Lecture Courses

Inference for Stochastic Processes: Professor BLS Prakasa Rao to give the Distinguished Lecture Series from May 21, 2007. More ...

TemperleyLieb algebras, Professor V. S. Sunder, IMSc Chennai.
Abstract: The title of this series of lectures refers to the one parameter family of towers TL(\tau) = TL_n(\tau) of algebras. Their
dependence on the parameter \tau becomes most striking when one espouses the C^* point of view. We shall describe Wenzl's theorem and
the essentially unique C^* quotient (the socalled Jones quotient) of
these algebras. In the final lectures, depending on the availability of
time, we shall introduce the rudiments of planar algebras, among the
most primitive of which are the TLalgebras. The first five lectures will more or less follow the Masters' Thesis of S. Sundar.
A more detailed abstract of Professor Sunder's course is available here.
Professor Sunder is visiting the department in the last two weeks of April 2007. The exact timings of the lectures will be announced shortly.

Quillen's proof of Serre' conjecture, Professor S.M. Bhatwadekar, TIFR Mumbai.
The aim of this course would be to present Quillen's proof
(with all the details) of the celebrated Serre's conjecture which asked
whether projective modules over polynomial rings over a field are free.
The conjecture was proved independently by Quillen and Suslin in 1976.
The proof of Quillen is quite elementary and accessible to those having
a basic course in commutative algebra.
Professor Bhatwadekar will give three lectures as part of this course on the
13^{th}, 20^{th} and the 27^{th} of April 2007.
Time: 3:30  5:30 pm
Venue: Ramanujan Hall, Dept of Mathematics

Spectral stochastic methods, Professor Didier Lucor, UPMC Universite de Paris6.
The lectures will address the issue of numerical solutions of differential equations with random inputs. The source of
random inputs can include uncertainty in system parameters, boundary and initial conditions, material properties,
source and interaction terms, geometry, etc. Such types of uncertainty are ubiquitous in engineering applications
and are often modeled as random fields. Uncertainty quantification requires the propagation of uncertainty through
the model and affects all stages of the numerical simulation. Nonstatistical stochastic approaches are available to
efficiently treat stochastic partial differential equations. Spanos & Ghanem (1989) pioneered the computational use
of the Polynomial Chaos (PC) expansion method, which is based on the homogeneous chaos theory of Wiener
(1938) and is well suited to solving stochastic differential equations. The PC representation is a spectral
decomposition of a secondorder random process in terms of orthogonal basis functions. The spatial and temporal
evolutions of the basis coefficients provide quantitative estimates of the modeled random process solution. The
recently developed generalized Polynomial Chaos (gPC) method has the advantage that gaussian and non
gaussian random processes can be optimally represented. The efficiency of this approach depends crucially on the
judicious choice of coordinates in probability space. PCbased methods have been applied to different flow problems
such as porous media flows, thermal problems and combustion. However, fewer studies exist that deal with full
stochastic incompressible NavierStokes equations. The mathematical framework and derivation of the key steps of
the gPC method will be presented. In particular, the intrusive and nonintrusive numerical approaches will be
distinguished. Then, the method will be adapted for the treatment of the stochastic NavierStokes equations. Finally,
applications to the case of stochastic flowstructure interactions with random inflow boundary conditions and the
sensitivity study of LargeEddy Simulations to parametric uncertainty in the subgridscale model will also be
presented in detail.
The lecture slots are as follows:
Wednesday, March 28, 2007 Time : 5:00 pm
Monday, April 2, 2007 Time : 5:00 pm
Tuesday, April 3, 2007 Time : 5:00 pm
Wednesday, April 4, 2007 Time : 5:15 pm
Venue: Ramanujan Hall, Dept of Mathematics

Commutative Algebra Seminars, Professor Tony J. Puthenpurakal, Indian Institute of Technology Bombay.

An introduction to Analytic Number Theory, Professor D. Suryaramana, HarishChandra Research Institute, Allahabad.
Professor D. Suryaramana of the HarishChandra Research Institute of
Allahabad will be visiting the Department of Mathematics, IIT Bombay from
January 23 to February 1. During his stay he will give a short course (6
to 8 hours) of lectures titled "An introduction to analytic number theory"
aimed at research scholars and advanced undergraduate students. The course
will assume very little  only a small amount of complex analysis (say, MA
204) will be required as a prerequisite. The main topics will centre
around the distribution of prime numbers and the zeros of the Riemann zeta
function.
An organisational meeting will be held on Wednesday, January 24 at 12:35
p.m. in Room 114 of the mathematics department to decide on when and where
the lectures are to be held.
Professor Surya Ramana is visiting our department as an extension of our
activities for the Special Year in elliptic curves, automorphic forms and
Lfunctions.

Generalisations of the Riemann Hypothesis and their analogues over finite fields, Professor Jasbir S. Chahal, Brigham Young University, Provo, Utah, USA.
This course will discuss the Riemann Hypothesis and its generalisations (the Grand Riemann Hypothesis or GRH) and their connection to the distribution of prime numbers. We will attempt to explain why the Riemann Hypothesis is perhaps the most central unresolved question in number theory today (it is one of Millenium Prize Problems of the Clay Foundation) with profound consequences for many different areas of mathematics. The course will also discuss analogues of GRH for curves over finite fields. In this latter context, GRH (also known as the Weil Conjectures) has actually been established by P. Deligne around 1970. The finite field analogues also have profound consequences for number theory, notably in the resolution of the celebrated Ramanujan Conjecture. Hopefully the students will see how this problem brings algebra, number theory, analysis and topology together.
The course is aimed at Research Scholars in the Department of Mathematics (who are strongly urged to attend!) and should not require more background than training at the Master's level. The initial part of the course will be accessible to any student with a basic course in complex analysis (for instance, MA 204).
The first (organisational) meeting for the course will be held on Thursday, January 11, 2007 at 4:00 p.m. in the Ramanujan Hall, Department of Mathematics, IIT Bombay.

Topology, Professor K. Varadarajan, University of Calgary, Canada.
Professor K. Varadarajan will be giving a lecture course on topology twice a week. The first meeting of the course will be on Friday, November 3 at 11:30 a.m. in Room 114 of the Department of Mathematics. Timings for subsequent lectures will be decided at that meeting.
The topics to be covered will be decided in consultation with the audience. Possible topics include vector bundles, the homotopy classification theorem, Kgroups and associated cohomology theory, homotopy groups of some classical groups and Complex Bott periodicity.
Professor Varadarajan is visiting the Department of Mathematics for a period of six weeks.
