Pedro Ribeiro (Porto, Portugal) - Thursday, Feb 22, 2024 - 4:00 PM (IST)

 Dear all,

This week's talk is by Pedro Ribeiro of the University of Porto, Portugal. We apologize for the late notification. The talk announcement follows. 

Talk Announcement: 

Title:  Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook

Speaker:  Pedro Ribeiro (University of Porto, Portugal) 
When: Feb 22, 2024, 4:00 PM- 5:00 PM IST (10:30 am Western European Time (WET)) 

Where: Zoom:

Live Link: https://youtube.com/live/yq4SVSlTD2o?feature=share

Abstract

In his lost notebook, Ramanujan recorded beautiful identities. These include earlier versions of Guinand's formula for the divisor function and the transformation formula for the logarithm of Dedekind's -function. 

In our presentation we will describe some generalizations of these formulas using a beautiful theory due to the forgotten mathematician N. S. Koshliakov. Our work will be presented under the point of view initiated by A. Dixit and R. Gupta, the first mathematicians of our century who have extended Koshliakov's theory in several directions. 

This talk is based on joint work with Semyon Yakubovich. 

Arvind Ayyer (IISc.) - Thursday, Feb 8, 2024 - 4:30 PM (IST)

 Dear all,

The next talk is by Arvind Ayyer of the Indian Institute of Science, Bangalore, India. The talk announcement follows. Please note that the talk is half an hour later than our usual meeting time. 

Talk Announcement: 

Title: A new combinatorial formula for the modified Macdonald polynomials

Speaker:  Arvind Ayyer (IISc, Bangalore, India)
When: Feb 8, 2024, 4:30 PM- 5:30 PM IST (Note special time) 

Where: Zoom: Please write to the organisers for the link

Live Link: https://youtube.com/live/udOYGrz0zaI?feature=share

Abstract

Macdonald polynomials are a remarkable family of symmetric
functions that are known to have connections to combinatorics, algebraic
geometry and representation theory. The modified Macdonald polynomials 
are obtained from the Macdonald polynomials using an operation called 
plethysm. A combinatorial formula for the latter was given by Haglund, 
Haiman and Loehr in a celebrated work (JAMS, 2004). We will give a new 
combinatorial formula (ALCO 2023).

Recently, a formula for the symmetric Macdonald polynomials was given by 
Corteel, Mandelshtam and Williams in terms of objects called multiline 
queues, which also compute probabilities of a statistical mechanics 
model called the multispecies ASEP on a ring. It is natural to ask 
whether the modified Macdonald polynomials can be obtained using a 
combinatorial gadget for some other statistical mechanics model. We 
answer this question in the affirmative via a multispecies totally 
asymmetric zero-range process (TAZRP) in (arXiv:2209.09859).

These are joint works with J. Martin and O. Mandelshtam.

Ramanujan Special: Frank Garvan (Florida) - Thursday, Jan 25, 2024 - 7:30 PM (IST)

 Happy new year. 

The first talk of the year (on January 25, 2023) is a ``Ramanujan Special". This year's speaker is Frank Garvan. Please note that the talk will be later than usual. A report on the activities of this seminar in 2023 appears in the SIAM newsletter OPSFNET. We hope this year is equally exciting for our group. Please consider the seminar to present your latest preprint. 

Talk Announcement: The 2024 Ramanujan Special

Title: Identities for Ramanujan's Mock Theta Functions and Dyson's Rank

Function

Speaker: Frank Garvan (University of Florida, USA)
When: Jan 25, 2024, 7:30 PM- 8:30 PM IST (9 AM EST) (Note special time) 

(EST= IST - 10:30)

Where: Zoom: Please write to the organisers for the link.

Live LInk: https://youtube.com/live/KWUjDWfjFwg?feature=share

Abstract

In Ramanujan's Lost Notebook there are identities connecting Ramanujan's fifth order mock theta functions and Dyson's rank mod 5. We extend these connections to Zagier's higher order mock theta functions. We consider Dyson's problem of giving a group-theoretic structure to

the mock theta functions analogous to Hecke's theory of modular forms. From this much surprising symmetry and q-series identities arise in joint work with Rishabh Sarma and Connor Morrow.

David Bradley (Maine) - Thursday Dec 7, 2023 - 6:30 PM (IST) (NOTE. Special Time)

 Dear all,

The talk this week is by David Bradley of the University of Maine. Please note the special time. Since Professor Bradley is located in the US, we are starting later than usual. 

This will be the final talk of the year. We will come back next year with a Ramanujan special, and hope that we get an opportunity to meet in person in the upcoming conference and holiday season. We wish you happy holidays and a great new year. 

