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: TBA
Abstract: TBA

Date: 22 March 2021
Speaker: Peter Varju
Title: TBA
Abstract: TBA

Date: 29 March 2021
Speaker: TBA
Title: TBA
Abstract: TBA

Date: 5 April 2021
Speaker: Han Yu
Title: TBA
Abstract: TBA

Date: 12 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: 19 April 2021
Speaker: Asaf Katz
Title: TBA
Abstract: TBA

Date: 26 April 2021
Speaker: Tushar Das
Title: TBA
Abstract: TBA

Date: 3 May 2021
Speaker: Pratyush Sarkar
Title: TBA
Abstract: TBA

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.