Combinatorics Seminar: Ideals in enveloping algebras of affine Kac-Moody algebras
Speaker: Rekha Biswal (NISER, Bhubaneswar)
Date: Fri, 17 Nov 2023
Time: 11:30 am
Venue: LH-3, Mathematics Department
In this talk, I will discuss about the structure of ideals in enveloping algebras of affine Kac–Moody algebras and explain a proof of the result which states that if $U(L)$ is the enveloping algebra of the affine Lie algebra $L$ and “$c$” is the central element of $L$, then any proper quotient of $U(L)/(c)$ by two sided ideals has finite Gelfand–Kirillov dimension. I will also talk about the applications of the result including the fact that $U(L)/(c-\lambda)$ for non zero $\lambda$ is simple. This talk is based on joint work with Susan J. Sierra.
Title: PhD Thesis colloquium: Dominating surface-group representations via Fock-Goncharov coordinates
Speaker: Pabitra Barman (IISc Mathematics)
Date: Fri, 17 Nov 2023
Time: 11:15 am
Venue: LH-1, Mathematics Department
Let $S$ be an oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S)$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher rank representations.
For a representation $\rho : \pi_1(S) \rightarrow PSL_n(\mathbb{C})$ where $n >2$, the Hilbert length of a curve $\gamma\in \pi_1(S)$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock–Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$-representations. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.
Let $ X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \ell_{X_n}(A)= \sum_{i=1}^{n}\log (\sigma_i(A))^2, \end{equation} where $\sigma_i(A)$ are the singular values of $A$. We show that the same $j$ dominates $\rho$ in the translation length spectrum as well. Lindström’s Lemma for planar networks is one of the key ingredients of the proof.
In both cases, if $S$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.
Title: Eigenfunctions Seminar: Slides Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Speaker: Hariharan Narayanan (TIFR, Mumbai)
Date: Fri, 17 Nov 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
Associated to two given sequences of eigenvalues is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as the number of eigenvalues tends to infinity.
Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdös–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. This is joint work with Scott Sheffield and Terence Tao.
Title: Algebra & Combinatorics Seminar: Aldous-type spectral gap results for the complete monomial group
Speaker: Subhajit Ghosh (Bar-Ilan University, Ramat-Gan, Israel)
Date: Wed, 15 Nov 2023
Time: 3 pm
Venue: LH-1, Mathematics Department
Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form
- $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or
- $(g)^{(w)}:=(\dots,\mathbf{e},g,\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w \gt 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$,
such that $\{(u,v)_G,(g)^{(w)} : x_{u,v} \gt 0,\; y_w\alpha_g \gt 0,\;1\leq u \lt v \leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u \lt v\leq n$ and $g^{(w)},g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous’ spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.
Title: APRG Seminar: Pseudo-differential calculi and entropy estimates with Orlicz spaces and Orlicz modulation spaces
Speaker: Joachim Toft (Linnaeus University, Vaxjo, Sweden)
Date: Wed, 15 Nov 2023
Time: 11 am
Venue: LH-1, Mathematics Department
A convex function $\Phi$ from $[0,\infty]$ to $[0,\infty]$ with properties \begin{equation} \Phi (0)=0,\qquad \lim_{t\to \infty}\Phi (t)=\Phi (\infty )=\infty , \end{equation} is called a Young function. For any Young function $\Phi$, the Orlicz space $L^\Phi$ is a Banach space, and consists of all measurable functions $f$ such that $\Phi (t\cdot |f|)\in L^1$ for some $t>0$. By choosing $\Phi$ in suitable ways we gain the definition of any (Banach) Lebesgue space $L^p$, as well as sums of such spaces like $L^p+L^q$, $p,q\in [1,\infty ]$. In particular, the family of Orlicz spaces contain any Lebesgue space.
The Orlicz modulation space $M^{\Phi}$ is obtained by imposing $L^\Phi$ norm conditions of the short-time Fourier transforms of the involved functions and distributions. In the same way we may discuss Orlicz modulation spaces $M^{\Phi ,\Psi}$ of mixed normed types. Again, by choosing the Young functions $\Phi$ and $\Psi$ in suitable ways, $M^{\Phi ,\Psi}$ becomes the classical Feichtinger’s modulation space $M^{p,q}$.
In the talk we explain some basic properties and give some examples on interesting Orlicz spaces and Orlicz modulation spaces. We also explain some classical results on pseudo-differential operators acting on Lebesgue or modulation spaces, and give examples on how such results can be extended to the framework of Orlicz spaces and Orlicz modulation spaces.