The announcement is as follows. 

Talk Announcement: 

Title:    On Fractal Subsets of Pascal's "Pyramid" and the Number of  Multinomial Coefficients Congruent to a Given Residue Modulo a Prime

Speaker: David Bradley (University of Maine, USA)

When: Thursday, Thursday Dec 7 23, 2023 - 6:30 PM (IST) (8AM EST/ 2PM (CET))

Where: Zoom: Ask the organisers for a link

Live Link: https://youtube.com/live/jXpB3U41ELs?feature=share 

Abstract. 

We obtain an explicit formula and an asymptotic formula for the number of multinomial coefficients which are congruent to a given residue modulo a prime, and which arise in the expansion of a multinomial raised to any power less than a given power of that prime. Each such multinomial coefficient can be associated with a certain Cartesian product of intervals contained in the unit cube. For a fixed prime, the union of these products forms a set which depends on both the residue and the power of the prime. In the limit as the power of the prime increases to infinity, the sequence of unions converges in the Hausdorff metric to a non-empty compact set which is independent of the residue. We calculate the fractal dimension of this limiting set, and consider its monotonicity properties as a function of the prime.  To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first p rows of Pascal’s "pyramid" which are congruent to a given nonzero residue r modulo the prime p. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes p, the number of binomial coefficients in Pascal's triangle which are congruent to r mod p for r = 1, 1 < r < p−1, and r = p−1 is well approximated by the respective linear functions of p given 

by 3p, p/2, and p. In particular, for large primes p there are approximately six times as many occurrences of the residue 1 in the first p rows of Pascal’s triangle reduced modulo p than there are of any other residue r in the range 1 < r < p − 1, and three times as many as r = p − 1. On the other hand, if we let the nonnegative integer k vary while keeping the prime p fixed, and look at the relative frequency of various residue classes that occur in the first pk rows, the seemingly substantial differences in frequency between r = 1, 1 < r < p−1, and r = p−1 when k = 1 are increasingly dissipated as k grows without bound. We show that in the limit as k tends to infinity, all nonzero residues are equally represented with asymptotic proportion 1/(p − 1).

Sonika Dhillon (ISI, Delhi) - Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST)

 Dear all,

The talk this week is by Sonika Dhillon, ISI, Delhi. The announcement is as follows. 

Talk Announcement: 

Title:   Linear independence of numbers
Speaker: Sonika Dhillon (ISI, Delhi)
When: Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST) 

Where: Zoom: Write to the organisers for the link

Live Link: https://youtube.com/live/JSqZ8FLhKdU?feature=share

Abstract. 

Let $\psi(x)$ denote the digamma function that is the logarithmic derivative of $\Gamma$ function.
In 2007, Murty and Saradha studied the linear independence of special values of  digamma function $\psi(a/q)+\gamma$ over some specific numbers fields which also imply the  non-vanishing of $L(1,f)$ for any rational-valued Dirichlet type function $f$.  In 2009, Gun, Murty and Rath studied the non-vanishing of $L'(0,f)$ for even Dirichlet-type periodic $f$ in terms of $L(1,\hat{f})$ and established that this is related to the linear independence of logarithm of gamma values.  In this direction, they made a conjecture which they call it as a variant of Rohrlich conjecture concerning the linear independence of logarithm of gamma values. In this talk, first we will discuss the  linear independence of digamma values over the field of algebraic numbers. Later, we provide counterexamples
to this variant of  Rohrlich conjecture.

Sagar Shrivastava (TIFR, India) - Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST)

 Dear all,

The talk in the coming week is by Sagar Shrivastava, School of Mathematics, Tata Institute of Fundamental Research (TIFR).

Talk Announcement: 

Title:   Representations, Determinants and Branching rules
Speaker: Sagar Shrivastava (TIFR, India)
When: Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST) 

Where: Zoom: Please write to the organisers for the link

Live Link: https://youtube.com/live/S4DQRvK46no?feature=share

Abstract. 

Branching rules/laws (restriction of representations) also known as symmetry breaking in physics has been an active area of research since the onset of the topic by Herman Weyl in 1950. In this talk, I would give a brief description of Highest weight theory and the determinantal form of the Weyl character formula. I would proceed to talk about branching from $GL_n$ to $GL_{n-1}$ and give an idea about the other classical groups. 

Seamus Albion (Vienna, Austria) - Thursday Oct 26, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Seamus Albion of the University of Vienna. The announcement is as follows.

Talk Announcement: 

Title:   An elliptic $A_n$ Selberg integral
Speaker: Seamus Albion (Vienna, Austria)
When: Thursday, Thursday Oct 26, 2023 - 4:00 PM (IST) (12:30 PM CEST)

Where: Zoom: Write to the organisers to get the link

Live Link: https://youtube.com/live/S4DQRvK46no?feature=share

Abstract. 

