In-person meetings are held in AP&M 7321 and Zoom meetings are held here.

If you would like to be included or removed from our email announcements, please email Brandon Seward.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward

Spring 2024 Speakers Past Speakers

- April 4: Sam Freedman (Brown University)

Location: Zoom

Title: Periodic points of Veech surfaces

Abstract: We will consider the dynamics of automorphisms acting on highly-symmetric flat surfaces called Veech surfaces. Specifically, we'll examine the points of the surface that are*periodic*, i.e., have a finite orbit under the whole automorphism group. While this set is known to be finite for primitive Veech surfaces, for applications it is desirable to determine the periodic points exactly. In this talk we will classify periodic points for the case of minimal Prym eigenforms, certain primitive Veech surfaces in genera 2, 3, and 4.

video

- April 11: Tariq Osman (Brandeis University)

Location: Zoom

Title: Limit Theorems for Theta Sums

Abstract: We define theta sums as exponential sums of the form S^f_N(x; \alpha, \beta) := \sum_{n \in \mathbb Z} f(n/N) e((1/2 n^2 + \beta n)x + \alpha n), where e(z) = e^{2 \pi i z}. Such sums arise naturally in problems of number theory and mathematical physics. If \alpha and \beta are fixed real numbers, and x is chosen randomly from the unit interval, then it is possible to use methods of homogeneous dynamics to show that N^{-1/2} S^f_N possesses a limiting distribution as N goes to infinity, provided f is sufficiently regular. In particular, F. Cellarosi and J. Marklof showed that if at least one of \alpha or \beta is irrational, then this limiting distribution is heavy tailed, and independent of \alpha and \beta. Later, in joint work with F. Cellarosi, we complemented this result by showing that if instead both \alpha and \beta are rational, then the limit distribution is either compactly supported or heavy tailed, depending on the choice of \alpha and \beta. We provide an overview of the key elements in the proofs of these results. Time permitting, we also discuss work in progress with J. Griffin and J. Marklof on extensions of these ideas to the study of the limit distribution of appropriately normalised exponential sums of the form \sum_{n \in \mathbb Z^d} f(\frac{1}{N} n) e^{2 \pi i Q(n) x}, where x is randomly sampled from the unit interval, and Q is a fixed generic, positive definite, irreducible quadratic form.

video

- April 18:

- April 25:

- May 2: Albert Artiles Calix (University of Washington)

Location: Zoom

Title: 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:

- Higher dimensional Diophantine approximation
- 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.

video

- May 9:

- May 16: Qingyuan Chen (UCSD)

Location: AP&M 7321

Title: Shannon Orbit Equivalences Preserve Kolmogorov-Sinai Entropy

Abstract: We will consider the behavior of the Kolmogorov-Sinai entropies of amenable group actions under a Shannon orbit equivalence. Although dynamical entropy is in general not invariant under orbit equivalences, recent works have shown that various notions of restricted orbit equivalences will preserve entropy. We focus on the case where the orbit equivalence is Shannon, and both groups are finitely generated amenable. In this talk, we will present a proof for our main result.

- May 23: Joshua Bowman (Pepperdine University)

Location: AP&M 7321

Title: Horocycle flow on H(2) and the gap distribution for slopes of saddle connections

Abstract: Saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of SL_2(R) on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a Poincaré section for the horocycle flow. In this talk, we will describe a Poincaré section for horocycle flow in the smallest nontrivial stratum $H(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.

- May 30: Carlos Ospina (University of Utah)

Location: AP&M 7321

- June 6: