## Thursday, November 19, 2020

### Manjil Saikia (Cardiff University) November 26, 2020: 3:55 PM to 5:00 PM (IST)

The next talk is by Manjil Saikia of Cardiff University.  The remaining two talks of the year will be on December 10 and December 17.

We will begin the new year with a special public lecture by Wadim Zudilin. More details as we come nearer 2021.

Talk announcement

Title: Refined enumeration of symmetry classes of Alternating Sign Matrices

Speaker: Manjil P. Saikia  (Cardiff University)

When: November 26, 2020 - 3:55 PM - 5:00 PM (IST)

ABSTRACT:

The sequence $1,1,2,7,42,429, \ldots$ counts several combinatorial objects, some of which I will describe in this talk. The major focus would be one of these objects, alternating sign matrices (ASMs). ASMs are square matrices with entries in the set {0,1,-1}, where non-zero entries alternate in sign along rows and columns, with all row and column sums being 1. I will discuss some questions that are central to the theme of ASMs, mainly dealing with their enumeration. In particular we shall prove some conjectures of Fischer, Robbins, Duchon and Stroganov. This talk is based on joint work with Ilse Fischer and some ongoing work.

## Thursday, November 12, 2020

### Sanoli Gun (IMSc, Chennai) Nov 12, 2020 - 3:55 PM - 5:00 PM

We skipped a week. The next talk is by Sanoli Gun. The details are given below.

Some members have complained about announcements of other groups on this mailing list. So we have now made a policy not to forward any other announcements to this mailing list. We are still slow to bring all the talks available online. But recordings of talks are available. In case you want please send us an email.

Talk Announcement

Title: Large values of $L$-functions

Speaker: Sanoli Gun (IMSc, Chennai)

When: November 12, 2020 - 3:55 PM - 5:00 PM (IST)

ABSTRACT:

In this lecture, we will give an overview of a method of Soundararajan and show that how this method can be used to produce large values of $L$-functions in different set-ups.

## Thursday, October 22, 2020

### Michael Schlosser (University of Vienna) - October 22, 2020 - 3:55 PM-5:00 PM (IST)

The next talk in the seminar is by Michael Schlosser.  This talk is a longer version of a talk he presented recently in an AMS regional meeting. Please note that this time the talk will be on Zoom.

Talk announcement

Title: Basic hypergeometric proofs of two quadruple equidistributions of Euler-Stirling statistics on ascent sequences

Speaker: Michael Schlosser (University of Vienna, Austria).

When: Thursday October 22, 3:55 PM-5:00 PM

Where: ZOOM this time. Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Abstract:
In my talk, I will present new applications of basic hypergeometric series to specific problems in enumerative combinatorics. The combinatorial problems we are interested in concern multiply refined equidistributions on ascent sequences. (I will gently explain these notions in my talk!) Using bijections we are able to suitably decompose some quadruple distributions we are interested in and obtain functional equations and ultimately generating functions from them, in the form of explicit basic hypergeometric series. The problem of proving equidistributions then reduces to applying suitable transformations of basic hypergeometric series. The situation in our case however is tricky (caused by the fact how the power series variable $r$ appears in the base $q=1-r$ of the respective basic hypergeometric series; so being interested in the generating function in $r$ as a Maclaurin series, we are thus interested in the analytic expansion of the nonterminating basic hypergeometric series in base $q$ around the point $q=1$), as none of the known transformations appear to directly work to settle our problems; we require the derivation of new identities. Specifically, we use the classical Sears transformation and apply some analytic tools to establish a new non terminating ${}_4\phi_3$ transformation formula of base $q$, valid as an identity in a neighborhood around $q=1$. We use special cases of this formula to deduce two different quadruple equidistribution results involving Euler--Stirling statistics on ascent sequences.  One of them concerns a symmetric equidistribution, the other confirms a bi symmetric equidistribution that was recently conjectured in a paper (published in JCTA) by Shishuo Fu, Emma Yu Jin, Zhicong Lin, Sherry H.F. Yan, and Robin D.B. Zhou.

