Thursday, June 18, 2020

Arvind Ayyer (IISc., Bangalore)

The next talk in the Seminar on Special Functions and Number Theory is by Arvind Ayyer of IISC, Bangalore. The information appears below.

As announced in the last talk, Atul Dixit is now a co-organizer of this Seminar. The Zoom access for this talk has been kindly provided by Ashoka University. So this seminar series is now co-organized by Ashoka University, IIT, Gandhinagar and JNU.

We are experimenting with various systems so that we can give options to speakers. Please get in touch with any of Atul, Krishnan or myself if you wish to try out zoom before the talk. It requires some installation. Zoom may be suitable for a webinar in case we organize a public lecture, which we plan to do at some point.

Title: The Monopole-Dimer Model
Speaker: Arvind Ayyer, IISc. (Bangalore)
When: Thursday, June 18, 2020: 3:55-5:00 pm
Where: On Zoom: Link (available on request). Please send email to sfandnt@gmail.com

Here is a link to the talk.

Abstract:

The dimer model is a model which arose in statistical physics as a study
of adsorption. We will first define the model and state Kasteleyn's
groundbreaking result expressing the partition function of the model as
a Pfaffian for planar graphs. We develop a new model of monopoles and
dimers whose partition function is a determinant for any planar graph.
We then apply this to the rectangular grid and obtain a generalization
of Kasteleyn's miraculous product formula. Lastly, we study the
thermodynamic limit and obtain formulas for the free energy and entropy.
Some interesting special functions show up in this limit. Time
permitting, we will also show that in some special cases, the partition
function becomes a perfect square. This work is based on arXiv:1311.5965
(Mathematical Physics, Analysis and Geometry, 2015) and arXiv:1608.03151
(to appear in Annals of Combinatorics).

Graduate students are welcome. We plan to include a friendly introduction.

Thursday, June 4, 2020

Atul Dixit

We are back in business, after a short break due to the lockdown. We decided to begin the seminar online. There is a great advantage to this. First of all, many people in Delhi who were not in a position to travel to JNU or ISI can now attend from home. Secondly we have greatly expanded the ability to get good speakers for the seminar. Finally, people working in related areas around the country, and perhaps in Europe and Singapore can now contribute. We wish at present to keep the timings as per our original calendar (suitable to our schedule in India) but if we get a great speaker, we may modify as per their convenience. We will continue to meet once every two weeks.

As our first talk of the new season, we are delighted to have as a speaker Atul Dixit from IIT, Gandhinagar. Atul has developed a small group of students and post-docs in Gandhinagar, doing interesting mathematics related to the themes covered in our seminar, and we hope they will join this group.

Title: Superimposing theta structure on a generalized modular relation\\

Speaker: Atul Dixit, IIT, Gandhinagar

When: Thursday, June 4, 2020; 3:55-5:00 pm IST (GMT+5:30)

Abstract: By a modular relation for a certain function $F$, we mean a relation governed by the map $z\to -1/z$ but not necessarily by $z\to z+1$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha)=F(iw, \beta)$ or the Ramanujan-Guinand formula $F(z, \alpha)=F(z, \beta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $\operatorname{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$ obtained by Kesarwani, Moll and the speaker, that is, one can superimpose theta structure on it.
In 2011, the speaker obtained a generalized modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on the generalized modular relation? While answering this question affirmatively, we were led to a surprising new generalization of $\zeta(z, a)$. We show that this new zeta function, $\zeta_w(z, a)$, satisfies a beautiful theory. In particular, it is shown that $\zeta_w(z, a)$ can be analytically continued to Re$(z)>-1, z\neq1$. We also prove a two-variable generalization of Ramanujan's formula which involves infinite series of $\zeta_w(z, a)$ and which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.