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Dynamical systems on surface group representations
Automorphism groups of fundamental groups of surfaces (such as "mapping class groups") are mysterious groups which arise in many mathematical contexts. These groups act on spaces of representations of surface groups, preserving natural symplectic or Poisson geometries and invariant smooth measures. Depending on the representations, the mapping class group actions display diverse dynamical behavior. Complicated dynamics of the mapping class group action accompanies complicated topology of the moduli space. Two of the most basic spaces in the theory of Riemann surfaces represent extreme cases: (1) The Teichmuller space is contractible with proper action of the mapping class group. (2) The Jacobi variety is a closed manifold, with a chaotic (ergodic) action of the mapping class group. In general, the dynamics intermediates between these two extremes.
Thursday, September 15, 2005, 245 Altgeld Hall, 4:00 p.m.
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