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Research interests

My general research interest are mathematical systems with symmetries. Under the following headlines I describe various past and present groups of projects. The descriptions contain references to my list of publications

Resonances and scattering theory for locally symmetric spaces

Resonances of dynamical systems appear in two different forms: As poles of meromorphic families of resolvent operators and as poles of meromorphic families of scattering matrices. In the case of quantum systems one also talks about quantum resonances, for classical systems the corresponding resonances are known as Pollicott-Ruelle resonances. I am mostly interested in resonances for geodesic flows on  locally symmetric spaces and their quantizations. In the study of such resonances one can use methods from harmonic analysis and representation theory, but also microlocal analysis. There are close connections with trace formulae and dynamical zeta functions. 

[A56, A70, B23, D23, D25, D27, D28]

Quantum chaos

In this research area one deals with the question whether classical chaotic behavior is reflected in its quantizations. This is not obvious as the description of chaos in classical systems (high sensitivity with respect to initial data) cannot be transferred to the mathematical formalisms of quantum mechanics (linear operators). On the other hand there is evidence (in special examples and by numerical experiments) that classiscal chaos is correlated with spectral properties of  the corresponding quantum systems. As toy models one studies systems which are well understood in the classical as well as the quantum context such as geodesic flows, in particular on locally symmetric spaces such, for example  the modular surface. The methods used for such example come from number theory (e.g. zeta functions and continued fraction expansions), differential geometry (e.g. symplectic and spectral geometry), dynamical systems (e.g. Markov decompositions or dynamical Lefschetz formulae),  statistical mechanics (e.g. transfer operators), scattering theory (e.g. resonances) and harmonic analysis (e.g. Plancherel and trace formulae).

Conversely, there have been attempts to apply the methods assembled in this context to purely mathematical questions such as the Riemann hypothesis (see some work of Connes and Deninger). More down to earth one finds connections with representation theory and harmonic analysis on symmetric spaces.

[A48, A50, A51, A52, A53, A54, A62, B15, B16, B17]

 

Harmonic analysis on symmetric (super-)spaces

A. Causal Symmetric Spaces

The original motivation for the study of spherical functions on ordered symmetric spaces came from a diagonalization procedure for the Bethe-Salpeter equation from scattering theory. The spherical functions give characters for algebras of invariant integral kernels satisfying causality conditions. Meanwhile there is a rich theory which parallels Harish-Chandra's theory for Riemannian symmetric spaces. In particular, for many examples one has integral representations and asymptotic estimates for spherical functions. There are interesting related questions about Laplace transforms, positive definiteness and connections with unitary representations.

[A17, A23, A34, A37, A38, A40, A41, A42, A43, A44, B04, C04]

 

B. Symmetric Super spaces

Some parts of Harish-Chandra's harmonic analysis for Riemannian symmetric spaces has been extended to symmetric super spaces in the sense of Zirnbauer. In particular one has results on the relevant invariant theory and Berezin integration. Moreover, necessary preparatory work on singular superspaces has been done.

[A58, A59, A63, A65, A67, A68]

 

C. Radon transforms on homogenen spaces

These are spezial integral operators which can be described via geometric substructures of given homogeneous spaces. A typical goal in this context is to establish inversion formulas.

[A47, A49, B06, B18]

 

D. Wiener-Hopf Operators

The starting point for this topic was the study of  integral and differential equations satisfying causality and symmetry conditions (e.g. Lorentz invariance). One use the interplay of order and symmetry to derive properties of solutions. One of the problems is to build an index theory for Wiener-Hopf operators (certain integral operators taking the causal structure of the underlying manifold into account) on ordered symmetiric spaces. This requires a structure theory for the C*-algebra of all Wiener-Hopf operators. This algebra is a quotient of a well-understood groupoid C*-algebra. Trying to determine the ideal one has to factor out naturally leads Radon transforms on homogeneous spaces.

[A27, A29, B08, C04]

Geometric realisations of unitary representations

A) Minimal Representations

The simplest way to give a geometric realization of a unitary representation is to consider the regular representation (i.e. translation in the argument) on a G-manifold with invariant measure. Not all representations admit such a realization, whose derived representation is then given by vector fields. For derived representations of singular representations the Lie algebra may act by higher order differerential operators. One measure for the singularity of a representation is its Gelfand-Kirillov dimension, which is minimized by minimal representations. 

[A57, A60, A61, A64, A66]

 

B) Highest Weight Representations

All unitarizable highest weight modules admit holomorphic extensions to complex semigroups. For such representations with scalar lowest K-type geometric quantization yields a geometric realization on nilpotent coadjoint orbits. For higher dimensional K-Typen there is still no unified construction. The attempt to realize as many as possible of the singular highest weight modules in a geometric way one is lead to Jordan algebras and positive definite operator-valued reproducing kernels.

[A15, A17, A20, A24, A32, A33, A35, A37, A42, A44, A45, B12, C04]

Hamiltonian actions

A G-manifold M with invariant symplectic form is called hamiltonian if the derived actioncan be lifted to a homomrphism from the Lie algebra of G to the Poisson-Lie algebra of M. In this case one obtains a G-equivariant moment map form M into the dual of the Lie algebra of G. For a umber of classes of examples one can prove convexity properties of the moment map. These have applications e.g. in convergence proofs for integral representations of spherical functions. Different applications of moment maps can be found in the study of conserved quantities of hamiltonian flows or branching laws of representations.

[A21, A22, A24, A31, A36, A57, A69, B07, B10, B16]

Geometry of causal structures

Motivated by problems from control theory and cosmology one studies order structures on manifolds which are defined by integral curves of vector fields whose directions are restricted by inequalities (e.g. via the forward light cone of a Lorentzian metric). In presence of additional symmetries this leads to a study of subsemigroups of Lie groups and cones in Lie algebras. In this context there arise a number of natural problems concerning structure and classification. The answers can often be applied in control theory and Lorentzian geometry.

 

[A01, A02, A03, A04, A05, A06, A07, A09, A10, A11, A12, A13, A14, A16, A18, A19, A25, A26, B01, B02, B03, B05, B09, B11, C01, C03, C04]

Math education

The University for the Information Society