The next talk in the Special Functions and Number Theory is as below. The speaker, Debashis Ghoshal, is one of the most regular attendees and long term supporters of this Seminar. We have long asked him to show how special functions arise in his work and to perhaps tell us about some problems.
Talk announcement
Speaker: Debashis Ghoshal (School of Physical Sciences, JNU)
Title: Two-dimensional gauge theories, intersection numbers and special functions
Where: Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance. (The link will be open at 3:30 PM for the organizers to test their systems)
Tea or Coffee: Please bring your own.
Abstract: The partition function of two dimensional Yang-Mills theory contains a wealth of information about the moduli space of connections on surfaces. We study this problem on a special class of surfaces ofinfinitegenus, which are constructed recursively. While the results are suggestive of an underlying geometrical structure, we use it as a prop to efficiently compute results for finitegenussurfaces. Riemann zeta function, confluent hypergeometric function and its truncations show up in explicit computations for the gauge group SU(2). Much of the corresponding results are open for other groups.
Welcome to the new academic year. To inject some much needed positivity in what continues to be an unusual year, we begin our semester with two talks by Apoorva Khare on total positivity. As usual, students are welcome.
The details of the two talks are as follows.
Talk 1: An introduction to total positivity
Speaker: Apoorva Khare (IISc., Bangalore)
When: Thursday, September 10, 3:55 PM - 5:00 PM IST (GMT + 5:30)
Where: TBA Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance. (The link will be open at 3:30 PM for the organizers to test their systems)
Tea or Coffee: Please bring your own.
Abstract:
I will give a gentle introduction to total positivity and the theory of Polya frequency (PF) functions. This includes their spectral properties, basic examples including via convolution, and a few proofs to show how the main ingredients fit together. Many classical results (and one Hypothesis) from before 1955 feature in this journey. I will end by describing how PF functions connect to the Laguerre-Polya class and hence Polya-Schur multipliers, and mention 21st century incarnations of the latter.
Talk 2: Totally positive matrices, Polya frequency sequences, and Schur polynomials
Speaker: Apoorva Khare (IISc., Bangalore)
When: Thursday, September 24, 3:55 PM - 5:00 PM IST (GMT + 5:30)
Where: TBA Please write to sfandnt@gmail.com to get the link to the talk. It is best to write a day in advance. (The link will be open at 3:30 PM for the organizers to test their systems)
Tea or Coffee: Please bring your own.
Abstract:
I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.
The next talk in the Special Functions and Number Theory Seminar is below. As usual, we have requested the speaker to give something suitable for students and non-experts, so please feel free to give this announcement to interested students and colleagues.
The regular semester is beginning soon. In case you have anything to present which you are currently excited about, please do volunteer for a talk in this seminar. Further, we would like to include an agenda for some expository talks, perhaps a series leading to a research problem. Any ideas in these directions are welcome!
Talk announcement
Speaker: Amritanshu Prasad, IMSc., Chennai.
Title: Character polynomials and their moments
When: Thursday, August 27, 3:55 PM - 5:00 PM IST (GMT + 5:30)
Where: Zoom: Please write to sfandnt@gmail.com to get the link to the zoom talk. It is best to write a day in advance. (The link will be open at 3:30 PM for the organizers to test their systems)
Tea or Coffee: Please bring your own.
Abstract:
A polynomial in a sequence of variables can be regarded as a class function on every symmetric group when the $i$th variable is interpreted as the number of $i$-cycles. Many nice families of representations of symmetric groups have characters represented by such polynomials.
We introduce two families of linear functionals of this space of polynomials -- moments and signed moments. For each $n$, the moment of a polynomial at $n$ gives the average value of the corresponding class function on the nth symmetric group, while the signed moment gives the average of its product by the sign character. These linear functionals are easy to compute in terms of binomial bases of the space of polynomials.
We use them to explore some questions in the representation theory of symmetric groups and general linear groups. These explorations lead to interesting expressions involving multipartite partition functions and some peculiar variants.
