Jeremy Lovejoy (CNRS, Paris) -- Thursday, Nov 21, 2024, 4: 00 PM IST (11:30 AM CET)

 Dear all,

Our talk this week is by Jeremy Lovejoy of  CNRS. Here is the announcement. 

Talk Announcement: 

Title: Bailey pairs, radial limits of q-hypergeometric false theta functions, and a conjecture of Hikami

Speaker:  Jeremy Lovejoy (CNRS, Paris)

When: Nov 21, 2024, 4:00 PM- 5:00 PM IST (11:30 AM CET)

Where: Zoom: Please write to the organisers

Live LInk:  https://youtube.com/live/PCvF5pykt9c?feature=share

Abstract: In the first part of this talk we describe a conjecture of Hikami on the values of the radial limits of a family of q-hypergeometric false theta functions. Hikami conjectured that the radial limits are obtained by evaluating a truncated version of the series. He proved a special case of his conjecture by computing the Kashaev invariant of certain torus links in two different ways. We show how to prove the full conjecture using Bailey pairs. In the second part ofthe talk we explain how the framework of Bailey pairs leads to further results of this type. The talkis based on joint work with Rishabh Sarma.

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

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Shashank Chorge (IIT, Gandhinagar) -- Thursday, Oct 24, 2024, 5:00 PM IST

 Talk Announcement: 

Title:  Voronoi summation formulas for certain arithmetic functions and related identities and omega bounds on error terms

Speaker:  Shashank Chorge (IIT, Gandhinagar)
When: Oct 24, 2024, 5:00 PM- 6:00 PM IST

Where: Zoom: Write to the organisers

Live LInk: https://youtube.com/live/CG81zYmTeIk?feature=share

Abstract:The Voronoi summation formulas for the divisor function and $r_2(n)$ are well-known. Not only do these formulas have interesting structure, but they have also been used to improve the error term in the Dirichlet divisor problem and the Gauss circle problem respectively. 

In this work we derive Voronoi summation
formulas for some other functions related to the generalized divisor function-$d^2(n)$ and Liouville Lambda function-$\lambda(n)$ and Mobius function-$\mu(n)$. We also make use of Vinogradov-Korobov zero free region for the Riemann zeta function to obtain the results.

We also derive beautiful analogues of Cohen's identity and the Ramanujan-Guinand formula associated to these functions.

We also derive certain Omega-bounds for the weighted sums of $d^2(n)$, $\lambda(n)$ and $\mu(n)$ assuming the Linear Independence conjecture.

This is a joint work with Atul Dixit.

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

Hjalmar Rosengren (Chalmars) -- Thursday, Oct 10, 2024, 4:00 PM IST (12:30 PM CEST)

 Dear all,

Our talk this week is by Hjalmar Rosengren (Chalmers). Here is the announcement.

Talk Announcement: 

Title:  String amplitudes and partial fractions: A mathematician's perspective

Speaker:  Hjalmar Rosengren (Chalmers University of Technology and University of Gothenburg)
When: Oct 10, 2024, 4:00 PM- 5:00 PM IST/ 12:30 PM (CEST)

Where: Zoom: Zoom

Live LInk: https://youtube.com/live/4Afu09q0LBE?feature=share

Abstract:

Using physics methods, Saha and Sinha recently obtained many intriguing expansions of string amplitudes (see Sinha's talk in this seminar 3 October). From a mathematical perspective, they are very unusual deformations of classical hypergeometric identities. One very special case gives a deformation of Madhava's famous series for pi, which received a lot of media attention. I will discuss these identities from a mathematical perspective. They can be derived from partial fraction expansions for symmetric rational functions, which may have some independent interest. 

The talk is based on the preprint https://arxiv.org/abs/2409.06658. 

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

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Ramanujan Explained Playlist

Aninda Sinha (IISc, Bangalore) -- Thursday, Oct 3, 2024, 4:00 PM IST

Dear all,

We are back to our usual time with a talk by Aninda Sinha of IISc, Bangalore. 

Next week (on Oct 10), we will have a follow-up talk by Hjalmar Rosengren on the topic: String amplitudes and partial fractions: A mathematician's perspective. The formal announcement will be made on the weekend. But please mark your calendars. 

Talk Announcement: 

Title:  Field theory expansions of string theory amplitudes

Speaker:  Aninda Sinha (Indian Institute of Science, Bangalore)
When: Oct 3, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: 

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

Abstract:

This talk is based on: Phys. Rev. Lett. 132 (2024) 22, 221601 (e-print: 2401.05733 [hep-th])

I will explain the reasons (both physics and maths) for trying to find a new formula, satisfying certain physics inspired criteria, for the Euler-Beta and related functions. I will give a sketch of the derivation which relies on a novel dispersion relation. Towards the end, time permitting, new results from the Bootstrap will be presented. 

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

George Andrews (Penn State University) -- Thursday, Sept 19, 2024, 5:30 PM IST (8:00 AM EDT)

 Dear all,

We are excited to begin after the summer break with a talk by Professor George Andrews, Atherton Professor of Mathematics, Penn State University. 

Please note the unusual time for the seminar. 

