Quantitative homotopy theory (22 years after Gromov's Princeton lecture)

I will discuss the geometry of spaces of maps between simplyconnected finite complexes. The Lipschitz constant of a map provides a natural notion of "geometric complexity" and we study the Morse landscape of this functional, whose shape turns out to be closely controlled by rational homotopy theory. Concrete questions include: how many homotopy classes have representatives with Lipschitz constant $\leq L$ (what is the growth of $\pi_0$ of the mapping space?) Given two homotopic $L$-Lipschitz maps, how hard is to deform one to another? (does the landscape have deep valleys?) Some of this is joint work with Shmuel Weinberger, Greg Chambers, and Dominic Dotterrer.