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Workshop: Several Complex Variables, Analysis on Complex Lie groups and Homogeneous Spaces

Zhejiang University, Hangzhou, Oct. 17-29, 2005

The workshop “Several Complex Variables, Analysis on Complex Lie groups and Homogeneous Spaces” will be held at Zhejiang University, Hangzhou from October 17 to 29, 2005.

It is organized by Zhejiang University, Chinese Academy of Science, Beijing and University of Angers, France (with support from the French Embassy in

P.R. China). Scientific direction: J. Hilgert (University of Padderborn, Germany),

A. Huckleberry (University of Bochum, Germany), J.-J. Loeb (University of Angers, France), G. Roos (St Petersburg, Russia), A. Sergeev (Steklov Insti­tute, Moscow, Russia), Zhou Xiangyu (Institute of Mathematics, AMSS, CAS, Beijing and Zhejiang University).

 

Organization: Zhou Xiangyu (Institute of Mathematics, AMSS, CAS, Bei­jing and Zhejiang University).

Local organizers: Chen Jiechen (Faculty of Sciences, Zhejiang Univer­sity), Xu Hongwei (Center of Mathematical Science, Zhejiang University), Zhou Xiangyu (Institute of Mathematics, AMSS, CAS and Zhejiang University).

The main purpose of the workshop is to provide access to recent develop­ments in the field, for doctoral students and young researchers. There will be five series of lectures of 8h each, whose contents is summarized below. Pre­requisites for the lectures are basic knowledge in Several Complex Variables, Differential Geometry and Lie Groups. The lectures will occupy approxima­tively 4h per day. The remaining time will be devoted to discussion sessions or talks by participants.

The working language will be English. A text of the lecture notes will be available for participants, in printed or electronic form, at the beginning of the workshop.

List of lectures:

1. J. Hilgert (University of Padderborn, Germany) Representation theory in complex geometric settings I.

2. A. Huckleberry (University of Bochum, Germany) Representation theory in complex geometric settings II.

3. J.-J. Loeb (University of Angers, France) On complex automorphisms and holomorphic equivalence of domains.

4. G. Roos (St Petersburg, Russia) Exceptional bounded symmetric domains, Jack polynomials, Dunkl operators.

5. A. Sergeev (Steklov Institute, Moscow, Russia) Seiberg-Witten equations and pseudo-holomorphic curves.

 

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Contents of lectures

1. and 2. These two strongly integrated courses will provide foundational results, describe basic guiding problems and indicate certain recent develop­ments. The following outline gives a rough indication of the contents.

Introduction to Hardy spaces and to the geometry of actions on flag man­

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ifolds.

Realization of unitary representations on Hardy and weighted Bergman

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spaces.

Uniform realization of all holomorphic discrete series representations as

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the irreducible components of a Hardy space.

Representations associated to cycle spaces of orbits in flag manifolds.

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3. The following topics will be covered

Generalization of the classical Schwarz lemma to bounded domains in Cⁿ .

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Extension to Kobayashi hyperbolic manifolds.

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Role of pseudo-convexity.

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Using pseudo-convexity in the case of non hyperbolic manifolds. Only a

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few results are known in this direction and there are many open questions.

4. We give a presentation of exceptional bounded symmetric domains using the Albert algebra and exceptional Jordan triple systems. In the last chapter, we give some facts about Jack polynomials and associated differential-difference operators. We then give their explicit form in the case of exceptional symmetric domains.

5. The main content is the relation between Seiberg-Witten equations on 4-dimensional symplectic (and though almost complex) manifolds and pseudo­holomorphic curves. This relation is established through a limiting construc­tion, due to Taubes, and has a non-trivial 3-dimensional analogue, related to Ginzburg-Landau equations, arising in the superconductivity theory.

 

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