This is joint work with Emma Yu Jin. For full results (and further ones), see https://arxiv.org/abs/2010.01435

## Thursday, October 8, 2020

### Debashis Ghoshal (SPS, JNU) - October 8, 2020 - 3:55 pm - 5 PM (IST)

The next talk in the Special Functions and Number Theory is as below. The speaker, Debashis Ghoshal, is one of the most regular attendees and long term supporters of this Seminar. We have long asked him to show how special functions arise in his work and to perhaps tell us about some problems.

Talk announcement

Speaker: Debashis Ghoshal (School of Physical Sciences, JNU)

Title: Two-dimensional gauge theories, intersection numbers and special functions

Where: Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Abstract: The partition function of two dimensional Yang-Mills theory contains a wealth of information about the moduli space of connections on surfaces. We study this problem on a special class of surfaces of infinite genus, which are constructed recursively. While the results are suggestive of an underlying geometrical structure, we use it as a prop to efficiently compute results for finite genus surfaces. Riemann zeta function, confluent hypergeometric function and its truncations show up in explicit computations for the gauge group SU(2). Much of the corresponding results are open for other groups.

## Thursday, September 3, 2020

### Apoorva Khare (IISc., Bangalore) September 10 and 24 at 3:55-5:00 IST

Welcome to the new academic year. To inject some much needed positivity in what continues to be an unusual year, we begin our semester with two talks by Apoorva Khare on total positivity. As usual, students are welcome.

The details of the two talks are as follows.

Talk 1: An introduction to total positivity

Speaker: Apoorva Khare (IISc., Bangalore)

When: Thursday, September 10, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Where: TBA Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Abstract:

I will give a gentle introduction to total positivity and the theory of Polya frequency (PF) functions. This includes their spectral properties, basic examples including via convolution, and a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how PF functions connect to the Laguerre-Polya class and hence Polya-Schur multipliers, and mention 21st century incarnations of the latter.

Talk 2: Totally positive matrices, Polya frequency sequences, and Schur polynomials

Speaker: Apoorva Khare (IISc., Bangalore)

When: Thursday, September 24, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Where: TBA Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Abstract:

I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.

## Thursday, August 27, 2020

### Amritanshu Prasad (IMSc, Chennai): August 27, 2020 at 3:55-5:00 PM (IST)

The next talk in the Special Functions and Number Theory Seminar is below. As usual, we have requested the speaker to give something suitable for students and non-experts, so please feel free to give this announcement to interested students and colleagues.

The regular semester is beginning soon.  In case you have anything to present which you are currently excited about, please do volunteer for a talk in this seminar. Further, we would like to include an agenda for some expository talks, perhaps a series leading to a research problem. Any ideas in these directions are welcome!

Talk announcement

Title: Character polynomials and their moments

When: Thursday, August 27, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Where: Zoom: Please write to sfandnt@gmail.com to get the link to the zoom talk. It is best to write a day in advance.  (The link will be open at 3:30 PM for the organizers to test their systems)

Abstract:

A polynomial in a sequence of variables can be regarded as a class function on every symmetric group when the $i$th variable is interpreted as the number of $i$-cycles. Many nice families of representations of symmetric groups have characters represented by such polynomials.

We introduce two families of linear functionals of this space of polynomials -- moments and signed moments. For each $n$, the moment of a polynomial at $n$ gives the average value of the corresponding class function on the nth symmetric group, while the signed moment gives the average of its product by the sign character. These linear functionals are easy to compute in terms of binomial bases of the space of polynomials.

We use them to explore some questions in the representation theory of symmetric groups and general linear groups. These explorations lead to interesting expressions involving multipartite partition functions and some peculiar variants.

Gaurav Bhatnagar (Ashoka University), Atul Dixit (IIT, Gandhinagar), Krishnan Rajkumar (JNU)

## Thursday, August 13, 2020

### Fatma Cicek (IIT, Gandhinagar), Aug 13, 3:55 PM - 5:00 PM (IST)

The next talk in the Special Functions and Number Theory Seminar is by Fatma Cicek.  Fatma has just joined IIT, Gandhinagar (virtually) as an Assistant Research Professor. We welcome her to India!