The next talk in the Special Functions and Number Theory Seminar is by Fatma Cicek. Fatma has just joined IIT, Gandhinagar (virtually) as an Assistant Research Professor. We welcome her to India!
Speaker: Fatma Cicek
Title: On the Logarithm of the Riemann Zeta-function Near the Nontrivial Zeros
When: Thursday, August 13, 3:55 PM - 5:00 PM IST (GMT + 5:30)
Where: Zoom. Please write to sfandnt@gmail.com to get a link a few hours ahead of time.
Tea or Coffee: Please bring your own.
Abstract:
Selberg's central limit theorem is one of the most significant probabilistic results in analytic number theory. Roughly, it states that the logarithm of the Riemann zeta-function on and near the critical line has an approximate two-dimensional Gaussian distribution.
In this talk, we will talk about our recent result which states that the distribution of the logarithm of the Riemann zeta-function near the sequence of the nontrivial zeros has a similar central limit theorem. Our results are conditional on the Riemann Hypothesis and/or suitable zero-spacing hypotheses. They also have suitable generalizations to Dirichlet $L$-functions.
The next talk is by Alok Shukla of Ahmedabad University. Alok has recently moved from the University of Manitoba and we welcome him back to India and this group Please note that we are now back into our usual time, one hour later than our previous talk. The talk is from 3:55 pm. The details are as follows.
Title: Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities
When: Thursday, July 30, 3:55 PM - 5:00 PM IST (GMT + 5:30)
Where: Zoom. Please write to sfandnt@gmail.com to get a link a few hours ahead of time.
Tea or Coffee: Please bring your own.
Abstract:
The Jacobi triple product identity and Rogers-Ramanujan identities are among the most famous $q$-series identities. We will present elementary combinatorial "tiling proofs" of these results. The talk should be accessible to a general mathematical audience.
The next talk in the Topics in Special Functions and Number Theory seminar is by Ali Uncu. This talk will again be on zoom. In case you wish to try out the software before the talk, please get in touch with one of the organizers. Further, requests for links should be made well before time. Some people did not get the link in time. In case you wish to be placed in the mailing list, please send an email to sfandnt@gmail.com.
Title: The Mathematica package qFunctions for q-series and partition theory applications
UPDATE: Please visithttps://akuncu.com/qfunctions/ for a version of this talk (video) and materials mentioned in the talk. Speaker: Ali Uncu, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria When: Thursday, July 16, 2020, 2:55 PM - 4:00 PM IST (GMT+5:30)
Where: On zoom (link available on request). Please write to sfandnt@gmail.com for the link a few hours before the talk.
Tea or Coffee: Please bring your own.
Abstract
In this talk, I will demonstrate the new Mathematica package qFunctions while providing relevant mathematical context. This implementation has symbolic tools to automate some tedious and error-prone calculations and it also includes some other functionality for experimentation. We plan to highlight the four main tool-sets included in the qFunctions package: (1) The q-difference equation (or recurrence) guesser and some formal manipulation tools, (2) the treatment of the method of weighted words and automatically finding and uncoupling recurrences,
(3) a method on the cylindrical partitions to establish sum-product identities, (4) fitting polynomials with suggested well-known objects to guess closed formulas.
This talk is based on joint work with Jakob Ablinger (RISC).
The next talk in the Topics in Special Functions and Number Theory seminar is by Hjalmar Rosengren on the Kanade-Russel identities. Please note that the talk will be one hour earlier than usual. This talk will again be on zoom. In case you wish to try out the software before the talk, please get in touch with one of the organizers. Title: On the Kanade-Russell identities
Speaker: Hjalmar Rosengren, Chalmers University of Technology and University of Gothenburg, Sweden
When: Thursday, July 2, 2020: 2:55-4:00 pm
Where: On Zoom: Link (available on request). Please send email to sfandnt@gmail.com a few hours before the talk.
Tea or Coffee: Please bring your own.
