Topology
Geometric topology is often split into low dimensional (4 or less) and high dimensional. This split is based upon the techniques employed, the kinds of question that can be answered, and the state of knowledge. There were enormous advances in high dimensional topology during the 60’s including the solution of the high dimensional Poincare conjecture, and a good understanding of how differentiability enters into the picture, for example through the existence of exotic smooth structures on spheres.
Today a considerable effort is being made to better understand manifolds of dimensions 3 and 4. The techniques, conjectures and outlooks in these two areas are very different, although there have also been hints of various unifying themes. In the 80’s it was discovered by Donaldson, Freedman and Casson that Euclidean space has exotic smooth structures only in 4 dimensions.
The theory of 3 dimensional manifolds was revolutionized in the late 70’s by Thurston’s Geometrization Conjecture. There are eight geometries (homogeneous Riemannian metrics) which (appear to) play a similar role in 3 dimensions to the three constant curvature geometries in two dimensions. Some problems in 3-dimensions are best studied through combinatorial and topological techniques using surfaces and their generalizations. Many problems in knot theory are of this type.
There are many connections to number theory, Riemannian geometry, geometric group theory and dynamical systems to name only a few.
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Faculty
- Stephen Bigelow
- PhD: University of California, Berkeley, 2000
- Interests: Braid groups
- Office: Room 6514
- email: bigelow@math.ucsb.edu
- Daryl Cooper
- PhD: University of Warwick, 1982
- Interests: Hyperbolic geometry and topology of three-manifolds
- Office: Room 6704
- email: cooper@math.ucsb.edu
- Michael Freedman
- PhD: Princeton, 1973
- Office: Room 6713
- Interests: Topology and quantum computing
- email: mfreedman@math.ucsb.edu
- Darren Long
- PhD: Cambridge University, 1983
- Interests: Low-dimensional topology
- Office: Room 6519
- email: long@math.ucsb.edu
- Fedya Manin
- PhD: University of Chicago, 2015
- Interests: Connections between topology, metric geometry, theory of computation, and probability
- Office: Room 6718
- email: manin@math.ucsb.edu
- Jon McCammond
- PhD: University of California, Berkeley, 1991
- Interests: Geometric Group Theory
- Office: Room 6711
- email: mccammon@math.ucsb.edu
- Ken Millett
- PhD: University of Wisconsin, 1967
- Interests: Knot theory and its applications in the natural sciences
- Office: Room 6512
- email: millett@math.ucsb.edu
- Zhenghan Wang
- PhD: UC San Diego, 1993
- Interests: Topology and quantum computing
- Office: Room 6713
- email: zhenghwa@math.ucsb.edu