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: