Student Talks 2020

Table of Contents

All expository talks given by students of ISI Bangalore in the year 2020 at our Math Club are listed below.

Spring 2020

EP2004: Classification of Curves on the Plane #

Speaker: Balarka Sen (B. Math, 2021)
Abstract: We shall consider smooth curves on the plane, possibly with self-intersection, but no “backtracking” or “haults”; these are said to be closed immersed curves on the plane. We say two such curves are regularly homotopic if one can be “smoothly deformed” to the other without “pinching parts of the loop infinitely tight”. We shall prove a surprising theorem of Whitney that two closed immersed curves on the plane are regularly homotopic if and only if the “turning numbers” of the curves agree. Two separate proofs will be given: a combinatorially inclined one and a purely analytic one. If time permits we shall mention Smale’s work on similar theorems in higher dimensions.
Date and Time: Sunday, 16th February 2020, 2:00 PM - 2:45 PM
Venue: G-26 Classroom, Academic Building

Speaker: Supravat Sarkar (M. Math, 2021)
Abstract: Quiver representations is a very important topic in representation theory. The problem of classifying representation of certain quivers vastly generalizes the problem of classifying linear transformation in linear algebra. Gabriel’s theorem classifies representations of certain quivers, called ‘finite type’ quivers, which are the quivers having only finitely many indecomposable representations. Interestingly, the classification is given by the positive roots of a root system. A root system is a combinatorial object arising unexpectedly in many seeming unrelated areas of representation theory. We shall talk about quiver representations, introduce root systems and finally prove one direction of Gabriel’s theorem.
Pre-requisites: I shall define everything, but it would be very helpful to have a look at root system and Weyl group.
Date and time: Saturday, 8th February 2020, 2:00 PM - 4:00 PM
Venue: G-26 Classroom, Academic Building

Speaker: Amartya Priyadarshan Muthal (B. Math, 2022)
Abstract: What properties of mathematical structures, e.g. groups, rings, graphs, are (in)expressible by a first-order formula? In this talk we address this question in the case when structures are finite. The two main techniques used for proving inexpressibility results are
(i) Ehrenfeucht-Fraïssé games
(ii) Locality properties
Guiding examples will be graphs and their properties, e.g. connectivity, colourability etc. The talk will include proofs of inexpressibility of graph connectivity by first-order logic using both the techniques.
References:
- Elements of finite model theory – Leonid Libkin
- Finite model theory – Heinz-Dieter Ebbinghaus & Jörg Flum
Date and Time: Sunday, 26th January 2020, 8:00 PM – 10:00 PM
Venue: G-26 Classroom, Academic Building

Speaker: Suvadip Sana (M. Math, 2021)
Abstract: We introduce the famous Galton-Watson Branching process. We try to understand the model and it’s various interesting features like probability generating function (pgf) associated with it, extinction probability, growth rate, etc. We will prove a very interesting fact that the extinction probability is the fixed point of the pgf of the progeny distribution.
Analtytic tools to be used
- Probability generating functions.
- Conditional probability and expectations.
- Wald’s equation (to be used in proofs).
- concept of transience and recurrence in markov chains (optional).
- Martingale convergence theorem (optional).
Pre-requisites: Knowledge of Probability I would be sufficient to understand most part of the talk. Knowledge of Markov chains and martingale theory would help you to understand some remarks (hand waving stuffs) that I will make without proofs. Mostly we will be dealing with discrete random variables taking values in \(\mathbb{N} \cup \{0\}\).
Date and Time: Sunday, 2nd January 2020, 3:30 PM - 5:00 PM, with a coffee break at 4:15 PM
Venue: G-26 Classroom, Academic Building