Lecture Series

### Special Semester on Analysis

Chennai Mathematical Institute announces a Special Semester on Analysis to be held during January–April, 2010. As part of this programme, there will be visitors to CMI throughout the semester delivering a series of lectures on various topics in Analysis.

All lectures will start at the beginning and the lecturers will make sure that they build a body of basic theory required to understand their subsequent lectures. The target audience for this activity will be students from final year BSc Mathematics, students from both years of MSc Mathematics and students enrolled for PhD.

#### Oustation participants

Outstation participants interested in visiting CMI for part or whole of the semester to attend lectures in this activity should apply with their CV and a recommendation letter from a faculty member of their parent institution to shrihari@cmi.ac.in at least a month before of their proposed visit. The application should specifically state if they require travel support and / or local hospitality.

#### Programme for January, 2010

All lectures are from 2:00 pm–3:15 pm, in Lecture Hall 3.

•  Speaker: Prof P Veeramani, IIT Madras Title: Best Approximation in Normed Linear Spaces Dates: January 4, 5, 7

Abstract

If A is a nonempty subset of a metric space (X, d) and x in X, y in A are such that d (x, y) = d(x, A) = inf {d(x, a) : a in A}, then we say that y is a best approximation to x from the set A. Also for each x in X, P(x) = {a in A : d(x, a) = d(x, A)}, known as the metric projection supported on A, is a subset of A. It is aimed to discuss

1. the existence and uniqueness of best approximation of a given point x in X;
2. some properties of the metric projection supported on a nonempty set A;
3. the role of best approximation and metric projection in the study of fixed point theory.

•  Speaker: Prof Michael G Cowling, University of Birmingham Title: Mappings of groups with geometric properties Dates: January 11, 12

Abstract

Mappings of groups and of homogeneous spaces of groups arise in various areas of mathematics, including algebra, analysis, geometry and topology. In some cases, apparently simple hypotheses lead to strong restrictions on the possible mappings. For instance, there are classical results that mappings of an open set in the plane that send circular arcs to circular arcs or line segments to line segments have to be analytic, and come from well-known Lie groups. And Mostow's celebrated strong rigidity theorem relies on analysis of quasiconformal mappings of nilpotent groups and on Tits "fundamental theorem of projective geometry", which are generalizations of these classical results. These two talks discuss and develop these theories. In the first talk we consider mappings of groups with metric properties: for instance, mappings which send metric balls to metric balls, and in the second we consider mappings of groups which preserve cosets of subgroups.

•  Speaker: Prof TSSRK Rao, ISI Bangalore Title: Some applications of the principle of local reflexivity Dates: January 18, 19, 21

Abstract

The Principle of Local Reflexivity, an important tool in Banach space theory, allows one to transfer information about finite dimensional subspaces of the second dual of a space to the space itself, in an almost isometric way. In modern terms it says that the second dual of a space is finitely representable in the space itself.

Assuming only metric space theory, we will develop all the necessary tools to understand a simple proof of this theorem and give some of its applications.

•  Speaker: Prof VS Sunder, IMSc Chennai Title: An invitation to free probability Dates: January 25, 27, 28

Abstract

These lectures will try to introduce some basic notions and convey a flavour of the theory of free probability. The three lectures may, in turn, be titled as follows:

1. Catalan numbers ;
2. Non-commutative probability spaces - free independence and free cumulants ;
3. The central limit theorems - classical and free versions.

The lectures will attempt to slowly graduate from the very elementary to the fairly sophisticated.

#### Programme for February, 2010

All lectures are from 2:00 pm–3:15 pm, in Lecture Hall 3.

•  Speaker: Prof BV Rao, CMI Title: Excursions into probability Dates: February 1, 3, 5

Abstract

We try to understand the following phenomena.

Lecture 1: A drunkard, while performing random walk, must surely return to the starting point, while a drunk bird has some chance of never returning to the starting point. So comments Mark Kac (Ukrainian / Polish / American 1914 - 1984).

Lecture 2: A 'typical' number between zero and one, in its decimal (or any other base) expansion exhibits randomness as well as regularity in the following sense. At each place, the digit pops up at random, independent of what happens at other places. At the same time, these digits organise in such a way that each (finite) pattern of digits appears with the correct frequency. So say Emile Borel (French, 1871 - 1956); Andrei Kolmogorov (Russian, 1903 - 1987). (Do not ask me to show a typical number, I have not seen one.)

Lecture 3: If you mix hot milk and cold water, the mixture reaches an equilibrium state (steady state temperature) eventually. So says Ludwig Boltzmann (Austrian, 1844 - 1906). Assuming we have a closed system; you will be able to see --- an unending number of times --- milk nearly as hot as it was originally and water nearly as cold as it was originally. So says Henri Poincare (French, 1854 - 1912). These two statements are not contradictory. So says Paul Ehrenfest (Austrian / Dutch, 1880 - 1933) and Tatiana Ehrenfest (Russian / Dutch, 1876 - 1964).

