Amita Malik (Max Plank Institute, Bonn) - Thursday, May 19, 2022 - 4:00 PM - 5:00 PM (IST)

 Our next speaker is Amita Malik of the Max Plank Institute. The talk announcement is below.

Talk Announcement:

Title: Partitions into primes in arithmetic progressions

Speaker: Amita Malik (Max Plank Institute, Bonn)

 

When: Thursday, May 19, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Tea or Coffee: Please bring your own.

Abstract:  

 In this talk, we discuss the asymptotic behavior of the number of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula.

Murali K Srinivasan (IIT Bombay) - Thursday, May 5, 2022

Our next speaker is Murali Srinivasan of IIT Bombay (at Mumbai). The talk announcement is below.

Talk Announcement:

Title: A q-analog of the adjacency matrix of the n-cube.

Speaker: Murali K Srinivasan (IIT Bombay)

 

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

Where: Zoom. 

Live Link: https://youtu.be/eRPAXQprb_E

Tea or Coffee: Please bring your own.

Abstract:  

Random walk on the $n$-cube gives rise to two matrices, the adjacency matrix and the Kac matrix, that have an elegant spectral theory with many applications.

We define a random walk on subspaces that yields q-analogs of the adjacency and Kac matrices with an equally elegant spectral theory (though more difficult to prove). We give an application to tree counting.

This is joint work with Subhajit Ghosh.

Gaurav Bhatnagar - Talk 2 on Thursday, April 21, 2022

On the coming Thursday, Atul Dixit (IIT, Gandhinagar) and Gaurav Bhatnagar (Ashoka University) will present the talks they presented at the recently concluded JMM meeting held online. Each talk will be approximately 20 minutes followed by 10 minutes for questions. The titles and abstracts are below.

Here is the announcement for the second talk of the day.

Title: An easy proof of  Ramanujan's famous mod $5$ congruences

Speaker: Gaurav Bhatnagar (Ashoka University)

When: Thursday, April 21, 2022 - 4:30 PM - 5:00 PM

Where: Zoom. 

Live LInk: https://youtu.be/IB3FiBM2j5s

Abstract 

We give an elementary proof of Ramanujan's famous congruences $p(5n+4) \equiv 0 \mod 5$ and $\tau(5n+5)\equiv 0 \mod 5$. The proof requires no more than Euler's techniques and a result of Jacobi. The proof extends to embed the congruences into 4 infinite families of congruences for rational powers of the eta function. 

This is joint work with Hartosh Singh Bal.

The video below is the same as the one in Atul's talk. It just begins in the middle. 

Atul Dixit (IIT Gandhinagar) - Talk 1 on Thursday April 21, 4:00 PM - 4:30 PM

 Dear all,

On the coming Thursday, Atul Dixit (IIT, Gandhinagar) and Gaurav Bhatnagar (Ashoka University) will present the talks they presented at the recently concluded JMM meeting held online. Each talk will be approximately 20 minutes followed by 10 minutes for questions. The titles and abstracts are below.

We should say that during the summer, the talks in the seminar may be less as we are finally able to catch up with colleagues in person. 

Here is the announcement for Atul's talk.

Title:  Combinatorial identities associated with a bivariate generating function for overpartition pairs

Speaker: Atul Dixit (IIT, Gandhinagar)

When: Thursday, April 21, 2022 - 4:00 PM - 4:30 PM

Where: Zoom. 

Live LInk:  https://youtu.be/a9wVp7t4K14

Abstract

We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just the partition function $p(n)$. Using a result of Shimura we also relate a certain double series with a weight 7/2 theta series. This is joint work with Ankush Goswami.

Soumyarup Banerjee (IIT, Gandhinagar) - Thursday, Mar 24, 2022 - 3:55 PM

Our next speaker is Soumyarup Banerjee of IIT, Gandhinagar. The talk announcement is below.

Talk Announcement:

Title: Finiteness theorems with almost prime inputs

Speaker: Soumyarup Banerjee (IIT, Gandhinagar)

 

When: Thursday, Mar 24, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/Fd2gNRyK04I

Tea or Coffee: Please bring your own.

Abstract:  

The Conway–Schneeberger Fifteen theorem states that a given positive definite integral quadratic form is universal (i.e., represents every positive integer with integer inputs) if and only if it represents the integers up to 15. This theorem is sometimes known as “Finiteness theorem" as it reduces an infinite check to a finite one. In this talk, I would like to present my recent work along with Ben Kane where I have investigated quadratic forms which are universal when restricted to almost prime inputs and have established finiteness theorems akin to the Conway–Schneeberger Fifteen theorem.

