Let A be a commutative Noetherian ring and P be a finitely generated projective A-module. We say that P has a unimodular element or P splits off a free summand of rank 1 if P is isomorphic to A⊕Q. If A = R[X, Y ±1] is Laurent polynomial ring and rank of P is > d = dimension of R, then it is well known that P has a unimodular element. In each case below P has a unimodular element.

The broad area of my research is commutative algebra. I also use some non-commutative algebra which is useful in my work. I work in the following four areas:

My primary interests centre around characterising L-functions by means of their analytic properties and in establishing the desired analytic properties for L-functions associated to automorphic forms. Recent work has involved the exterior square L-functions associated to GLn (over local and global fields) and primitivity results for the L-functions of cusp forms associated to GL3 in the extended Selberg class.

My current research focuses on Hilbert functions of admissible filtrations of ideals in Cohen-Macaulay rings. These rings are central objects of study in commutative algebra. Through the knowledge of their Hilbert polynomials we can obtain information about various blow-up algebras such as Rees algebras, associated graded rings and the ring itself. Information about these algebras is often useful in resolution of singularities as demonstrated in the works of Abhyankar, Hironaka, Sally, Zariski and others.

Study of special classes of operators on Hilbert spaces such as subnormal operators using Complex Analysis and Homological Algebra.

My current research area of interest is operator theory. To be more precise, my most recent works are based on dilation of operators. Dilation is a mathematical tool which largely used in different branches of Mathematics to understand the underlying object better. With no exception, dilation of operators is used to understand operators which are not normal. A brief summary of my recent works in this area are given below.

Based on a careful analysis of functional models for contractive multi-analytic operators we have established a one-to-one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi-analytic operators. This allows an effective construction and classification of all such liftings with given defects. G. Popescu’s theory of characteristic functions of completely non-coisometric row contractions is obtained as a special case satisfying a Szeg ̈o condition.

We introduced, for any set S, the concept of K-family between two Hilbert C∗-modules over two C∗-algebras, for a given completely positive definite (CPD-) kernel K over S between those C∗-algebras and obtained a factorization theorem for such K-families. If K is a CPD-kernel and E is a full Hilbert C∗-module, then any K-family which is covariant with respect to a dynamical system (G, η, E) on E, extends to a K ̃-family on the crossed product E×ηG, where K ̃ is a CPD-kernel.

The theory and application of integral equations is an important subject within applied mathematics. Integral equations are used as mathematical models for many physical situations. I am interested in a numerical solution of the Fredholm integral equation of the second kind. The integral operator T is either a linear integral operator or a nonlinear Urysohn integral operator. The eigenvalue problem associated with a linear integral operator T is of equal interest.

H. Wu constructed a new Hermitian metric that provides an interesting interpolant between the Finsler (like the Carath ́eodory and Kobayashi metrics) and the K ̈ahler invariant metrics. The Wu metric raises several intriguing questions some of which I investigated. One of them is closely related to a problem posed by Kobayashi in 1970: It is well known that if M is a Hermitian manifold with negative holomorphic curvature, then M is Kobayashi hyperbolic.

We have studied complex geometry and operator theory on polynomially convex and inhomogeneousdomains in C3 namely the tetrablock and the symmetrized tridisc which in literature are denoted by E and Γ3 respectively and are defined in the following way

I work primarily in extremal and probabilistic combinatorics. Most problems of an extremal nature involve a set (_nite) with certain constraints (which are of a combinatorial nature) and one is then interested in how large/how small such a family could be. In classical problems of extremal combinatorics, the extremal families also admit short (low complexity) descriptions.

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.