Seminars

By Year

Venue: LH-1, Mathematics Department

The analysis for Yang-Mills functional and in general, problems related to higher dimensional gauge theory, often requires one to work with weak notions of principal G-bundles and connections on them. The bundle transition functions for such bundles are not continuous and thus there is no obvious notion of a topological isomorphism class.

In this talk, we shall discuss a few natural classes of weak bundles with connections which can be approximated in the appropriate norm topology by smooth connections on smooth bundles. We also show how we can associate a topological isomorphism class to such bundle-connection pairs, which is invariant under weak gauge changes. In stark contrast to classical notions, this topological isomorphism class is not independent of the connection.

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Venue: LH-5, Mathematics Department

It is commonly expected that $e$, $\log 2$, $\sqrt{2}$, $\pi$, among other “classical” numbers, behave, in many respects, like almost all real numbers. For instance, they are expected to be normal to base 10, that is, one believes that their decimal expansion contains every finite block of digits from ${0, \ldots , 9}$. We are very far away from establishing such a strong assertion. However, there has been some small recent progress in that direction. After surveying classical results and problems on normal numbers, we will adopt a point of view from combinatorics on words and show that the decimal expansions of $e$, of any irrational algebraic number, and of $\log (1 + \frac{1}{a})$, for a sufficiently large integer $a$, cannot be ‘too simple’, in a suitable sense.

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Venue: LH-1, Mathematics Department

We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class having polynomial Euler product and satisfying Selberg’s orthonormality condition. We show that on every vertical line $s=\sigma+it$ in the complex plane with $\sigma \in(1/2,1)$, these $L$-functions simultaneously take “large” values inside a small neighborhood.

This is joint work with Kamalakshya Mahatab and Lukasz Pankowski.

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Venue: LH-1, Mathematics Department

Let $G$ be an algebraic group defined over a finite field $\mathbb{F}_q$ and let $m$ be a positive integer. Shintani descent is a relationship between the character theories of the two finite groups $G(\mathbb{F}_q)$ and $G(\mathbb{F}_{q^m})$ of $\mathbb{F}_q$ and $\mathbb{F}_{q^m}$-valued points of $G$ respectively. This was first studied by Shintani for $G=GL_n$. Later, Shoji studied Shintani descent for connected reductive groups and related it to Lusztig’s theory of character sheaves. In this talk, I will speak on the cases where $G$ is a unipotent or solvable algebraic group. I will also explain the relationship with the theory of character sheaves.

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Venue: LH-1, Department of Mathematics

We discuss a set of purely sequential strategies to estimate an unknown negative binomial mean $\mu$ under different forms of loss functions. We develop point estimation techniques where the thatch parameter $\tau$ may be known or unknown. Both asymptotic first-order efficiency and risk efficiency properties will be elaborated. The results will be supported by an extensive set of data analysis carried out via computer simulations for a wide variety of sample sizes. We observe that all of our purely sequential estimation strategies perform remarkably well under different situations. We also illustrate the implementation of these methodologies using real datasets from ecology, namely, weed count data and data on migrating woodlarks. (This is a Skype talk.)

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Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesday, January 29)

Lecture 4. Multivariable Scattering Theory (Wednesday, February 5)

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Venue: LH-1, Mathematics Department

The quintic threefold (the zero set of a homogeneous degree 5 polynomial on CP^4) is one of the most famous examples of a Calabi Yau manifold. It is one of the most studied in the field of Enumerative Geometry. For example, how many lines are there on a Quintic threefold? In this talk we will explain some approaches to count curves on the Quintic threefold. In particular, we will try to explain the following idea: If Y is a submanifold of X, and we understand the Enumerative Geometry of X, how can we answer questions about the Enumerative Geometry of Y? We will try to explain the idea used by Andreas Gathman to compute all the genus zero Gromov-Witten invariants of the Quintic Threefold.

The talk will be self contained and will not assume any prior knowledge of Enumerative Geometry or Gromov-Witten Invariants.

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Venue: LH-3, Mathematics Department

For a finite abelian group $G$ and $A \subset [1, \exp(G) - 1]$, the $A$-weighted Davenport Constant $D_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$-weighted zero-sum subsequence. The original motivation for studying Davenport Constant was the problem of non-unique factorization in number fields. The precise value of this invariant for the cyclic group for certain sets $A$ is known but the general case is still unknown. Typically an extremal problem deals with the problem of determining or estimating the maximum or minimum possible cardinality of a collection of finite objects that satisfies certain requirements. In a recent work with Prof. Niranjan Balachandran, we introduced an Extremal Problem for a finite abelian group related to Weighted Davenport Constant. In this talk I will talk about the behaviour of it for different groups, specially for cyclic group.

