Active Areas of Mathematics Research at Stony Brook

Algebra and Representation Theory

Areas of current research are Lie groups, Lie algebras and their representations; Kac-Moody algebras and their representations; quantum groups and their representations.

Algebraic Geometry

Areas of current research include the topology of algebraic varieties, algebraic cycles, the McKay correspondence via derived categories, homological mirror symmetry.

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); real-analytic methods in one complex variable (harmonic measure, Brownian motion); theta functions and their applications to combinatorics and number theory.

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; mathematics related tro string theory and mirror symmetry.

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); quantum invariants of knots and 3-manifolds.