Algebra, Algebraic Groups, Algebraic Geometry, Analytic Number Theory, Arithmetic Geometry, Moonshine, Modular forms, Vertex Algebra

Differential geometry, applied analysis, partial differential equations, complex analysis, geometric function theory, spectral theory, and mathematical physics.

Probabilistic combinatorics, structural and spectral graph theory, Ramsey theory, hypergraphs, ordered combinatorial and algebraic structures, theory of computation, circuit complexity, geometric algorithms, approximation algorithms

Development of methods for labeling anatomic and micro-anatomic structures found in images, systems software and techniques for management and analysis of very large image data, and techniques and software tools for sharing this labeling information in a computational grid architecture. Computational methods for large-scale image processing, in particular, image deblurring, image reconstruction, image registration.

Numerical Linear Algebra, Numerical Optimization, Numerical Partial Differential Equations, Numerical Methods for Inverse Problems, Image Processing, Geophysics, Computational Fluid Dynamics, Numerical Methods for Complex Networks, Parallel and Distributed Computing.