Selberg's multivariate extension of the beta integral appears 
all over mathematics: in random matrix theory, analytic number theory,  multivariate orthogonal polynomials and conformal field theory. The goal of my talk will be to explain a recent unification of two important generalisations of the Selberg integral, namely the Selberg integral associated with the root system of type A_n due to Warnaar and the elliptic Selberg integral conjectured by van Diejen and Spiridonov and proved by Rains. The key tool in our approach is the ellipticinterpolation kernel, also due to Rains. This is based on joint work with Eric Rains and Ole Warnaar.

David Wahiche (Univeriste de Tours, France) - Thursday, October 12, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by David Wahiche of the University of Tours, France. The title and abstract is below. 

Talk Announcement: 

Title:   From Macdonald identities to Nekrasov--Okounkov type formulas
Speaker: David Wahiche (Universite' de Tours, France)
When: Thursday, Oct 12, 2023, 4:00 PM- 5:00 PM IST 

Where: Zoom: Ask the organisers for a link

Live Link: https://youtube.co/live/9BNeHB2umCM?feature=share

Abstract.

Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.

This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).

Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.

In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.

At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.

Seema Kushwaha (IIIT, Allahabad) - Thursday Sept 28, 2023 - 4:00 PM (IST)

Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week. 

Talk Announcement: 

Title:   Farey-subgraphs and Continued Fractions
Speaker: Seema Kushwaha (IIIT, Allahabad)
When: Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST 

Where: Zoom: Please send email to the organisers for a link.

Live Link: https://youtube.com/live/UzJEHaj123Y?feature=share

Abstract. 

Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n}
\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.
\end{equation*}   
The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

  Let $\mathcal{X}$ be a  subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a  notion which is analogous to best rational approximation. 

An element  $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.  
    
In  this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties. 

Shashank Kanade (University of Denver) - Thursday Sept 14, 2023 - 6:00 PM (IST)

 Dear all,

We are back after an extended summer break. I hope many of us had an opportunity to meet each other and further our research goals. 

The next talk is by Shashank Kanade, University of Denver. It is a little later in the evening from our usual time.

Talk Announcement: 

Title:  On the $A_2$ Andrews--Schilling--Warnaar identities

Speaker: Shashank Kanade (University of Denver)

When: Thursday, Sept 14, 2023, 6:00 PM- 7:00 PM IST (6:30 AM MDT)

Where: Zoom: please write to the organisers for the link

Live Link: https://youtube.com/live/jnsJGm69sjM?feature=share

Abstract 

I will give a description of my work with Matthew C. Russell on the $A_2$

Andrews--Schilling--Warnaar identities. Majority of our single variable

sum=product conjectures have been proven by S. O. Warnaar; I will also

explain what remains. Bi-variate versions of our conjectures are largely open. 

Michael Schlosser (Vienna, Austria) - Thursday May 25, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Michael Schlosser of the University of Vienna, Austria. 

After this talk we will be taking a break for the summer. Hopefully, we will get an opportunity to meet in person during this time. 

Talk Announcement: 

Title: Bilateral identities of the Rogers-Ramanujan type

Speaker: Michael Schlosser (University of Vienna, Austria)
When: May 25, 2023, 4:00 PM- 5:00 PM IST (12:30 PM CEST)

Where: Zoom. Please write to the organisers for a link

Live Link: https://youtube.com/live/VO3hTqh8TSw?feature=share

Abstract 

The first and second Rogers-Ramanujan (RR) identities have a prominent history. They were originally discovered and proved in 1894 by Leonard J. Rogers, and then independently rediscovered by the legendary self-taught Indian mathematician Srinivasa Ramanujan some time before 1913. They were also independently discovered and proved in 1917 by Issai Schur. About the RR identities Hardy remarked

`It would be difficult to find more beautiful formulae than the
``Rogers-Ramanujan'' identities, ...'

Apart from their intrinsic beauty, the RR identities have served as a stimulus for tremendous research around the world. The RR and related identities have found interpretations in
various areas including combinatorics, number theory, probability theory, statistical mechanics, representations of Lie algebras, vertex algebras, knot theory and conformal field theory.

In this talk, a number of bilateral identities of the RR type will be presented. We explain how these identities can be derived by analytic means using identities for bilateral basic hypergeometric series. Our results include bilateral extensions of the RR and
of the Göllnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater.

Corresponding results for multiseries are given as well, including multilateral extensions of the Andrews-Gordon identities, of Bressoud's even modulus identities, and others.

