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. Click Here for Selected Topology Puzzles Click Here for Detailed Topology Group Interests
 RESEARCH FIELDS Algebra Analysis Applied Math Geometry Number Theory PDE Topology

Faculty

 Stephen Bigelow PhD: University California at Berkeley, 2000 Interests: Braid groups Office: Room 6514 bigelow@math.ucsb.edu Daryl Cooper PhD: University of Warwick, 1982 Interests: Hyperbolic geometry and topology of three-manifolds Office: Room 6704 cooper@math.ucsb.edu Darren Long PhD: Cambridge University, 1983 Interests: Low-dimensional topology Office: Room 6519 long@math.ucsb.edu Jon McCammond PhD: University of California, Berkeley, 1991 Interests: Geometric Group Theory Office: Room 6711 mccammon@math.ucsb.edu Ken Millett PhD: University of Wisconsin, 1967 Interests: Knot theory and its applications in the natural sciences Office: Room 6512 millett@math.ucsb.edu Marty Scharlemann PhD: University of California at Berkeley, 1974 Interests: Low-dimensional topology Office: Room 6718 mgscharl@math.ucsb.edu

Visiting Faculty