Symmetries have played an important rôle in the study of mathematical structures since the end of the 19th century. This pertains to structures motivated by geometry, number theory or algebra as well as to structures coming directly from physics. For a long time symmetries were described more or less exclusively in terms of group theory, but nowadays one has a number of important generalizations and variations, for example quantum groups or groupoids and structures derived from them. The development of the theory of unitary representations and the related harmonic analysis of function spaces on sets with group actions started in the 1930s. Its initial motivation was the construction of models for quantum physics.
Later, the efforts of understanding number theoretic reciprocity theorems in a uniform way gave a further impulse to this subject. The original goals, which are the classification of irreducible representations as well as the transfer of the classical results of Fourier analysis to the non-commutative setting, are still far from being complete. But the intensive studies in these directions have produced a lot of deep and useful information. Presently the research emphasis has shifted to a wide variety of applications in fields like partial differential equations, mathematical physics, number theory, ergodic theory, complex analysis, topology and non-commutative geometry on the one hand, and generalizations to infinite-dimensional geometry on the other.
The basic idea of the research program is to combine the expertise of neighboring but distinct fields like representation theory, harmonic analysis, operator algebras, and homological algebra. In this way it is possible to attack a variety of problems which were either too hard to be solved by techniques from a single field or, because of their interdisciplinary character, have failed to attract the attention they deserve.
Ph.D. projects could be in areas like
- Algebraically compact representations in Lie theory
- Description of the analytic vectors of irreducible Banach space representations of solvable Lie groups
- Geometry and analysis of loop spaces
- Geometry of infinite dimensional groups and homogeneous spaces
- Geometry of large N limits and large N dualities
- Growth of semigroups in L1(G) of solvable Lie groups G
- H-invariant harmonic analysis on Kε-spaces
- Hypergeometric functions associated with root systems and generalizations
- Index theory for Wiener-Hopf operators on symmetric spaces
- Integral transformations: Radon, Fourier-Laplace, Penrose
- Local index theory in Non-Commutative Geometry
- Properties of the Fourier algebra of a locally compact group
- Symplectic analogies: geometric quantization of symplectic actions
- Topology of the dual space of nilpotent Lie groups and Lie groups with cocompact nilradicals