The Mathematics Colloquium is a unique feature of Ashoka University. Distinguished mathematicians visit Ashoka and explain their research in terms that students can understand. Many of these talks are available online. In addition, many distinguished visitors offer courses in special topics. At present we meet almost every Tuesday at 3:15 PM IST.

Jitendra Rathore (Harish-Chandra Research Institute (HRI) , Prayagraj)

Date: 27th Feb 2024, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title: Brauer group and its arithmetic for curves over local fields.

Abstract:

The Brauer group is an arithmetic invariant associated with a field that classifies all finite-dimensional central division algebras over the given field. In this talk, we will start by providing some motivation and the definition of the Brauer group of fields. We will then extend this definition to more general objects such as algebraic varieties. For varieties of dimension one (referred to as “curves”), the Brauer group is closely connected to their Picard group. We will explore this connection in this talk and present some new results on this topic, which are part of a joint work with Amalendu Krishna and Samiron Sadhukhan.

Previous Talks in the Department of Mathematics

Trishen Gunaratnam (University of Geneva)

Date: 15th Feb 2024, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-009

Title: The mathematics of 2d lozenge tilings.

Abstract:

In this talk I will give a gentle introduction to the mathematics of lozenge tilings (closely related to domino tilings). My main goal is to discuss Thurston’s theorem concerning when a domain is tileable or not. However, along the way we shall see that this is just the start of a beautiful story. The study of limit shapes of random Lozenge tilings led to ideas that played a pivotal role in some of the major breakthroughs in probability theory in the 2000s. I hope to give a glimpse into this rich and fascinating area.

Trishen Gunaratnam (University of Geneva)

Date: 13th Feb 2024, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title: Tricritical phenomena in the Blume-Capel model.

Abstract:

The Blume-Capel model is a ferromagnetic spin model that was introduced in the ’60s to model an exotic phase transition in uranium oxide. Mathematically speaking, it is an Ising model coupled to a site percolation, combining two of the most beautiful models in statistical physics. It has a line of critical points – the Curie temperatures whereby the magnetisation-demagnetisation transition occurs. Along this critical line, the model is expected to undergo a further phase transition at the so-called tricritical point. Despite many fascinating physics conjectures concerning the tricritical universality class, there are few rigorous results. In this talk, I will discuss these conjectures and touch upon recent results joint with Dmitry Krachun (Princeton University) and Christoforos Panagiotis (University of Bath) in establishing the existence of a tricritical phenomenon in all dimensions.

Gaurav Bhatnagar (Ashoka University)

Date: 30th Jan 2024, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title : A proof of two of Ramanujans congruences.

Abstract:

A partition is a way of writing a number as an unordered sum of numbers. For example, three of the five partitions of 4 are: 4, 3+1, 2+2. Can you find the remaining two? The partition function p(n) counts the number of partitions of n.

We give a proof of the simplest of Ramanujan’s congruences for the partition function, and a related result for the tau function. The proof rests on results of Euler, Gauss and Jacobi. The talk will be accessible to undergraduates as long as they are willing to believe a few things about infinite products. We prove an infinite family of congruences for powers of the eta function, which contain the two congruences in question. This is joint work with Hartosh Singh Bal.

Hoang Son Tran (National University of Singapore)

Date: 21st Nov 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title :Gaussian fluctuations for spin systems and point processes: Near-optimal rates via quantitative Marcinkiewicz’s theorem.

Abstract:

In this talk, He would like to introduce a broad, quantitative extension of the classical Marcinkiewicz’s theorem to establish central limit theorems (CLTs) with explicit rates of convergence. In particular, we obtain quantitative decay estimates on the Kolmogorov-Smirnov distance between a random variable X and a Gaussian under the condition that the characteristic function Ψ does not vanish only on a bounded disk. This leads to quantitative CLTs applicable to very general and possibly strongly dependent random systems with a near-optimal rate of convergence. Later, He will apply this method for two important classes of models in probability and statistical physics, namely, spin systems and α-determinantal processes. These applications demonstrate the significance of having to control the characteristic function only on a (small) disk, and lead to CLTs which, to the best of our knowledge, are not known in generality. This is a joint work with T.-C. Dinh and S. Ghosh.

Suraj Krishna (Technion – Israel Institute of Technology)

Date: 15th Nov 2023, Time : 01:45 PM- 02:45 PM, Venue: Lecture Room – AC04-302

Title :Cubulating hyperbolic mapping tori.

Abstract:

A group is cubulated if it acts properly and cocompactly on a CAT(0) cube complex, which is a generalisation of a product of trees. Some well-known examples are free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds.

Our setup consists of mapping tori of hyperbolic groups which are again hyperbolic. Two prominent examples are (1) mapping tori of fundamental groups of hyperbolic surfaces over pseudo-Anosov automorphisms, and (2) mapping tori of free groups over atoroidal automorphisms. Both these classes of groups are known to be cubulated.

Building on these two important results, and placing them in a unified framework, I will show that all such groups of our setup are cubulated, and explain a few consequences of the result.

This is joint work with François Dahmani and Jean Pierre Mutanguha.

Suraj Krishna (Technion – Israel Institute of Technology)

Date: 14th Nov 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title :Mapping tori in geometric group theory.

Abstract:

This talk will be a gentle introduction to geometric group theory through the concept of mapping tori. I will introduce the relevant topics with pictures and examples and give a glimpse of some of the spectacular progress made by stalwarts like Bill Thurston. I will then mention some recent progress in the field, along with my own forays.

Mahan MJ (Tata Institute of Fundamental Research, Mumbai)

Date: 10th Nov 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title:Commensurate.

Abstract: A group is said to commensurate a subgroup , if for all , is of finite index in and , where denotes the conjugate of by . The commensuration action of on can be studied dynamically. And, we will discuss a range of theorems and conjectures in this context, starting with work of Margulis, and coming to the present day.

Sagar Shrivastava (Tata Institute of Fundamental Research, Mumbai)

Date: 7th Nov 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title:Representations, Determinants and Branching rules.

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.

This is a joint work with Prof. C.S. Rajan.

Somnath Jha (Indian Institute of Technology, Kanpur)

Date: 31st Oct 2023, Time : 03:30 PM- 04:30 PM, Venue: Lecture Room – AC04-302

Title:Cube sum problem.

Abstract: The classical Diophantine problem of determining which integers can be expressed as a sum of two rational cubes has a long history; it includes works of Sylvester, Selmer, Satgé, Leiman and the recent work of Alpöge-Bhargava-Shnidman-Burungale-Skinner. In this talk, we will use Selmer groups of elliptic curves and integral binary cubic forms to study some cases of the rational cube sum problem. This talk is based on joint works with D. Majumdar, P. Shingavekar and B. Sury.

Niladri Patra (Tata Institute of Fundamental Research, Mumbai)

Date: 17th Oct 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC04-302

Title:Irreducibility results in dynamical moduli spaces.

Abstract: Consider the moduli space, $\mathcal{M}_{d}$, of degree $d \geq 2$ polynomials over $\C$, with a marked critical point. Given $k \geq 0,\; p$ an odd prime, we show that the set $\Sigma_{k,1,p}$ of conjugacy classes of degree $p$ polynomials, for which the marked critical point is strictly $(k,1)$-preperiodic, is an irreducible set under Zariski topology on $\mathcal{M}_{p}$. Irreducibility of these sets was conjectured by Milnor, and has been proved for $p=3$ by Buff, Epstein and Koch.

We prove that the subspaces of $\Sigma_{k,1,p}$, that arise by varying the ramification index of the marked critical point all the way up to the unicritical case, are all irreducible subvarieties. Finally, using the irreducibility of $\Sigma_{k,1,p}$ we give a new and short proof of the fact that the set of all unicritical points of $\Sigma_{k,1,p}$ form one Galois orbit under the action of absolute Galois group of $\Q$.

