# Research Areas

The Mathematics Research unit is composed of five research areas.

**- Algebra and Number Theory:** The research group carries out fundamental research on number theoretic aspects of Galois theory, modular forms, elliptic curves and abelian varieties, especially concerning Galois representations (modularity, congruences, local properties, images, compatible systems, independence), modular forms and the inverse Galois problem. A particular feature is the development and the use of computer algebra tools for explicit study of examples and the compilation of databases...

**- Geometry and the Mathematical Theory of Quantisation: **Geometry is one of the basic pillars of mathematics. At the RMATH we study different directions of modern geometry. A selection of topics under considerations is given by the following list (more can be found at the different teams): algebraic geometry, complex manifolds, differential geometry, symplectic geometry, non-commutative geometry, moduli space problems, higher structures in geometry, algebraic topology, algebraic aspects in geometry, supergeometry, infinite-dimensional Lie algebras, conformal field theory, algebraic aspects of quantum field theory...

**- Harmonic and Geometric Analysis**: We focus mostly on the geometric and analytic study of Riemannian and Lorentzian symmetric spaces. Among our main research directions are: harmonic analysis on geometrically finite groups, including scattering theory and trace formulas, the structure of pseudo-Riemannian symmetric spaces, the geometry of 3-dimensional hyperbolic and anti-de Sitter manifolds, the relations between 3-dimensional geometry and Teichmüller theory, discrete and polyhedral geometry in relation with constant curvature spaces.

**- Probability Theory and its Applications, Mathematical Finance**: The topics in this research area lie at the intersection of several branches of modern probability theory, namely: stochastic analysis on manifolds, geometry of stochastic differential equations, functional inequalities, combinatorics, limit theorems, non-commutative probability and stochastic geometry. Various applications to finance, physics and statistics are actively developed.

**- Mathematical Modelling**: The research groups focus on theoretical research on various aspects of mathematical modeling, including linear inverse problems with convex constraints, constrained statistical inference, Malliavin calculus of variations, Stein's method, functional inequalities, random matrices, rough paths theory, decision making, game theory, aggregation function theory, operations research, computer science, logic, and system reliability theory. The main areas involved in this research include abstract and universal algebras, combinatorics, probability theory, stochastic analysis, free probability, concentration of measure, high dimensional geometry, analysis and in particular functional equation theory.