Venue: LH-1
Nov. 24 (Thursday) Nov. 25 (Friday)
SessionTime Chair Speaker Chair Speaker
I10:00-10:45 Dipnit
Biswas
Rishabh
Agnihotri
10:45-11:30
11:30-11:45 Coffee Coffee
11:45-12:30
12:30-2:00 Lunch
II2:00-2:45 Agniva
Chatterjee
Renjith
Thazhathethil
2:45-3:30
3:30-3:45 Coffee
III3:45-4:30 Arpita
Mal
Gargi
Ghosh
4:30-5:15
Click on the speaker's name to navigate to their title and abstract.
Some tips for giving good math talks are placed at the end of this page.

### Titles & Abstracts

Annapurna Banik

Local continuous extension to the boundary of proper holomorphic maps

Abstract. We shall discuss a couple of results on local continuous extension of proper holomorphic maps $F: D \rightarrow \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on their boundaries $\partial{D}$ and $\partial{\Omega}$. The first result allows us to have much lower regularity, for the pieces of $\partial D, \partial{\Omega}$ that are relevant, than in earlier results in the literature. The second result is in the spirit of a result by Forstnerič–Rosay. In this talk, we will mostly discuss the motivations for the above results. Lastly, we shall look at an outline of the proof of the first result.

Sudeshna Bhattacharjee

Behaviour of intersecting geodesics in exponential last passage percolation model on $\mathbb{Z}^2$

Abstract. Last passage percolation (LPP) model on $\mathbb{Z}^2$ is an example of random growth models. In this model we assign i.i.d. random weights on each vertices of $\mathbb{Z}^2$ and one is usually interested in questions about maximum collected weights between two vertices (known as last passage time) and maximum weight attaining paths between two vertices (known as geodesics). LPP model is believed to belong to a large class of random growth models, known as Kardar-Parisi-Zhang (KPZ) universality class, for a general class of weights. However, it is rigorously known only in a few cases including the special case of exponentially distributed weights that we focus on. In this talk we derive some results on the intersection of a pair of geodesics. We sketch the idea how one can obtain such results using planarity and transversal fluctuation of geodesics arguments. No pre-knowledge on LPP shall be assumed and we introduce the background needed on asymptotic of the last passage times, transversal fluctuations and coalescence of geodesics.

Mainak Bhowmik

Inner functions on the symmetrized polydisc and approximation

Abstract. It is well known that the rational inner functions on the open unit disc in the complex plane are the finite Blaschke products. Rudin gave a structure of rational inner functions in the polydisc. In this talk, we shall see a canonical structure of rational inner functions on the symmetrized polydisc. Then we shall discuss the existence of a rational iso-inner or coiso-inner solution of a solvable Nevanlinna–Pick problem with initial nodes in the symmetrized bidisc, $\mathbb{G}$, and final nodes in the operator norm unit ball of $M \times N$ matrices. As an application of this result, we shall see a Caratheodory-type approximation which says that any matrix-valued holomorphic function on $\mathbb{G}$ with sup-norm no greater than one, can be approximated by rational iso-inner or coiso-inner functions, uniformly on compact subsets of $\mathbb{G}$.

A Fisher-type approximation on the symmetrized polydisc will also be discussed. If time permits, we shall see some factorization results for Schur-class functions in $\mathbb{G}$.

This is a joint work with Poornendu Kumar.

Sumanta Das

Strong topological rigidity of non-compact orientable surfaces

Abstract. We show that if a homotopy equivalence between two non-compact orientable (finite or infinite-type) surfaces is proper, then it is properly homotopic to a homeomorphism, provided surfaces are neither the plane nor the punctured plane. Thus all non-compact orientable surfaces, except the plane and the punctured plane, are topologically rigid in a strong sense.

Hiranya Kishore Dey

Descents, excedances and alternating-runs in positive elements of Coxeter groups

Abstract. The Eulerian polynomial $A_n(t)$ is one of the most interesting and well-studied polynomials in algebraic combinatorics. The $k$-th coefficient of the $n$-th Eulerian polynomial counts the number of permutations in the symmetric group $\mathfrak{S}_n$ with exactly $k$ descents. The coefficient vectors of these polynomials satisfy all of combinatorists' favorite distributional properties: palindromicity, gamma positivity, unimodality etc.

In this talk, we will firstly see some approaches towards proving these above-mentioned properties. We may see a group-action based proof by Foata and Strehl which shows that the Eulerian polynomials are Gamma positive. We will then consider the descent enumerating polynomials over the alternating group. We derive recurrences for these polynomials and show various results regarding gamma positivity of them.