Sayani Bera (Indian Association for the Cultivation of Science, Kolkata)

Title :Fatou-Bieberbach domains and attracting basins.

Abstract:

Fatou-Biberbach domains in $\mathbb{C}^k, k \ge 2$ are proper subdomains of $\mathbb{C}^k, k \ge 2$ that are biholomorphic to $\mathbb{C}^k, k \ge 2$. Such domains clearly do not exist in $\mathbb{C}$ and, further arise naturally from dynamics of automorphisms of $\mathbb{C}^k, k \ge 2$, admitting an attracting fixed point. The goal of this talk is to discuss a few pathological examples of such domains, and also their relevance in the context of geometry, complex function theory and dynamics.

Further, we will introduce non-autonomous (dynamical) families and their basins attractions, and discuss an affirmative solution to a long standing open problem, called the Bedford’s conjecture, related to the (non-autonomous) dynamical origins of a Fatou-Bieberbach domain. This is a joint work with Dr. Kaushal Verma.

Sayani Bera (Indian Association for the Cultivation of Science, Kolkata)

We introduce the setup of discrete dynamics of a (polynomial) map both in real line and in the complex plane, and introduce the notion of ‘conjugacy’ classes among them. In the context of dynamics of quadratic maps in complex plane, we discuss the “Mandelbrot set” and its connection to dynamics.

Finally in the last part of the talk we introduce the class of polynomial automorphisms of $\mathbb{C}^2$, called the Hénon maps and discuss a few results related to the rigidity of Julia sets associated to such maps.

Soumya Ganguly (University of California, San Diego)

Title :Bergman-Einstein metrics on two-dimensional Stein spaces

Abstract: We show that the Bergman metric of the ball quotients $\mathbb{B}^2/\Gamma$, where $\Gamma$ is a finite and fixed point free group, is Kähler-Einstein if and only if $\Gamma$ is trivial. As a consequence, we characterize the unit ball $ \mathbb{B}^2$ , among 2 dimensional Stein spaces with isolated normal singularities, proving an algebraic version of Cheng’s conjecture for 2 dimensional Stein spaces.

Rajendra Bhatia (Department of Mathematics, Ashoka University)

Date: 29th Aug 2023, Time : 03:15 PM- 04:15 PM, Venue: Lecture Room – AC02-008

Title :Positivity and Matrix Inequalities

Abstract:

I will begin with a few reminiscences of K. R. Parthasarathy, an outstanding mathematician and a great teacher. Then I will describe the principal ideas in some work I did with him on norm inequalities for positive definite matrices.

Jaikumar Radhakrishnan (Tata Institute of Fundamental Research, Mumbai)

Date: 27th April 2023, Time : 02:00 PM- 03:00 PM, Venue: Lecture Room – AC02-214

Title : Sensitivity and computational complexity of Boolean functions

Abstract:

Let f:{0,1}^n -> {0,1} be a Boolean function. We will discuss the connection between various notions of complexity associated with f: its sensitivity, block sensitivity, degree, approximate degree, and decision tree depth. We will present the recent breakthrough of Hao Huang (2019) who showed that if f cannot be represented by a polynomial of low degree then it must be highly sensitive. Finally, we will show a new upper bound on the size of the deterministic decision tree computing f in terms of the size of the corresponding randomized decision tree. The last result is from a recent joint work with Arkadev Chattopadhyay, Yogesh Dahia, Nikhil Mande and Swagato Sanyal. The talk will be accessible to a general scientific audience; we will assume nothing beyond basic linear algebra and combinatorics.

Rajendra Bhatia (Department of Mathematics, Ashoka University)

Date: 18h April 2023, Time : 3:15 PM- 4:15 PM, Venue: Lecture Room – AC04-407

Title :Perturbation of eigenvectors

Abstract:

If two Hermitian operators are close to each other, how far apart can their eigenvectors be? This is a question of fundamental importance in physics and in computational mathematics. We will discuss some answers. This talk is a tribute to Chandler Davis (1926-2022), a pioneer in this subject, from whom I learnt much.

Kapil Hari Paranjape (IISER, Mohali)

Date: 11th April 2023, Time : 3:15 PM- 4:15 PM

Title: Beyond Abel and Jacobi

Abstract:The classical theorem of Abel and Jacobi characterises the zeroes and poles of a meromorphic function on a compact Riemann surface (or smooth projective variety) using a map which is nowadays called the Abel-Jacobi homomorphism.

Following the work of many algebraic geometers over the past 150 years, this result has been successively refined and studied for higher-dimensional varieties. The most sophisticated version of this refinement is via the theory of “motives” as first proposed by Grothendieck. The most elaborate conjectural refinements of the Abel-Jacobi theorem are those of Bloch and Beilinson.

As with most conjectures, there have been many attempts to construct counter-examples. One method is to construct “minimal” instances of these conjectures for “well-understood” varieties where the conjectures do not (as yet) follow from known results. Such examples were constructed by Griffiths, Mumford-Roitman and many others which led to the more precise formulations of the conjectures that we see today.

Debanjana Kundu (Fields Institure, Toronto)

Date: 4th April 2023, Time : 3:15 PM- 4:15 PM

Title: Iwasawa Theory and Arithmetic Statistics

Abstract: In this talk I will give a brief overview of my current research on questions that lie in the intersection of Iwasawa theory and arithmetic statistics.

The first part of this talk will be expository in nature, where I will give an introduction to Iwasawa theory and explain the main questions that drive a lot of the research in this area. In the second part of the talk, I will discuss some of the progress made towards questions on the average behaviour of Iwasawa invariants – both the classical Iwasawa invariants and Iwasawa invariants of Selmer groups of elliptic curves.

Yashonidhi Pandey (Indian Institute of Science Education and Research, Mohali)

Title: On the degree of hypersurfaces with given singularities.

Abstract:

Let $n$, $t$ be positive integers and $S$ be a finite set of points in $\C^n$. We denote by $\omega_t(S)$ the least degree of a nonzero polynomial vanishing with multiplicity at least $t$ at each point of $S$. The sequence $(\omega_t(S)/t)_{t\ge 0}$ has a limit $\Omega(S)$ as $t$ tends to infinity. This invariant was introduced in 1975 for the proof of a Schwarz Lemma in several variables which occurs in the solution by Bombieri in 1970 of a conjecture of Nagata dealing with a generalization of a transcendence result of Schneider and Lang. The same invariant occurs in connection with another conjecture that Nagata introduced in 1959 in his work on Hilbert’s 14th problem. It is closely related with Seshadri’s constant.

Bibhas Chakraborty ( National University of Singapore)

Title:Designing Micro-randomized Trials for Mobile Health Studies via Thompson Sampling

Abstract:

Mobile health (mHealth) or digital health interventions are becoming increasingly common in tandem with advances in technology. In this talk, we will present an innovative trial design arising in mHealth, namely, the micro-randomized trial (MRT) that involves sequential, within-person randomization over many instances. MRTs often aim to estimate the proximal effects of “push”-type mHealth interventions, e.g., motivational text-messages to promote physical activity or other healthy behaviors. The basic MRT design can be further improved to make it adaptive, thereby enabling it to learn from accumulated data as the trial progresses. This is appealing from an ethical perspective since the adaptive learning tends to make better interventions available to the trial participants. Adaptive learning is often operationalized via contextual bandit algorithms. Specifically, we will discuss the role of a particular contextual bandit algorithm, namely, Thompson sampling, in designing adaptive MRTs with count-type outcomes. Simulation results will be shown to validate the proposed design approach. mHealth case studies will be discussed in detail.

