Spring 2024 Titles and Abstracts

Date: 29 February 2024
Speaker: Lei Yang
Title: Incidence geometry and effective equidistribution in homogeneous dynamics
Abstract: I will explain my proof of an effective version of Ratner’s equidistribution theorem for unipotent orbits in SL(3,R)/SL(3,Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the proof is based on a bootstrapping argument improving the local dimension of measures generated by unipotent orbits. The key is to relate the behavior of the unipotent orbits to a Kakeya model.

Date: 7 March 2024
Speaker: Gaurav Aggarwal
Title: Joint Equidistribution of Approximates
Abstract: The distribution of integer points on varieties has occupied mathematicians for centuries. In the 1950’s Linnik used an “ergodic method” to prove the equidistribution of integer points on large spheres under a congruence condition. As shown by Maaß, this problem is closely related to modular forms. Subsequently, there were spectacular developments both from the analytic as well as ergodic side. I will discuss a more refined problem, namely the joint distribution of lattice points in conjunction with other arithmetic data. An example of such data is the “shape” of an associated lattice, or in number theoretic language, a Heegner point. In a completely different direction, a “Poincaré section” is a classical and useful tool in ergodic theory and dynamical systems. Recently, Shapira and Weiss, constructed a Poincaré section for the geodesic flow on the moduli space of lattices to study joint equidistribution properties. Their work in fact is very general but crucially uses the fact that the acting group has rank one. In joint work with Anish Ghosh, we develop a new method to deal with actions of higher rank groups. I will explain this and, if time permits, some corollaries in Diophantine analysis.

Date: 14 March 2024
Speaker: Shreyasi Datta
Title: Bad is null via constant invariance
Abstract: The set of badly approximable vectors in Diophantine approximation plays a significant role. In a recent work with Victor Beresnevich, Anish Ghosh, and Ben Ward, we developed a general framework to show a `constant invariance’ property for a large class of limsup sets of neighbourhoods of subsets of a metric measure space. As a consequence, we get that the set of badly approximable points has measure zero in a metric space equipped with certain natural measures. In particular, given any C^2 manifold, we show almost every point is not badly approximable.

Date: 11 April 2024
Speaker: Carsten Peterson
Title: Quantum ergodicity on the Bruhat-Tits building for PGL(3) in the Benjamini-Schramm limit
Abstract: Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which “converge” to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.

Date: 18 April 2024
Speaker: Nikolay Moshchevitin
Title: Bounded ratios and badly approximability
Abstract: We will discuss relatively new criteria of badly approximability in terms of ratios of best approximations. Let qν be convergents of continued fractions to real irrational α. It is well known that

α is badly approximable   iff   supν qν+1/qν is finite   iff   infν||qν+1α||/||qνα||>0.

We will discuss how this property may be generalised to Diophantine Approximation in higher dimensions. The answer seems to be rather non-trivial. Some of the related properties may be expressed in terms of Parametric Geometry of Numbers recently developed by Schmidt, Summerer, Roy and the others. Also we discuss some properties of ratios under the consideration in accordance with the study of multidimensional Dirichlet spectra.