This proposal deals with topological problems in a way that keeps track of sizes of solutions to the problems. These include measurements of displacement (bounded topology), of Lipschitz constants and distortions, and related metric invariants. Applications of such ideas arise within pure topology, stratified spaces, homology manifolds, Riemannian geometry, geometric group theory, and theoretical computer science. The proposal involves a mix of problems from these areas, some related to singularities and group actions which involve refining and applying previously developed tools, while others are motivated by rigidity theory, and conjectures of Gromov that seem to require new methods.
Topology is ordinarily thought of as an area of mathematics that is qualitative in nature. Progress on a number of current central problems requires more precise quantitative control of the solutions of related problems that were solved in earlier generations by powerful but inexplicit algebraic methods. Such control brings topology in closer interaction with applications in other fields of mathematics as well as computer science and engineering. This proposal suggests an attack on some of these problems.