Snehajit Misra ( Institute of Mathematical Sciences, Chennai)

Title:On existence of Weak Zariski decomposition on projective bundles

Abstract:

Zariski introduced a decomposition on surfaces to dismantle orthogonally (pseudo-)effective divisors into positive and negative parts, both of which enjoy the useful geometric properties. Many attempts has been made to generalise this decomposition on higher dimensional varieties, for instance the Fujita-Zariski decomposition and the CKM-Zariski decomposition. Recently, a weeker notion of this decomposition has been suggested by Birkar. In this talk, we will discuss about existence of weak Zariski decomposition of pseudo-effective divisors on projective bundles over higher dimensional varieties in order to answer a question posed by Birkar.

Nabanita Ray( Institute of Mathematical Sciences, Chennai)

Title: Weyl and Zariski chambers on projective surfaces

Abstract:

Let X be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of X. In this talk I describe these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. Moreover we see when a Weyl chamber can intersect non-trivially with a Zariski chamber.

Title: The Korányi Spherical Maximal Function on Heisenberg groups

Abstract:

In this talk, we discuss the problem of obtaining sharp L^p\to L^q estimates for the local maximal operator associated with averaging over dilates of the Korányi sphere on Heisenberg groups. This is a codimension one surface compatible with the non-isotropic Heisenberg dilation structure. I will describe the main features of the problem, some of which are helpful while others are obstructive. These include the non-Euclidean group structure (the extra “twist” due to the Heisenberg group law), the geometry of the Korányi sphere (in particular, the flatness at the poles) and an “imbalanced” scaling argument encapsulated by a new type of Knapp example, which we shall describe in detail.

Shraddha Srivastava, Indian Institute of Science, Bengaluru

Date: 7th Feb 2023, Time: 3:15 PM – 4:15 PM (IST)

Title : Stable multiplicities in the representation theory of symmetric groups.

Abstract:

A fundamental question in the representation theory of the symmetric group S_n is to determine the multiplicity of an irreducible representation of S_n in a given representation of S_n. Many interesting representations of S_n, for each n, fit into a natural sequence such that multiplicities of irreducible representations exhibit a stability phenomenon along the sequence.

We will discuss concrete results on stable multiplicities for sequences arising from restricting (some) Weyl representations of the general linear group to the subgroup of permutation matrices (based on joint work with Sridhar Narayanan, Digjoy Paul, and Amritanshu Prasad). We will also address a result on stable multiplicities for sequences arising from tensor products of irreducible representations of the symmetric group (based on joint work with Volodymyr Mazorchuk).

Shraddha Srivastava, Indian Institute of Science, Bengaluru

Date: 6th Feb 2023, Time: 5:00 PM – 6:00 PM (IST)

Title : Symmetric functions and representations of the general linear group

Abstract:

Symmetric functions are a generalization of symmetric polynomials to infinitely many variables. In this talk, we will define symmetric functions and see various examples of them. One of the goals will be to discuss a connection between symmetric functions and representations of the general linear group.

The composition of representations of the general linear groups corresponds to the plethysm (or substitution) of symmetric functions. Time permitting, we will see a connection between the plethysm of symmetric functions and the restriction problem from the general linear group to the symmetric group.

No prior knowledge of representation theory is assumed.

Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer s, there exists a positive integer n so that for every partition of the edges of the complete graph on n vertices into two classes, one of the classes must contain a complete subgraph on s vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest n that satisfies this property.

For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that “pseudo-randomness” and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

Pedro Ribeiro, University of Porto, Portugal

Date: 22nd Nov 2022, Time: 3:30 PM to 4:30 PM

Venue: AC04 310

Title : Euler’s formula for $\zeta(2n)$ and beyond.

Abstract:

In this talk I will broadly speak about different ways to reach Euler’s famous formula for. I will discuss how this formula has been important to my academic path and, from a certain historical perspective, to the theory of summation formulas. Many aspects of this study happened during my undergraduate studies so I expect that the talk will be accessible to undergraduate students in math and allied subjects.

Haruzo Hida, University of California, Los Angeles

Date: 7th Nov 2022, Time: 5:00 PM – 6:00 PM (IST)

Title : Back-ground of modular p-adic deformation theory and a brief outline

Abstract :

The deformation theory of modular forms is increasingly attracting many researchers in arithmetic geometry as it has been an important step in the proof of Fermat’s last theorem by Wiles (and Taylor) and supplied an effective tool for the study of the p-adic Birch and Swinnerton Dyer conjecture in the proof by Skinner-Urban of divisibility of the characteristic power series of the Selmer group of a rational elliptic curve by its p-adic L-function under appropriate assumptions. I try to give my back-ground motivation of creating the theory and describe an outline of the theory.

Debopriya Mukherjee, ISI, Bangalore.

Date: 18th October 2022 ; 3:30 PM to 4:30 PM

Title : Theory of Ferromagnetism and Chemotaxis

Abstract:

In my talk, I will address several mathematical questions about the constrained Stochastic Partial Differential Equations (SPDEs) arising from (i) Dynamics of ferromagnetism, (ii) Chemotaxis. Landau-Lifshitz-Gilbert equations (LLGEs) arise in the theory of the magnetization of ferromagnetic materials below a certain temperature (called the Curie temperature) and the constraint arises from the physical “saturation” property of the material. Here one usually seeks a “solution” to this problem such that the so-called “solution” lies on the three-dimensional unit sphere for all time. Next, I will specify the mechanisms for spatial pattern formation in systems of coupled reaction-diffusion equations with underlying chemotaxis. This system will be perturbed by a stochastic noise term, modeling neglected fluctuations or random perturbations from outside. Analysis of these kinds of PDEs with constraints under random perturbation is non-trivial and we adopt non-conventional but intriguing methods.

Nishant Chandgotia, TIFR-CAM, Bangalore.

Date: 4th October 2022 ; 3:30 PM to 4:30 PM

Title : The Dimer Model in 3 dimensions

Abstract:

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000).

This is joint work with Scott Sheffield and Catherine Wolfram and no prior background is required to understand the talk.

Manami Roy, Fordham University.

Date: 13th September 2022 ; 3:30 PM to 4:30 PM

Title: Dimensions of spaces of Siegel cusp forms of degree 2

Abstract:

Computing dimension formulas for the spaces of Siegel modular forms of degree 2 is of great interest to many mathematicians. We will start by discussing known results and methods in this context. The dimensions of the spaces of Siegel cusp forms of non-squarefree levels are mostly unavailable in the literature. This talk will present new dimension formulas of Siegel cusp forms of degree 2, weight k, and level 4 for three congruence subgroups. Our method relies on counting a particular set of cuspidal automorphic representations of GSp(4) and exploring its connection to dimensions of spaces of Siegel cusp forms of degree 2. This work is joint with Ralf Schmidt and Shaoyun Yi.

Aftab Pande,

Universidade Federal do Rio de Janeiro (UFRJ), Brasil

Date: September 06, 2022; 3:30 PM – 04:30 PM

Title: On slopes and families of modular forms.

Abstract: Congruences between modular forms have provided many interesting arithmetic properties e.g. the Ramanujan congruence. These congruences led Serre to define the notion of a p-adic family of non-cuspidal modular forms. Hida provided the first example of families of slope zero (ordinary) cuspidal modular forms which have had many important applications e.g FLT and Iwasawa main conjecture. For non-zero slope Gouveá and Mazur made some very precise conjectures which serve as a necessary condition for the existence of p-adic families of modular forms. In this talk, we will give a brief introduction to these concepts and report on current work in progress with Cláudio Velasque related to the Gouveá-Mazur conjecture and some applications.

Debapratim Banerjee International Centre for Theoretical Sciences, Bengaluru

Date: April 19, 2022; 05:00 PM – 06:00 PM

Title: New combinatorial approach for edge universality of Wigner matrices

Abstract: In this talk we shall consider a classical problem of random matrix theory i.e. the edge universality. We shall present a new combinatorial approach to solve the problem. Unlike the existing combinatorial approach this approach doesn’t need the entries of the matrix to be symmetrically distributed around zero. In course of the talk we shall discuss some applications of the method. Basic concepts about random matrix theory like the semicircular law, Dyck paths, Catalan numbers etc. will assumed. I shall try my best to make the talk understandable to a large audience.

