Holomorphic Foliation on complex manifold

Speaker:  Sahil Gehlawat

Date: 23rd January 2018 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk, I’ll recall the definition of complex manifold and then define holomorphic foliation on a complex manifold. We will see some methods of finding holomorphic foliations using some basic theorems of differential geometry and workout some examples. In the end, I will give a brief introduction to singular holomorphic foliation.

Prerequisites: Basic definitions and facts from Differential Geometry and Complex Analysis.

Area: Complex Analysis, Differential Geometry

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Proofs and Applications of the CSB (Cauchy-Schwarz-Bunyakowsky) and AM-GM (arithmetic mean-geometric mean) Inequalities

Speaker:  Raghavendra Tripathi

Date: 16th January 2018 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk I will revisit two very basic gems of modern analysis, namely the CSB inequality and the AM-GM inequality. I will present an elegant proof of the AM-GM inequality which uses induction.
  As a consequence of these two simple inequalities I shall prove three seemingly unrelated results. (Two from analysis (the first result concerns the positions of roots of monic polynomial with real roots; the second concerns the bounds on tangential triangle and rectangle of monic polynomials which are positive in (-1,1)) and one from Graph Theory, namely a special case of Turan’s theorem).

Area: Analysis

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

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