## Gap Distribution and Homogeneous Dynamics

### Time

Dec 6 2012 – 2:30pm – 3:30 pm

### Location

CH240

### Speaker

Jayadev Athreya (University of Illinois, Urbana-Champaign)

### Abstract

The Farey sequence F(Q) is the collection of fractions between [0,1] whose denominator (when written in lowest terms) is at most Q. As Q grows, these points become uniformly distributed in the interval, so in some sense, look `random’. However, when you look at the gaps between them, they do not behave like those for uniformly distributed random variables, but instead follow an unusual law known as Hall’s Distribution. We will explain a proof of this result that uses horocycle flow on the space of lattices SL(2,R)/SL(2, Z), and discuss how this picture can be generalized to explicitly computing the gap distribution between directions of saddle connections on Veech surfaces, which we had used computer experiments to approximate. This talk will include elements from joint work with Y. Cheung, joint work with J. Chaika, and joint work with J. Chaika and S. Lelievre.