Monday, September 25, 2023

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.


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. 

Thursday, September 14, 2023

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


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. 


Sunday, May 21, 2023

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


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.).


Sunday, May 7, 2023

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:

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.



Tuesday, April 25, 2023

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.


Friday, April 7, 2023

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
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.

Wednesday, March 29, 2023

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.

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.

Saturday, March 11, 2023

Bruce Berndt (UIUC) - Thurs Mar 16 - 6:00 PM (Note Special Time)

 Dear all,


We are happy to announce our next talk is by Professor Bruce Berndt, the world's biggest authority on Ramanujan's mathematics and related areas. 

Please note the special time. It is two hours later than usual. Please circulate this announcement in your department. 

Please see a further announcement below.

Talk Announcement: 

Title: Finite Trigonometric Sums: Evaluations, Estimates, Reciprocity Theorems

Speaker: Bruce Berndt (University of Illinois at Urbana Champaign)
When: Mar 16, 2022, 6:00 PM- 7:00 PM IST (7:30 AM - 8:30 AM (CDT))
Where: Zoom. Please write to the organisers for the link.

Abstract 
First, motivated by a theorem in Ramanujan's lost notebook, Martino Fassina, Sun Kim, Alexandru Zaharescu, and the speaker developed representations for finite sums of products of trig functions for which we provided theorems and several conjectures.   

Second, a paper of Richard McIntosh served as motivation.  First, he made a very interesting conjecture, which was recently proved by Likun Xie, Zaharescu, and the speaker.  Second, he examined a particular trigonometric sum, which inspired Sun Kim, Zaharescu, and the speaker to evaluate in closed form several classes of trigonometric sums, and find reciprocity theorems for others.  



New conference announcement
A new conference on algebraic combinatorics has been announced by Arvind Ayyer. It is called 
Meru Annual Combinatorics Conference
Dates: 29th to 31st May, 2023




Sunday, February 26, 2023

B. Ramakrishnan, ISI, Tezpur - Thursday, Mar 2, 2023 - 4:00 PM

The next talk is by B. Ramakrishnan (popularly known as Ramki), formerly of HRI, Allahabad, and now in ISI, Tezpur. 


Talk Announcement: 

Title: An extension of Ramanujan-Serre derivative map and some applications.

Speaker: B. Ramakrishnan (Indian Statistical Institute North-East Center, Tezpur)
When: Mar 2, 2022, 4:00 PM- 5:00 PM IST 
Where: Zoom: Please write to organisers for the link.

Abstract 

In this talk, we present a simple extension of the Ramanujan-Serre derivative map and 
describe how it can be used to derive a general method for explicit evaluation of convolution sums  of the divisor functions. We provide explicit examples for four types of convolution sums.

This is a joint work with Brundaban Sahu and Anup Kumar Singh.  


Saturday, February 11, 2023

Galina Filipuk, University of Warsaw - Thursday, Feb 16, 2023 - 4:00 PM


We are back to our usual time with a talk by Galina Filipuk all the way from Warsaw, Poland. Please note that we will be open to changing the time, since speakers from the US find this time to be very inconvenient, and we surely would like speakers from the US. The discussions in the previous talk went quite late into the night (for New Zealand) and we thank Shaun Cooper for a very nice talk. 

Talk Announcement: 

Title: (Quasi)-Painleve equations and Painleve equivalence problem

Speaker: Galina Filipuk (University of Warsaw, Poland)
When: Feb 16, 2022, 4:00 PM- 5:00 PM IST (11:30 CET in Warsaw)
Where: Zoom: Please write to sfandnt@gmail.com for the link.

Abstract 

Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications (e.g., in the theory of orthogonal polynomials) but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem.
The so-called geometric approach may help in many cases.

In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.


Saturday, January 28, 2023

Ramanujan Special: Shaun Cooper (Massey University) - Thursday, Feb 2, 2023 - 2:30 PM

 Happy new year. 


The first talk of the year (on Feb 2, 2023) is a "Ramanujan Special". This year's speaker is Shaun Cooper. Please note that the talk will be earlier than usual.  

The last year was quite exciting for our group with many talks as well as a mini course. We hope this year is equally exciting. Please consider the seminar to present your latest preprint. 

