|Geometric structures on surface and 3-manifolds from the triangulation point of view|
|Time：||10:10-12:00 Tue./Thu. (Mar. 29: 14:00-15:50)|
|Instructor：||Feng Luo [Rutgers University]|
|Place：||Conference Room 3, Floor 2, Jin Chun Yuan West Building|
We will introduce some of the recent developments of constructing geometric structures on surfaces and 3-manifolds starting from triangulations. Every Riemann surface has a constant Gaussian curvature metric within its conformal class and each 3-manifold can be canonically decomposed into pieces which support geometric structures. Main issue is to construct geometric structures algorithmically. The course will focus on using triangulations as initial input and finite dimensional variational principles as tools to find geometric structures.
The topics to be covered are:
1. 2-dimensional triangulations and its relationship to hyperbolic metrics, measured laminations, and quadratic differentials. In particular, we will discuss Thurston’s coordinate for Teichmuller space, Dehn’s parameterization of space of curves and Strebel differentials.
2. 3-dimensional triangulations and its relationship to Haken’s normal surface theory, Thurston's gluing equation for hyperbolic structure and discrete SL(2,C) Chern-Simons theory. In particular, we will discuss the volume optimization approach to find solutions to Thurston's equation.
This course can be considered as a continuation of Professor H. Rubinstein course on low dimensional manifolds with emphasis on geometry.
Beginning courses in algebraic topology, differential topology and differential geometry would be very helpful.
Reference for the course:
 F. Luo, Variational Principles on Triangulated Surfaces, arXiv:0803.4232
 F, Luo, Triangulated 3-Manifolds: from Haken's normal surfaces to Thurston's algebraic equation, arXiv:1003.4413