Speaker: Fatma Cicek

Title: On the Logarithm of the Riemann Zeta-function Near the Nontrivial Zeros

When: Thursday, August 13, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Abstract:

Selberg's central limit theorem is one of the most significant probabilistic results in analytic number theory. Roughly, it states that the logarithm of the Riemann zeta-function on and near the critical line has an approximate two-dimensional Gaussian distribution.

In this talk, we will talk about our recent result which states that the distribution of the logarithm of the Riemann zeta-function near the sequence of the nontrivial zeros has a similar central limit theorem. Our results are conditional on the Riemann Hypothesis and/or suitable zero-spacing hypotheses. They also have suitable generalizations to Dirichlet $L$-functions.

## Saturday, July 25, 2020

The next talk is by Alok Shukla of Ahmedabad University. Alok has recently moved from the University of Manitoba and we welcome him back to India and this group

Please note that we are now back into our usual time, one hour later than our previous talk. The talk is from 3:55 pm.

The details are as follows.

Title: Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities

When: Thursday, July 30, 3:55 PM - 5:00 PM IST (GMT + 5:30)

Abstract:
The Jacobi triple product identity and Rogers-Ramanujan identities are among the most famous $q$-series identities. We will present elementary combinatorial "tiling proofs" of these results. The talk should be accessible to a general mathematical audience.

## Thursday, July 16, 2020

### Ali Uncu (RICAM, Austrian Academy of Sciences, Linz, Austria)

The next talk in the Topics in Special Functions and Number Theory seminar is by Ali Uncu.  This talk will again be on zoom. In case you wish to try out the software before the talk, please get in touch with one of the organizers. Further, requests for links should be made well before time. Some people did not get the link in time. In case you wish to be placed in the mailing list, please send an email to sfandnt@gmail.com.

Title: The Mathematica package qFunctions for q-series and partition theory applications

UPDATE: Please visit https://akuncu.com/qfunctions/ for a version of this talk (video) and materials mentioned in the talk.

Speaker: Ali Uncu, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria

When: Thursday, July 16, 2020, 2:55 PM - 4:00 PM IST  (GMT+5:30)

Where: On zoom (link available on request).  Please write to sfandnt@gmail.com for the link a few hours before the talk.

Abstract

In this talk, I will demonstrate the new Mathematica package qFunctions while providing relevant mathematical context. This implementation has symbolic tools to automate some tedious and error-prone calculations and it also includes some other functionality for experimentation. We plan to highlight the four main tool-sets included in the qFunctions package:

(1) The q-difference equation (or recurrence) guesser and some formal manipulation tools,

(2) the treatment of the method of weighted words and automatically finding and uncoupling recurrences,

(3) a method on the cylindrical partitions to establish sum-product identities,
(4) fitting polynomials with suggested well-known objects to guess closed formulas.

This talk is based on joint work with Jakob Ablinger (RISC).

## Thursday, July 2, 2020

### Hjalmar Rosengren (Chalmers University of Technology and University of Gothenburg)

The next talk in the Topics in Special Functions and Number Theory seminar is by Hjalmar Rosengren on the Kanade-Russel identities. Please note that the talk will be one hour earlier than usual.

This talk will again be on zoom. In case you wish to try out the software before the talk, please get in touch with one of the organizers.

Speaker: Hjalmar Rosengren, Chalmers University of Technology and University of Gothenburg, Sweden

When: Thursday, July 2, 2020: 2:55-4:00 pm

Where: On Zoom: Link (available on request). Please send email to sfandnt@gmail.com a few hours before the talk.

Kanade and Russell conjectured several Rogers-Ramanujan-type identities for triple series. Some of these conjectures are related to characters of affine Lie algebras, and they can all be interpreted combinatorially in terms of partitions. Many of these conjectures were settled by Bringmann, Jennings-Shaffer and Mahlburg. We describe a new approach to the Kanade-Russell identities, which leads to new proofs of five previously known identities, as well as four identities that were still open. For the new cases, we need quadratic transformations for q-orthogonal polynomials.

## Thursday, June 18, 2020

### Arvind Ayyer (IISc., Bangalore)

The next talk in the Seminar on Special Functions and Number Theory is by Arvind Ayyer of IISC, Bangalore. The information appears below.