Kanade and Russell conjectured several Rogers-Ramanujan-type identities for triple series. Some of these conjectures are related to characters of affine Lie algebras, and they can all be interpreted combinatorially in terms of partitions. Many of these conjectures were settled by Bringmann, Jennings-Shaffer and Mahlburg. We describe a new approach to the Kanade-Russell identities, which leads to new proofs of five previously known identities, as well as four identities that were still open. For the new cases, we need quadratic transformations for q-orthogonal polynomials.
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
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.
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)
Please log in 5 minutes before 4:00 pm at the appointed date. Krishnan and I are available if you wish to test your systems. Finally, please write to us if you wish to be included in the announcement list. Please feel free to distribute the attached announcement to your students and other interested parties.
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.
I am sorry to inform you that as per JNU administration requirements, no meetings are allowed. So we have decided to postpone the seminars to April. In case the situation permits in April, I hope we will meet every week and pursue our agenda.
The agenda, which got clarified since my previous email, is as follows:
a. Circle Method by Manoj Verma
b. Intersection Numbers using Yang-Mills Theory on Riemann Surfaces, by Debashis Ghoshal
c. Mini Course on Continued Fractions 1: Basic Moves by Gaurav Bhatnagar
d. Mini Course on Continued Fractions 2: Convergence Theorems by Krishnan Rajkumar
So please keep your tuesdays free in April (and maybe beyond)
Meanwhile, this is a wonderful opportunity to stay quarantined, not meet anyone, and focus all our efforts on our research for a couple of weeks! I suppose I should have said "I am happy to inform you..."
Best wishes,
Gaurav Bhatnagar and Krishnan Rajkumar
March 11, 2020
Dear all,
We hope you had a safe and happy holi. The last couple of months of the academic year are upon us, and we thought we will do something special before we close for the summer.
The next talk in our Topics in Special Functions and Combinatorics Seminar is on March 17, 2020. It is by Manoj Verma and it is on Waring's problem and the circle method. The abstract appears below.
The talk after that is by Debashis Ghoshal on March 31.
In April, there will be 2-4 talks on the topic of Continued fractions by the two organizers of this seminar. These talks will comprise a mini-course on the subject of continued fractions. We plan to cover the basic moves as well as the convergence theory. Since this topic is not covered in most graduate courses, we hope this will be something unique and useful, especially for those with interests in Combinatorics, Number Theory and Special Functions. We will come back with details soon; please keep your tuesday afternoons free if you are interested, and kindly let interested students know about this.
In all of the above, the lecturers have promised a focus on techniques and will provide (optional) exercises, so that those of us who are interested can actually get our hands dirty and learn something which we can use in our work.
Best wishes,
Krishnan Rajkumar and Gaurav Bhatnagar
TALK ANNOUNCEMENT
Speaker: Manoj Verma (SPS, JNU)
Title: Waring's problem and the circle method
When: Tuesday, March 17, 2020, 4 pm.
Where: Seminar Room, School of Physical Sciences (SPS), Dr. CV Raman Marg, JNU.
(TO BE CONFIRMED)
Abstract: This talk will be an introduction to the circle method for those with no previous familiarity with the circle method. I shall introduce the method using Waring's problem as the prototype. For a positive integer $k$, let $G(k)$ denote the smallest positive integer $s$ such that every sufficiently large positive integer is a sum of s kth powers of integers. We shall sketch a proof of the fact that $G(k)$ is less than or equal to $2^{k} + 1$.
Title: Multiplicity one theorems in representation theory
Speaker: Shiv Prakash Patel (IIT, Delhi)
Date: Tuesday, February 25, 2020
Time: 4:00 pm
Venue: Seminar Room, School of Physical Sciences (SPS)
ABSTRACT
A representation $\pi$ of $G$ is called multiplicity free if the dimension of the vector space $Hom_{G}(\pi, \sigma)$ is at most 1 for all irreducible representation $\sigma$ of $G$. Let $H$ be a subgroup of $G$ and $\psi$ an irreducible representation of $H$. The triple $(G,H, \psi)$ is called Gelfand triple if the induced representation $Ind_{H}^{G} (\psi)$ of $G$ is multiplicity free. There is a geometric way to prove if some triple is Gelfand triple, which is called Gelfand's trick. Multiplicity free representations play an important role in representations theory and number theory, e.g. the use of Whittaker models. We will discuss Gelfand's trick and its use in a simple cases for Whittaker models of the representations of the group $GL_{n}(R)$ where $R$ is finite local ring.
