A natural and important question representation theory is to understand how a representation of group restricts to a given subgroup. It was realized, almost three decades ago, that closely related questions in the context of p-adic and adelic groups have significant number theoretic aspects. In the case of p-adic group, one is interesting in finding out which representation of group admit non-trivial linear forms that are invariant under the fixed group of an involution.

Following results were proved: 1. For any non-trivial action of (C, +) on the polynomial ring C[X,Y,Z] the ring of invariants C[X,Y,Z]Ga is isomorphic to a polynomial ring in two variables.

2. (Jointly with Avinash Sathaye) Let (C,+) act regularly and non-trivially on a smooth affine 3-fold X with trivial integral homology, Then the quotient X/(C,+) is smooth. The proof of (1) is much more accessible than the original proof of M. Miyanishi.

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