Title: Topologically nontrivial counterexamples to Sard's theorem
Speaker: Piotr Hajlasz (University of Pittsburgh)
Abstract: I will discuss the following result: If $n=2,3$ and $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ is not homotopic to a constant map, then there is an open set $\Omega\subset \mathbb{S}^{n+1}$ such that ${\rm rank}\, df=n$ on $\Omega$ and $f(\Omega)$ is dense in $\mathbb{S}^n$, while for any $n\geq 4$, there is a map $f\in C^1(\mathbb{S}^{n+1}, \mathbb{S}^n)$ that is not homotopic to a constant map and such that ${\rm rank} df<n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth. I will also discuss an application of the result to a solution of a recent conjecture of Jacek Galeski. In particular I will show that there is a $C^1$ mappings in $\mathbb{R}^5$ with the derivative of rank at most $3$ that cannot be uniformly approximated by $C^2$ mappings with the derivative of rank at most 3. The methods use analysis, algebraic topology and geometric measure theory. The talk will be accessible to graduate students. The presentation is based on my two joint papers. One with P. Goldstein and one with P. Goldstein and P. Pankka.
Biosketch: Piotr Hajlasz is a full professor at University of Pittsburgh (PhD, Warsaw University 1994). Piotr is author of 60+ publication with over 1900 citations, His main focus is on the theory of Sobolev spaces with applications to various areas like the theory of quasiconformal mappings, calculus of variations, regularity of nonlinear elliptic PDEs, and Carnot-Caratheodory spaces.
Colloquium URL: https://web.math.osu.edu/colloquium/