Talk Announcement: Ramanujan Special

Title: Apéry-like sequences defined by four-term recurrence relations: theorems and conjectures

Speaker: Shaun Cooper (Massey University, Auckland, New Zealand)

When: Feb 2, 2022, 2:30 PM- 3:30 PM IST (Note special time) (IST= GMT - 5:30)

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

Abstract 

The Apéry numbers are famous for having been introduced and used by R. Apéry to prove that~$\zeta(3)$ is irrational. They may be defined by the recurrence relation
$$
(n+1)^3A(n+1)=(2n+1)(17n^2+17n+5)A(n)-n^3A(n-1),
$$
with the single initial condition $A(0)=1$ being enough to start the recurrence. The Apéry numbers are all integers, a fact not obvious from the recurrence relation, and they satisfy interesting congruence properties. The generating function
$$
y=\sum_{n=0}^\infty A(n)w^n
$$
has a splendid parameterisation given by
$$
y = \prod_{j=1}^\infty \frac{(1-q^{2j})^7(1-q^{3j})^7}{(1-q^{j})^5(1-q^{6j})^5}
\quad
\mbox{and}
\quad
w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})^{12}}{(1-q^{2j})^{12}(1-q^{3j})^{12}}.
$$
In this talk I will briefly survey other sequences defined by three-term recurrence relations that have properties similar to those satisfied by the Apéry numbers described above. I will also introduce some sequences defined by four-term recurrence relations and describe some of their properties.

Several conjectures will be presented.

Here are the slides of the talk.


Saturday, November 12, 2022

Nishu Kumari (IISc, Bangalore) - Thursday Nov 17, 2022 - 4:00 PM to 5:00 PM

 Dear all, 


The next talk is by Nishu Kumari, a graduate student in IISc, Bangalore. The announcement is below.

Talk Announcement

Title: Factorization of Classical Characters twisted by Roots of Unity

Speaker: Nishu Kumari (IISc, Bangalore)

When: Thursday, November 17, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Tea or Coffee: Please bring your own.

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 $t$'th root of unity \omega, Schur polynomials evaluated at elements $\omega^{k} x_i$ 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)


Tuesday, November 1, 2022

Nicholas Smoot (RISC, Johannes Kepler University, Linz, Austria) - Thursday Nov 3, 2022 - 4:00 PM to 5:00 PM

The talk this week is by Nicholas Smoot from the Research Institute of Symbolic Computation (RISC) at Johannes Kepler University (JKU), Linz, Austria. The announcement is below.


Talk Announcement:

Title:  Partitions, Kernels, and Localization

Speaker: Nicholas A. Smoot (RISC at JKU, Austria)
 
When: Thursday, November 3, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom. Please write to the organisers if you require the link (sfandnt@gmail.com)

Tea or Coffee: Please bring your own.

Abstract:  

Since Ramanujan's groundbreaking work, a large variety of infinite congruence families for partition functions modulo prime powers have been discovered. These families vary enormously with respect to the difficulty of proving them. We will discuss the application of the localization method to proving congruence families by walking through the proof of one recently discovered congruence family for the counting function for 5-elongated plane partitions. In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are members of the kernel of a certain linear mapping to a vector space over a finite field. We believe that this approach holds the key to properly classifying congruence families.


Sunday, October 16, 2022

Sunil L. Naik (IMSc, Chennai) - Thursday, October 20, 2022 - 4:00 PM - 5:00 PM (IST)

Our next speaker is Sunil Naik, a grad student in IMSc, Chennai. 

Talk Announcement:

Title:  Prime factors of non-zero Hecke eigen values

Speaker: Sunil L. Naik (IMSc, Chennai)
 
When: Thursday, October 20, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom. Please ask the organisers for a link

Tea or Coffee: Please bring your own.

Abstract:  

The non-vanishing as well as primitivity of the values of the Fourier  
coefficients of non-CM Hecke eigen forms, in particular the Ramanujan  
$\tau$ function is a deep and mysterious theme in Number theory.
In this talk, we will report on our recent work on the number of  
distinct prime factors of the values of the Fourier coefficients of  
non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function.


Saturday, September 17, 2022

Kaneenika Sinha (IISER, Pune) - Mini Course - Thursday, Sept 22 and Oct 6 - 4-5:00 PM (IST)

We are happy that Kaneenika Sinha (IISER, Pune) has consented to give a mini-course on Central limit theorems in number theory. The course will comprise two lectures. The announcement is below. Graduate students who are interested in number theory are especially welcome to hear Professor Sinha. 

Mini-course announcement

Title:  Central Limit theorems in Number Theory

Speaker: Kaneenika Sinha (IISER, Pune)

Abstract:
The goal of these lectures is to review a theme that binds the study of different types of arithmetic functions, namely central limit theorems.  After reviewing the "prototype" theorem in this theme, namely the classical Erdos-Kac theorem about the prime-omega function, we will survey different types of central limit theorems in the context of zeroes of zeta functions, eigenvalues of Hecke operators acting on spaces of cusp forms and eigenvalues of regular graphs.

Where: Zoom. Please ask the organisers for a link

Talk 1:  Thursday, September 22, 2022 - 4:00 PM - 5:00 PM (IST) 


Talk 2:  Thursday, October 6, 2022 - 4:00 PM - 5:00 PM (IST) 


Tea or Coffee: Please bring your own.


Talk 1


Talk 2