## Department Colloquium

### Friday, March 19th, 2021

**Time:** 2:30 p.m. **Place:** Online (via Zoom)

**Speaker:** Mike Hill (UCLA)

**Title:** Counting exotic spheres.

**Abstract: ** The circle, surfaces, and three manifolds have essentially one smooth structure on them: there is a unique way to "do calculus" on these. For dimensions at least 5, ordinary Euclidean space does too. In 1956, Milnor shocked the mathematical community by showing that this is not the case for spheres: the 7-sphere has "exotic" smooth structures! In this talk, I will discuss the question "how many distinct smooth structures are there on a given sphere?'' In particular, I will describe some work addressing when the only smooth structure on a sphere is the usual one.

**Mike Hill** is a Professor at the University of California, Los Angeles. His research focus is in algebraic topology. Prof. Hill completed his Ph.D. at the Massachusetts Institute of Technology in 2006. Prior to joining UCLA in 2015, he was a faculty member at the University of Virginia. He is an editor for Mathematische Zeitschrift, Documenta Mathematica and the Transactions of the American Mathematical Society.