This is a proposal that aims to study the sizes of solutions to (families of) topological problems with respect to various norms. The purely metric aspect of this goes under the name of "controlled topology" and has had numerous applications to topological rigidity, stratified spaces (especially orbifolds), and homology manifolds and these will continue to be studied. However, other norms involving fewer or more derivatives are important for applications to analysis (especially variational problems) and differential geometry (e.g. systoles) --- these will be studied simultaneously with their applications.
In general, topology, when applied traditionally, has been used mainly as a qualitative tool. However, it is possible to mix topological ideas with analytic ones, and fashion tools that measure not only whether things are possible or impossible, but rather how difficult they are. In so doing, topology makes contact with other branches of mathematics: notably analysis and probability. It is hoped that this will thereby enrich all these fields.
Many of the problems ordinarily considered by topologists have a qualitative nature, answered by a yes or no: Can one object (a coffee cup) be deformed into another (a donut)? (Yes). The themes of this project are to examine more quantitative aspects of such problems: e.g. when a deformation exists, what can we say about the way in which the objects are distorted, and then to apply such more refined quantitative information to mathematical and other scientific problems. There are two sides to the question of understanding the quantitative nature of topological problems: the first is finding examples where the deformations are provably extreme. The second is finding conditions where one can show that not ony are the objects (or functions) deformable to each other, but that one can control how complicated the deformations are. We are not close to understanding the contours of this question. Joint work with Nabutovsky (University of Toronto) gave examples of the first phenomenon related to the revolutionary work of Godel in Mathematical logic from the early part of last century. It suggested one theme that has been recurrent throughout the project and which seems likely to continue in the future: a robust interaction with logic and theoretical computer science. The positive results were joint with Steve Ferry (Rutgers University) and were related to earlier joint work with Sasha Dranishnikov (University of Florida) on the Novikov conjecture. The Novikov conjecture is a problem relating the closed curves on a space to all of the higher dimensional geometry of that space. It has been a central problem in topology and geometry for over forty years, and has been solved in a great many cases – although it remains open in complete generality. The closed curves on a space (up to deformation) have the mathematical structure of a group. Geometric group theory uses geometric methods to study these objects, and the Novikov conjecture then provides a way of getting information to go in the reverse direction. Essentially, results about sizes of deformations give information about the large scale structure of certain groups. It is worth noting that the obstacles to solving the Novikov conjecture are related to distortions of embeddings, a problem and theme important for completely different reasons in theoretical computer science. Three areas where the quantitative ideas of this project have found fruit are in the theory of aspherical manifolds (manifolds, which contain no essential spheres of dimension larger than (1) where a forty year old problem of Conner and Raymond was disproved (in joint work with Sylvain Cappell of the Courant Institute and Min Yan of the Hong Kong University of Science and Technology), (2) the classification of certain spaces with singularities that come from spaces with symmetry (the axis of a symmetry provides a singularity, because, like a black hole, its nature is different from that of the typical point around it) and (3) towards the development of robust geometric methods for the analysis of large data sets. During the period of this grant, the principal investigator gave a number of lectures explaining the ideas of quantitative topology and how they can be applied in mathematics and outside of it. Transparencies of some of these lectures are available on his web page. He also gave a number of outreach lectures at the college, high school, and elementary school level in Chicago, Berkeley, and Pennsylvania. Through interdisciplinary courses at the University of Chicago and lectures at non-mathematical scientific meeting, he has reached out to the scientific and statistical communities. He has had 6 graduate students during this time; one completed his Ph.D. and the other five are working on a range of projects in pure and applied topology.
