The Department aims to promote close collaboration between various mathematical disciplines and with other applied areas. The areas of current research are:
Schemes over symmetric monoidial categories, bivariant Chow theory. Hochschild and cyclic cohomology. Algebraic cycles, algebraic K-theory, $A^1$-homotopy theory of varieties, arithmetic geometry, and $K$-theory of algebraic stacks.
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 their $p$-adic families) and representations, mod $p$ modular and Hilbert modular forms and their Hecke algebras, Galois representations, L-functions, Iwasawa theory, Arithmetic Geometry, Langlands programme, Analytic Number theory, Algebraic K-theory.
Existence and regularity/singularity of minimizers and/or critical points of possibly nonconvex, noncoercive functionals, weak continuity and weak lower semicontinuity, compensated compactness and concentration compactness, differential inclusions.
Homogenization of partial differential equations, controllability, viscosity solutions. Numerical methods for partial differential equations. Existence and regularity of nonlinear elliptic partial differential equations and systems.
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/kernels and functions operating on them, (total) positivity preservation, structured matrices and kernels.
Holomorphic mappings, holomorphic interpolation, Ohsawa-Takegoshi type extension theorems. Invariant metrics: estimates, metric geometry of hyperbolic domains. Domains in $\mathbb C^n$: convexity and finite-type conditions, optimal and random polyhedral approximations, integral representations and holomorphic function spaces. Convexity properties of real submanifolds in complex spaces. 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. Higher dimensional Gauge theory, Yang–Mills equation. Harmonic, $p$-harmonic and polyharmonic maps.
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.