Weinberger plans to develop both quantitative versions of known theorems of algebraic and geometric topology, extending the basic ideas of controlled topology into directions of both more and less smoothness. In part, this interacts with logic and computer science, as in the analysis of Dehn functions, and applications of Kolmogorov complexity arguments to variational problems, but the goal is to get more realistic upper and lower bounds on specific (solvable) topological problems. A subsidiary task is to understand the "information based complexity" of geometric problems, and understand the rates of growth of various sets arising in classification problems via the philosophy of the Goodwillie calculus.
The motivation for this comes from a number of directions, both applied and theoretical. In applications of topology to experimental sciences, one rarely has as explicit space to study: one has large data sets, and the investigator must infer an underlying space. The features that often arise as hypotheses in mathematical theorems cannot simply be seen: they must be detected. One problem to be studied is how to determine, say, the dimension of such an underlying space. After that come problems of implementing software and understanding how much data one needs to make reliable estimates. Another sort of problem comes from the existence of singularities in many equations of applied mathematics and geometry. As topology usually confines itself to the study of continuous maps, the implication of discontinuities of various sorts could potentially be of use. Finally, in many problems of pure geometry, the detailed structure of solutions which have only been proved to exist, but which have never been "seen" would be of great value. Reproving their existence with quantitative estimates should lead to a better understanding of their nature.
