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.

My research is in algebraic topology and in particular stable homotopy theory. Algebraic topology is the study of topological space through algebraic invariants. The stable homotopy category is a tensor triangulated category where all these classical invariants are representable. Recent developments in model categories and higher category theory has provided the language to carry out a program of doing algebra and algebraic geometry in the stable category. Some topics that specifically interest me are chromatic homotopy theory, higher K-theories, topological modular forms, infinity categories, higher descent, algebraic K-theory, derived algebraic geometry and motives.

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.

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.

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.

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.

My research interests lie in group action on surfaces and studying related algebraic properties. More specifically, for a finite group G, a genus spectrum sp(G) of G is a collection of non-negative integers g such that there is a Riemann Surface X with genus g, on which G acts faithfully via orientation preserving diffeomorphisms. My primal research focus is to give a description of the map that sends G to sp(G), which is known as Hurwitz Problem. Computing sp(G) for interesting infinite class of finite groups is a challenge. This is known for a few classes of finite p-groups and in general, partial results are known for even finite cyclic groups. Using the computing aid by GAP, these are also known for quite a few finite simple groups. Based on these results, it is known that the map is not injective in general. However it is yet to be understood if there are only finitely many finite groups that share a spectrum set. Secondly, towards computing the spectrums certain algebraic class of finite p-groups called Gorenstein-Kulkarni groups became important especially for p=2. A part of my research goal is towards an algebraic classification of these.

Number Theory, with special focus on modular forms, Jacobi forms and Siegel modular forms. Specifically, Fourier coefficients and Dirichlet series associated to them. Also interested in studying certain special correspondences among various kinds of modular forms, and their applications.

My research interests lie in the general area of Harmonic analysis with a particular emphasis on harmonic analysis on Euclidean spaces. My current research is devoted to study the following topics

- Bilinear multiplier operators and Littlewood-Paley square functions
- Multi-frequency Calderon-Zygmund operators with applications to Bochner-Riesz means
- Weighted estimates for one-sided Hardy-Littlewood maximal functions.

Algebraic geometry, more specifically Brill-Noether theory, Vector bundles, Higgs Bundles and abelian variety. I am also interested in Coding Theory and Cryptography and right now understanding applications of algebraic geometry in these areas.

Operator Algebras and their K-theory: I am primarily interested in the classification program for C* algebras. Specifically, I am trying to understand the equivariant E-theory of continuous fields of C* algebras, and UCT-type results that help compute the E-theory group using K-theoretic invariants. I am also interested in non-commutative notions of dimension, and its use in determining the CuntzSemigroups and related structures for various C* algebras. Mapping Class Groups of Surfaces: I have also recently begun learning about Mapping Class groups of surfaces, and the problem of determining if an element of the mapping class group is primitive or not.

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