Saturday, April 30, 2022

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

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.

Monday, April 18, 2022

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

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

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.