Bhumika Mittal

UG’24, Ashoka University

Date: April 12, 2022; 05:00 PM – 06:00 PM

Title: Existence of Balanced Generalized DE BRUIJN Sequences

Abstract: A balanced generalized de Bruijn sequence with parameters (n, l, k) is a cyclic sequence of n bits such that (a) the number of 0’s equals the number of 1’s, and (b) each substring of length l occurs at most k times. We will determine necessary and sufficient conditions on n, l, and k for the existence of such a sequence and see its application with a magical card trick.

Jyotishman Bhowmick

Indian Statistical Institute, Kolkata

Date: April 5, 2022; 05:00 PM – 06:00 PM

Title: Differential calculus for noncommutative rings

Abstract: Noncommutative geometry is the study of noncommutative rings by imitating techniques from differential geometry. Many rings which will be of interest to us appear as a parametrized family of deformations of commutative rings of smooth functions on level surfaces. More precisely, if S is a level surface in an Euclidean space ( zeroes of a bunch of polynomials satisfying a regularity condition ), then our rings will be of the form A_q ( for q between 0 and 1 ) such that A_1 is isomorphic to a ring of smooth functions on S.

We will prepare the ground by defining the basic objects from geometry and then give a family of examples of noncommutative rings which are of interest to us. Finally, we will see what we mean by a geometric data on a noncommutative ring along with the example of the noncommutative torus.

Krishna Maddaly

Professor, Department of Mathematics, Ashoka University

Date: March 29, 2022; 05:00 PM – 06:00 PM

Title: Local statistics for some random band matrices

Abstract: Random matrices were considered by Wigner to model large atoms. These are real-symmetric or Hermitian matrices of dimension N whose independent entries are mutually independent random variables. An additional feature of the problem is to restrict the matrix to be non-zero in a band around the diagonal (the width of the band is W). These are called random band matrices. There are several questions of interest when the matrices become large, namely when N goes to infinity. The density of eigenvalues of all these matrices converges to the famous Wigner semicircle Law. In addition, in such a limit there are mainly two conjectures on how the eigenvalues behave. One of them is that the eigenvalues are correlated in the regime W > \sqrt(N) and the other is that the eigenvalues are themselves independent random variables, in the regime W < \sqrt{N}. While there are some results in the former case, there were none in the latter case and we give a theorem that shows the independence of eigenvalues when W < N^{1/7}, when the matrix entries are normally distributed. This is joint work with Peter Hislop.

Naina Praveen

ASP, Department of Mathematics, Ashoka University

Date: March 22, 2022; 5:00 PM- 6:00 PM

Title: Restricted Invertibility of Continuous Matrix Functions

Abstract: In 1987, Bourgain and Tzafriri proved the Restricted Invertibility Theorem, which roughly states that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which it is well-invertible. This bound happens to be optimal upto universal constants. We prove that the Restricted Invertibility Theorem can further be extended from matrices to continuous matrix functions satisfying similar hypotheses.

Akshaa Vatwani

Department of Mathematics, Indian Institute of Technology Gandhinagar

Date: March 15, 2022; 5:00 PM- 6:00 PM

Title: Limitations to equidistribution in arithmetic progressions

Abstract: We will discuss the distribution of prime numbers lying in an arithmetic progression $a$ modulo $q$. Dirichlet observed that any such sequence contains infinitely many primes and that the proportion of primes in each such progression is the same. As $q$ varies, more questions can be asked regarding how far such equidistribution persists. We will explore this theme and discuss some variants and applications. This talk is based on joint work with Aditi Savalia.

U. K. Anandavardhanan

Department of Mathematics, IIT Bombay

Date: March 1, 2022; 5:00 PM- 6:00 PM

Title: Orthogonality of invariant vectors

Abstract: This talk is about finite groups and their representation theory. Given a group G and two Gelfand subgroups H and K of G, associated to an irreducible representation \pi of G, there is a notion of H and K being correlated with respect to \pi in G. This notion was defined by Benedict Gross in 1991. Towards the end of the talk, we’ll present some recent results regarding this theme (which are joint with Arindam Jana)

Neelam Saikia

Postdoc, Department of Mathematics, University of Virginia

Date: February 15, 2022; 5:00 PM- 6:00 PM

Title: AGM and jellyfish swarms of elliptic curves over finite fields

Abstract: The classical $\AGM$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. In this talk we discuss a finite field analogue $\AGM_{\F_q}$ over finite fields $\F_q,$ with $q\equiv 3\pmod 4,$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a {\it jellyfish swarm}, as the 3D renderings of the connected components resemble {\it jellyfish} (i.e. tentacles connected to a bell head).

These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\F_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ many jellyfish.

Moreover, this interpretation gives a description of the {\it class numbers} of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish. This is a joint work with Michael J. Griffin, Ken Ono and Wei-Lun Tsai.

Arindam Biswas

Postdoc, Department of Mathematics, University of Copenhagen

Date: February 1, 2022; 5:00 PM- 6:00 PM

Title: On Cheeger type inequalities for graphs

Abstract: The discrete Cheeger inequality is a well-known inequality in spectral graph theory, which relates the second largest eigenvalue of the adjacency operator to the Cheeger constant. In this talk, we shall discuss recent results on lower Cheeger inequalities involving the smallest eigenvalue, for certain classes of algebraic graphs.

Antar Bandyopadhyay

Professor, Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi

Date: January 18, 2022; 5:00 PM- 6:00 PM

Title: A classical random reinforcement model viewed differently!

Abstract: In this talk, we will start by explaining a classical random reinforcement model introduced by Pólya back in 1920 which is widely known as the Pólya’s balanced urn scheme. We will then indicate several unrelated examples can be thought as following Pólya’s model provided we allow ourselves a novel infinite dimensional generalization. After indicating the difficulties which arise due to absence of good algebraic techniques to deal with infinite dimensional matrices (or appropriate linear operators), we will explain a new probabilistic technique to handle the infinite dimensional generalization. We shall define a novel stochastic process which will turn out to be Markov with respect to a new type of “time” parameter, which instead of deterministic and unidirectional, will be random and have a tree-type structure. We shall show that this new process and the infinite dimensional generalization of Pólya’s balanced urn scheme are same in law. This will enable us to derive fairly general scaling limits for the infinite dimensional scheme and show that classical results can be easily derived without much difficulties and completely avoiding algebraic techniques. Moreover, we will show that apparently unrelated problems arising from a variety of different fields such as combinatorics, statistical physics and statistics, can be solved by using this new generalization.

[The talk will be based on several works done in the last few years. Some of which are joint work with Debleena Thacker, University of Bath, UK and Svante Jonson, Uppsala University, Sweden]

Manjil Saikia

Postdoctoral Research Associate, School of Mathematics, Cardiff University

Date: November 23, 2021; 4:30 PM- 5:30 PM

Title: Parity Biases in Partitions and Restricted Partitions

Abstract: Recently, Kim, Kim & Lovejoy (2020) proved that partitions with more odd parts than even parts are more in number than partitions with more even parts than odd parts (for all n>2). This, they called as parity bias in integer partitions. We prove that this is true even if we restrict the partitions under consideration to that of distinct parts partitions (for all n>19). We also show that parity bias is reversed if we restrict the smallest part that can occur in a partition to 2 (for all n>7). Some other results of similar flavour can be proved for partitions where we restrict the set of allowed parts. All of these results are proved combinatorially. Using analytical techniques some of the inequalities can be further strengthened, we will discuss this as well as some related results for other classes of partitions, if time permits. This talk is based on joint work with K. Banerjee, S. Bhattacharjee, M. G. Dastidar & P. J. Mahanta as well as on a work in progress with P. J. Mahanta & A. Sarma.

