Mathematische Physik
Mathematical physics deals with a broad spectrum of interwoven mathematics, such as differential and algebraic geometry (of moduli spaces of geometric structures), low-dimensional topology (surfaces, 3-manifolds, knot theory, ...), combinatorics and enumerative geometry, algebraic structures (infinite-dimensional Lie groups, quantum groups, Hopf algebras, vertex operator algebras, operads, higher structures, etc.) and their representation theory, differential equations coming from geometry (integrable systems, variations of Hodge structures, etc.), functional analysis and asymptotic analysis (rigorous statistical physics, random matrix theory, etc.). Besides their pure mathematical interest, they can be used to consolidate the understanding of quantum field theories (supersymmetric gauge theories, conformal field theory, 2d quantum gravity, etc.), and string theory, and can take inspiration from the insights of theoretical physics to discover new mathematics and establish relations between apparently distant problems.