Data: 20 stycznia 2026, godzina 10:30-12:00
Miejsce: A1 33 (sala Rady Wydziału)
Prelegent: Paweł Goldstein (Uniwersytet Warszawski)
Abstract: Consider the following natural question, posed in 2017 by Jacek Gałęski:
Let \(f\) be a \(C^1\) map into \(\mathbb{R}^n\) defined on an open subset of \(\mathbb{R}^n\), with rank of \(Df(x)\) at most \(k\), \(k<n\), for all \(x\). Can \(f\) be approximated (uniformly, locally uniformly) by smooth mappings \(g_j\) with the same condition on the rank of the derivative, i.e., \(\mathrm{rank}\, Dg_j(x)\le k\) everywhere?
This is, in fact, an infinite number of conjectures, for varying \(n\) and \(k\). Together with Piotr Hajłasz, we obtained in the past years both positive and negative answers to these conjectures, using explicit constructions, tools from algebraic topology, analysis on metric measure spaces and geometric analysis. I shall talk about these results and closely related problems on the boundary of geometric analysis and algebraic topology.