Agnid Banerjee

TIFR CAM (Centre for Applicable Mathematics)

Title: Strong unique continuation for heat operator with Hardy type potential

Abstract: I will talk about strong unique continuation for the heat operator with Hardy type potential. This is. based on a recent joint work with Nicola Garofalo and Ramesh Manna.

Gaurav Bhatnagar

Visiting Associate Professor, Ashoka University

Title: Telescoping continued fractions for the error term in Stirling’s formula

Abstract: We calculate the error term $r_n$ in Stirling’s approximation

$$ n! = undefinedsqrt{2undefinedpi}n^{n+1/2}e^{-n}e^{r_n},$$ giving one approach to a problem considered by Stirling in 1730. This is an extension of Robbins’ (1955) idea and shows how to better previous lower bounds for $r_n$ given by Cesundefined`{a}ro (1922), Nanjundiah (1959), Maria 1965), and Popov (2008). Our approach is elementary and algorithmic and uses continued fractions.

This is ongoing joint research with Krishnan Rajkumar (JNU).

In this talk, we will first outline Robbins’ proof of Stirling’s formula. The talk should be accessible to students.

R B Bapat

Visiting Professor of Mathematics, Ashoka University

Title: A glimpse of spectral graph theory

Abstract: Spectral graph theory is the study of the interplay between the spectrum of the adjacency matrix of a graph and the properties of the graph. We present a selection of results from spectral graph theory. These include a result on non-isomorphic cospectral trees, a problem on decomposing the complete graph on ten vertices by copies of the Petersen graph, and characterization of nonsingular trees. We conclude by presenting a path-breaking recent proof of the sensitivity conjecture by Huang.

Rajeeva Karandikar

Professor Emeritus, Chennai Mathematical Institute

Date: September 28, 2021

Title: Power and Limitations of Opinion Polls

Abstract: How can obtaining the opinion of, say 20000 voters be sufficient to predict the outcome of an election in a country with over 80 million voters?

Do the opinion polls conducted say a month before the election accurately predict what is to happen on the voting day

I will answer these questions and share my own experiences with opinion polls and exit polls in India over the last 2 decades.

Amritanshu Prasad

Professor of Mathematics, The Institute of Mathematical Sciences, Chennai

Date: September 14, 2021

Title: Generating Functions Associated to Species of Structures

Abstract: Species of structures were introduced by André Joyal and his group in Québec in the 1980s. They provide a way of organizing classes of labeled combinatorial objects that elevate the art of studying their generating functions to a science.

Combinatorial relationships realized bijectively among such classes are transformed into functional relationships of their generating functions. For example, from the combinatorial interpretation of a set partition as a set of non-empty sets, the exponential generating function for Bell numbers exp(exp(z)-1) becomes blindingly clear; exp(z) is the generating function of sets and exp(z)-1 that of non-empty ones.

I will discuss species of structures and some generating functions that are associated with them. I will explain how algebraic operations on generating functions can be seen to arise from set-theoretic operations on species. I will introduce the Frobenius characteristic generating function of a species of structures, which is a simple variation of the cycle index generating function, landing us in the world of symmetric polynomials.

School of Mathematics, Tata Institute of Fundamental Research, Mumbai

Date: August 31, 2021

Title: From Clay tablets to Clay Prize: Journey of the local-global principle in number theory.

YouTube

Abstract: Please click here to view the title and abstract

Rajendra Bhatia

Professor of Mathematics, Ashoka University

Date: April 20, 2021

Title: Loewner Matrices

Abstract: Given a function f on R, a positive integer n and a choice of points p_1,….,p_n, the Loewner matrix L(f, p) is the nxn matrix whose (i,j) entry is the divided difference [f(p_i) – f(p_j)] / [p_i – p_j]. These matrices arise in several contexts, one of them being the celebrated Loewner’s theory of matrix monotone functions. Some of this theory was discussed in Siddharth Mulherkar’s talk on 6 April. We will talk some more of these matrices. The talk is intended for and will be accessible to, students who have taken a second course on Linear Algebra. The first part will be on the blackboard introducing all concepts involved.

Shanta Laishram

Professor, Indian Statistical Institute Delhi

Date: April 13, 2021

Title: On a Conjecture of Erdos on Squares in Arithmetic Progression

Abstract: A remarkable result of Erdos and Selfridge states that a product of a two or more consecutive integers is never a perfect power. Erdos conjectured that if a product of $k$ consecutive terms of an arithmetic progression is a perfect power, then $k$ is bounded explicitly. In this talk, I will give an overview of the problem with emphasis on the squares case and present some new results and related problems.

Siddharth Mulherkar

ASP, Department of Mathematics, Ashoka University

Date: April 6th, 2021

Title: Operator Monotone Functions and Applications

Abstract: For any two Hermitian matrices $A,B$, we shall say that $A \leq B$ if $B-A$ is positive semi-definite. Note that $\leq$ is a partial order on the space of positive definite matrices. For any interval I \subseteq \R we say that $f:I \rightarrow \R$ is \matrix monotone of order $n$ if for all $n \times n$ Hermitian matrices $A,B$ we have $f(A) \geq f(B)$ whenever $A \geq B$ and $\sigma(A) \cup \sigma(B) \subset I$. If $f$ is matrix monotone for all $n$ then we say that $f$ is matrix monotone or operator monotone. We give various characterizations of operator monotone functions and outline some applications to matrix inequalities. This talk will be expository in nature.

Guhan Venkat

Chinese Academy of Sciences, Beijing, PRC

Date: March 23, 2021

Title: Stark-Heegner cycles and arithmetic

Abstract: Stark-Heegner points were first defined by Henri Darmon (McGill U.) about twenty years ago. These are p-adic rational points on elliptic curves defined over the rational numbers and they have consequences towards the Birch and Swinnerton-Dyer conjecture. In this seminar, I will report on the recent generalisations and progress in this theory, in particular focusing on the construction of Stark-Heegner cycles for automorphic forms for the group GL(2).

Rishideep Roy

Assistant Professor, Decision Sciences, IIM Bangalore

Date: March 16, 2021

Title: Multinomial data with randomly varying probabilities

Abstract: We consider a sequence of multinomial data, with multiple classes for each trial. We assume that the probabilities associated with these classes vary randomly over time. We show that under suitably chosen prior distribution on these probabilities, there is posterior consistency. We further consider an application of this method in calling elections, with voting data coming in multiple rounds.

Aditi Dandapani

Date: March 5, 2021, 10:00-11:00 AM

Title: From Quadratic Hawkes Processes to Super Heston Rough Volatility

Abstract: Using microscopic price models based on Hawkes processes, it has been shown that under some no-arbitrage conditions, the high degree of endogeneity of markets together with the phenomenon of metaorders splitting generates rough Heston-type volatility at the macroscopic scale. One additional important feature of financial dynamics, at the heart of several influential works in econophysics, is the so-called feedback or Zumbach effect. This essentially means that past trends in returns convey significant information on future volatility. A natural way to reproduce this property in microstructure modeling is to use quadratic versions of Hawkes processes. We show that after suitable rescaling, the long-term limits of these processes are refined versions of rough Heston models where the volatility coefficient is enhanced compared to the square root characterizing Heston-type dynamics. Furthermore, the Zumbach effect remains explicit in these limiting rough volatility models.

S.M. Srivastava

NBHM Visiting Professor, Indian Association for the Cultivation of Science, Kolkata

Date: March 2, 2021

Title: The Birth of Set Theory

Abstract: Please click here to view the Abstract

Jishnu Ray

Postdoctoral Researcher at CRM, Université de Montréal

Date: February 16, 2021

Title: Selmer groups of elliptic curves and Iwasawa algebras

Abstract: The Selmer group of an elliptic curve over a number field encodes several arithmetic data of the curve providing a p-adic approach to the Birch and Swinnerton Dyer, connecting it with the p-adic L-function via the Iwasawa main conjecture. Under suitable extensions of the number field, the dual Selmer group becomes a module over the Iwasawa algebra of a certain compact p-adic Lie group over Z_p (the ring of p-adic integers), which is a completed group algebra.

