Derivation modules of curves and hypersurfaces, monomial curves, complete intersections and set-theoretic complete intersections, intersection theory of algebraic varieties. Schemes over symmetric monoidial categories, bivariant Chow theory. Hochschild and cyclic cohomology.
Representations of: finite groups and finite dimensional associative algebras; p-adic groups and other linear groups with entries from local rings; finitely generated discrete groups. Representations of Lie algebras: semisimple, Kac–Moody, Borcherds, and current algebras; quantum groups; algebras with triangular decomposition.
Analysis on the Heisenberg group and generalisations such as H-type groups, analysis on symmetric spaces of non-compact type and on semisimple Lie groups, spectral multipliers of Laplcians and sub-Laplacians on these spaces, integral geometry on homogeneous spaces and relations with complex analysis.
Algebraic and enumerative combinatorics, random geometric graphs, graph limits, symmetric functions, Schur polynomials, determinantal identities,
Coxeter groups, root systems, structure theory of Borcherds–Kac–Moody algebras and connections to algebraic graph theory, lattice polytopes and polyhedra.
Combinatorial manifolds, PL-manifolds, minimal triangulation of manifolds, triangulation of spheres and projective planes with few vertices, pseudomanifolds with small excess, equivelar polyhedral maps.
Manifolds of positive curvature, Einstein manifolds, conformal geometry, Teichmüller theory, Kähler geometry, complex Monge–Ampere type equations, special metrics on vector bundles, degenerations of canonical metrics and connections, applications of differential geometry in computer science.
Random matrix theory, zeroes of analytic functions. Diffusion and related topics: first passage time problems for anomalous diffusion, measure-valued diffusion, branching processes. Stochastic dynamic games, stability and control of stochastic systems, applications to manufacturing systems. Stochastic differential equations. Stochastic geometry, random geometric graphs.