Saturday, April 27, 2024

Ramanujan Explained - Lecture 2 - May 2, 2024, 4:00 PM- 5:00 PM IST

 Dear all,

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

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:


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:

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

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


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.

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.