# Geometric Analysis and Topology Seminar

#### Homotopical Effects of K-Dilation

Speaker: Larry Guth, MIT

Location: Warren Weaver Hall 805

Date: Monday, April 21, 2014, 2 p.m.

Synopsis:

The $$k$$-dilation of a map measures how much it stretches $$k$$-dimensional areas. If $$\mathrm{Dil}_k f < L$$, then it means that for any $$k$$-dimensional submanifold $$S$$ in the domain, $$\mathrm{Vol}_k (f(S))$$ is at most $$L \mathrm{Vol}_k(S)$$. We discuss how the $$k$$-dilation restricts the homotopy type of a map. Our main theorem concerns maps between unit spheres, from $$S^{m}$$ to $$S^{m-1}$$. If $$k > (m+1)/2$$, then there are homotopically non-trivial maps $$S^m$$ to $$S^{m-1}$$ with arbitrarily small $$k$$-dilation. If $$k$$ is at most $$(m+1)/2$$, there every homotopically non-trivial map from $$S^m$$ to $$S^{m-1}$$ has $$k$$-dilation at least $$c(m) > 0$$. In this talk, I want to focus on the case $$k$$ at most $$(m+1)/2$$. The non-trivial homotopy type of a map $$S^m$$ to $$S^{m-1}$$ is detected by a certain Steenrod square. The main issue is how to connect Steenrod squares with quantitative estimates about $$k$$-dimensional volumes. This involves a mix of topology and geometry -- on the geometrical side the tools are related to isoperimetric inequalities/geometric measure theory. (We don't assume familiarity with Steenrod squares.).