A short decription of my ongoing research is here. This document describes all of my research activities in more detail. Portions of this document are excerpted from a successful proposal for a CAREER award from the National Science Foundation.
The following is a broad classifications of the various areas that I have worked on. They are not mutually exclusive. Rather, most problems that I have worked on have aspects from more than one of these areas.
Multiple scale analysis
Analysis of Nonlinear PDEs
Collective behavior in systems with
many degrees of freedom
Geometry and Topology
Applications of dynamical systems
Stochastic dynamics
A common theme for all of my ongoing research is the analysis of systems that display multiple scale behavior. The problems in this category include
Nonconvex variational problems
Singularities and micro-structure in thin elastic
sheets
Coarsening and defect dynamics in
extended systems
Nonequilibrium phase transitions
Blowup in nonlinear parabolic PDE
``Homogenization'' and the dynamics of
micro-structure
Nonconvex variational problems
Singularities and micro-structure in thin elastic
sheets
Blowup in nonlinear parabolic PDE
Fekete Points: Distribution of charges on a 2-D conductor
Coarsening and defect dynamics in extended systems
Nonequilibrium phase transitions
Singularities and micro-structure in thin elastic
sheets
Sound propagation in crumpled sheets
Memories in Charge density wave conductors
Crumpling in more than 3 dimensions
Front propagation in systems with
coexisting ordered and chaotic behaviors
Patterns in vibrated granular layers
Underdetermined PDEs and convex integration
Isometric Immersions
Event horizons of black holes
Anomalous diffusion in planetary type flows
Dynamical systems with invariant manifolds
Nonequilibrium phase transitions
Anomalous diffusion in planetary type flows
Dynamical systems with invariant manifolds