## On Beurling-Lax-Halmos theorem

Speaker: Monojit Bhattacharjee

Date: 31 March, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

In this talk we will present to you the proof of the “Beurling-Lax-Halmos” theorem. Let $T$ be a $C_{0}$ contraction on a Hilbert space $\mathcal{H}$ and $\mathcal{S}$ be a non-trivial closed subspace of $\mathcal{H}$. We prove that $\mathcal{S}$ is a $T$-invariant subspace of $\mathcal{H}$ if and only if there exists a Hilbert space $\mathcal{D}$ and a partial isometric operator $\Pi:H^2_{\mathcal{D}} \rightarrow \mathcal{H}$ such that $\Pi \mathcal{M}_{z}= T \Pi$ and $\mathcal{S}$ = ran $\Pi$. As a corollary of this theorem we will prove the “Beurling-Lax-Halmos” theorem. This theorem characterizes the shift-invariant subspaces of the vector-valued Hardy space.

Area: Functional Analysis, Operator Theory

## An introduction to homogenization

Speaker: Bidhan Chandra Sardar

Date: 17 March, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract: The theory of homogenization helps to understand the multi-scale phenomena which has applications, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. In the process, various methods were developed to study homogenization problems. Some of them are: multi-scale expansion, test function method, compensated compactness, two-scale and multi-scale convergence, bloch wave analysis, method of unfolding and so on. In this talk we will discuss multi-scale expansion and two-scale convergence.
Area: Differential Equations

## To be or knot to be

Speaker: T.V.H. Prathamesh

Date: 10 March, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:
This talk attempts to introduce and survey some of the interesting results in knot theory. Knots are closed loops in a 3 dimensional space. Knot theory, as a discipline, goes back to the 19th century when knots were used as a model to study atoms. Today, it is an important area in topology, with deep connections to areas ranging from operator theory to real algebraic geometry.
We will introduce basic definitions and further go on to sketch various attempts that are made to distinguish a knotted loop from an unknotted loop.  We will describe some of the important invariants and, if time permits, discuss some of the applications of knot theory to statistical physics and biology.
Area: Knot Theory & Topology

## Watson-Crick pairing, the Heisenberg group and Milnor invariants

Speaker: Siddharth Sabharwal

Date: 3 March, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract: The seminar will deal with ideas from the paper (arXiv:0809.3110) with the same title by Siddhartha Gadgil. We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict allosteric structures for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson-Crick pairs found.

Area: Topology

## Multispecies Juggling Probabilities

Speaker: François Nunzi

Date: 24 February, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

We consider refined versions of Markov chains related to juggling. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities and closed-form expressions for the normalization factor. We also refine and generalize enriched Markov chains on set partitions. Lastly, we generalize the construction to juggling with different kind of balls.

Area: Probability Theory

## The Weierstrass Preparation and Division Theorems.

Speaker : Samrat Sen.

Date : 10 February, 2015 (Tuesday)

Time : 09:15 pm – 10:15 pm

Venue : Lecture Hall I, Department of Mathematics

Abstract: The Weierstrass Preparation and Division Theorems are the main tools to understand the structure of the local ring consisting of the germs of holomorphic functions in several variables, at a given point $P$. According to the Weierstrass Preparation Theorem, any non-zero germ is, up to a non-singular linear change of coordinates and multiplication by a germ of a function not zero at $P$, a Weierstrass Polynomial in one variable of certain degree. On the other hand, the Weierstrass Division theorem says that, given any germ $f$ and a suitable one-variable Weierstrass Polynomial $g$ of degree $k>0$, there exists a unique pair of germs $h$ and $j$ such that $f=g \cdot h+j$, where $j$ is a Weierstrass Polynomial of degree less than $k$. We’ll prove both of the theorems using elementary several complex variables techniques in the seminar and lastly we’ll see that these are the extended forms of the Implicit Function Theorem and the Division Algorithm on polynomial rings respectively.

Area: Complex Analysis, Several Complex Variables

## The Looman-Menchoff Theorem.

Speaker: Anwoy Maitra.

Date: 3 February, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract: In this talk, we will present a full proof of a result one hears about often during a first course in complex analysis, but a proof of which is very rarely given. It is the Looman-Menchoff theorem, which asserts that if $D$ is an open subset of $\mathbb{C}$ and if $f: D \to \mathbb{C}$ is a continuous function such that all the first order partial derivatives of $f$ exist and such that the Cauchy-Riemann equations are satisfied, then $f$ is holomorphic. This result, while simple-sounding, is quite nontrivial!

Area: Complex Analysis