University of California, San Diego.

Date  Speaker  Topic 
April 16 
Dubi Kelmer
Boston College 
Shrinking target problems, homogenous dynamics and Diophantine approximations
Abstract: The shrinking target problem for a dynamical system tries to answer the question of how fast can a sequence of targets shrink so that a typical orbit will keep hitting them indefinitely. I will describe some new and old results on this problem for flows on homogenous spaces, with various applications to problems in Diophantine approximations. 
April 30 
Diaaeldin Taha
University of Washington 
On Cross Sections to the Horocycle and Geodesic Flows on Quotients of \({\rm SL}(2, \mathbb{R})\) by Hecke Triangle Groups \(G_q\), \(G_q\)BCZ Map, and Symmetric \(G_q\)Farey Map.
Abstract: In this talk, we explore explicit cross sections to the horocycle and geodesic flows on \(\operatorname{SL}(2, \mathbb{R})/G_q\), with \(q \geq 3\). Our approach relies on extending properties of the primitive integers \(\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}\) to the discrete orbits \(\Lambda_q := G_q (1, 0)^T\) of the linear action of \(G_q\) on the plane \(\mathbb{R}^2\). We present an algorithm for generating the elements of \(\Lambda_q\) that extends the classical SternBrocot process, and from that derive another algorithm for generating the elements of \(\Lambda_q\) in planar strips in increasing order of slope. We parametrize those two algorithm using what we refer to as the symmetric \(G_q\)Farey map, and \(G_q\)BCZ map, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on \(\operatorname{SL}(2, \mathbb{R})/G_q\) to particular cross sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the \(G_q\)BCZ and symmetric \(G_q\)Farey maps. This talk is selfcontained, and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics. 
May 28 
Michael Shulman
University of San Diego 
All \((\infty, 1)\)toposes have strict univalent universes
Abstract: We prove the conjecture that any Grothendieck \((\infty,1)\)topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to \((\infty,1)\)toposes, just as higherorder logic is used for 1toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2monad theory. 
Wednesday, June 5, 3:304:30p APM 6402 
Asaf Katz
University of Chicago 
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 noncompact, 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, leading to a quantified equidistribution result for a large class of points, akin to the results of A. Strombergsson regarding the SL2 case. In particular we deduce a fully effective quantitative equidistribution statement for horospherical trajectories of lattices defined over number fields. 
Thursday, June 6, 4:005:00p APM 7218 
Asaf Katz
University of Chicago 
Quantitative disjointness of nilflows and horospherical flows.
Abstract: In his influential disjointness paper, H. Furstenberg proved that weaklymixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh's disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by GreenTaoZiegler. If time permits, I will discuss applications of such theorems in sparse equidistribution problems and number theory. 