Research
Pure and applied dynamical systems. The topics include quantitative and qualitative study of attractors on smooth surfaces, dynamics of complex systems as networks of coupled dynamical systems, applications of dynamical systems and graph theory to control theory problems. Also interested in open dynamical systems (maps with holes) and its connections with number theory.
My primary research focus is the representation theory of p-adic groups. I am interested in understanding the ordinary and twisted signs associated to representations of such groups. My current research focuses on computing the sign attached to self-dual Iwahori spherical representations of the symplectic group.
Differential geometry and geometric mechanics; more specifically, principal and affine connections and jet bundle geometry. Let M be a Riemannian manifold with a Riemannian metric that is invariant under the action of a Lie group G which acts freely and properly on M. Let D be a G-invariant distribution. We study the trajectories which satisfy the geodesic equation (for the Levi-Civita connection) and whose tangent vectors lie in the given distribution. It turns out that the trajectories satisfying the constraint are actually geodesics of a certain affine connection on M. Our objective is to study geodesic spray of this connection in the presence of a principal connection on the principal bundle with M as the total space and M/G as the base. In the case when D is G-invariant and non-integrable, this will lead to reduction of a nonholonomic system with symmetry. The problem can be generalised by considering an arbitrary affine connection instead of the Riemannian connection.
My primary research interest lies in tensor products of C*-algebras and their associated order theoretic (approximation) properties like the factorisation property and weak expectation property. Recently, I have also been interested in operator algebraic properties of discrete/compact quantum groups and matrix convexity.
Broadly, differential geometry and geometric analysis. In particular, various aspects of Ricci flow and Riemannian functionals with an aim to understand rigidity of standard homogeneous spaces. This includes study of various differential operators (e.g., rough Laplacian, Lichnerowicz Laplacian, etc.) and analyzing their spectra.
Within the broader context of commutative algebra, my present research is focussed on infinite free resolutions of modules over local rings. Iterated approximations of a module by free modules give rise to a free resolution. The shape of the minimal free resolution of the residue field of a local ring is reflected in the sequence of Betti numbers (ranks of free modules in the resolution) and carries useful information about the ring. In my current research, I study the Betti sequence by considering it's generating function, viz. Poincare series and try to understand the connection between asymptotic behaviour of the Betti sequence and properties of the underlying local ring.
Harmonic Analysis on the Euclidean space and the Heisenberg group. For example, uncertainty principles; study of the Schrodinger representations of the Heisenberg group; the lattice point counting problems; mapping properties of various integral operators, such as the maximal operator, multi-linear maximal operators; and Bochner-Riesz means. Subsidiary interest is on topics in Partial Differential Equations where Harmonic Analysis plays an important role, such as Riesz potentials and various Hardy type inequalities, just to name a few.
I work with Operator Algebras, specifically, C*-algebras. My primary interest has been the interplay between C*-algebras and topology. The topological objects I use to study C*-algebras are topological groupoids equipped with Haar systems. Such groupoids can be assigned full or reduced C*-algebras which carry information about the groupoids and vice-versa. Plenty of useful C*-algebras have natural groupoids associated with them. A groupoid can be thought of as a generalised group or a generalised space. However, for us, more precisely, groupoids represent topological dynamical systems and their C*-algebras represent the C*-algebras of the dynamical systems. This ideology can be traced back to Heisenbergs study of the hydrogen atom spectrum and the matrix algebras.
I am mainly focusing on nonlinear inverse problems, in particular, p-Laplace type problems. Especially, I am working on inclusion detection, interior uniqueness for the conductivity, construction of complex geometrical optics solutions in 2D, boundary uniqueness for the higher order normal derivatives of the conductivity for weighted p-Laplace equation.
Broadly Geometry and Topology. In particular, Symplectic and Contact Topology. It is a young field which has seen rapid growth in recent years. Symplectic and Contact structures arise in various other fields such as Partial Differential Equations, Classical Mechanics, Geometric Optics, Thermodynamics. In the last decade, symplectic and contact structures have been used to understand problems in low dimensional topology, namely topology of three and four dimensional manifolds.
Differential Geometry with special emphasis on Riemannian geometry. Specifically, eigenvalues of the Laplacian on compact Riemannian manifolds with non-negative curvature, harmonic functions on complete Riemannian manifolds with non-negative Ricci curvature. Subsidiary interests in partial differential equations and ordinary differential equations.
Low-dimensional topology: Problems pertaining to the mapping class groups of surfaces. In particular, investigating the primitivity of elements in the mapping class group of a closed oriented surface, and understanding the geometric realizations of its finite subgroups. Graph theory: Analyzing the spectral and topological properties of Cayley graphs both from a theoretical, and an applied perspective.
I am primarily interested in the variation of arithmetic objects in p-adic families. Hida's construction of ordinary families was extended to the non-ordinary forms by Coleman and Mazur. Since then, several p-adic families were constructed by Buzzard, Ash and Stevens, Emerton, Urban. This provides ample scope to study variation in families.