As announced in the last talk, Atul Dixit is now a co-organizer of this Seminar. The Zoom access for this talk has been kindly provided by Ashoka University. So this seminar series is now co-organized by Ashoka University, IIT, Gandhinagar and JNU.

We are experimenting with various systems so that we can give options to speakers. Please get in touch with any of Atul, Krishnan or myself if you wish to try out zoom before the talk. It requires some installation. Zoom may be suitable for a webinar in case we organize a public lecture, which we plan to do at some point.

Title: The Monopole-Dimer Model
Speaker: Arvind Ayyer, IISc. (Bangalore)
When: Thursday, June 18, 2020: 3:55-5:00 pm
Where: On Zoom: Link (available on request). Please send email to sfandnt@gmail.com

Here is a link to the talk.

Abstract:

The dimer model is a model which arose in statistical physics as a study
of adsorption. We will first define the model and state Kasteleyn's
groundbreaking result expressing the partition function of the model as
a Pfaffian for planar graphs. We develop a new model of monopoles and
dimers whose partition function is a determinant for any planar graph.
We then apply this to the rectangular grid and obtain a generalization
of Kasteleyn's miraculous product formula. Lastly, we study the
thermodynamic limit and obtain formulas for the free energy and entropy.
Some interesting special functions show up in this limit. Time
permitting, we will also show that in some special cases, the partition
function becomes a perfect square. This work is based on arXiv:1311.5965
(Mathematical Physics, Analysis and Geometry, 2015) and arXiv:1608.03151
(to appear in Annals of Combinatorics).

Graduate students are welcome. We plan to include a friendly introduction.

## Thursday, June 4, 2020

### Atul Dixit

We are back in business, after a short break due to the lockdown. We decided to begin the seminar online. There is a great advantage to this. First of all, many people in Delhi who were not in a position to travel to JNU or ISI can now attend from home. Secondly we have greatly expanded the ability to get good speakers for the seminar. Finally, people working in related areas around the country, and perhaps in Europe and Singapore can now contribute. We wish at present to keep the timings as per our original calendar (suitable to our schedule in India) but if we get a great speaker, we may modify as per their convenience. We will continue to meet once every two weeks.

As our first talk of the new season, we are delighted to have as a speaker Atul Dixit from IIT, Gandhinagar. Atul has developed a small group of students and post-docs in Gandhinagar, doing interesting mathematics related to the themes covered in our seminar, and we hope they will join this group.

Title: Superimposing theta structure on a generalized modular relation\\

Speaker: Atul Dixit, IIT, Gandhinagar

When: Thursday, June 4, 2020; 3:55-5:00 pm IST (GMT+5:30)

The Presentation is available.

Please log in 5 minutes before 4:00 pm at the appointed date. Krishnan and I are available if you wish to test your systems. Finally, please write to us if you wish to be included in the announcement list. Please feel free to distribute the attached announcement to your students and other interested parties.

Abstract: By a modular relation for a certain function $F$, we mean a relation governed by the map $z\to -1/z$ but not necessarily by $z\to z+1$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha)=F(iw, \beta)$ or the Ramanujan-Guinand formula $F(z, \alpha)=F(z, \beta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $\operatorname{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$ obtained by Kesarwani, Moll and the speaker, that is, one can superimpose theta structure on it.

In 2011, the speaker obtained a generalized modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on the generalized modular relation? While answering this question affirmatively, we were led to a surprising new generalization of $\zeta(z, a)$. We show that this new zeta function, $\zeta_w(z, a)$, satisfies a beautiful theory. In particular, it is shown that $\zeta_w(z, a)$ can be analytically continued to Re$(z)>-1, z\neq1$. We also prove a two-variable generalization of Ramanujan's formula which involves infinite series of $\zeta_w(z, a)$ and which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.

## Sunday, March 15, 2020

### Cancellations etc.

March 14, 2020 (Pie Day!)

Dear all,

I am sorry to inform you that as per JNU administration requirements, no meetings are allowed. So we have decided to postpone the seminars to April. In case the situation permits in April, I hope we will meet every week and pursue our agenda.

