**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.

**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.

**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.

**Shraddha Srivastava**

**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**

(Terry Tao: https://terrytao.wordpress.com/career-advice/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.

(A. Kercheval:
https://www.ams.org/journals/notices/201910/rnoti-p1650.pdf)

*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**

(https://www.ams.org/journals/notices/201910/rnoti-p1647.pdf, Satyan Devadoss)

*The best method in helping perfect your timing is practice, practice, practice, either in front of others or by yourself. Ideas that look reasonable in notes or on
slides often don't work when said out loud. Giving voice to the written word also reveals new and better ways to frame and articulate your mathematics. *

(Terry Tao:
https://terrytao.wordpress.com/career-advice/talks-are-not-the-same-as-papers/)

Note: this is the same blog-post that was referenced in Item 1

*If you have to give the very first talk of your career, it may help to practice it, even to an empty room, to get a rough idea of how much time
it will take and whether anything should be put in, taken out, moved, or modified to make the talk flow better. *