Focus of Prof. Singhi's research has been problems in theory of designs, _nite geometries, codes, families of _nite sets, graphs and hypergraphs etc. Algebraic methods are among the main tools, used for studying some of these combinatorial problems, in his papers.

I work primarily in enumerative combinatorics and linear algebraic graph theory with a strong bias towards q-analogues. I am also interested in the enumerative aspects of posets, graphs and polytopes. In linear algebraic graph theory, I am interested in getting graph theoretic information by applying linear algebraic methods to matrices associated with graphs. In particular, I have been studying properties of various distance matrices and laplacian matrices associated to graphs.

My initial work was in Combinatorial optimization and Sperner theory. Over the last several years I have been primarily interested in explicit diagonalization and block diagonalization of several natural *-algebras of matrices occurring in combinatorics

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.