Thursday, October 24, 2024

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



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)

Monday, October 7, 2024

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)



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)

Thursday, October 3, 2024

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 


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)

Friday, September 13, 2024

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)

Sunday, April 14, 2024

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

 Dear all,

Sunday, March 31, 2024

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

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.

Saturday, March 16, 2024

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. 

Saturday, March 2, 2024

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 $x$. 


Monday, February 19, 2024

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

 Dear all,

Tuesday, February 6, 2024

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

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.



Friday, January 19, 2024

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.


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.



Monday, December 4, 2023

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


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 < p1, and r p1 is well approximated by the respective linear functions of p given 
by 3pp/2, and pIn 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 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 < p1, and p1 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).


Sunday, November 19, 2023

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


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.


Sunday, November 5, 2023

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


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. 

Sunday, October 22, 2023

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

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.