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05/07/2005, 16:00 — 17:00 — Amphitheatre Pa2, Mathematics Building

Harald Upmeier, *University of Marburg, Germany*

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Quantization of Complex Domains

In flat Minkowski space, the theory of quantization (coherent states) is fundamental for mathematical physics. In the complex wave representation (Fock space), methods from complex analysis such as Toeplitz operators on Hilbert spaces of holomorphic functions play a basic role. In my talk I will concentrate on two important generalizations: First, the incorporation of curved manifolds, such as symmetric Kaehler manifolds and bounded hermitian symmetric domains, in the theory of quantization, leading to the construction of a unified calculus on Bergman spaces generalizing the Toeplitz-Berezin calculus and the Weyl calculus. Second, the more recent ideas concerning super-symmetry, where one considers super-Hilbert spaces of Grassmann analytic functions and still has a deep theory of quantizations invariant under (super) Lie groups.