Title: A generalization of the 3d distance theorem Speaker: Manish Mishra, IISER Pune Where: Seminar Room, School of Physical Sciences (SPS), C V Raman Marg, JNU When: Tuesday, February 11, 2020, 4 PM Abstract:
Let P be a positive rational number. Call a function f : R → R to have finite gaps property mod P if the following holds: for any positive irrational α and positive integer M, when the values of f(mα), 1 ≤ m ≤ M, are inserted mod P into the interval [0,P) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant kf which depends only on f. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non- differentiable points has finite gaps property mod P. We also show that if f is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_f ≤ 6$. This is a joint work with Amy Philip.
Title: The von Neumann-Day problem for finitely presented groups. Speaker: Yash Lodha, EPFL, Lausanne Where: Seminar Room, School of Physical Sciences (SPS), C V Raman Marg, JNU When: Tuesday, January 28, 2020, 4 PM Abstract: The so called von Neumann-Day problem asks if there exist nonamenable groups that do not contain nonabelian free subgroups. I will describe a recent solution to the problem for the class of torsion free finitely presented groups (joint with Justin Moore). The groups emerge as groups of homeomorphisms of the real line, and an encoding by means of continued fractions is used to prescribe a certain combinatorial structure. This structure allows us to establish that the groups are finitely presented.
Where: Seminar Room, Indian Statistical Center, 7 SJS Sansanwal Marg, Delhi 110016
ABSTRACT
It is exactly a hundred years since Ramanujan died at an early age of 32 years, but his mathematical legacy lives on. One of the things he was famous for was his work on continued fractions. We will give a brief introduction to Ramanujan's life and the material available on his story and his work. We will show Ramanujan's $q$-continued fractions and show how an elementary idea used by Euler can be used to prove many of Ramanujan's continued fractions. We expect to briefly mention some topics of current research interest. Students are welcome; much of the talk will be suitable for a general audience.
***
A page from Ramanujan's Lost Notebook with some continued fractions
It was a long, hard, grim winter. I was working a sixty-hour week at Bomber Command. The bomber losses which I was supposed to analyze were growing steadily higher. The end of the war was not in sight. In the evenings of that winter I kept myself sane by wandering in Ramanujan's garden, reading the letters I was receiving from Bailey, working through Bailey's ideas and discovering new Rogers--Ramanujan identities of my own.
This was written by Freeman J. Dyson in his paper celebrating the centenary of Ramanujan's birth. This year, 2020, is the centenary of Ramanujan's passing. So our first talk of the year is on Ramanujan and some of his work. Students are welcome.
We hope this year our seminar will thrive and build on what we have just begun. In addition, we hope we will get an opportunity to listen to the next generation of budding mathematicians in topics related to special functions and number theory.
We wish you a happy and productive 2020, with many new ideas and papers.
Gaurav Bhatnagar and Krishnan Rajkumar
2020: New Year Message
UPDATE (4 June 2020):
This was a new year message we wrote a few months ago, before a talk was held in ISI, Delhi (rather than JNU). But it seems appropriate for the year we are going through. The seminar was stopped for a few months due to a lockdown. Now we are starting again, with an expanded agenda.
Atul Dixit has joined as a co-organiser and we hope to make this seminar an international group. However, at present, we wish to keep timings as per our our Indian schedule, so as not to lose focus on those whom we wish to benefit the most. However, we will on occasion change timings to suit our international speakers.
The above quote is an appropriate one for many reasons. If you scan through the talks in this seminar, Ramanujan's mathematics plays a large role in almost all of them. This also represents the three organizer's own interests.
Finally, a word on what we request of speakers. We would like to learn techniques which we can use in our own problems; and sometimes learn about problems which we can solve using our techniques. Many students and people with a variety of interests come for the talks, so make sure you have something everyone can take home.
June 4, 2020
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar
