The Department aims to promote close collaboration between various mathematical disciplines and with other applied areas. The areas of current research are:
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.
Automorphic forms and representations, Galois representations, L-functions, Iwasawa theory, Arithmetic Geometry, Langlands programme, Analytic Number theory, Algebraic K-theory.
Homogenization of partial differential equations, controllability, viscosity solutions. Numerical methods for partial differential equations.
Hilbert modules, multivariable operator theory, complex geometry.
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.
Positive matrices and functions operating on them, positivity preservation, structured matrices.
Holomorphic mappings, holomorphic interpolation, Ohsawa-Takegoshi type extension theorems. Invariant metrics: estimates, metric geometry of hyperbolic domains. Convexity, finite-type domains. Pluripotential theory and holomorphic dynamical systems.
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 aspects of simplicial complexes, Tessellation and tiling problems.
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, and degenerations of canonical metrics and connections.
Topology of three-manifolds and smooth four-manifolds, hyperbolic geometry, geometric group theory, Heegaard Floer theory and its relations to geometric topology.
Homotopy type theory and its applications to automated theorem proving.
Statistical mechanics, exactly solvable models, partial differential equations arising from string theory and general relativity.
Couded dynamical systems, Synchronization, Turing patterns, applications of Lie algebraic methods to nonlinear Hamiltonian systems, fractal dimensional analysis, generalized replicator dynamics.
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.
Time series and long-memory processes. Application of time series analysis techniques to neuroscience, especially to brain-machine interface; applications to geophysics.
Option pricing, portfolio optimization, interest rate models, credit risk models.