2:30 pm Wednesday, November 4, 2020
Junior Topology : Flat structures on surfaces, the Hodge bundle, and the Eskin-Mirzakhani-Mohammadi Theorem by Hunter Vallejos in Zoom
Let S be any surface; then, we can endow S with any flat metric so long as we allow for singularities. It turns out that generic flat metrics are very hard to understand, and many basic questions remain unanswered. Most progress so far is restricted to the case where the metric has trivial linear holonomy. A translation surface is a flat surface with trivial holonomy; given some fixed parameters on singularities, we can build a moduli space of translation surfaces which is a complex orbifold. There is a natural GL(2, R)-action on this moduli space which has some fantastic dynamical properties, most notable of which is stated by the Eskin-Mirzakhani Mohammadi [EMM] Theorem: every closed orbit of this action is a finite union of nicely immersed manifolds! We will build up some background about translation structures in an effort to understand the statement of the EMM theorem, discuss some of its implications, and state some other cool geometric facts about translation surfaces and their orbits (there is a clear connection to Riemann surfaces as well). This talk will be more "surface" level since it is less than 30 minutes.
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