Department of Mathematics

Abstract:The Narasimhan-Seshadri Theorem is one of the spectacular theorems from India in the past 50 years or so. The theorem is more than a deep result but is in a way a philosophy or correspondences and symmetries. The theorem has had an impact on several aspects of mathematics. The theorem has also led to developments along lines that are similar but by themselves are also deep and central. Since the talk is for a general audience, I plan to give an overview of the theorem, a few of its big impacts on topology and geometry, and a few of its ramifications in terms of generalizations.

Abstract:On August 8, 1900, David Hilbert delivered his famous lecture about 23 open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced the decision of a recently formed Clay Mathematical Institute (CMI) to announce the seven Millennium Prize Problems in the CMI Millennium Meeting held on May 24, 2000. One such problem is the theme of the present talk. Now it is widely accepted that the motion of an incompressible viscous fluid with moderate velocity is described by the Navier-Stokes Equations. Although these equations were written down in the 19th century, the existing mathematical results are not adequate to unfold the secrets hidden in the Navier-Stokes equations. In this talk, I shall concentrate on:

• a brief description of this problem
• mathematical model
• a quick look at history
• what is known at this point
• some important approaches
• What is possibly needed

Finally, I conclude the talk with a note on the present state of Indian applied mathematics and whether we are ready to contribute towards this millennium problem.

Speaker: Prof. Ratnasingham Shivaji, University of North Carolina Greensboro, US

Abstract:

Since long, oceans have been serving mankind in various ways and of course, at the same time the waves have also shown their fury at times by claiming lives and creating huge destruction. In order to understand the oceans and the strength they display through waves, it becomes very pertinent to study ocean waves, their behaviour and effects with respect to wave propagation, scattering, damping, trapping etc., which influence a number of issues in connection with marine structures and coastal regions. The comfortable situation to work with ocean waves is by considering simple conditions thereby neglecting many important aspects such as the porosity of the structures that are installed in ocean for various activities, porous and elastic effects of the sea-bed and also the unevenness of the sea-bed. However, the real situation demands that some or all of these be taken into account while modelling a problem so that the problem is formulated for a more realistic scenario.

In order to reduce the wave impact on marine structures (i.e., to dissipate the wave effects), it is essential that the structures (such as cylinders, barriers, caissons) possess in them the ability to reflect the wave and help in attenuating the wave energy. This brings into fore the utility of porous structures which can be used as breakwaters in the coastal and offshore regions for various applications. Further, it is practically impossible to find sea-beds which are flat and do not possess any porosity or elasticity. In other words, a more realistic formulation can be carried out by considering structures possessing porosity, and sea-beds being uneven and possessing porosity and/or elasticity.

The objective of this talk is to discuss scattering of water waves by various types of porous structures placed on flat or uneven sea-bed which may possess porosity and/or elasticity. The study focuses on how reflection varies when different important parameters are changed. The observations made from this study will allow one to design structures which will be effective enough to reduce wave impact on the structures. In this talk, some results will be displayed which portray the physical scenario. An attempt will be made to present some practical problems which take into account a number of important properties and parameters so that the results become practical to be followed by relevant people for various activities with regard to protecting certain regions and structures from harsh ocean wave action. It will mainly emphasize the significant role that porosity and elasticity play with regard to ocean wave propagation and various related issues. The emphasis will be on the modelling of the problems which will establish the essence of mathematics. A brief and friendly introduction to water wave propagation will precede the main components of the talk to give some general idea to the audience.

Abstract:

Navier-Stokes and Euler equations play an important role in studying the flow of incompressible fluids. Weak solutions to these equations can be obtained by Galerkin method but the uniqueness is a big open problem. It is a big challenge to obtain an extra condition for the class of functions, so that in this class obtain the existence and uniqueness. In order to understand this phenomenon, it is better to look at a one-dimensional case where the equation turns out to be viscous Burger’s equation or Burger’s equation with non-linearity is of quadratic order. In this talk, we will restrict to Burger’s type equations called the scalar conservation laws in one space dimension with strict convex flux. Way back in the 50’s, this equation was studied by Lax and Olenik and obtained an explicit formula for a solution. Olenik showed that this satisfies an extraction called the “entropy condition: and then showed that in this class the solution is unique. Later Kruzkov, in an ingenious way, generalized this to obtain a unique solution for scalar conservation law in higher dimensions and Lipschitz fluxes.

This result was taken up by Zuazua and his collaborators who studied the Optimal Controllability for Burgers equation. They showed the existence of optimal control and to capture this, they derived a numerical algorithm whose convergence is still open. In a different direction, this was attacked, and the problem was completely solved. Getting the optimal solution is via projection method in a Hilbert space. Recently, this was extended in a non-trivial way to conservation laws with convex discontinuous flux. In this talk, I will explain the main ideas of this work.

Abstract:Groups are encountered early on in one’s education. Many groups that one learns already in undergraduate studies are in fact matrix groups with coefficients which can be numbers from finite fields to real and complex numbers or these days p-adic numbers. Such groups, called finite groups of Lie type or real Lie groups or p-adic groups as the case may be, have been playing a dominant role in many parts of mathematics. The lecture will be a gentle introduction to some of these topics.