In this talk, we give an explicit ring-theoretic presentation, by generators and relations, of Iwasawa algebras and explore the structure of Selmer groups over non-commutative Lie extensions.

K Ramasubramanian

Professor, Indian Institute of Technology, Bombay

Date: February 2, 2021

Title: Construction of 4 × 4 Pandiagonal Magic Squares with Turagagati

Abstract: In India, magic squares seem to have been known for more than two millennia. However, among the extant texts, a systematic introduction to the principles governing their construction can be found only in the work of Nārāyaṇa Paṇḍita (c. 1356 CE). He has dedicated one full chapter of his Gaṇitakaumudī to describe Bhadragaṇita, namely, methods for constructing magic squares of different orders. The focus of this talk would be to present the algorithm propounded by Nārāyaṇa Paṇḍita for constructing pan-diagonal magic squares of order 4 using only turagagati or horse-moves. Earlier studies by a few scholars starting with Datta and Singh have discussed this algorithm, showing how consecutive pairs get placed in horse-moves. Whereas, in our presentation, we shall demonstrate that the construction of the entire square can be made by employing only horse-moves. We shall also touch upon the properties exhibited by such pan-diagonal squares.

Atul Dixit

Assistant Professor of Mathematics, Indian Institute of Technology Gandhinagar

Professor, Indian Statistical Institute, Bangalore

Date: November 17, 2020

Title: Invariants

Abstract: Here is a simple puzzle: Start with a rectangular 3cm x 5cm piece of paper. Cut it down into smaller rectangular pieces and re-arrange to have a square of size 4cm x 4cm. Without wasting any paper, a little bit of thought should tell you that this is impossible as the original area is 15 cm^2 and any rearrangement would have the same area whereas we are asked to get a square of area 16 cm^2. Here `area’ is an `invariant’. It is an obstruction to realize the transformation asked for. The notion of invariants is widely used in mathematics to classify objects and to detect obstructions in transforming systems from one state to another. It also has many practical applications. In this talk, we will describe the concept of invariants through various puzzles and some mathematical problems.

Krishna Maddaly

Professor, Ashoka University

Date: November 10, 2020

Title: Wavelets – Are these small waves?

Abstract: Are wavelets small waves? This is the first question that comes to mind if one has never heard of them. In this talk, we will explain what they are, why they appeared in mathematics, how they quickly took root, and how they silently form part of our lives without our ever realizing the fact.

Tirthankar Bhattacharyya

Professor, Indian Institute of Science

Date: November 3, 2020

Title: Dilation and von Neumann’s inequality for matrices

Abstract: We shall show some easy matrix techniques to come up with interesting results like the maximum modulus principle and the von Neumann’s inequality. This involves forming polynomials of matrices. So, we shall talk about the functions of matrices. Suppose T is an n by n matrix with the largest singular value not larger than 1. The von Neumann’s inequality is a fundamental result which states that for a polynomial p and a matrix T as above, the largest singular value of p(T) is not larger than 1. Interestingly, this has a relation with complex analysis. The method of proof of von Neumann’s inequality produces a new proof of the maximum modulus principle as well.

Sourav Ghosh

PostDoc, Universite du Luxembourg

Date: October 27, 2020

Title: Tilings of Spaces

Abstract: Tilings are found in nature in the forms of crystals and honeycombs. Tilings have also been used throughout human history for both practical and decorative purposes. In this talk, we will discuss the mathematical theory behind the tilings of spaces.

Riddhipratim Basu

Faculty, International Centre for Theoretical Sciences of the Tata Institute of Fundamental Research (ICTS-TIFR), Bengaluru

Date: October 20, 2020

Title: A story of universality in random interface growth

Abstract: One-dimensional interfaces growing in time (consider, for example, the top envelope of the configuration in the game of TETRIS) are ubiquitous in nature. I shall describe a class of stochastic models for interface growth that are believed to, asymptotically, share the same universal characteristics observed in many naturally occurring interfaces, and sketch, in parts, an ongoing story of the fascinating mathematics developed over the last twenty years with a view to understand such interfaces rigorously.

Nishad Kothari

Assistant Professor, Department of Computer Science and Engineering, I.I.T. Madras

Professor, School of Engineering, Trinity College Dublin, Ireland

Date: 15th September 2020,

Title: On a three-dimensional nonlinear model of Pacific equatorial ocean dynamics

Abstract: This talk focuses on some investigations into a recently developed non-linear, three-dimensional Pacific equatorial model for ocean dynamics. The development of the model had been motivated by observations and the model is able to capture some essential properties of the flow in the Pacific equatorial region. Analysis of velocity field and flow paths indicate that several known and unknown features (which are essentially non-linear and three dimensional such as upwelling/downwelling, cellular flow structures, divergence of flow from the equator and extra-equatorial flows, subsurface ocean ‘bridge’ in the equatorial direction and sharp change in gradient of the flow path) exist and can be simulated by the model.

Chandan Singh Dalawat

Professor H+, Harish-Chandra Research Institute, Allahabad

03rd March 2020

Title: Two footnotes to Galois’s Memoirs

Abstract: We will review the history of the solvability of polynomial equations by radicals, concentrating on the two Memoirs of Evariste Galois. We will show how the first Memoir allows us to determine all equations of prime degree which are solvable by radicals, and the second Memoir similarly leads to the determination of all primitive equations which are solvable by radicals. All these terms will be explained.

The talk will be meant for a general audience. We will take a historical point of view, starting from the work of the Italians on the cubic and the quartic, mentioning the work of Abel and Galois’s First Memoir, and illustrating the whole theory with the toy case of extensions of prime degree.

T.S.S.R.K. Rao

FASc, FNASc, Visiting Professor-Ashoka University

25th February 2020

Title: Some Min-Max formulas in Optimization

Abstract: An important technique in Optimization theory is to find formulas that equate the minimum of a class of objects with the maximum from a different class. When such a formula can be found, Min-Max problems can be more easily handled since one can consider more convenient side of the equation to proceed further. It is known that linear analysis (Functional Analysis) is a good set up to solve optimization problems. In this talk, we present a recently obtained, such a formula for continuous linear maps which allows for global optimization from local conditions.

Biswajyoti Saha

School of Mathematics and Statistics, University of Hyderabad

18th February 2020

Title: Prime numbers and the Riemann zeta function

Abstract: Prime numbers are the building blocks for the whole numbers. Just by starting from the infinitude of primes and their distribution, one is led to many questions that have been foundational for research in number theory. For instance, the study of the Riemann zeta function is intimately connected to the distribution of prime numbers. Our lecture will be devoted to exploring some of these connections and it would be accessible to the undergraduate students in mathematics.

Inder Bir Passi

Visiting Professor of Mathematics, Ashoka University

Professor Emeritus, Panjab University Chandigarh Honorary Professor, IISER Mohali

11th February 2020

Title: Mathematics and Society: Voting Systems

Abstract: For arriving at decisions in a democratic manner, it is common to take recourse to vote. In this talk, a mathematical view of various commonly used voting systems will be discussed.

Yashonidhi Pandey

IISER Mohali

04th February 2020

Title: Some results on the geometry of curves and surfaces

Abstract: We will see a proof of Pythagoras theorem on the plane and another on the sphere. Consider the paths traced by the front and rear wheel of a bicycle that returns to its initial position. We will discuss an interesting fact about the signed area between the tracks.

If time permits, we will discuss another result on the sphere relating angles and area of a triangle. (Message to the experts: the theme of the talk will be to introduce the audience to the geometry around the Gauss-Bonnet theorem).

