Abstract: A Lipschitz map from ℝ^n to an n-manifold M has positive asymptotic degree if, roughly speaking, it wraps efficiently around M. M is elliptic in the sense of Gromov if it admits such a map. Similarly, M is quasiregularly elliptic if it admits a quasiregular map from ℝ^n: a map with geometric properties similar to those of holomorphic functions. Gromov in his book Metric Structures suggested that there may be a connection or even equivalence between ellipticity and quasiregular ellipticity. This is supported by the fact that the known obstructions to ellipticity and quasiregular ellipticity for closed manifolds are exactly the same (for example, in both cases the fundamental group has to be abelian). On the other hand, we know much less about constructing quasiregular maps than Lipschitz maps of positive asymptotic degree. Moreover, for open manifolds, a host of simple examples shows that these two properties are quite different. I will discuss joint work with Berdnikov and Guth and with Prywes and highlight a number of open problems.

Zoom Meeting ID: 181 155 584