# Courses

Dmitry Dolgopyat (University of Maryland), Title: "Stochasticity of circle rotations",

Abstract: We review the basic tools for studying statistical properties of deterministic systems and then apply these tools for the study of circle rotations which are the most regular dynamical systems. Multidimensional extensions and open questions will also be presented.

Maksym Radziwill (Northwestern University), Title: "Fine distribution of sequences modulo 1",

Lecture 1. We will focus on a number of techniques for establishing the equidistribution of sequences modulo one. Specifically Weyl differencing, Poisson summation and expansion in Fourier series.

Lecture 2. We will discuss the inherent limitations of the above techniques when it comes to problems about the gap distribution of sequences modulo 1. We will also introduce a number of elementary techniques are be useful in the gap distribution context, including a variant of the circle method.

Lecture 3. We will tie together the outcome of Lecture 1 and 2 by establishing a result of Elkies and McMullen on the gap distribution of square roots of integers modulo 1. (Based on joint work with Nicolas Technau).

Andreas Strömbergsson (Uppsala University), Title: "Effective equidistribution in homogeneous dynamics via techniques from analytic number theory",

Lecture 1. Equidistribution of long closed horocycles on hyperbolic surfaces. Let Γ ⊂ SL(2, R) be a cofinite Fuchsian group, and let M = Γ\H2 be the corresponding hyperbolic surface. I will discuss how techniques using the spectral expansion of L2(M ), involving Maass waveforms and Eisenstein series, lead to precise results on the equidistribution of long closed horocycles, and subsegments thereof.

Lecture 2. Representation theoretic techniques I will discuss how representation theoretic techniques can be used to prove stronger and more general equidistribution results than those discussed in Lecture 1. I will focus on the case of SL(2, R), but will also briefly survey the case of general Lie groups.

Lecture 3. Equidistribution in the space of 2-dimensional tori with k marked points. Let G = SL(2, R) ⋉ (R2)⊕k and Γ = SL(2, Z) ⋉ (Z2)⊕k. I will discuss results on the effective equidistribution of unipotent orbits in Γ\G, the proofs of which build on bounds on exponential sums and the circle method.

Matthew Welsh (University of Maryland), Title: "The dynamics of theta sums".

Abstract: Exponential sums with a quadratic phase, which we call theta sums, have a long history in mathematics and their importance continues today. Gauss and Jacobi established their importance to number theory and later work, by Weyl in particular, saw their importance to physics. Today, theta sums are a fundamental tool for understanding general exponential sums, the spectra of tori, Schrodinger operators, and many other objects of interest. In this mini-course, we will approach theta sums from a modern perspective by relating them to dynamics on the symplectic group by means of the oscillator, or Shale-Weil, representation. In lecture one, we review some of the subject's classical origins and begin our discussion of the oscillator representation in one variable. In lecture two, we finish this discussion and revisit some of the classical results from this new perspective. In lecture three, we outline the theory in several variables and discuss some applications.