Lecture Courses

Inference for Stochastic Processes: Professor BLS Prakasa Rao to give the Distinguished Lecture Series from May 21, 2007.      More ...
Temperley-Lieb 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 so-called 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 TL-algebras. 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 13th, 20th and the 27th of April 2007.
    Time: 3:30 - 5:30 pm
    Venue: Ramanujan Hall, Dept of Mathematics
Spectral stochastic methods, Professor Didier Lucor, UPMC Universite de Paris-6.
    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. Non-statistical 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 second-order 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. PC-based 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 Navier-Stokes equations. The mathematical framework and derivation of the key steps of the gPC method will be presented. In particular, the intrusive and non-intrusive numerical approaches will be distinguished. Then, the method will be adapted for the treatment of the stochastic Navier-Stokes equations. Finally, applications to the case of stochastic flow-structure interactions with random inflow boundary conditions and the sensitivity study of Large-Eddy Simulations to parametric uncertainty in the subgrid-scale 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, Harish-Chandra Research Institute, Allahabad.
    Professor D. Suryaramana of the Harish-Chandra 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 L-functions.
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, K-groups 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.