Moving on, we will consider the polynomial that enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. Here we will see that when enumerated with sign taken into account, one gets a neat formula. As a consequence of the signed enumeration, we get a near refinement of a result of Wilf on the exponent of $(1+t)$ when it divides the alternating runs enumerating polynomial in the alternating group.

Time permitting, we will see log-concavity results and central limit theorems involving descents and excedances over positive elements of Coxeter groups.

Shivani Goel

Resistance matrices of balanced directed graphs

Abstract. Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The Laplacian matrix of $G$ is then the matrix (not necessarily symmetric) $L:=D-A$, where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal matrix such that the row sums and the column sums of $L$ are equal to zero. Let $L^\dagger=[l^{\dagger}_{ij}]$ be the Moore–Penrose inverse of $L$. We define the resistance between any two vertices $i$ and $j$ of $G$ by $r_{ij}:=l^{\dagger}_{ii}+l^{\dagger}_{jj}-2l^{\dagger}_{ij}$. Some interesting properties of the resistance and the corresponding resistance matrix $[r_{ij}]$ will be discussed in the talk.

The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. Numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e., $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In the talk we will mention that this inequality holds for all directed cactus graphs. This is a joint work with Professor R. Balaji and Professor R. B. Bapat.

Irfan Habib

Demazure crystals for specialized non-symmetric Macdonald polynomials

Abstract. Sanderson proved that for the type $A$ affine Lie algebra $\widehat{\mathfrak{sl}_{n+1}},$ the specialized non-symmetric Macdonald polynomials at $t=0$ are the characters of Demazure modules using certain recursion relations given by Knop. In this talk, we shall give a positive formula for specialized nonsymmetric Macdonald polynomials at $t=0$ as the non-negative $q$-graded sum of Demazure characters. We shall define certain combinatorial object called 'semi-standard key tableaux' on which Kashiwara raising and lowering operators are defined. We shall show that under this action, each component of the crystal graph is a Demazure crystal. This is a work by Sami Assaf and Nicolle González.

Aakanksha Jain

The reduced Bergman kernel and its properties

Abstract. The Bergman kernel is well-known and has proved to be an essential tool in the function theory of several complex variables. An important space closely related to the Bergman space is the space of all holomorphic functions whose derivatives are square-integrable with respect to the area measure. It is a reproducing kernel Hilbert space, and its kernel is called the reduced Bergman kernel.

Here, we shall talk about the history of the reduced Bergman kernel in brief and then discuss some of its properties. In particular, we shall look at the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences and for the weighted reduced Bergman kernel under proper holomorphic maps. Furthermore, we look at Ramadanov-type theorems, localization, and boundary behavior of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these higher-order kernels under biholomorphisms.

Finally, if time permits, I will discuss some (open) questions that originated during our study of the reduced Bergman kernel. (Joint work with Sahil Gehlawat and Amar Deep Sarkar)

Shankar K

Homogenization of Stokes systems in an oscillating domain

Abstract. We consider the Stokes system in a domain with inhomogeneous boundary data on the oscillating boundary part of the domain. After establishing the existence and uniqueness of the solutions due to the transformation from evolving domain techniques, we set up a suitable framework to study the homogenization process. Finally, we derive the corrector results and corresponding error estimates.

Nishu Kumari

Factorization of classical characters twisted by roots of unity

Abstract. Schur polynomials are the characters of irreducible representations of classical groups of type A parametrized by partitions. For a fixed integer $t\geq 2$ and a primitive $t^{\text{th}}$ root of unity $\omega$, Schur polynomials evaluated at elements $\omega^kx_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, were considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). They characterized partitions for which the specialized Schur polynomials are nonzero and showed that if the Schur polynomial is nonzero, it factorizes into characters of smaller classical groups of type A.

In this talk, I will present a generalization of the factorization result to the characters of classical groups of type B, C and D. We give a uniform approach for all cases. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. This is joint work with A. Ayyer and is available here. (Preprint: https://arxiv.org/abs/2109.11310.)

Srijan Sarkar

Pure contractions on Hilbert spaces of analytic functions

Abstract. A contraction $T$ on a Hilbert space $\mathcal{H}$ (that is, $\|T\| \leq 1$) is said to be pure if the sequence $\{T^{*n}\}_n$ converges to $0$ in the strong operator topology. It is evident that this condition is motivated by the behaviour of a unilateral shift. In fact, the fundamental Sz.-Nagy–Foias's analytic model shows that such contractions are indeed unitary equivalent to certain compressions of the shift operator on vector-valued Hardy spaces on the unit disc.

