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
 the existence and uniqueness of best approximation of a given
point x in X;
 some properties of the metric projection supported on a nonempty
set A;
 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 wellknown 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:
 Catalan numbers ;
 Noncommutative probability spaces  free independence and free
cumulants ;
 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 dtuples of contractions;
 Dilations of one parameter semigroups.

Speaker: 
Prof Somesh Bagchi, ISI Kolkata 
Title: 
HardyLittlewood Maximal Function 
Dates: 
March 3, 5, 8, 9 
Abstract
A key concept, and historically the first of its kind, in classical
analysis is the HardyLittlewood 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 straightforward
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 ndimensional
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 ndimensional
Euclidean space and the upper halfspace in the n+1dimensional
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 Noncommutative 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 nonabelian
compact groups and the PeterWeyl theorem. SL (2, R) and harmonic
analysis on the upperhalf plane.

Speaker: 
Prof R Radha, IIT Madras 
Title: 
TimeFrequency 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 timefrequency 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 onevariable 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 Amodule 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 BorelCantelli 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
$\int_0^{\infty}r_t^{n1}dt$ 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.
