# Seminars & Events for Minerva Lectures

##### Episodes from Quantitative Topology: 1. Variational problems, Morse and Turing.

This lecture will begin the series of discussing how effective solutions of topological problems are: and in particular, how large solutions to geometric topological problems are with various measures of complexity. Lecture one will show how one can use basic results about computability, algorithmic undecidability, and more general complexity measures to prove the existence of many solutions to certain variational problems. (This is largely based on joint work with Alex Nabutovsky.)

##### Episodes from Quantitative Topology: 2. Quantitative Nullcobordism

In the 50's, Rene Thom solved the problem of determining when a closed smooth manifold bounds a compact manifold. Subsequent work of Milnor and Wall solved the analogous oriented problem. These works comprise an important early example of the fundamental method of geometric topology via reduction to algebraic topology. The basic question (posed by Gromov) is: given a complexity measure for manifolds, how complicated must a nullcobordism of a given manifold be? I won't solve this problem, but I will explain (based on joint work with Steve Ferry, and with Greg Chambers, Dominic Dotterer, and Fedor Manin) some recent estimates.

##### Episodes from Quantitative Topology: 3. Gromov-Hausdorff space and homeomorphisms

Gromov-Hausdorff space is a metric space of compact metric spaces and is useful in many areas of geometry. Motivated by Cheeger's thesis, there are a number of results proving that for many pre-compact sets in GH space, there are only finitely many homeomorphism types of manifolds. I will explain some work with Sasha Dranishnikov and Steve Ferry that shows infinite dimensional phenomena arise in certain effective versions of this result. This indirectly leads to a certain kind of metric-topological rigidity that holds for all manifolds whose fundamental groups are lattices in real Lie groups (or are word-hyperbolic), but not shared by all those whose fundamental groups are linear.