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Current ResearchResearch Areas

The Department aims to promote close collaboration between various mathematical disciplines and with other applied areas. The areas of current research are :

Algebraic and Combinatorial Topology
Combinatorial manifolds, PL-manifolds, minimal triangulation of manifolds, triangulation of spheres and projective planes with few vertices, pseudomanifolds With small excess, equivelar polyhectral maps.

Commutative Algebra and Algebraic Geometry
Study of derivation modules of curves and hypersurfaces, connection with Zariski - Lipman conjecture, monomial curves, complete intersections and set theoretic complete intersections, intersection theory of algebraic varieties, minimal number of generators for ideals and modules. Study of certain algebraic surfaces.

Differential Equations
Homogenization of partial differential equations, controllability, viscosity solutions.

Differential Geometry
Manifolds of positive curvature (Ricci, scalar and isotropic), Einstein manifolds, conformal geometry (Weyl curvature and the Yamabe invariant), Gromov hyperbolic spaces.

Finite fields and Coding Theory
Classification of permutation polynomials, study of PAPR of families of codes, construction of codes with low PAPR.

Functional Analysis and Operator Theory
Hilbert modules, multivariable operator theory, indefinite inner product spaces.

Harmonic Analysis
Analysis on the Heisenberg group and generalisations such as H-type groups, analysis on symmetric spaces of non compact type and on semisimple Lie groups, spectral multipliers of Laplcians and sub-Laplacians on these spaces, integral geometry on homogeneous spaces and relations with complex analysis.

Low Dimensional Topology
Topology of three-manifolds and smooth four-manifolds, Geometric group theory, Heegaard Floer theory and its relations to geomtric topology.

Nonlinear Waves, Hyperbolic Equations and Numerical Analysis
Physical phenomena associated with a class of nonlinear waves governed by a hyperbolic systems of quasilinear partial differential equations and hyperbolic conservation laws. Application of ray methods to study successive positions of a curved wave front and a shock front. Relation between kinematical conservation laws and level set theory. Theoretical (i.e., study of approximate equations), numerical (i.e., computation of solutions with discontinuties) and applied (sonic boom, extension of Fermat's principle) problems.

Nonlinear Dynamics
Couded dynamical systems, Synchronization, Turing patterns, applications of Lie algebraic methods to nonlinear Hamiltonian systems, fractal dimensional analysis, generalized replicator dynamics.

Probability and Stochastic Processes
Stability and control of stochastic systems, diffusion and related topics, stochastic dynamic games, applications to manufacturing systems, first passage time problems for anomalous diffusion, long memory processes, branching particle systems, stochastic differential equations, queuing theory.

Several Complex Variables
Holomorphic mappings, convexity and its applications to function theory, finite-type domains, pluripotential theory, and complex dynamical systems.

Time Series Analysis
Application of time series analysis techniques to neuroscience esp. brain-machine interface, applications to geophysics.

 


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