Order structure in normed spaces and operator spaces (matricially normed spaces)
Theory of operator ideals (Geometry of Banach Spaces).
Incidence Geometry
Supercongruences: 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.
Differential Operators: 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.
Convolution sums and applications: 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.
Representations of numbers by quadartic forms: We find number of representations of an integer by certain quadratic forms by computing modular forms/ theta series.
Moduli of Curves, Teichmuller Theory
Symmetric ciphers, Algebraic Attack, Boolean Functions, Combinatorics.
Modular forms, L-functions.
Transcendental number theory, Modular forms and Multiple zeta values
Combinatorics
Algebraic Graph Theory
Discrete Mathematics
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 user research interest is on the application of probabilistic tools to analyze these stochastic models.
My Primary research areas are Operator algebra ( 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
Representation Theory
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.
