1. Algebraic Geometry
We work on problems arising from the study of moduli spaces (stacks) of geometric objects like vector bundles, principal bundles, Higgs bundles etc on algebraic varieties.
Techniques employed to study these problems include those from deformation theory, homological algebra, Geometric invariant theory etc.
Dr. Rohith Varma
2. Information Theory, Statistics, and Probability
In statistics, our broad research area is ‘inference based on distance functionals’. We study projection theorems of distance functionals and estimating equations that arise in robust inference problems. This study has helped us solve the underlying estimation problems and discover new statistical models that are of power-law in nature. The other area of research in statistics is ‘information geometry’, where we study the geometry of statistical models with respect to distance functionals. These geometric properties many a times enable us to discover new results concerning the underlying estimation problems and also help solve them.
In information theory, we are interested in physical (information-theoretic) problems where optimal solutions are obtained in terms of some information measure such as Shannon entropy, Renyi entropy, etc.
Dr. Ashok Kumar M.
3. Functional Analysis
One of the active topics of research in functional analysis is the geometry of Banach spaces. Within the general framework of Banach spaces, we focus on the theory of best approximation, which is applicable in a variety of problems arising in nonlinear functional analysis. For instance, in fixed point theory, Gaussian quadrature etc. The problem of best approximation amounts to the minimization of a distance, which permits us to use geometric intuitions. The main techniques employed in this study are intersection properties of balls and techniques from M-structure theory. We use these properties to study existence of projections of norm one and to understand the structure of real Banach spaces.
- Prof. S. H. Kulkarni
- Dr. C. R. Jayanarayanan
4. Several Complex Variables
Research in Multidimensional Complex Analysis / Several Complex Variables (SCV), is centered around ‘mapping problems’; this is a subfield of central importance in SCV, with an obvious question which is still unresolved, namely, the problem of determining whether a given pair of domains are biholomorphically equivalent are not. This has both, given rise to several branches of research within SCV as well as to interactions with other areas of mathematics. Techniques of particular importance in tackling such ‘mapping problems’ are those of biholomorphically invariant metrics and distances, which are of interest here as well.
Dr. G. P. Balakumar
5. Partial Differential Equations
Current research in partial differential equations focuses on the existence, uniqueness and multiplicity of solutions to various semilinear elliptic boundary value problems. We also study qualitative properties of these solutions such as positivity and regularity. Some of these works are motivated by certain models used in spatial ecology. The other major area of interest is the eigenvalue problems for the p-Laplace operator. There is a vast trove of open problems in this area, starting with the fundamental challenge of finding a complete characterization of the eigenvalues. The standard approaches to studying these problems employ tools from topological and variational methods, such as degree theory, monotone methods, bifurcation theory and minimax methods.
- Dr. Lakshmi Sankar
- Sarath Sasi