The agenda, which got clarified since my previous email, is as follows:

a. Circle Method by Manoj Verma
b. Intersection Numbers using Yang-Mills Theory on Riemann Surfaces, by Debashis Ghoshal
c. Mini Course on Continued Fractions 1: Basic Moves by Gaurav Bhatnagar
d. Mini Course on Continued Fractions 2: Convergence Theorems by Krishnan Rajkumar

Meanwhile, this is a wonderful opportunity to stay quarantined, not meet anyone, and focus all our efforts on our research for a couple of weeks! I suppose I should have said "I am happy to inform you..."

Best wishes,

Gaurav Bhatnagar and Krishnan Rajkumar

March 11, 2020

Dear all,

We hope you had a safe and happy holi. The last couple of months of the academic year are upon us, and we thought we will do something special before we close for the summer.

The next talk in our Topics in Special Functions and Combinatorics Seminar is on March 17, 2020. It is by Manoj Verma and it is on Waring's problem and the circle method. The abstract appears below.

The talk after that is by Debashis Ghoshal on March 31.

In April, there will be 2-4 talks on the topic of Continued fractions by the two organizers of this seminar. These talks will comprise a mini-course on the subject of continued fractions. We plan to cover the basic moves as well as the convergence theory. Since this topic is not covered in most graduate courses, we hope this will be something unique and useful, especially for those with interests in Combinatorics, Number Theory and Special Functions. We will come back with details soon; please keep your tuesday afternoons free if you are interested, and kindly let interested students know about this.

In all of the above, the lecturers have promised a focus on techniques and will provide (optional) exercises, so that those of us who are interested can actually get our hands dirty and learn something which we can use in our work.

Best wishes,

Krishnan Rajkumar  and Gaurav Bhatnagar

TALK ANNOUNCEMENT

Speaker: Manoj Verma (SPS, JNU)

Title: Waring's problem and the circle method

When: Tuesday, March 17, 2020, 4 pm.

Where: Seminar Room, School of Physical Sciences (SPS), Dr. CV Raman Marg, JNU.
(TO BE CONFIRMED)

Abstract: This talk will be an introduction to the circle method for those with no previous familiarity with the circle method. I shall introduce the method using Waring's problem as the prototype. For a positive integer $k$, let $G(k)$ denote the smallest positive integer $s$ such that every sufficiently large positive integer is a sum of s kth powers of integers. We shall sketch a proof of the fact that $G(k)$ is less than or equal to $2^{k} + 1$.

Students are welcome.

## Friday, February 21, 2020

### Shiv Prakash Patel (IIT, Delhi)

Title:  Multiplicity one theorems in representation theory

Speaker: Shiv Prakash Patel (IIT, Delhi)

Date: Tuesday, February 25, 2020

Time: 4:00 pm

Venue: Seminar Room, School of Physical Sciences (SPS)

ABSTRACT

A representation $\pi$ of $G$ is called multiplicity free if the dimension of the vector space $Hom_{G}(\pi, \sigma)$ is at most 1 for all irreducible representation $\sigma$ of $G$. Let $H$ be a subgroup of $G$ and $\psi$ an irreducible representation of $H$. The triple $(G,H, \psi)$ is called Gelfand triple if the induced representation $Ind_{H}^{G} (\psi)$ of $G$ is multiplicity free. There is a geometric way to prove if some triple is Gelfand triple, which is called Gelfand's trick. Multiplicity free representations play an important role in representations theory and number theory, e.g. the use of Whittaker models. We will discuss Gelfand's trick and its use in a simple cases for Whittaker models of the representations of the group $GL_{n}(R)$ where $R$ is finite local ring.

## Monday, February 10, 2020

### Manish Mishra, IISER, Pune

Title: A generalization of the 3d distance theorem

Speaker: Manish Mishra, IISER Pune

Where: Seminar Room, School of Physical Sciences (SPS), C V Raman Marg, JNU

When: Tuesday, February 11, 2020, 4 PM

Abstract:

Let be a positive rational number. Call a function → to have finite gaps property mod if the following holds: for any positive irrational α and positive integer M, when the values of f(), 1 ≤ ≤ M, are inserted mod into the interval [0,P) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant kwhich depends only on f. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non- differentiable points has finite gaps property mod P. We also show that if is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_≤ 6$. This is a joint work with Amy Philip.