Wenguang Zhai (China Inst. of Mining and Tech, Beijing, PRC) - Thursday, Mar 10, 3:55 PM (IST)

Our next speaker of Wenguang Zhai of the China Institute of Mining and Technology, Beijing, who will speak on recent work with Xiadong Cao (Beijing Institute of Petro-Chemical Technology, Beijing) and Yoshio Tanigawa (Nagoya, Japan).

Talk Announcement:

Title: The MC-algorithm and continued fraction formulas involving ratios of Gamma functions

Speaker: Wenguang Zhai (China Institute of Mining and Technology, Beijing, PRC)

 

When: Thursday, Mar 10, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/C6jzvD2NPuo

Tea or Coffee: Please bring your own.

Abstract:  

Ramanujan discovered many continued fraction expansions about ratios of the Gamma functions. However, Ramanujan left us no clues about how he discovered these elegant formulas. In this talk, we will explain the so-called MC-algorithm. By this algorithm, we can not only rediscover many of Ramanujan's continued fraction expansions, but also find some new formulas.

Krishnan Rajkumar (JNU) - Thursday Feb 24, 2022 - 3:55 PM (IST)

The next talk is by Krishnan Rajkumar. This talk will be recorded but not available online immediately.  

Talk Announcement:

Title: The Binet function and telescoping continued fractions

Speaker: Krishnan Rajkumar (Jawaharlal Nehru University (JNU))

 

When: Thursday, Feb 24, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Tea or Coffee: Please bring your own.

Abstract:  

The Binet function $J(z)$ defined by the equation $\Gamma(z)  = \sqrt{2 \pi} z^{z-\frac{1}{2}}e^{-z} e^{J(z)}$ is a well-studied function. The Stirling approximation comes from the property $J(z) \rightarrow 0$ as $z\rightarrow \infty$, $|arg z|<\pi$. In fact, an asymptotic expansion $J(z) \sim z^{-1} \sum_{k=0}^{\infty} c_k z^{-2k}$ holds in this region, with closed form expressions for $c_k$ and explicit integrals for the error term for any finite truncation of this asymptotic series. 

In this talk, we will discuss two different classical directions of research. The first is exemplified by the work of Robbins (1955) and Cesaro (1922), and carried forward by several authors, the latest being Popov (2018), where elementary means are used to find rational lower and upper bounds for $J(n)$ which hold for all positive integers $n$. All of these establish inequalities of the form $J(n)-J(n+1) > F(n)-F(n+1)$ for an appropriate rational function $F$ to derive the corresponding lower bounds by telescoping.

The second direction is to use moment theory to derive continued fractions of specified forms for $J(x)$. For instance, a modified S-fraction of the form $\frac{a_1}{x \ +} \frac{a_2}{x \ +}\frac{a_3}{x \ +} \cdots$ can be formally derived from the above asymptotic expansion using a method called the qd-algorithm. The resulting continued fraction can then be shown to converge to $J(x)$ by the asymptotic properties of $c_k$ and powerful results from moment theory. There are no known closed-form expressions for the $a_k$.

We will then outline what we call the method of telescoping continued fractions to extend the elementary methods of the first approach to derive the modified S-fraction for $J(x)$ obtained in the second by a new algorithm. We will describe several results that we can prove and some conjectures that together enhance our understanding of the numbers $a_k$ as well as provide upper and lower bounds for $J(x)$ that improve all known results.

This is joint work with Gaurav Bhatnagar.

 ---

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

Surbhi Rai (IIT, Delhi) - Thursday, Feb 10, 2022 - 3:55 PM IST

 Dear all,

The next talk is by Surbhi Rai, a graduate student in IIT, Delhi. The announcement is as follows. 

Talk Announcement:

Title: Expansion Formulas for Multiple Basic  Hypergeometric Series Over Root Systems

Speaker: Surbhi Rai (IIT, Delhi)

 

When: Thursday, Feb 10, 2022 - 4:00 PM - 5:00 PM (IST) 

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

 

Live Link: https://youtu.be/3WlOJivytek

Tea or Coffee: Please bring your own.

