__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-Käehler manifolds; linear series on higher-dimensional varieties and multiplier ideals; geometric questions in commutative algebra.

**Analysis**

Probability; Analytic Number Theory; Geometric Function Theory; Geometric Measure Theory; Harmonic Analysis; Computational Geometry; Metric Geometry; Quasiconformal and Quasisymmetric Mappings; Several Complex Variables; and Partial Differential Equations related to fluids.

**Complex Analysis**

Areas of current research include Riemann surfaces (Kleinian groups, Teichmüller theory, relations with 3-dimensional topology); complex manifold theory (emphasizing 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; Conformal Mappings (including algorithmic aspects).

**Differential Geometry**

Areas of current research include comparison geometry; Gromov-Hausdorff convergence; minimal submanifolds and geometric measure theory; Einstein manifolds; Kähler geometry; manifolds of special holonomy; geometry and topology of low-dimensional manifolds; spin geometry; twistor theory.

Areas of current research include Julia and Mandelbrot sets for polynomial maps in one and several complex variables; Tecihmüller theory and Kleinian groups; Transcendental Dynamics (iteration of entire functions); Renormalization theory.

**Mathematical Physics**

Areas of current research are integrable systems, conformal field theories, and gauge theories; mathematics related to string theory and mirror symmetry.

**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.