Title: From topology to algebraic geometry and back again

We will present applications of secant varieties in topology
through k-regular embeddings. An embedding of a variety in an affine space
is called k-regular if any k points are mapped to linearly independent
points. Numeric conditions for the existence of such maps are an object of
intensive studies of algebraic topologists dating back to the problem posed
by Borsuk in the fifties. Current world record results were obtained by
Pavle Blagojevic, Wolfgang Lueck and Guenter Ziegler. Our results relate
k-regular maps to punctual versions of secant varieties. This allows us to
prove existence of such maps in special cases. The main new ingredient is
providing relations to the geometry of the punctual Hilbert scheme and its
Gorenstein locus. The talk is based on two joint works: with Jarosław
Buczynski, Tadeusz Januszkiewicz and Joachim Jelisiejew and with Christopher