活动地点：

活动类型：学术报告

主讲人：Prof. Feng Luo

活动时间：

活动内容：

Abstract:

This is an introduction to  topology of geometry of 3-manifolds. The theory was revolutionized by William Thurston in 1978 who proposed the Geometrization Conjecture which implies the Poincare conjecture. The recent work of Perelman and Hamilton using Ricci flow may have solved the conjecture. The goal of the lecture is  to cover the basic material in the topological theory and Thurston theory with many examples from historical point of view.   The course intends to be self-contained and should be understood by first and second year graduate students with some knowledge of manifolds and topology.


Lecture Plan

Part 1.  Basic Examples
1. introduction to 3-manifolds (the geometrization
conjecture, 2-dimensional analogy)
2. examples: manifolds, manifolds with geometric
structures, quotient topology, group action on
manifolds and quotient spaces, geometries  S^n, E^n,
H^n.
3. examples of Seifert Fibered spaces, Nil manifolds
and how to see the geometric structures of them
4. Dehn surgery, gluing, knot complements
5. triangulations, classification of surfaces and the
hyperbolic surfaces
6. triangulation of 3-manifolds and Seifert theorem
(i.e., a triangulated psuedo 3-manifold is a
3-manifolds iff euler characteristic is 0)
7. the example of Poincare homology sphere and its
hyperbolic analogy (=the first known closed hyperbolic
3-manifold)
8. the trefoil knot complement and SL(2,R)/SL(2,Z)

Part 2. Introduction to hyperbolic geometries in dim 2 and 3

Part 3. Basic topological tools in 3-manifolds
     
1. submanifolds, embedding, immersion, transversality,
isotopy, regular neighborhood, irreducible manifolds,
incompressibility of surfaces
2. the proof of Jordan curve theorem in dim=2
3. the proof of Schoenflies theorem in dim =3
4. applications: irreducibility, prime, etc.
5. normal surfaces theory
6. Kneser¡¯s theorem on the finiteness of
decompositions (Haken¡¯s finiteness theorem)
7. the loop theorem
8. the sphere theorem

Part 4. Hyperbolic 3-manifolds
1. figure 8 knot and how Thurston discovered the
hyperbolic structure of it.
2. basic topology of hyperbolic 3-manifolds

Part 5. Variational approachs to find geometric structures on triangulated  surfaces and 3-manifolds
1. the work of Rivin and Leibon on geometric
structures on triangulated surfaces
2. a discrete curvature flow on compact 3-manifolds
3. geometric triangulations of 3-manifolds.