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

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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

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