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Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3. Automorphic Forms in Berezin Quantization and von Neumann Algebras (Wednesday, January 29)

Lecture 4. Multivariable Scattering Theory (Wednesday, February 5)

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Venue: LH-1, Mathematics Department

In 1976, E.M. Stein proved $L^p$ bounds for spherical maximal function on Euclidean space. The lacunary case was dealt on later by C.P. Calderon in 1979. In a recent paper, M. Lacey has proved sparse bound for these functions and $L^p$ bounds will follow immediately as a result.

In this talk, we will look at various maximal functions corresponding to spherical averages and find sparse bounds for those functions. We will also observe some weighted and unweighted estimates that will follow as a consequences.

First, we will show sparse bound for lacunary spherical maximal function on Heisenberg group . Next we move on to full spherical maximal function. Then we study lacunary maximal function corresponding to the spherical average on product of Heisenberg groups. Finally, we will revisit generalized spherical averages on Euclidean space and prove sparse bounds for the related maximal functions.

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Venue: LH-1, Mathematics Department

Solitons are solutions of a special class of nonlinear partial differential equations (soliton equations, the best example is the KdV equation). They are waves but behave like particles. The term “soliton” combining the beginning of the word “solitary” with ending “on” means a concept of a fundamental particle like “proton” or “electron”.

The events: (1) sighting, by chance, of a great wave of translation, “solitary wave”, in 1834 by Scott–Russell, (2) derivation of KdV equation by Korteweg de Vries in 1895, (3) observation of a very special type of wave interactions in numerical experiments by Kruskal and Zabusky in 1965, (4) development of the inverse scattering method for solving initial value problems by Gardener, Greene, Kruskal and Miura in 1967, (5) formulation of a general theory in 1968 by P.D. Lax and (6) contributions to deep theories starting from the work by R. Hirota (1971-74) and David Mumford (1978-79), which also gave simple methods of solutions of soliton equations, led to the development of one of most important areas of mathematics in the 20th century.

This also led to a valuable application of solitons to physics, engineering and technology. There are two aspects of soliton theory arising out of the KdV Equation:

• Applied mathematics – analysis of nonlinear PDE leading to dynamics of waves.
• Pure mathematics – algebraic geometry. It is surprising that each one of these can inform us of the other in the intersection that is soliton theory, an outcome of the KdV equation.

The subject is too big but I shall try to give some glimpses (1) of the history, (2) of the inverse scattering method, and (3) show that an algorithm based on algebraic-geometric approach is much easier to derive soliton solutions.

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Venue: LH-1, Mathematics Department

This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about a problem on certain holomorphic differential operators on the space of Jacobi forms.

In the first part we will briefly discuss how the statistics of signs of newforms determine them (work of Matomaki-Soundararajan-Kowalski) and then introduce certain ‘Linnik-type’ problems (the original problem was concerning the size of the smallest prime in an arithmetic progression in terms of the modulus) which ask for the size of the first negative eigenvalue (in terms of the analytic conductor) of various types modular forms, which has seen a lot of recent interest. Also specifically we will discuss the problem in the context of Yoshida lifts (a certain subspace of the Siegel modular forms), where in the thesis, we have improved upon the previously known result on this topic significantly. We will prove that the smallest $n$ with $\lambda(n)<0$ satisfy $n < Q_{F}^{1/2-2\theta+\epsilon}$, where $Q_{F}$ is the analytic conductor of a Yoshida lift $F$ and $0<\theta <1/4$ is some constant. The crucial point is establishing a non-trivial upper bound on the sum of Hecke eigenvalues of an elliptic newform at primes away from the level.

We will focus on a similar question concerning the first negative Fourier coefficient of a Hilbert newform. If ${C(\mathfrak{m})}_{\mathfrak{m}}$ denotes the Fourier coefficients of a Hilbert newform $f$, then we show that the smallest among the norms of ideals $\mathfrak{m}$ such that $C(\mathfrak{m})<0$, is bounded by $Q_{f}^{9/20+\epsilon}$ when the weight vector of $f$ is even and $Q_{f}^{1/2+\epsilon}$ otherwise. This improves the previously known result on this problem significantly. Here we would show how to use certain ‘good’ Hecke relations among the eigenvalues and some standard tools from analytic number theory to achieve our goal.

Finally we would talk about the statistical distribution of the signs of the Fourier coefficients of a Hilbert newform and essentially prove that asymptotically, half of them are positive and half negative. This was a breakthrough result of Matomaki-Radziwill for elliptic modular forms, and our results are inspired by those. The proof hinges on establishing some of their machinery of averages multiplicative functions to the number field setting.

