Degrees of maps and multiscale geometry

I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper? That is, for what M is there a 1-Lipschitz map $\mathbb R^n \to M$ with positive asymptotic degree? Gromov called such manifolds elliptic. We show that, for example, the connected sum of k copies of CP^2 is elliptic if and only if k ≤ 3. I will try to explain the intuition behind this example, how it extends to a more general dichotomy governed by the de Rham cohomology of M, and why ellipticity is central to the program of understanding the relationship between topology and metric properties of maps.

If I have time, I'll also explain why for a non-elliptic M, a maximally efficient map $\mathbb R^n \to M$ must have components at many different frequencies (in a Fourier-analytic sense), and even then it's at best logarithmically far from having positive asymptotic degree. This is joint work with Sasha Berdnikov and Larry Guth.