As another example we discuss suitable Orlicz modulation spaces and the entropy functional $f\mapsto E_\phi (f)$ with $\phi$ as the coherent state, considered by E. H. Lieb when discussing kinetic energy in quantum systems. Here we find an Orlicz modulation space $M^\Phi$ which satisfies \begin{equation} M^{p_1}\subsetneq M^\Phi \subsetneq M^{p_2},\qquad p_1<\frac 12,\ p_2\ge \frac 12 \end{equation} for which $E_\phi$ is continuous on $M^{p_1}$ and $M^\Phi$, but discontinuous on $M^{p_2}$. We hope that this should shed some light on how to find suitable Banach spaces when dealing with non-linear functionals.
The talk is based on joint works with A. Gumber, E. Nabizadeh Morsalfard, N. Rana, S. Öztop and R. Üster.
Title: APRG Seminar: Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates
Speaker: Joachim Toft (Linnaeus University, Vaxjo, Sweden)
Date: Tue, 14 Nov 2023
Time: 3:30 pm
Venue: LH-1, Mathematics Department
Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. Especially we extend some results in our recent works and those of Bhimani, Cordero, Gröchenig, Manna, Thangavelu, and others. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially we show that fractional Fourier transforms of any complex order can be defined, and that these transforms are continuous on any Pilipovic space and corresponding distribution space, which are not Gelfand–Shilov spaces. (The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.)
Title: PhD Thesis colloquium: Geometry of Normed Linear Spaces in Light of Birkhoff-James Orthogonality
Speaker: Babhrubahan Bose (IISc Mathematics)
Date: Fri, 10 Nov 2023
Time: 2:30 pm
Venue: Hybrid - Microsoft Teams (online) and LH-1, Mathematics Department
Two points $x$ and $y$ in a normed linear space $\mathbb{X}$ are said to be Birkhoff-James orthogonal (denoted by $x\perp_By$) if $|x+\lambda y|\geq|x|~~\text{for every scalar}~\lambda.$ James proved that in a normed linear space of dimension more than two, Birkhoff-James orthogonality is symmetric if and only if the parallelogram law holds. Motivated by this result, Sain introduced the concept of pointwise symmetry of Birkhoff-James orthogonality in a normed linear space.
In this talk, we shall try to understand the geometry of normed spaces in the light of Birkhoff-James orthogonality. After introducing the basic notations and terminologies, we begin with a study of the geometry of the normed algebra of holomorphic maps in a neighborhood of a curve and establish a relationship among the extreme points of the closed unit ball, Birkhoff-James orthogonality, and zeros of holomorphic maps.
We next study Birkhoff-James orthogonality and its pointwise symmetry in Lebesgue spaces defined on arbitrary measure spaces and natural numbers. We further find the onto isometries of the sequence spaces using the pointwise symmetry of the orthogonality.
We shall then study the geometry of tensor product spaces and use the results to study the relationship between the symmetry of orthogonality and the geometry (for example, extreme points and smooth points) of certain spaces of operators. Our work in this section is motivated by the famous Grothendieck inequality.
Finally, we study the geometry of $\ell_p$ and $c_0$ direct sums of normed spaces ($1\leq p<\infty$). We shall characterize the smoothness and approximate smoothness of these spaces along with Birkhoff-James orthogonality and its pointwise symmetry. As a consequence of our study we answer a question pertaining to the approximate smoothness of a space, raised by Chmeilienski, Khurana, and Sain.
Title: Geometry & Topology Seminar: The Einstein-Bogomolnyi metrics on Riemann sphere
Speaker: Chengjian Yao (Shanghai Tech)
Date: Mon, 06 Nov 2023
Time: 04:00 pm
Venue: MS Teams (online)
Einstein-Bogomolnyi metrics, which physically models the Cosmic Strings, solve the Einstein’s Fields Equation coupled with an Abelian gauge field and a Higgs field. In this talk, I will present a general existence and uniqueness theorem for Einstein-Bogomolnyi metrics on Riemann sphere. I will also discuss the behaviors of the metrics as the volume approaches the lower bound and infinity respectively, and the moduli space problem. Part of this talk is based on the joint work with Luis-Alvarez, Garcia-Fernandez, Garcia-Prada and Pingali.
Title: Eigenfunctions Seminar: Representations of $p$-adic groups over close local fields
Speaker: Radhika Ganapathy (IISc Mathematics)
Date: Fri, 03 Nov 2023
Time: 3 – 5 pm (with a 15 minute break in between)
Venue: LH-1, Mathematics Department
In the first part of the talk, we will discuss the main statement of local class field theory that describes the abelian extensions of a non-archimedean local field $F$ in terms of the arithmetic of the field $F$. Then we will discuss the statement of the local Langlands conjectures, a vast generalization of local class field theory, that gives a (conjectural) parametrization of the irreducible complex representations of $G(F)$, where $G$ is a connected, reductive group over $F$, in terms of certain Galois representations.