# 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.

Areas of current research include applications of Hodge theory to several areas of algebraic geometry; intersection homology and the topology of algebraic maps; theta functions, modular forms and their applications to moduli spaces; period mappings, GIT and compactifications of moduli spaces; D-modules, derived categories and the geometry of irregular varieties; rational curves on varieties and rational connectedness; the arithmetic and geometry of varieties over function fields; hyper-Kaehler manifolds; linear series on higher-dimensional varieties and multiplier ideals; geometric questions in commutative algebra.

**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.