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I will talk about the following question of Gromov: what closed manifolds can be efficiently wrapped with Euclidean wrapping paper?Ìý That is, for what M is there a 1-Lipschitz map $\mathbb R^n \to M$ with positiveÌýasymptotic degree?Ìý Gromov called such manifoldsÌýelliptic.Ìý We show that, for example, the connected sum of k copies of CP^2 is elliptic if and only if k ≤ 3.Ìý 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.ÌýIf 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.Ìý

This is joint work with Sasha Berdnikov and Larry Guth.

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