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