Abstract:  In a series of works, Zhi-Guo Liu extended some of the central summation and transformation formulas of basic hypergeometric series. In particular, Liu extended Rogers' non-terminating very-well-poised  $_{6}\phi_{5}$  summation formula, Watson's transformation
formula, and gave an alternate approach to the orthogonality of the Askey-Wilson polynomials. These results are helpful in number-theoretic contexts too. All this work relies on three expansion formulas of Liu. 

This talk will present several infinite families of extensions of Liu's fundamental formulas to multiple basic hypergeometric series over root systems. We will also discuss results that extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. Subsequently, we will look at an application based on the expansions of infinite products. These extensions have been obtained using the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from $A_n$, $C_n$ and $D_n$ basic hypergeometric series theory. We will observe how this approach brings Liu's expansion formulas within the Bailey transform methodology. 

This talk is based on joint work with Gaurav Bhatnagar. (https://arxiv.org/abs/2109.02827)

 ---

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

www.sfnt.org

sfandnt@gmail.com

Ramanujan Special: Alan Sokal (University College, London and New York University) Thursday, January 27, 2022, 4-5 PM (IST)

Dear all, 

Welcome to 2022. We begin the year with a Ramanujan Special talk by Alan Sokal. The talk announcement is below. 

We encourage you to distribute this announcement to friends and colleagues in your department or otherwise, so that they come to know of our seminar. 

Talk Announcement:

Title: Coefficientwise Hankel-total positivity

Speaker: Alan Sokal (University College London and New York)

 

When: Thursday, January 27,  2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/wUo4QcAS3mQ

Click here for the presentation

Tea or Coffee: Please bring your own.

Abstract:  

  A matrix $M$ of real numbers is called totally positive
   if every minor of $M$ is nonnegative.  Gantmakher and Krein showed
   in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$
   of real numbers is totally positive if and only if the underlying
   sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence.
   Moreover, this holds if and only if the ordinary generating function
   $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type
   continued fraction with nonnegative coefficients:
$$
   \sum_{n=0}^{\infty} a_n t^n
   \;=\;
   \cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 -  \cfrac{\alpha_3 t}{1- \cdots}}}}
$$
   (in the sense of formal power series) with all $\alpha_i \ge 0$.
   So totally positive Hankel matrices are closely connected with
   the Stieltjes moment problem and with continued fractions.

   Here I will introduce a generalization:  a matrix $M$ of polynomials
   (in some set of indeterminates) will be called
   coefficientwise totally positive if every minor of $M$
   is a polynomial with nonnegative coefficients.   And a sequence
   $(a_n)_{n \ge 0}$ of polynomials will be called
   coefficientwise Hankel-totally positive if the Hankel matrix
   $H = (a_{i+j})_{i,j \ge 0}$  associated to $(a_n)$ is coefficientwise
   totally positive.  It turns out that many sequences of polynomials
   arising naturally in enumerative combinatorics are (empirically)
   coefficientwise Hankel-totally positive.  In some cases this can
   be proven using continued fractions, by either combinatorial or
   algebraic methods;  I will sketch how this is done.  In many other
   cases it remains an open problem.

   One of the more recent advances in this research is perhaps of
   independent interest to special-functions workers:
   we have found branched continued fractions for ratios of contiguous
   hypergeometric series ${}_r \! F_s$ for arbitrary $r$ and $s$,
   which generalize Gauss' continued fraction for ratios of contiguous
   ${}_2 \! F_1$.  For the cases $s=0$ we can use these to prove
   coefficientwise Hankel-total positivity.

   Reference: Mathias P\'etr\'eolle, Alan D.~Sokal and Bao-Xuan Zhu,
   arXiv:1807.03271

Koustav Banerjee (RISC, JKU, Linz) - Thursday Nov 25, 2021 - 3:55 PM (IST)

The final talk of the year will be by Koustav Banerjee of RISC. After this we will take a break and return next year. 

Talk Announcement:

Title: Inequalities for the modified Bessel function of first kind and its consequences.

Speaker: Koustav Banerjee (RISC, Johann Kepler University, Linz, Austria)

 

When: Thursday, Nov 25, 2021 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom:

 

Live Link: https://youtu.be/kmpLf-lzSsk

Tea or Coffee: Please bring your own.

