Speaker: Vitaly Bergelson (Ohio State)

Title: Translations on Nilmanifolds and Uniform Distribution of Generalized Polynomials

Abstract: "Generalized polynomials" are functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition, and multiplication. After briefly reviewing the connection between the generalized polynomials and dynamical systems on nilmanifolds, we will discuss a recent joint work with Inger Haaland Knutson and Younghwan Son on a generalization of the classical Weyl polynomial equidistribution result to generalized polynomials.

Speaker: Florin Boca (University of Illinois at Urbana-Champaign)

Title: Periodic points of some Gauss type shifts

Abstract: Many ergodic properties of regular continued fractions are captured by the Gauss shift, a well-studied transformation in ergodic theory. Its periodic points coincide with the reduced quadratic irrational numbers, known to be related with the Pell equation and with closed geodesics on the modular surface. Classical work of Pollicott proved uniform distribution of these numbers with respect to the Gauss probability measure, when ordered by their associated closed primitive geodesic length. An effective version was established more recently by Ustinov. This talk will discuss the distribution of periodic points of Gauss type shifts arising from other types of continued fractions. This is joint work with M. Siskaki.

Speaker: Daniel El-Baz (TU Graz)

Title: Applications of (effective) joint equidistribution of rational points on expanding horospheres.

Abstract: This talk is based on a joint work with Bingrong Huang and Min Lee as well as ongoing work with Min Lee and Andreas Strömbergsson. I will focus on the applications of our main theorems to the following seemingly disparate topics: the typical value of Frobenius numbers, the typical value of certain metric parameters attached to some Cayley graphs, the distribution of 'shapes' of unimodular lattices and a precise counting result concerning the number of small solutions to systems of linear congruences.

While I will present the main theorems, details about the methods involved in the proof will be given in Min Lee's talk on Wednesday.

Speaker: Alexander Gorodnik (University of Zürich)

Title: Diophantine approximation on semisimple groups

Abstract: We discuss the problem of counting solutions of Diophantine inequalities on a semisimple groups and explain how this problem can be studied using harmonic analysis on homogeneous spaces. It turns out that such Diophantine approximation results are intimately connected with the spectral gap property of the automorphic representation. In particular, this provides a new approach to establishing the spectral gap property.

Speaker: Dmitry Kleinbock (Brandeis)

Title: Exceptional orbits on homogeneous spaces and their applications.

Abstract

Speaker: Min Lee (Bristol)

Title: Effective joint equidistribution of rational points on expanding horospheres

Abstract: In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016). Their proof uses techniques from homogeneous dynamics and relies particularly on measure-classification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger's formula, and the spectral theory of automorphic functions. Our methods yield an effective estimate on the rate of convergence for a specific horospherical subgroup in any dimension.

This is a joint work with D. El-Baz, B. Huang, J. Marklof and A. Strömbergsson.

Speaker: Seonhee Lim (Seoul National University)

Title: Effective Hausdorff dimension of epsilon bad sets for function fields

Abstract: In this talk, we will introduce the inhomogeneous Diophantine approximation over the completion of a global function field (over a finite field) for a discrete valuation. We obtain an effective upper bound for the Hausdorff dimension of the set of epsilon-badly approximable targets for a fixed matrix using an effective version of entropy rigidity in homogeneous dynamics for appropriate diagonal action on the space of grids. We further characterize matrices for which the epsilon-bad set has full Hausdorff dimension for some epsilon by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances. The talk is mainly based on joint work with Frederic Paulin and Taehyeong Kim.

Speaker: Jens Marklof (Bristol)

Title: Geodesic random line processes and the roots of quadratic congruences

Abstract: In 1963 Christopher Hooley showed that the roots of a quadratic congruence mod m, appropriately normalized and averaged, are uniformly distributed mod 1. In this lecture, which is based on joint work with Matthew Welsh (Bristol), we will study pseudo-randomness properties of the roots on finer scales and prove, for instance, that the pair correlation density converges to an intriguing limit. A key step in our approach is to translate the problem to convergence of certain geodesic random line processes in the hyperbolic plane, which in turn exploits equidistribution properties of horocycle flows.

Speaker: Andreas Strömbergsson (Uppsala)

Title: The low-density Lorentz gas in a union of grids

Abstract: The Lorentz gas describes an ensemble of noninteracting point particles in an infinite array of spherical scatterers. In recent joint work with Jens Marklof, we developed a framework for proving, for a given deterministic scatterer configuration, the convergence of the particle dynamics to a limiting transport process, in the limit of low scatterer density. In the present talk I will discuss joint work with Matthew Palmer on the particular case when the scatterer configuration is an arbitrary finite union of (possibly shifted) Euclidean lattices.

Speaker: Jimmy Tseng (Exeter)

Title: Shrinking target horospherical equidistribution via translated Farey sequences

Abstract: Consider a finite-volume space with cusps and a geometric object that, under a flow, equidistributes on that space. Shrinking target equidistribution, if it is possible, is the equidistribution of such an object on a target shrinking into the cusps. For the space of unimodular lattices SL(d, Z) \ SL(d, R) and a certain diagonal flow, we will discuss shrinking target horospherical equidistribution (STHE). Here, the object is the horosphere or a translated horosphere.

The proofs of STHE for the horosphere rely on properties of Farey sequences, especially the equidistribution of Farey sequences on distinguish sections, and on a renormalization technique. Likewise, the proofs of STHE for a translated horosphere rely on the renormalization technique and on the equidistribution of an analogous sequence, which we define and refer to as a translated Farey sequence. These translated Farey sequences generalize the Farey sequence and have similar properties such as equidistribution on the same distinguished sections, and it is this equidistribution that makes the renormalization technique possible.

These results extend results (referred to as shrinking target horocycle equidistribution) for periodic horocycles on L \ PSL(2, R) where L is any cofinite Fuchsian group with at least one cusp.

Speaker: Barak Weiss (Tel-Aviv)

Title: Geometric and arithmetic aspects of approximation vectors

Abstract: I will describe several natural questions that arise in connection with the sequence of best approximation vectors. This sequence, which I will introduce in the talk, is a higher dimensional generalization of continued fraction convergents (which I will also discuss briefly). The questions involve the statistical behavior of certain observables, and the means to understand them involve a mix of number theory, ergodic theory, and elementary geometry. The talk will be based on joint work with Uri Shapira.

Speaker: Shucheng Yu (Uppsala)

Title: The light cone Siegel transform, its moment formulas and applications

Abstract: The classical Siegel transform is a transform which takes functions on the Euclidean space to functions on the space of lattices. In this talk I will discuss a new type of Siegel transform where the role of the Euclidean space is replaced by the light cone of a certain indefinite integral quadratic form. In this setting one can use the spectral theory of incomplete Eisenstein series to prove explicit first and second moment formulas for this transform, generalizing the classical results of Siegel and Rogers. I'll also discuss some applications of our moment formula to various counting problems, including one on intrinsic Diophantine approximations on spheres. This is work in progress with Dubi Kelmer.