We investigate the classification of compact topological and smooth manifolds, as well as embedding problems - knot theory. We use both new and classical homological, geometric and analytic techniques such as algebraic L-and K-theory, controlled topology, operator theory and Von Neumann signatures. One problem attacked, from differential geometry, is to determine whether almost flat manifolds bound compact manifolds. New knot invariants, defined during the prior grant, are continuing to reveal new and unexpected knotting phenomena and will be used to further investigate the classification of topological four manifolds within a fixed homology type. These invariants will also be applied to further compute the topological concordance group of knots, and to the rich structure of three manifolds as seen through descending series of the fundamental group. Another problem is to compute and apply characteristic classes for combinatorial vector bundles, an initial step in understanding a new category of manifolds - the combinatorial differentiable manifolds of Gelfand-MacPherson. A final topic is the computation of algebraic K- and L-theory using controlled topology and applying these computations to manifolds with fixed fundamental group.
A n-dimensional manifold is a set of points which is locally modeled on an n-dimensional linear space. For instance a surface is a locally 2-dimensional linear space, such as a sphere, or the surface of a donut (torus). The principal problem of geometric topology is the classification of manifolds. In differential topology we also ask how many ways we may do calculus on a given manifold, addressing the uniqueness question. The fundamental group of a manifold gives an algebraic structure to the collection of all loops in the manifold, thus providing a pathway to algebra. The researchers focus on both high dimensional and low dimensional manifolds. Four dimensional manifolds are of particular interest as the results and techniques in this dimension manifest the harmony of the algebraic techniques which have found success in high dimensional manifold topology and the geometric techniques which prevail in dimension three. Investigating the foundations of manifold and knot theory fundamentally underpins our understanding of geometry, algebra, physics and differential equations. Additionally, knot theory plays a growing role in string theory, quantum field theory and the study of DNA.
Forwarded message Date: Fri, 11 May 2001 14:27:02 -0500 (EST) From: "James F. Davis"
Kent and I were very happy to here that you plan to recommend that our NSF proposal be granted. However, some of circumstances have changed since we wrote the proposal, so I thought I should discuss them with you in case the budget is affected. Sorry for the delay getting back to you, I was visiting the University of Chicago this week. We asked for summer money to support three graduate students: Karl Bloch, Diarmuid Crowley, and Tae-Hee Kim. Well, I am happy to report that Diarmuid Crowley has been named a Clay Mathematical Institute Lift-Off Mathematician
and will receive some money from them this summer (provided some visa problems work out.) He is suggesting to Clay that they award him the money from June 13 to July 27. After Aug 1, I believe he will be supported by Max Planck Institute in Bonn. However, he could still use NSF money, for example, he is attending the 3-week School on high-dimensional manifold topology in at the ICTP in Trieste, Italy, mostly at his own expense. Please advise us if it is necessary to rebudget and return the money allotted to Diarmuid.
The second change is with regard to my proposed REU student, Matt Cecil. Matt Cecil worked with me as an REU student last summer (partially supported by NSF funds), and continued throughout the academic year. He is attending UCSD as a math graduate student in the fall. However, due to mathematical reasons and personal reasons, he will not be an REU student with me this summer. So the question again is - what to do with the money allotted to him? I talked to our chair, Dan Maki, and he said that he could use the money to help undergraduate research projects connected with industry. Dan has many industrial contacts throughout the state of Indiana and had lead such projects for several years. The money would primarily go to student travel expenses.
So, Chris, please advise on whether we should give some money back to the NSF and rebudget, or just go with the modifications outlined above.