The goal of a Theoretical Computer Scientist is to understand the power of computation: can computers perform the tasks one is interested in? Can they do so efficiently, with constraints on resources such as time, space, non-determinism, parallelism, randomness, etc.? The "right" constraint might depend on the application at hand: algorithmists often want linear-time algorithms for their problems; logicians are sometimes satis_ed to prove that their algorithms halt in _nite time; complexity theorists of different flavours look at various notions of efficiency.

I have made major contributions to the study of the dynamical behavior of flows on homogeneous spaces of Lie groups and applications of the theory to problems in Diophantine approximation. One of the problems concerns the set of values of quadratic forms, evaluated at points with integer coordinates. Conditions for the set to be dense, and estimates for the number of solutions with values in prescribed intervals, have now been understood, through the work of various authors including myself, in the case when the number of variables is at least 3.

In a current project with Ritwik Mukherjee and Martijn Kool, we attempt to prove a variant of the Gottsche conjecture by looking at the number of δ-nodal degree d planar curves in P 3 intersecting the right number of generic lines. We wish to show that for d ≥ δ this number is given by a universal polynomial of some determined degree involving d. We attempt to do this by performing a relative version over the Grassmannian Gr(3, 4) ∼ = P 3∗ of the computations of Kool-Thomas-Shende in their proof of the Gottsche conjecture.

Joyal’s species constitute a good framework for the study of algebraic structures associated to com-binatorial objects. In joint work with Marcelo Aguiar, we studied the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. We constructed examples from combinatorial and geometric data inspired by ideas of Rota and Tits’ theory of Coxeter complexes.

My research interests lie in stable homotopy theory and topological combinatorics. Computing good lower bounds for chromatic number of graphs is an important problem in combinatorics. Kozlov, Bab-son, Dochtermann and several others have proved results in this context which hinge on understanding the connectivity of certain simplicial complexes called Hom complexes. Several motions from homotopy theory of topological spaces like homotopy equivalances, etc.  have analogues in the category of graphs.

The monodromy groups of the n-th order hypergeometric differential equations are characterized as the subgroups of the general linear group GL(n), which are generated by the companion matrices of two monic co-prime polynomials f (x) and g(x) of degree n.

My research interest lies in the field of linear and nonlinear wave propagation. More specifically, my interest is in the study of partial differential equations of hyperbolic type. Most of the systems of PDEs (of practical interest) are not solvable exactly to get closed form solutions and therefore an approximation procedure needs to be adopted. Also, since the solution spaces of these systems are often vast (for instance the space of BV functions in the case of hyperbolic system of conservation laws), any qualitative analysis becomes more challenging.

"Plates are plane structural elements with a small thickness compared to the planar dimensions. De-pending on the length to thickness ratios, plates are classified into moderately thick, thin and very thin plates. One of my primary research interests has been to employ finite element methods to determine approximations to deformation and stresses in thin and very thin plates when they are subject to loads. A Kirchoff model is used for the thin elastic plates and this leads to fourth order elliptic equations with the transverse displacement as the unknown variable.

(1) Asymptotic analysis of PDEs: Most of the physical phenomena are modeled by Hyperbolic PDEs and in many of them feature multiple characteristics. Roughly it corresponds to studying crossings of eigenvalues of operators parametrized by a vectorial parameter. We are studying some of such situations.

(2) Conservation laws: We are studying the question of convergence generalized viscous approximations to entropy solutions of hyperbolic conservation laws.

Extension of Recent Contributions for Wider Applications in Computational Biology/Drug Discovery/Immunoinformatics/Automated Pulse-Diagnostics

(1) ParaDes: AI Software for Epitope-Paratope Designing (Copyright: GoI No. SW-698/2002) deploying knowledge-based correlation mapping on Hopfield Network.

(2) Efficient ab-initio Protein Structure Prediction using AI & Nonparametric Statistics.[Web-server: www.math.iitb.ac.in/~epropainor/ ]

My research area is Reliability Theory. Currently, I have been working on research problems that deal with Multi-state coherent systems. These systems are such that their components and systems themselves can be in one of (M + 1) possible states 0, 1, 2, • • • , M at any given time, where the extreme states 0 and M represent completely failed and completely working states respectively, and others are intermediate states that decrease in performance level as the system or a component makes a transition from state i to (i − 1), i = M, (M − 1), • • • , 1.

Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables Y and X equals the marginal measure of association. We have discussed the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure.