Often rapid and precise identification of composition of raw and finished products (on and off the site) are vital in several industries like mining, pharmaceutical, petrochemical, agriculture and food processing, etc. Our laboratory offers solutions for in situ analysis of materials in such challenging environments using reflectance, emission (optical) and X-ray fluorescence spectroscopic techniques.

Black hole is a region exhibiting such strong gravitational pull that nothing, including particles and light can escape from it. It is formed when the inward gravitational collapse of a massive star cannot be halted. It grows by capturing matter, energy, or by merging with other black holes.

Event horizon of black hole: Surface around black hole through which, matter and light can only pass inward. Therefore, if event occurs within this boundary, then the information from the event cannot reach an outside observer; due to which, it is impossible to determine if the event occurred.

Spin transfer torque magnetic random access memory (STT-MRAM) is a new memory technology based on electron spin. MRAM promises to bring non- volatile, low-power, low cost and high speed memory and has the potential to replace FLASH, DRAM and even hard-discs. The basic building cell of the STT-MRAM is the magnetic tunnel junction (MTJ) device.

Newton’s laws of motion are based on calculus. Einstein unified gravity with Special Theory of Relativity using Riemannian geometry. We see that mathematics provides a language by means of which physical laws can be formulated to explain nature. Einstein’s theory of General Relativity inspired developments in the area of Differential Geometry. This interplay of mathematics applied to physics and physics inspiring development in mathematics has intensified in the last five decades.

Systems of partial differential equations (PDEs) are usually solved by nu- merical methods. As opposed to this standard practice, in this project, we have been building tools that are capable of solving PDEs symbolically with the help of algebraic computations. The main motivation behind this endeav- our is the quest for exact solutions of PDEs.

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