## Dynamics, Geometry, & Groups - Anthony Sanchez (U Washington)

### Thursday, March 4th, 2021

**Time:** 1:30 p.m** Place:**

**Speaker:** Anthony Sanchez (University of Washington, Seattle)

**Title:** Gaps of saddle connection directions for some branched covers of tori.

**Abstract:** Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. In this talk, we consider the following question: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.