ACTIVE AREAS OF RESEARCH AT STONY BROOK


Algebra and Representation Theory: Areas of current research are complex Algebraic geometry; Lie groups, Lie algebras and their representations; Kac-Moody algebras and their representations.
Complex Analysis: Areas of current research include Riemann surfaces (Kleinian groups, Teichmuller theory, relations with 3-dimensional topology); complex manifold theory (emphasisizing links with Riemannian geometry, symplectic topology, and algebraic geometry); CR manifolds (cohomology; pseudoconvavity/convexity); and real-analytic methods in one complex variable (harmonic measure, Brownian motion).
Differential Geometry: Areas of current research include comparison geometry; Gromov-Hausdorff convergence; minimal submanifolds and geometric measure theory; Einstein manifolds; Kaehler geometry; manifolds of special holonomy; geometry and topology of low-dimensional manifolds; spin geometry; twistor theory.
Dynamics: Areas of current research include Julia and Mandelbrot sets for polynomial maps in one and several complex variables; Tecihmuller theory and Kleinian groups.
Mathematical Physics: Areas of current research are integrable systems, conformal field theories, and gauge theories.
Partial Differential Equations: Areas of current research include harmonic analysis; several complex variables; non-linear elliptic systems; integral equations; complexes of partial differential equations; tangential Cauchy-Riemann operators; conservation laws; continuum mechanics.
Topology: Areas of current research include symplectic topology; high-dimensional manifolds (surgery theory, topological rigidity); topology of complex projective varieties; 4-manifolds (Seiberg-Witten theory); 3-manifolds (hyperbolic 3-manifolds, geometrization conjecture).