Pure contractive multiplication operators on Hilbert spaces of analytic functions serve as an important connection between several topics like operator theory, function theory, and several complex variables. For instance, $(a)$ these operators play an important role in finding models for abstract operators on Hilbert spaces and $(b)$ in a seminal work by Agler and McCarthy, the authors showed that by using this kind of operators on vector-valued Hardy spaces, one could explicitly describe distinguished varieties in the bidisc which are important in the context of extremal Pick interpolation problems for the bidisc. Thus, it is evident that these operators play significant roles in several questions involving operator theory. However, in the literature there hasn't been a detailed study on the following question: when does a multiplication operator become pure contractive? In this talk, we will answer this question by obtaining a complete characterization for pure contractive multiplication operators on several Hilbert spaces of analytic functions on both the unit polydisc and the unit ball in $\mathbb{C}^n$. If time permits, we will look at some applications of the results.

Mihir Sheth

On the admissibility of irreducible representations of $p$-adic groups

Abstract. I will begin by discussing what $p$-adic groups are and why their (smooth) admissible representations are important in number theory. I will then show that irreducible representations over a certain class of rings are automatically admissible. This is a generalization of Jacquet's classical theorem for irreducible complex representations and is based on the finiteness result of Dat, Helm, Kurinczuk, and Moss for Hecke algebras of $p$-adic groups.

Schur–Weyl dualities and tensor structures

Abstract. The classical Schur–Weyl duality provides an interplay between representations of the symmetric group $S_d$ and polynomial representations of the general linear group $GL_n$ of fixed degree $d$. Over the years, there have been many variants of the classical Schur–Weyl duality. Among these variants, there is a such duality, due to Jones and Martin, between a diagrammatic algebra namely the partition algebra and the symmetric group $S_n$. A fundamental question in representation theory is to understand the tensor product of two representations. For symmetric group, the canonical tensor product is also known as the Kronecker product. In this talk, we discuss Schur–Weyl dualities. We also describe a tensor product on polynomial representations of $GL_n$ of fixed degree, which via the classical Schur–Weyl duality corresponds to the Kronecker product for symmetric group.

Vijaya Kumar U

The joint spectrum for a commuting pair of isometries

Abstract. The study of a pair $(V_1,V_2)$ of commuting isometries is a classical theme. We shine new light on it by using the defect operator $$C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+V_1V_2V_2^*V_1^*.$$ We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.

### Some tips for the symposium speakers

You can find a useful compilation of tips on giving good mathematical talks and common pitfalls to avoid on the University of Michigan's website: https://sites.lsa.umich.edu/math-graduates/best-practices-advice/giving-talks/.

Here are some highlights from these resources. Use these tips as general guidelines but develop your own style.

1. Talks are not the same as papers

One should avoid the common error of treating a talk like a paper, with all the attendant details, technicalities, and formalism.

A good talk should also be "friendly" to non-experts by devoting at least the first few minutes going over basic examples or background, so that they are not completely lost even from the beginning. Even the experts will appreciate a review of the background material.

2. A slide talk versus a blackboard talk

(Jordan Ellenberg: https://quomodocumque.files.wordpress.com/2010/09/talktipsheet.pdf)

For a half-hour talk, the time-saving that comes with slides usually makes them a better choice.

3. Time is of the essence, and less is more

A good rule of thumb: you should allow between 1 minute and 2 minutes per slide. Do not pack your slides with information and try to artificially achieve the goal of 1-2 minutes per slide by speaking quickly. In fact, consider having as little information on each slide as possible.

If you check the clock during your presentation and say, "Uh oh, I'd better speed up!" this angers the gods.

4. Do not try to impress the audience with your brilliance.

"Making the talk complicated so that your work appears profound is a great sin."

(https://montrealnumbertheory.org/qvntsspeakers)

As a corollary, do not present your proof "in all its details, paying fond attention to what happens when p=2 and when the spectral sequence fails to degenerate after the 17-th stage." Instead, focus on your exposition by presenting the big picture which provides the background and motivation for the mathematics that you have done.

5. Use the power of examples

(https://quomodocumque.wordpress.com/2009/11/16/what-to-do-in-talks/)

Give an example so easy that it is insulting. Then give an example that is slightly less insulting. Finally give an interesting example.

6. Practice, Practice, and Practice