Teichmueller theory of Riemann surfaces
This is a topic course on Teichmueller theory of Riemann surfaces and the related topics. These include the quantization of Teichmueller theory, the mapping class groups and discretization of Riemann surfaces
The following more specific topics will be covered:
1. uniformization theorem (will not be proved)
2. hyperbolic geometry on surfaces (geodesics, triangulations, Fenchel Nielsen coordinate)
3. Teichmuller space of hyperbolic structures on surfaces (Teichmuller theorem, quasiconformal maps, Teichmuller metrics)
4. Mapping class groups of the surfaces (Dehn twists, isometries of Teichmuller metrics, Thurston’s classification)
5. Quadratic differential on surfaces and cell structures on Teichmuller spaces
The above are more or less classical and can be found in John Hubbard's book on "Teichmuller theory".
We also plan to cover, if time permits, Konstvevich's solution of Witten conjecture on moduli spaces ( ribbon graphs, Feynman diagrams and matrix models), some TQFT (Turaev-Viro invariant of 3-manifold) and the volume conjecture in 3-dim and some of recent work on discrete uniformization on surfaces.
Basic differential geometry, topology and complex analysis