# Spring 2021 Titles and Abstracts

Date: 1 February 2021
Speaker: Cagri Sert
Title: Expanding measures and random walks on homogeneous spaces
Abstract: We will start by reviewing some recent works on random walks on homogeneous spaces. We will continue by discussing the notion of a H-expanding probability measure on a connected semisimple Lie group H, that we introduce inspired by these developments. As we shall see, for a H-expanding µ with H < G, on the one hand, one can obtain a description of µ-stationary probability measures on the homogeneous space G/Λ using the measure classification results of Eskin– Lindenstrauss, and on the other hand, the recurrence techniques of Benoist–Quint can be generalized to this setting. As a result, we will deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and some epimorphic subgroups of H. If time allows, we will see how, using an idea of Simmons–Weiss, this allows also us to deduce Birkhoff genericity of a class of fractal measures with respect to expanding diagonal actions. Joint work with Roland Prohaska and Ronggang Shi.

Date: 8 February 2021
Speaker: Barak Weiss
Title: Classification and statistics of cut-and-project sets
Abstract: We introduce a class of so-called “Ratner-Marklof-Strombergsson measures”. These are probability measures supported on cut-and-project sets in Euclidean space of dimension d>1 which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R) (affine or linear maps preserving orientation and volume). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets. Joint work with Rene Ruehr and Yotam Smilansky.

Date: 22 February 2021
Speaker: Tsviqa Lakrec
Title: Equidistribution of affine random walks on some nilmanifolds
Abstract: We consider the action of the group of affine transformations on a nilmanifold. Given a probability measure on this group and a starting point, a random walk on the nilmanifold is defined. We study quantitative equidistribution in law of such affine random walks on nilmanifolds. Under certain assumptions, we show that a failure to have fast equidistribution on a nilmanifold is due to a failure on some factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds, such as Heisenberg nilmanifolds.
This talk is based on joint works with Weikun He and Elon Lindenstrauss.

Date: 1 March 2021
Speaker: TBA
Title: TBA
Abstract: TBA

Date: 8 March 2021
Speaker: Minju Lee
Title: Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends
Abstract: This is joint work with Hee Oh. We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with Fuchsian ends. For $d = 3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\geq 1$,
(1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold;
(2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold;
(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\mathrm{core} M$ becomes dense in $M$.

Date: 15 March 2021
Speaker: Maxim Kirsebom
Title: Towards an extreme value law for the deepest cusp excursions of the unipotent flow
Abstract: The unipotent flow on the unit tangent bundle of the modular surface is a classic example of a homogeneous flow when understood through the identification with PSL_2(R)/PSL_2(Z). The ergodicity of the flow implies that almost every orbit is dense in the space and hence must eventually make excursions deeper and deeper into the cusp. We are interested in understanding the nature of these excursions. In the described setting, and more generally, Athreya and Margulis proved that the maximal excursions obey the logarithm law almost surely, meaning that their growth rate scales the logarithm of the time. In this work we focus on a more precise description of this behaviour, namely determining the probability that the deepest excursion fails to outperform the expected asymptotic behaviour by an additive amount. This question may be phrased in the language of extreme value statistics and we establish some results towards a complete extreme value law in this setting. The methods used are based on classical ideas from geometry of numbers. This is work in progress, joint with Keivan Mallahi-Karai.

Date: 22 March 2021
Speaker: Peter Varju
Title: On the dimension of self-similar measures
Abstract: Let f1,…,fn be a collection of contracting similarities on R, and let p1,…,pn be a probability vector. There is a unique probability measure mu on R that satisfies the identity
mu = p1 f1(mu) + … + pn fn(mu).
This measure is called self-similar. The maps f1,…,fn are said to satisfy the no exact overlaps condition if they generate a free semigroup (i.e. all compositions are distinct). Under this condition, the dimension of mu is conjectured to be the minimum of 1 and the ratio of the entropy of p1,…,pn and the average logarithmic contraction factor of the fi. This conjecture has been recently established in some special cases, including when n=2 and f1 and f2 have the same contraction factor. In the talk I will discuss recent progress by Ariel Rapaport and myself in the case n=3. In this case new difficulties arise as was demonstrated by recent examples of Baker and Barany, Kaenmaki of IFS’s with arbitrarily weak separation properties.

