## Special Colloquium

Wouter Van Limbeek

Michigan

Symmetry and self-similarity in Riemannian geometry

**Abstract:**In 1893, Hurwitz showed that a Riemann surface of genus $g \geq 2$ admits at most $84(g-1)$ automorphisms; equivalently, any 2-dimensional hyperbolic orbifold $X$ has $Area(X)\geq \pi / 42$. In contrast, such a lower bound on volume fails for the n-dimensional torus $T^n$, which is closely related to the fact that $T^n$ covers itself nontrivially. Which geometries admit bounds as above? Which manifolds cover themselves? In the last decade, more than 100 years after Hurwitz, powerful tools have been developed from the simultaneous study of symmetries of all covers of a given manifold, tying together Lie groups, their lattices, and their appearances in differential geometry. In this talk I will explain some of these recent ideas and how they lead to progress on the above (and other) questions.

Wednesday December 6, 2017 at 3:00 PM in SEO 636