##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Albert Artiles Calix

#### University of Washington

## Statistics of Minimal Denominators and Short Holonomy Vectors of Translation Surfaces

##### Abstract:

This talk will explore the connection between Diophantine approximation and the theory of homogeneous dynamics. The first part of the talk will be used to define and study the minimal denominator function (MDF). We compute the limiting distribution of the MDF as one of its parameters tends to zero. We do this by relating the function to the space of unimodular lattices on the plane.

The second part of the talk will be devoted to describing equivariant processes. This will give a general framework to generalize the main theorem in two directions:

1. Higher dimensional Diophantine approximation

2. Statistics of short saddle connections of Veech surfaces

If time allows, we will compute formulas for the statistics of short holonomy vectors of translation surfaces.

Host: Brandon Seward

### May 2, 2024

### 10:00 AM

Zoom ID 967 4109 3409

Research Areas

Ergodic Theory and Dynamical Systems****************************