Quantifying Distinctions / The Random Walk

Speaker:  Yash Mehta

Date: 14th November 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

The talk will basically consist of two different solutions of the random walk problem, and one of the key techniques used in one of the proofs would be used to provide a method to estimate the number of “types” of an entity in a region. Basic proofs will be outlined along the way (some might be skipped depending on the availability of time). Actual examples of its utilisation will be discussed at the end.

Prerequisites: Basic arithmetic, elementary ideas of calculus and limits.

Area: Probability Theory

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Gromov Hausdorff convergence, ultralimits, and some applications

Speaker:  Sayantan Khan

Date: 7th November 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk will outline a very useful generalization of Hausdorff convergence (which will also be defined in the talk), and will involve proofs of some essential properties of this form of convergence (some proofs will be sketched, and some easy ones will be skipped). We’ll also look at ultralimits, which are a nice way of abstracting out the diagonal argument used in many proofs in analysis. Finally, we’ll have a look at some applications of the theory of Gromov Hausdorff convergence, if time permits.

Prerequisites: Basic knowledge of analysis, familiarity with compactness, and basic topology, i.e. definition of metric spaces.

Area: Topology

Introduction to Game Theory (with examples)

Speaker:  Nidhi Rathi

Date: 31st October 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk is going to serve as an introductory session for Game Theory. I will touch upon some of its (many) basic yet interesting concepts. To begin with, I will introduce the Game of Life devised by the British mathematician John H. Conway in 1970. Moving on, I will talk about combinatorial Sperner’s lemma and its set covering analog, KKM lemma (they are both equivalent to Brouwer’s fixed point theorem). Time permitting, I would love to introduce The Game of Hex, invented independently by the mathematicians Piet Hein and John Nash.

Prerequisites: A curious mind 😛

Area: Game Theory

References:

1. Game Theory – Michael, Solan, Zamir
2. Algorithmic Game Theory – Nisan, Roughgarden et al

Introduction to Complex Dynamics in one variable

Speaker:  Mayuresh Londhe

Date: 24th October 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk is going to be very introductory. To begin with, I will give a motivation, by using a few examples. We will then see definitions of Fatou and Julia sets with a few historical notes. Towards the end, I will prove some theorems and state a few without proof.
Prerequisites: First course in one variable complex analysis and basics of covering spaces.

Area: Complex Dynamics

References:

1. Dynamics in one complex variable (Geometric viewpoint) – John Milnor
2. Complex Dynamics (Analytic viewpoint) – L. Carleson and T. Gamelin
3. Iteration of rational functions (for examples) – Alan Beardon
4. One hundred years of complex dynamics (Survey article) – Mary Rees

 

Cryptography: Encryption Schemes and Some Advanced Tools

Speaker:  Sruthi Sekar

Date: 3rd October 2017 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

This talk is aimed to be an introduction to some schemes in Cryptography. The talk is structured to include relevant history in Cryptography, specific encryption schemes like RSA and some demonstrations to explain advanced tools used in Cryptography like Zero Knowledge Proofs and Multi-party Computation.

Area: Cryptography and Number Theory

References:

  1. Introduction to Modern Cryptography
    Book by: Jonathan Katz and Yehuda Lindell
  2. The Code Book
    Book by: Simon Singh
  3. The Codebreakers
    Book by: David Kahn

How far is 1 from the subspace spanned by polynomials vanishing at the origin?

Speaker: Md. Ramiz Reza

Date: 21th April, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

I will talk about Szego’s Theorem. The set up for the theorem is as follows. Let \mu be a positive measure on the unit circle and we want to know in the Hilbert space L^2(\mu) what the distance from the constant function 1 is to the subspace spanned by polynomials which vanish at the origin. Szego’s theorem states that the square of the distance (infimum) above is equal to \exp \{ \frac{1}{2\pi} \int \limits _{0}^{2\pi} \log h d\theta \}, where h is the derivative of \mu w.r.t the normalized Lebesgue measure.

In this process we will also prove the famous theorem of F and M Riesz which tells us that if a complex Borel measure on the unit circle is analytic, i.e., \int \exp(i n \theta) d\mu(\theta) = 0 for all n\geq 1, then \mu is absolutely continuous w.r.t Lebesgue measure.

Area: Functional Analysis

Littlewood’s theorem for composition operators on Hardy space

Speaker: Vikramjeet Singh Chandel

Date: 7th April, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

Consider the Hardy space H^2 on the unit disc \mathbb{D} of square-summable power series coefficients. To each holomorphic function \varphi that takes \mathbb{D} into itself we associate composition operator C_{\varphi} defined by C_{\varphi}f = f \circ \varphi, where f \in H^2. I’ll first present Littlewood’s famous Subordination Principle which establishes that C_{\varphi} maps H^2 into itself. Based on this is Littlewood’s Theorem which essentially says that the composition operator is bounded. Further investigations into properties like compactness, spectra of composition operators lead to classical results in complex dynamics like the Denjoy-Wolff Iteration Theorem, Konig’s Linearization Theorem and the Koebe Distortion Theorem.

Area: Complex Analysis, Functional Analysis, Operator Theory