Werner Kirsch

Department of Mathematics,

Fern University, Hagen, Germany

28th January 2020

Title: Mathematics, Power and Democracy: A mathematical look at voting systems

Abstract: In this talk, we present some methods and results from the mathematical theory of voting. These results can directly be applied to real-world voting systems, like a Council of states (Rajya Sabha or the Council of Ministers of the European Union).

In particular, we will discuss how a fair representation of states in a union can be realized.

Meet a Mathematician

Peter Semrl

University of Ljubljana, Slovenia.

24th January 2020

The Mathematics Department organized an informal series of interactions between prominent mathematicians and students Meet a Mathematician

The first session was held on 24thJanuary 2020

Gaurav Bhatnagar

Visiting Faculty, School of Physical Sciences, JNU

21st January 2020

Title: Ramanujan and $q$-continued fractions

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 elementary ideas due to Euler can be used to prove some 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 Conference on Topics in Real Analysis

This Conference aims for the exchange of knowledge in advancements in the area of Mathematical Analysis broadly Matrix Analysis, the finite and infinite rank of a matrix.

18th December and 19th December

Speakers:

Barry Simon

Kumarjit Saha

Tanvi Jain

Ranjana Jain

Dhriti Ranjan Dolai

Sameer Chavan

Arup Pal

Priyanka Grover

Pankaj Jain

Ved Prakash Gupta

Peter Hislop

Barry Simon

IBM Professor of Mathematics, CALTECH

18th December 2019

A Conference on Topics in Real Analysis

Title: Tales of Our Forefathers

Abstract: This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems.

With vignettes and anecdotes, I’ll convince you they were also human beings and that, as the Chinese say, “May you live in interesting times” really is a curse.

Amber Habib

Head,

Department of Mathematics,

Shiv Nadar University

19th November 2019

A Conference on Topics in Real Analysis

Title: Episodes in Indian Mathematics

Abstract: Mathematics had a rich history in ancient and medieval India. Indian mathematicians made original contributions to algebra, number theory, and geometry. In this lecture, we cover some of the highlights of the emergence and development of mathematics and mathematical thoughts in India. We will focus especially on themes related to square roots and the Pythagoras theorem. While Class X mathematics is enough background to follow these topics, the ideas and creativity involved are non-trivial and exciting. It is also illuminating to see how our ancestors absorbed and improved knowledge from our neighbors, and in turn, influenced them.

Mahan Mj

Tata Institute of Fundamental Research, Mumbai

12th November 2019

Title: Percolation on Hyperbolic Groups

Abstract: We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an .i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about the asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the ‘velocity’ exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary.

For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along with word geodesic rays in every fixed boundary direction.

Anish Sarkar

Indian Statistical Institute, Delhi

29th October 2019

Title: Martingales and collision time of random walks

Abstract: In this talk, we will introduce martingales and stopped martingales. Using the stopped martingale, we will analyze some of the properties of the simple symmetric random walks. In particular, we will discuss the first collision times of two or three random walks and compute some of the basic distributional property (such as expectation) of the collision time.

A part of the talk is based on joint work with D. Coupier, K. Saha and Chi Tran.

Shivam Sahu

Ashoka University

22nd October 2019

Title: Decoding the Ashoka Logo

Abstract: Are you intrigued by the Logo of Ashoka University? These concentric circles tell us a lot about liberal arts and the circle of life. This talk will give you an introduction to the logo and its history and most importantly the beautiful derivation of the logo using Mathematics. Besides, we will see how you can write a code to generate a logo of a given size. You will get a chance to draw the logo and understand the logic of the intersecting circles and lines.

Pritam Ghosh

Ashoka University

15th October 2019

Title: Matrices from a geometric viewpoint.

Abstract: We will look into matrices from a geometric viewpoint and connect it to various topics in metric spaces like compactness and connectedness. What we will see is the introduction to a vastly studied and useful topic in mathematics called “Lie groups”.

The target audience for this talk are the students who have basic familiarity with the metric of R^n. No advanced background is required.

Tanvi Jain

Indian Statistical Institute

24th September 2019

Title: Spectra of Some Special Matrices

Abstract: Some special matrices related to the Hilbert and Cauchy matrices depict an intriguing spectral behavior. We investigate this behavior. In particular, we focus on computing the numbers of positive, negative and zero eigenvalues of these matrices.

T.S.S.R.K.Rao

FASc, FNASc, Visiting Professor-Ashoka University

17th September 2019

Title: An interaction between geometry and approximation theory

Abstract: This talk is intended for a general audience. Here we demonstrate by way of several examples, whereby techniques from the geometry of Banach spaces can be applied to solve some very practical problems in approximation theory.

Vaibhav Vaish

IISER Mohali

10th September 2019

Title: On the Ramanujan’s conjecture, and motives behind it.

Abstract: In 1916, Ramanujan came up with a series of conjectures about the coefficients of the $\tau$-function, a function that appears naturally in number theory. Last of these were eventually proved as a consequence of Deligne’s result (1974) on Weil conjectures in Algebraic geometry. The common motif connecting the two seemingly disparate topics is the so-called theory of motives as laid out by Grothendieck and his school. This talk will try to summarize this long journey, and time permitting, present some of our related recent results.

Kumarjit Saha

Ashoka University

03rd September 2019

Title: Scaling limit of a dynamical drainage network model and application to convergence to Brownian net in the non-crossing path set up.

Abstract: You can ignore all the jargon mentioned in the title. In the first part of the talk, we will discuss Skorohod topology on the Right Continuous Left Limit (RCLL) paths which give a complete separable metric space. In the second part of the talk, we will discuss how this topology helps to solve a difficult open problem in the area of the scaling limit of random graphs. This is a joint work with Ravishankar Krishnamurthy from the state university of New York.

Yash Lodha

EPFL-SNSF Ambizione fellow,

Lausanne, Switzerland.

27th August 2019

Title: The Banach-Tarski paradox

Abstract: The Banach-Tarski paradox is one of the most striking phenomena in all of mathematics. It is actually a mathematical fact, yet denoted as a “paradox” due to its counterintuitive nature. In this talk, he will describe the paradox and some of the underlying measure-theoretic and group-theoretic reasons behind it.

Krishna Maddaly

Ashoka University

24th April 2018

Title: Multiplicity bounds for Random Operators

Abstract: In this talk I will discuss a recent theorem proved on multiplicity of spectrum of random operators. In the case when the spectrum is R, coming from independent random variables, multiplicity on a set of positive Lebesgue measure for the set point spectrum away from the continuous one gives a lower bound for multiplicity of singular continuous spectrum everywhere.

Siddhartha Gadgil

Indian Institute of Science 17th April 2018

Title: Homogeneous length functions on Groups: A polymath adventure

Abstract: Terence Tao posted on his blog a question of Apoorva Khare, asking whether the free group on two generators has a length function l:F_2→R (i.e., a function satisfying the triangle in-equality) which is homogeneous, i.e., such that l(gn)=nl(g)l(gn)=nl(g). A week later, the problem was solved by an active collaboration of several mathematicians (with a little help from a comput-er) through Tao’s blog. In fact a more general result was obtained, namely that any homogeneous length function on a group G factors through its abelianization G/[G,G]. I will discuss the proof of this result and also the process of discovery (in which I had a minor role).

Dario Darji

Ashoka University

10th April 2018

Title: Shannon’s Entropy, Arithmetic Compression and Exotic functions

Abstract: In this simple and heuristic talk using Huffman and Arithmetic compression, we will give a motivation as to why Shannon’s entropy is defined as is. Then, we will show how arithmetic compression naturally gives rise to some of the exotic functions of real analysis and measure theory. This talk will be elementary and geared towards undergrads

Tulsi Srinivasan

Ashoka University

3rd April 2018

Title: Geometric Group Theory (part 2)

Abstract: This is the second of a two-part introduction to geometric group theory. This talk is on hyperbolic groups and their boundaries.

K R Parthasarathy

Indian Statistical Institute

27th March 2018

Title: How does one measure Information?