Date: 5 April 2021
Speaker: Nicolas Chevallier
Title: Minimal vectors in $\C^2$ and best constant for Dirichlet theorem over $\C$
Abstract: We study minimal vectors in lattices over Gaussian integers in $\C^2$.We show that the index of the sub-lattice generated by two consecutive minimal vectors in a lattice of $\C^2$, can be either $1$ or $2$.Next, we describe the constraints on pairs of consecutive minimal vectors. These constraints  make it possible to find the best constant for Dirichlet theorem about approximations of complex numbers by quotient of Gaussian integers.

Date: 12 April 2021
Speaker: Han Yu
Title: Rational numbers near self-similar sets
Abstract: We will discuss a problem on counting rational numbers near
self-similar sets. In particular, we will show that the set of rational
numbers is ‘reasonably well distributed’ around the middle $p$-th Cantor
set when $p$ is a large integer. Our approach is via Fourier analysis
and we will also discuss some problems on Fourier transform of
self-similar measures which are of independent interest. As a result, it
is possible to show that $p=5$ satisfies the previous statement. The
materials come from various working-in-progress projects with D. Allen,
S. Chow and P. Varju.

Date: 19 April 2021
Speaker: Tushar Das
Title: Using templates to study problems in dynamics and number theory
Abstract: Templates may be viewed as a combinatorial device that helps study
asymptotic properties of lattice successive minima. This simple idea,
introduced in joint work with Lior Fishman, David Simmons, and Mariusz
Urbanski, promises to be useful in several areas beyond our current
applications. The latter lie at the fertile interface along Dani’s
correspondence principle between Diophantine approximation and
homogeneous flows, deepened by Kleinbock & Margulis; and Schmidt &
Summerer’s parametric geometry of numbers, deepened by Roy. Templates
are at the heart of our variational principle (arXiv:1901.06602),
which provides a unified framework to compute the Hausdorff and
packing dimensions of a variety of sets of dynamical and
number-theoretic interest. We will introduce and give some flavor for
our project, hint at a few new directions, and hope to present several
open problems of varying depth to reward participants of this
wonderful seminar!

Date: 26 April 2021
Speaker: Asaf Katz
Title: An application of Margulis’ inequality to effective equidistribution
Abstract: Ratner’s celebrated equidistribution theorem states that the trajectory of any point in a homogeneous space under a unipotent flow is getting equidistributed with respect to some algebraic measure. In the case where the action is horospherical, one can deduce an effective equidistribution result by mixing methods, an idea that goes back to Margulis’ thesis. When the homogeneous space is non-compact, one needs to impose further “diophantine conditions” over the base point, quantifying some recurrence rates, in order to get a quantified equidistribution result. In the talk I will discuss certain diophantine conditions, and in particular I will show how a new Margulis’ type inequality for translates of horospherical orbits helps verify such conditions. This results in a quantified equidistribution result for a large class of points, akin to the results of A. Strombreggson dealing with SL2 case. In particular we deduce a fully effective quantitative equidistribution for horospherical trajectories of lattices defined over number fields, without pertaining to the strong subspace theorem.

Date: 3 May 2021
Speaker: Pratyush Sarkar
Title: Generalization of Selberg’s 3⁄16 theorem for convex cocompact thin subgroups of SO(n, 1)
Abstract: Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak’s breakthrough works initiated many recent developments to generalize Selberg’s theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

Date: 10 May 2021
Speaker: Seungki Kim
Title: Counting problems on a random lattice
Abstract: A random lattice is a random element of SL(n,Z) \ SL(n,R) equipped with the probability measure inherited from the Haar measure of SL(n,R). Analogous to the usual lattice point-counting, one tries to “count” — more precisely, study the statistics of — the random lattice points inside a ball or other shapes. I’ll give a gentle introduction to this topic, discussing the early works of Siegel, Rogers and Schmidt and some of the recent results, as well as their applications.