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, Algebraic Combinatorics
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.