Carathéodory’s approximation theorem

Speaker: Abhay Jindal (Int.PhD)

Date: 29th November 2022

Time: 9:15 PM-10:15 PM

Abstract: We shall give two proofs of the following approximation theorem due to Carathéodory. 

Any holomorphic function from the open unit disc to the closed unit disc can be approximated locally uniformly by finite Blaschke products. 

We’ll also discuss some generalizations of this theorem and recent advances. We’ll conclude the talk with some open problems.

Prerequisites: First Course in Complex Analysis

Landau-Siegel zeros

Speaker: Ashutosh Suresh Jangle (PhD)

Date: 22nd November 2022

Time: 9:15 PM-10:15 PM

Abstract: L-functions are meromorphic functions which are of great importance in number theory. Landau-Siegel zero is a possible zero of an L-function in the ‘critical strip’, which is ‘close’ to the Re(s)=1 line in the complex plane.

If such a zero exists, it is an obvious counter-example to the generalised Riemann hypothesis. It is an open conjecture that no such zero exists. In this talk, we introduce the relevant background needed to define a Landau-Siegel zero and discuss its implications in number theory. Finally, we discuss Yitang Zhang’s claimed bound and some other known bounds for the Landau-Siegel zero.

Prerequisites: Basic Complex Analysis

The Bergman Kernel

Speaker: Aakanksha Jain (PhD)

Date: 15th November 2022

Time: 9:15 PM-10:15 PM

Abstract: The Bergman space of a domain is a reproducing kernel Hilbert space. Its reproducing kernel is called the Bergman kernel, a critical tool in the function theory of several complex variables. In this talk, we shall start with the definition of the Bergman kernel and then prove its representation in terms of an orthonormal basis of the Bergman space. We shall use this representation to compute the Bergman kernel of the unit disc.

Further, we shall see the relationship between the Bergman kernel and the Riemann mapping function. Finally, we shall use this relation to compute the Bergman kernel of the unit disc in an alternative way.

Prerequisites: Complex Analysis and Functional Analysis

Limits and Colimits

Speaker: Subhajit Das (PhD)

Date: 08th November 2022

Time: 9:15 PM-10:15 PM

Abstract: Universal objects such as Initials, terminals, products, unions, intersections, kernels, cokernels etc, to name a few, are ubiquitous in mathematics. This 1 hour (+ some epsilon minutes, perhaps) talk is all about mapping them into two broad types of universal constructions, Limits and Colimits. We start off with a few concrete examples to motivate the general definitions of Products, Coproducts, Pullbacks, Pushouts, Equalizers, Coequalizers etc, in an arbitrary category and then move on to how these are special instances of Limits/Colimits. We end with a brief discussion of how adjoints behave with these constructions.

Prerequisites: The previous seminar, but mostly the definitions of Categories, Functors and Natural Transformations.

Introduction to Category Theory

Speaker: Arghan Dutta (Int.PhD)

Date: 25th October 2022

Time: 9:15 PM-10:15 PM

Abstract: We start with the observation that many properties of mathematical systems can be unified and simplified just by a presentation with diagrams of arrows.
In this talk or maybe a series of talks, we basically start from scratch. We introduce the notion of a category, and then go on to define a number of category theoretical terms, especially limits, colimits, adjunctions and to prove the Yoneda Lemma.

Prerequisites: A lot of patience.

Some Properties of Manifolds with an Inner Distance

Speaker: Rumpa Masanta (PhD)

Date: 18th October 2022

Time: 9:15 PM-10:15 PM

Abstract: A distance(d_M) on a Manifold(M) is said to be Inner, if it captures the notion of smallest length (with respect to d_M) between two points on the Manifold.

In this talk, we discuss properties of Manifolds with an Inner distance and prove necessary and sufficient conditions for the following:

1) Topology (Top\,d_M) generated by the distance (d_M) being the same as the topology of the Manifold .

2) (M,d_M) is Cauchy complete.

Prerequisites: Basic Differential Topology.

An Introduction to Random Matrices

Speaker: Sourish Parag Maniya (Int.PhD)

Date: 11th October 2022

Time: 9:15 PM-10:15 PM

Abstract: In this lecture, we will talk about random matrices, particularly the study of their spectrum.  

We will first discuss the origins of random matrices and then introduce a few important examples. Next, we state the Semicircle Law, the analogue of the Central Limit Theorem in Random Matrix Theory and provide a sketch for a couple of its proofs.  

The last part of the lecture will focus on the eigenvalue distribution for a more general class of random matrices, where we shall encounter the Marcenko-Pastur law.

Prerequisites: Basic Probability and Linear Algebra.

Geodesics in First and Last Passage Percolation 

Speaker: Sudeshna Bhattacharjee (PhD)

Date: 27th September 2022

Time: 9:15 PM-10:15 PM

Abstract: First and last passage percolation on $latex\mathbb{Z}^d$ are examples of random growth models which are very useful to study various growth models which arise in Physics and Biology etc. In this talk we define these models and various concepts like passage times, geodesics etc. There are many interesting questions one can ask about such models. Such as, the limiting behaviour of the passage times, the limiting shape of the growth models, behaviour of geodesics etc. In this talk we will mainly focus on geodesics. We discuss basic existence, uniqueness results and some basic geometric properties of finite and infinite geodesics in both first and last passage percolation models.

In the later part of the talk, if time permits, we explore the conjectures about the connection with the KPZ universality class. In particular, we talk about the wandering exponent for geodesics and mention few results from exponential last passage percolation on $latex\mathbb{Z}^2$, where this exponent is well known.

Prerequisites: Familiarity in basic probability and measure theory.

An Introduction to Classical Algebraic K-Theory

Speaker: Jitendra Rathore (PhD, TIFR Mumbai)

Date: 20th September 2022

Time: 9:15 PM-10:15 PM

Abstract:  In this lecture, we will try to give a basic introduction to classical algebraic K-theory, which roughly deals with the study of classical K-groups namely K_0, K_1 and K_2 of a commutative ring.
   These groups were first defined during 1950-60’s by A. Grothendieck, H.Bass , J. Milnor etc. and turned out to be very important invariants of a commutative ring. We will begin with the definitions and few explicit examples. We shall also try to see a relation of these groups with some groups, which quite often occur in number theory.

Prerequisites: Definition of commutative ring and Modules

Alternating Sign Matrices.

Speaker: Dipnit Biswas (Int. PhD)

Date: 6th September 2022

Time: 9:15 PM-10:15 PM

Abstract: Alternating Sign Matrices (ASMs) are square matrices with entries in {0, ±1}, with the sum of entries in each row and column 1 and the non-zero entries alternate in sign in each row and column. These are generalisations of the permutation matrices.

In this talk, we prove the Alternating Sign Matrix theorem which counts the number of ASMs of order n. Then we talk about other entities whose ‘enumeration sequence’ are the same as ASMs. Later if time permits, we will talk about possible connections between these entities.

Prerequisites: Familiarity with basic Combinatorics.