## 16:640:537 Selected Topics in Geometry II

## Fall 2020

### Feng Luo

### Subtitle:

Teichmueller theory of Riemann surfaces

### Course Description:

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 more specific following 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 theorem, quasiconformal maps, Thurston's classification)

5. Quadratic differential on surfaces and cell structures on Teichmuller spaces

The above are more or less classical can and can be found in John Hubbard's book on "Teichmuller theory".

We also plan to cover, if time permits, Konstvevivh'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 onjecture in 3-dim and some of recent work on discrete uniformization on surfaces.

### Textbook:

None

### Prerequisites:

Basic differential geometry, topology and complex analysis