Talk Announcement: 

Title:  Extensions of MacMahon's Generalization of Sums of Divisors

Speaker:  George Andrews (Penn State University)
When: Sept 19, 2024, 5:30 PM- 6:30 PM IST (8:00 AM EDT)

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

Abstract:

This talk is on joint work with Tewodros Amdeberhan and Roberto Tauraso. In our joint paper, Extensions of MacMahon’s Sums of Divisors (Research in the Math. Sciences, 11,8 (2024)), we examined the arithmetic properties of MacMahon’s original functions and their generalizations. In this talk, we consider further extensions of MacMahon’s work, and we look at both arithmetic and combinatorial aspects including a stunning identity for the generating function of the sum of the divisors of n.

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

Ramanujan Explained - Lecture 1 - April 18, 2024, 4:00 PM- 5:00 PM IST

 Dear all,

We have launched a course under the title of Ramanujan Explained. There will be a series of lectures, all given by Gaurav Bhatnagar, with accompanying notes and exercises. The goal is to cover (a large number of) Ramanujan's identities. The first talk in this series is in our next seminar slot. Kindly do share this announcement with students who may be interested in Ramanujan and his mathematics. The first few lectures will target $q$-hypergeometric series and special cases, and can serve as an introduction to basic hypergeometric series. We hope these lectures will serve as a useful supplement to the monumental work of Bruce Berndt  (Ramanujan's Notebooks I-V) and George Andrews and Bruce Berndt  (Ramanujan's Lost Notebook I-V).  

Lecture notes, slides, lecture videos and more: Ramanujan Explained Website

We will not be including the notifications of these talks on this website. 

Talk Announcement: 

Title:  Ramanujan Explained 1: How to discover the Rogers-Ramanujan identities

Speaker:  Gaurav Bhatnagar (Ashoka University)
When: April 18, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to the organisers (sfandnt at gmail dot com) for the link

Live LInk: https://youtube.com/live/_M4WbbAffIA?feature=share

Abstract:

About the Rogers-Ramanujan identities, Hardy famously remarked: "It would be difficult to find more beautiful formulae than the "Rogers-Ramanujan" identities... " In the first introductory lecture in the Ramanujan Explained course, we explain Askey's idea on how Ramanujan may have come across these identities. Continued fractions played an important part of Ramanujan's work, and Askey's explanation is all about the simplest $q$-continued fraction and how it naturally leads to the Rogers-Ramanujan identities.

Alexandru Pascadi (Mathematical Institute, Oxford) -- Thursday, Apr 4, 2024 - 4:00 PM (IST)

Dear all,

The next talk is by Alexandru Pascadi of the University of Oxford. The announcement is as follows. 

Talk Announcement: 

Title:  The dispersion method and beyond: from primes to exceptional Maass forms

Speaker:  Alexandru Pascadi (Mathematical Institute, University of Oxford)
When: April 4, 2024, 4:00 PM- 5:00 PM IST (11:30 AM BST)

Where: Zoom: Write to the organisers for the link

Live LInk: https://youtube.com/live/h9Nil7L8tOI?feature=share

Abstract:

The dispersion method has found an impressive number of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain exponential sums, using deep inputs from algebraic geometry, the spectral theory of GL2 automorphic forms, and GLn automorphic L-functions. We'll give a broad outline of this process, which combines various types of number theory; time permitting, we'll also discuss the key ideas behind some new results.

Gaurav Bhatnagar (Ashoka University) -- Thursday, Mar 21, 2024 - 4:00 PM (IST)

 Dear all,

The next talk is by Gaurav Bhatnagar of Ashoka University. The announcement is as follows. 

Talk Announcement: 

Title:  Elliptic enumeration and identities

Speaker:  Gaurav Bhatnagar (Ashoka University)
When: Mar 21, 2024, 4:00 PM- 5:00 PM IST 

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

Live LInk: https://youtube.com/live/cpqHK-R2oXg?feature=share

Abstract

Many of the ideas of $q$-counting and $q$-hypergeometrics are now being extended to the elliptic case. The approach is not very far from the $q$-case. In this talk, we show several examples to illustrate this idea. First we extend some Fibonacci identities using combinatorial methods. Many such identities can be found by telescoping, so we next use telescoping to find elliptic extensions of elementary identities such as the sum of the first  odd or even numbers, the geometric sum and the sum of the first  cubes. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. Finally, we introduce elliptic hypergeometric series and give an extension of some important identities of Liu. As applications, we find 5 double summations and 4 new elliptic transformation formulas. Again, these are new in the $q$-hypergeometric case, where the nome $p$ is 0. 

This is a report of joint work with Archna Kumari and Michael Schlosser. 

Shivani Goel (IIIT, Delhi) - Thursday, Mar 7, 2024 - 4:00 PM (IST)

 Dear all,

The next talk is by Shivani Goel, of the Indraprastha Institute of Information Technology (IIIT), Delhi. The announcement is as follows. 

Talk Announcement: 

Title:  Distribution and applications of Ramanujan sums

Speaker:  Shivani Goel (IIIT, Delhi)
When: Mar 7, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Ask the organisers for the link.

Live LInk: https://youtube.com/live/mQ9EiVeqimI?feature=share

Abstract

While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the $n^{th}$ power of the primitive $q^{th}$ roots of unity and denoted it as $c_q(n)$. These sums are now known as Ramanujan sums.

Our focus lies in the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev initially considered this problem. They estimated the first and second moments of Ramanujan sums.  Building upon their work, we extend the estimation of the moments of Ramanujan sums for cases where $k\ge 3$.  Apart from this, We derive a limit formula for higher convolutions of Ramanujan sums to give a heuristic derivation of the Hardy-Littlewood formula for the number of prime $k$-tuplets less than $x$. 

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.