This talk is based on the speaker's preprint arXiv:1806.011153v2 (which has been accepted for publication in Trans. Amer. Math. Soc.).

Bishal Deb (University College, London) - Thursday May 11, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Bishal Deb of University College, London. Here is the announcement. 

Talk Announcement: 

Title: The "quadratic family" of continued fractions and combinatorial sequences

Speaker: Bishal Deb (University College, London)
When: May 11, 2023, 4:00 PM- 5:00 PM IST (11:30 AM BST)

Where: Zoom:

Live Link: https://youtube.com/live/sl0ef-T5c3o?feature=share

Abstract 

We will begin this talk by introducing some combinatorial sequences whose Stieltjes-type continued fraction coefficients increase linearly. We briefly mention the work of Sokal and Zeng where they systematically studied multivariate generalisations of these continued fractions for factorials, Bell numbers and double factorials. 

Next, we will define the Genocchi and median Genocchi numbers and  introduce D-permutations, a class of permutations which enumerate these numbers. We mention some multivariate continued fractions counting various statistics on D-permutations.

Finally, we move to the secant numbers and introduce cycle-alternating permutations; these are another class of permutations  which enumerate the secant numbers. We mention some multivariate continued fractions counting various statistics on cycle-alternating permutations. We then describe the entries in the Jacobi-Rogers matrix of our continued fraction using alternating Laguerre digraphs, which are a class of directed graphs. If time permits, we will briefly state some remarks on the Jacobian elliptic functions.

This talk will be based on joint work with Alan Sokal.

Rahul Kumar (Penn State University) - Thursday, April 27 - 4:00 PM

The next talk is by Rahul Kumar, Fulbright-Nehru Postdoctoral Fellow, Penn State University.  The announcement is as follows. 

Talk Announcement: 

Title: Arithmetic properties of the Herglotz-Zagier-Novikov function

Speaker: Rahul Kumar (Penn State University)
When: Apr 27, 2023, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to sfandnt@gmail.com for a link

Live Link: https://youtube.com/live/5iJRbNZOksM?feature=share

Abstract

The Kronecker limit formulas are concerned with the constant term in the Laurent series expansion of certain Dirichlet series at $s=1$. Various special functions appear in Kronecker limit formulas; one of them is \emph{Herglotz function}. Recently, Radchenko and Zagier extensively studied the properties of the Herglotz function, such as its special values, connection to Stark's conjecture, etc. This function appeared in the work of Herglotz, and Zagier. After providing an overview of the history of this research area, we will discuss the arithmetic properties of a Herglotz-type function that appears in a Kronecker limit formula derived by Novikov. For example, we will present the two- and three-term functional equations satisfied by it along with its special values. This is joint work with Professor YoungJu Choie.

A. Sankaranarayanan (Hyderabad) - April 13, 2023 - 4:00 PM

 Dear all,

The next talk is by A. Sankaranarayanan of the School of Mathematics and Statistics, University of Hyderabad. The announcement is as follows. 

Talk Announcement: 

Title: On the Rankin-Selberg L-function related to the Godement-Jacquet L-function

Speaker: A. Sankaranarayanan (University of Hyderabad)
When: Apr 13, 2023, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to organisers for the link

Live Link: https://youtube.com/live/ijtihr4Bi14?feature=share

Abstract 

We discuss the Riesz mean of the Coefficients of the Rankin-Selberg L-function related to the Godement-Jacquet L-function.

This is a joint work with Amrinder Kaur and recently appeared in Acta Mathematica Hungarica.

Christophe Vignat (Tulane) - Thursday, Mar 30 - 4:00 PM

 The next talk is by Christophe Vignat.   It will be at our usual time of 4 PM IST. 

Talk Announcement: 

Title: Dirichlet Series Under Standard Convolutions: Variations on Ramanujan’s Identity for Odd Zeta Values 

Speaker: Christophe Vignat (Université Paris-Saclay, CentraleSupelec, Orsay, France and Tulane University)
When: Mar 30, 2023, 4:00 PM- 5:00 PM IST (1:30 PM EEST)

Where: Zoom. Write to the organisers for a link.

Live Link: https://youtube.com/live/nx60NHwUUjg?feature=share

Abstract 

I will show a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; this formula is inspired by a famous identity of Ramanujan.  Some specializations of this convolution formula produce new identities and allow to recover several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and his collaborators, or the convolution of Bernoulli numbers by Y. Komori and his collaborators. 

If time permits, I will also exhibit some matrix product representations for the Riemann zeta function evaluated at even and odd integers.

This is joint work with P. Chavan, S. Chavan and T. Wakhare.