The work of Perelman has resolved the geometrization conjecture of Thurston, thereby confirming that "most" closed 3-manifolds are hyperbolic. The expectation now is that, given Perelman's work, a great deal of the focus of 3-manifold topology will be on understanding the geometry and topology of finite volume hyperbolic 3-manifolds. Motivated by this the PI will study hyperbolic 3-manifolds, their fundamental groups and representations of their fundamental groups. This will involve the study of finite sheeted covering spaces, finite quotient groups, profinite completions of discrete groups, their connections with number theory, and expander families of graphs. The PI will also explore other discrete groups, like lattices in other Lie groups and Mapping Class Groups.

Three dimensional manifolds are locally like the space we live in and understanding these objects have been one of the central themes of research in the last 30 years. The importance of these objects extends far beyond their intrinsic interest, since their study connects to mathematical physics, mathematical biology and computer science. Various algebraic objects can be associated to a three dimensional manifold, one of which (a group) captures symmetries of the manifold and other manifolds related to it. Much of the proposal is aimed at exploring properties of these groups. For example, the PI will explore their connections to families of so-called "expanding graphs". These graphs are well-known in computer science because of their importance in building efficient networks. In addition another project connects the modern mathematical world of flexible geometry to a question in elementary number theory that goes back to the ancient Egyptians. A solution to this old question via the techniques suggested would be very interesting.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
1105002
Program Officer
Joanna Kania-Bartoszynska
Project Start
Project End
Budget Start
2011-06-01
Budget End
2014-05-31
Support Year
Fiscal Year
2011
Total Cost
$296,304
Indirect Cost
Name
University of Texas Austin
Department
Type
DUNS #
City
Austin
State
TX
Country
United States
Zip Code
78712