राष्ट्रीय विज्ञान शिक्षा एवंअनुसंधान संस्थान

National Institute of Science Education and Research

**Order structure in normed spaces and operator spaces (matricially normed spaces);
Theory of operator ideals (Geometry of Banach Spaces).**

Partial differential equations of evolution type, Inverse problems

Incidence Geometry

The numbers which occur in Ap\'{e}ry's proof of the irrationality of zeta(2) and zeta(3) have many interesting congruence properties.Work started with F. Beukers and D. Zagier, then extended by G. Almkvist, W. Zudilin and S. Cooper recently has complemented the Ap\'{e}ry numbers with set of sequences know as Ap\'{e}ry-like numbers which share many of the remarkable properties of the Ap\'{e}ry numbers. We study supercongruences properties of Ap\'{e}ry-like numbers.*Supercongruences:*

There are many interesting connections between differential operates and modular forms. Using Rankin-Cohen type differential operators on Jacobi forms/ Siegel modular forms we study certain arithmetic of Fourier coefficients.*Differential Operators:*

We compute convolution sums of divisor function using the theory of modular forms and quasi modular forms and apply those to find number of representations of an integer by certain quadratic forms.*Convolution sums and applications:*

**Representations of numbers by quadartic forms:**We

Moduli of Curves, Teichmuller Theory

**Specialisation: **Theoretical Computer Sciences, Coding Theory, Cryptology, Discrete Mathematics.

**Present Research Interests: **Symmetric ciphers, Algebraic Attack, Boolean Functions, Combinatorics.

Cowen-Douglas Class of operators, Hilbert modules over function algebra and Dilation theory.

Modular forms, L-functions

Transcendental number theory, Modular forms and Multiple zeta values

Algebraic Graph Theory, Discrete Mathematics

Intersecting Families, Hypergraphs, Graph Theory, Applications of Szemer\'{e}di Regularity Lemma, Applications of analysis and probability theory to combinatorics, Probabilistic methods in combinatorics, Random graphs.

Moduli space of bundles over curves and surfaces, Linear Series.

Partial Differential Equations

Disordered systems pops up quite often in physics (spin glass), biology (artificial neural network), social sciences (matching) and many other places. To analyze, usually these systems are identified with the stochastic models. My main research interest is on the application of probabilistic tools to analyze these stochastic models.

My Primary research areas are functional analysis and Ergodic Theory. My central research area is operator algebra and it evolves around operator algebra. I study one parameter family of endomorphisms on von Neumann algebras.

I also study structure theory of von Neumann algebras, Connes's classifications theory of type III factors and various others property of type III factors and verious dynamical system on von Neumann algebras.

Harmonic Analysis

Enumerative geometry of singular curves, using methods from Differential Topology.

I work on Harmonic Analysis on Euclidean Spaces and Heisenberg Groups.

At present my research interest is Spherical harmonics, Hermite and Laguere expansion and Dunkl Transform.

Probabilistic and Analytic Number Theory.

Schur multiplier and non-abelian tensor product of groups, Central extensions of groups, Projective representations of finitely generated groups, Structures of Twisted group rings.

Quantum Groups, Quantum Symmetry, Noncommutative Geometry, Operator Algebras.

**Finite group theory:**Classification of finite p-groups, Influence of conjugacy class sizes on the structure of finite groups, Automorphisms of finite groups, Word maps defined on groups, Graphs associated with groups.-
**Combinatorial group theory / Low dimensional topology:**Coxeter groups and Artin groups, Braid groups, Planar braid groups, their virtual analogous and pure subgroups. **(Nilpotent) Lie algebra:**Classification of finite dimensional nilpotent Lie algebras, Camina Lie algebras, Breadth of a Lie algebra.