Abstract:  

Study on asymptotics of modified Bessel functions dates back to 18th century.
In this talk, we will describe how from the study of asymptotics of modified Bessel function of first kind of non-negative order, one can comes up with a host of inequalities that finally leads to answer combinatorial properties, for example log-concavity, higher order Turán inequality of certain arithmetic sequences arising from Fourier coefficients of modular forms. In addition to that, we will discuss briefly on a result of Bringmann et al. and analyze with the work addressed above.

Kamalakshya Mahatab (CMI, Chennai) - Thursday, Nov 11, 2021 - 3:55 - 5:00 PM (IST)

Dear all, 

The next speaker in our seminar is Kamalakshya Mahatab of CMI, Chennai

Talk Announcement:

Title: Large oscillations of the argument of the Riemann zeta function.

Speaker: Kamalakshya Mahatab (CMI, Chennai)

 

When: Thursday, Nov 11, 2021 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom (please write to the organizers for the link)

 

Live Link: https://youtu.be/nFWdn5rNMa4

Tea or Coffee: Please bring your own.

Abstract: 

We will obtain large values of the argument of the Riemann zeta function using the resonance method. We will also apply the method to the iterated arguments. This is a joint work with A. Chirre.

Akshaa Vatwani (IIT, Gandhinagar) - Thursday October 28, 3:55 PM - 5:00 PM (IST)

 Dear all, 

The next speaker in our seminar is Akshaa Vatwani of IIT, Gandhinagar

Talk Announcement:

Title: Limitations to equidistribution in arithmetic progressions

Speaker: Akshaa Vatwani (IIT, Gandhinagar)

 

When: Thursday, October 28, 2021 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom:

 

Live Link: https://youtu.be/QzEOE5fhr3I

Tea or Coffee: Please bring your own.

Abstract:  

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions ``on average" for moduli  in the range  for any . In , building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli  is extended to   for any . We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.

Meesue Yoo (Chungbuk National University, Korea) - Thursday, October 14, 2021 - 3:55 PM (IST)

 The next speaker in our seminar is Meesue Yoo of Chungbuk National University, Korea. 

Talk Announcement:

Title: Elliptic rook and file numbers

Speaker: Meesue Yoo (Chungbuk National University, Korea)

 

When: Thursday, October 14, 2021 - 4:00 PM - 5:00 PM (IST)  (7:30 PM in S. Korea)

Where: Zoom: Please write to the organizers if you don't have the link.

 

Live Link: https://youtu.be/eVSPuJmgdrY

Tea or Coffee: Please bring your own.

Abstract:  

In this talk, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corresponding factorization theorems which in the classical case was established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind, and Abel numbers. 

We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.

This is joint work with Michael Schlosser.

Jaban Meher (NISER, Bhubaneswar) - Thursday Sept 30, 2021; 3:55-5:00 PM

 Dear all,

The next talk is by Jaban Meher of NISER, Bhubaneswar. The announcement is as follows.

Talk Announcement:

Title: Modular forms and certain congruences

Speaker: Jaban Meher (NISER, Bhubaneswar)

 

When: Thursday, September 30, 2021 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom: (Please write to sfandnt@gmail.com for the link)

Live Link: https://youtu.be/vYs5YGuS_L4

Tea or Coffee: Please bring your own.

Abstract:  

 In this talk we shall discuss about modular forms and certain types of congruences among the Fourier coefficients of modular forms. We shall also discuss about the non-existence of Ramanujan-type congruences for certain modular forms.

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

www.sfnt.org

sfandnt@gmail.com

Neelam Saikia (University of Virginia), Thursday, September, 16, 2021 - 3:55 PM (IST)

 Dear all, 

The next talk is by Neelam Saikia who is currently a post-doc with Ken Ono at the University of Virginia. 

Talk Announcement:

Title: Frobenius trace distributions for Gaussian hypergeometric functions

Speaker: Neelam Saikia (University of Virginia)

 

When: Thursday, September 16, 2021 - 4:00 PM - 5:00 PM (IST) 

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

 

Live Link: https://youtu.be/o7qNW8BhgJI

Tea or Coffee: Please bring your own.

Abstract:  

 In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. These functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss, Kummer and others. These functions have played important roles in the study of supercongruences, the Eichler-Selberg trace formula, and zeta-functions of arithmetic varieties. We study the distribution (over large finite fields) of the values of certain families of these functions. For the $_2F_1$ functions, the limiting distribution is semicircular, whereas the distribution for the $_3F_2$ functions is the more exotic \it{Batman distribution.}

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

www.sfnt.org

sfandnt@gmail.com