Abstract: This will be an extempore talk on how to measure information.

Inder B Passi

Ashoka University

20th March 2018

Title: Geometric Group Theory (part 1)

Abstract: Given a finitely generated group G and a finite subset S of its generators, the Cayley graph Γ(G, S) associates to G a metric space, and thus enables geometric methods to be applied for the study of the group G. I will discuss this approach; in particular, growth of groups and Gromov’s polynomial growth theorem.

Rhea Schroff,

Ashoka University

6th March 2018

Title: Some results in Group Theory

Abstract: In this talk I will talk about a few concepts I learned in group theory during my study at Ashoka. The talk will be quite elementary.

Michael Barany

Society of Fellows, Dartmouth College

27th February

Title: The Secret History of the Fields Medal.

Abstract: “First presented in 1936, the Fields Medal quickly became one of mathematicians’ most prestigious, famous, and in some cases notorious prizes. Because its deliberations are confidential, we know very little about the early Fields Medals: how winners were selected, who else was considered, what values and priorities were debated—all these have remained locked in hidden correspondence. Until now. My talk will analyze newly discovered letters from the 1950 and 1958 Fields Medal committees, which I claim demand a significant change to our understanding of the first three decades of medals. I will show, in particular, that the award was not considered a prize for the very best mathematicians, or even for the very best young mathematicians. Debates from those years also shed new light on how the age limit of 40 came about, and what consequences this had for the Medal and for the mathematics profession. These findings also offer a new perspective on the Medal’s role in mathematicians’ international relations in this period. I argue that 1966 was the turning point that set the course for the Fields Medal’s more recent meaning.”

Nikita Agarwal

IISER, Bhopal

20th February 2018

Title: Ergodic Properties of Open Dynamical Systems

Abstract: Ergodic Theory deals with the study of asymptotic behaviour of a dynamical system, which is either a group action, or a flow or a map on the state space. Dynamical systems can be broadly classified into closed and open systems. In closed systems, the orbit of a point lie in the state space for all time, whereas in open systems, the orbit of a point may escape from the state space through a hole. A classic example of this escape phenomenon is in the study of the motion of a billiard ball on the table with a hole (pocket). The first account of open dynamical systems is due to Pianigiani and Yorke in 1979, who were motivated by this example. This talk is a brief historical account of the development of the related theory and some potential questions. Ideas from symbolic dynamics and arithmetic dynamics will be presented in this context.

Jean-Marc Deshouillers

University of Bordeaux, France

13th February 2018

Title: Sums of few powers, a probabilistic approach

Abstract: “The usual methods for studying Waring’s problem (i.e. finding how many $s$-th powers does one need to represent all the integers?) require a rather large number of summands. Erd\H{o}s and R\’{e}nyi introduced in 1960 a probability model for studying sums of $s$ numbers which are $s$-th powers. Their model has been largely developed since then. After introducing the subject, the talk will focus on this probabilistic approach, up to the most recent contributions obtained jointly with the late Javier Cilleruelo. This is a colloquium talk where no specialised knowledge of number theory nor probability theory is required.”

Rajendra Bhatia

Ashoka University

30th January 2018

Title: Perturbation of eigenvalues

Abstract: When a matrix is changed (perturbed by a physicist or approximated by a computer) by how much can its eigenvalues change? I worked on this problem through the 1980’s and will recall what I learnt. Some problems that are still not solved will be mentioned

Arnab Mitra

Technion- Israeli Institute of Technology

21st November 2017

Title: Gelfand pair for compact groups.

Abstract: Let G be a compact topological group and H be a closed subgroup. Say that (G,H) is a Gelfand pair if for every irreducible representation V of G, the space of H-invariant linear forms on V has dimension at most one. We will see some basic properties and examples of Gelfand pairs, and if time permits, some applications.

Alok Goswami

Indian Statistical Institute, Kolkata

24th October 2017

Title: Young’s Old Theorem Revisited

Abstract: While reviewing a recent result in the context of stochastic processes, we stumbled upon a very old theorem of W.H.Young on real functions. We found a remarkable similarity between these two results which are otherwise in two completely different contexts. In this talk, I will describe a general result that we discovered, which provides a connecting link. I will also describe some of its other applications. This is a joint work with B. V. Rao.

Pierre Gilbert

Vienna

10th October 2017

Title: Hilbert tenth problem and variations.

Abstract: In 1900, Hilbert posed a famous list of 23 problems, which shaped the mathematics of the following century. We shall talk about Hilber tenth problem: “Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.” The problem can be modernized in the following way: Write a program which given as parameter a polynomial P(X1,…Xn) in multiple variable, with integers coefficient, find whether or not P(x1,…xn)=0 has a solution, where x1…xn are integers. A lot of work on the subject was done by Martin Davis, Julia Robinson, and Hilary Putnam. Finally Yuri Matiyasevich, in 1970, did the last step, proving that there is no solution to Hilbert tenth problem. To be more precise there is no programs that can work for all polynomials. The goal of this talk is to discuss the result and various related problems: Is a similar problem solvable if we want to find a root in the natural numbers? in the rational numbers? or in the real numbers?

Kumarjit Saha,

Ashoka University

19th September 2017

Title: Existence and coalescence of directed infinite geodesics in the percolation cone.

Abstract: For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We consider FPP on two-dimensional integer lattice with i.i.d. passage times distributed as Durrett-Liggett class of measures. For this model, we show that along any direction in a deterministic angular sector (known as percolation cone), starting from every lattice point there exists an infinite geodesic along that direction and such directed geodesics coalesce almost surely. We prove that for this model, bi-infinite geodesics exist almost surely. Our proof does not require any assumption on the limiting shape.

Dario Darji

University of Liouville, USA

12th September 2017

Abstract: In the early 1970’s Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.

Rajendra Bhatia

Ashoka University

August 29th 2017

Title: The Marvellous Number Pi

Abstract: Everyone has some idea about pi. It is 22/7 (approximately) when calculating areas and perimeters. While measuring angles it is in radians what 180 is in degrees. It is an irrational number, etc. Less appreciated is the fact that it is connected with many diverse subjects ranging from probability to number theory, and its study has a very rich history. In this talk we will discuss some interesting facts about this object. Most of the talk will be accessible to anyone with an interest in mathematics.

Nandini Neelakantan, Indian Institute of Technology, Kanpur 25th March 2017 Title: Platonic Solids and Euler Formula Geetha Venkataraman, Ambedkar University, Delhi 25th March 2017

Title: Groups and Symmetry

Abstract: We shall explore the historical connection between Symmetry and Groups and also try to address current research questions concerning symmetry groups. In 1829-30, a nineteen-year-old mathematician, Évariste Galois made an important breakthrough. Mathematicians for centuries had been wrestling with the problem solving equations of degrees higher than 2. The quest was to find a `nice formula’ expressing the roots of the equations in terms of the coefficients. In the 17th century, Italian mathematicians had shown that there are such formulae for the cubic and the quartic equation. Abel had in 1824 shown that such a general formula was not possible for a quintic. Galois, however saw the shapes hidden in the roots of equations. This enabled him to realise that the symmetries associated with certain quintic equations were very different to those of the quadratic, cubic or quartic and lead him to `invent’ groups and use their properties to show the impossibility of a general formula for roots of polynomial equations of degree 5 or higher. The lecture will also dwell on `indivisible symmetry shapes’ and connect them to `finite simple groups’. Some glimpses of the large research project of the 20th century in classifying the finite simple groups will be given and also of current research that falls broadly in this area.

Riddhi Shah,

School of Physical Sciences, Jawaharlal Nehru University, New Delhi 25th March 2017

Title: Dynamics of Distal Maps

Abstract: An automorphism T of a locally compact group is said to be distal if the closure of the T-orbit of any nontrivial element stays away from the identity of the group. We discuss some properties of distal automorphisms on groups. (This will be a survey talk and most of it would be accessible to students).