•  Speaker: Prof S Thangavelu, IISc Bangalore Title: On the role of special functions in harmonic analysis Dates: February 8, 10, 12

Abstract

We plan to illustrate by way of examples how special functions such as Bessel, Jacobi, Hermite and Laguerre functions, arise in harmonic analysis on Lie groups and how indispensable they are for solving certain problems.

#### Programme for March, 2010

All lectures are from 2:00 pm–3:15 pm, in Lecture Hall 3.

•  Speaker: Prof Rajaram Bhat, ISI Bangalore Title: Dilation Theory Dates: March 1, 2, 4

Abstract

In these lectures, we will study some basic ideas behind dilation theory. We plan to cover in the three lectures,

• Sz Nagy dilation of contractions to unitaries;
• Dilations of d-tuples of contractions;
• Dilations of one parameter semigroups.

•  Speaker: Prof Somesh Bagchi, ISI Kolkata Title: Hardy-Littlewood Maximal Function Dates: March 3, 5, 8, 9

Abstract

A key concept, and historically the first of its kind, in classical analysis is the Hardy-Littlewood maximal function along with its weak estimate. The aim of these lectures is to illustrate the use and power of this maximal function in simple situations. A straight-forward application will be made to obtain the Lebesgue Differentiation Theorem. In another direction, in the context of pointwise convergence of Fourier series, the maximal function will be exploited to prove an important result, the Fatou's theorem.

•  Speaker: Prof V Muruganandam, NISER Bhubaneswar Title: Harmonic functions and all that Dates: March 10, 11, 12

Abstract

We begin with the rudiments of harmonic functions over n-dimensional Euclidean space, study elementary properties including Mean value property, Harnack's theorem. We proceed to discuss the Dirichlet problem for the interior of the unit sphere in the n-dimensional Euclidean space and the upper half-space in the n+1-dimensional Euclidean space. We emphasize group theoretical methods and will touch upon Hardy spaces, F and M Reisz theorem if time permits. As far as possible, we make the course self contained.

•  Speaker: Prof Alladi Sitaram, IISc Bangalore Title: An Introduction to Non-commutative Harmonic Analysis Dates: March 15, 16, 18, 19

Abstract

The following topics will be covered in the lectures. Fourier series and Fourier transforms. Fourier series on non-abelian compact groups and the Peter-Weyl theorem. SL (2, R) and harmonic analysis on the upper-half plane.

•  Speaker: Prof R Radha, IIT Madras Title: Time-Frequency Analysis Dates: March 22, 24, 26

Abstract

The main aim of the talks is to study "Feichtinger algebra". It can be characterized as the smallest time-frequency homogeneous Banach space of continuous functions. We also study its other characterizations and its important role in Gabor analysis.

•  Speaker: Prof Gautam Bharali, IISc Bangalore Title: Analytic continuation in several complex variables Dates: March 29, 30, April 1

Abstract

The theory of functions in several complex variables is challenging because of several unexpected phenomena that have no analogues in one variable. One such phenomenon is that there exists domains in C^n, n>1, for which EVERY holomorphic function defined on this domain analytically continues past the boundary. Demonstrating this is quite easy for domains possessing a lot of symmetry. What is quite hard, in general, is to compute the largest domain to which all holomorphic functions simultaneously extend. In the first lecture, we shall recapitulate some one-variable results and see the surprising ways in which they can be used. The second lecture will be devoted to developing some of the tools that will be needed in the third lecture. The third lecture will concentrate on some new results on the theme of analytic continuation.

#### Programme for April, 2010

All lectures are from 2:00 pm–3:15 pm, in Lecture Hall 3.

•  Speaker: Prof Ajit Iqbal Singh, ISI Delhi Title: Uniformly continuous functions and some locally compact groups Dates: April 5, 6, 8, 9

Abstract

We commence our study with a review of uniformly continuous functions on the real line and metric spaces and their geometrical interpretations and approximations. Then we investigate their role in studying the group algebra, its dual A* considered as an A-module or its second dual A** considered as an algebra. We also discuss the concept for locally compact groups and illustrate the theory for some concrete matrix groups and other examples.

•  Speaker: Prof C S Aravinda, TIFR Bangalore Title: A dynamic Borel-Cantelli Lemma Dates: April 13, 15, 16

Abstract

The main result that we discuss in these lectures is the following. Let M be a finite volume real hyperbolic $n$-manifold. Let A be the set of geodesic rays from a fixed point p of M that, for arbitrarily large times t, intersect a decreasing family of balls of radius $r_t$. Then A has full or null measure depending whether the integral $\int_0^{\infty}r_t^{n-1}dt$ diverges or converges.

•  Speaker: Prof K Parthasarathy, RIASM Chennai Title: Fourier algebra Dates: April 20, 22, 23

Abstract

After taking a quick look at the (classical) algebra of absolutely convergent Fourier series, we shall pass on to nonabelian groups. In addition to the basics on the Fourier algebra of a nonabelian group (introduced by P.Eymard), we shall try to present some of the recent developments in the area.