In the second part of the talk we will introduce Jacobi forms and certain differential operators indexed by $\{D_{v}\}_{0}^{2m}$ that maps the space of Jacobi forms $J_{k,m}(N)$ of weight $k$, index $m$ and level $N$ to the space of modular forms $M_{k+v}(N)$ of weight $k+v$ and level $N$. It is also known that the direct sum of the differential operators $D_{v}$ for $v={1,2,…,2m}$ maps $J_{k,m}(N)$ to the direct sum of $M_{k+v}(N)$ injectively. Inspired by certain conjectures of Hashimoto on theta series, S. Bocherer raised the question whether any of the differential operators be removed from that map while preserving the injectivity. In the case of even weights S. Das and B. Ramakrishnan show that it is possible to remove the last operator. In the talk we will discuss the case of the odd weights and prove a similar result. The crucial step (and the main difference from the even weight case) in the proof is to establish that a certain tuple of congruent theta series is a vector valued modular form and finding the automorphy of the Wronskian of this tuple of theta series.

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Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

The date and time for the third and fourth lectures will be announced in due course.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3. Automorphic Forms in Berezin Quantization and von Neumann Algebras

Lecture 4. Multivariable Scattering Theory

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Venue: LH-1, Mathematics Department

This work has two parts. The first part contains the study of phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_{\lambda}$ in $\mathbb{R}^2$ of intensity $\lambda$. In the homogeneous RCM, the vertices at $x,y$ are connected with probability $g(\mid x-y\mid)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $\mid \cdot \mid$ is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal{P}_{\lambda}$ is endowed with weights that are non-negative independent random variables $W$, where $P(W>w)=w^{-\beta}1_{w\geq 1}$, $\beta>0$. Vertices located at $x,y$ with weights $W_x,W_y$ are connected with probability

for some $\eta, \alpha > 0$, independent of all else. The edges of the graph are viewed as straight line segments starting and ending at points of $\mathcal{P}_{\lambda}$. A path in the graph is a continuous curve that is a subset of the collection of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the midpoint of each line located at a distinct point of $\mathcal{P}_{\lambda}$. Intersecting lines then form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. The conditions for the existence of a phase transition has been derived. Under some additional conditions it has been shown that there is no percolation at criticality.

In the second part we consider an inhomogeneous random connection model on a $d$ -dimensional unit torus $S$, with the vertex set being the homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$. The vertices are equipped with i.i.d. weights $W$ and the connection function as above. Under the suitable choice of scaling $r_s$ it can be shown that the number of isolated vertices converges to a Poisson random variable as $s \to \infty$. We also derive a sufficient condition on the graph to be connected.

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Venue: LH-1, Department of Mathematics

The second part of the lectures on multi-variable automorphic forms emphasizes relations to harmonic analysis and operator algebras, together with some explicit constructions of automorphic forms.

The date and time for the third and fourth lectures will be announced in due course.

Lecture 1. Automorphic Forms on Semisimple Lie Groups (Tuesday, January 14)

Lecture 2. Construction of Automorphic Forms (Eisenstein series and Theta functions), relations to number theory in higher dimension (Thursday, January 16)

Lecture 3. Automorphic Forms in Berezin Quantization and von Neumann Algebras

Lecture 4. Multivariable Scattering Theory

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Venue: LH-1, Mathematics Department

Many models of one dimensional random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For a few such models, the limiting interface profile, after scaling by characteristic KPZ scaling exponents of one-third and two-third, is known to be the Airy_2 process shifted by a parabola. This limiting process is expected to be “locally Brownian”, and a recent result gives a quantified bound on probabilities of events under the Airy_2 process on a unit order interval in terms of probabilities of the same events under Brownian motion (of rate two). This comparison also holds in the prelimit for the particular model of Brownian last passage percolation. In this talk, we will introduce KPZ universality and discuss this result and a number of consequences, using last passage percolation as an expository framework.

Joint work with Jacob Calvert and Alan Hammond.

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Venue: LH-1, Mathematics Department

We will discuss the celebrated Kneser–Tits conjecture for algebraic groups and report on some recent results. We will keep the technicalities to the minimum.

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Venue: LH-1, Mathematics Department

We will discuss some work on the Ricci flow on manifolds with symmetries. In particular, cohomogeneity one manifolds, i.e. a Riemannian manifold M with an isometric action by a Lie group G such that the orbit space M/G is one-dimensional. We will also explain how this relates to diagonalizing the Ricci tensor on Lie groups and homogeneous spaces.

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Venue: LH-1, Mathematics Department

The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.

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Venue: LH-1, Mathematics Department

Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, we found that a certain averaging of the height function at the rough smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show after suitable centering and rescaling that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough. This is joint work with Kurt Johansson and Vincent Beffara.

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