UC San Diego Group Actions Seminar

In-person meetings are held in AP&M 7321 and Zoom meetings are held here.

If you would like to be included or removed from our email announcements, please email Brandon Seward.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward

Current Quarter      Past Quarters

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• September 23: Felipe García-Ramos (Universidad Autónoma de San Luis Potosí)
  
  Title: Local entropy theory and descriptive complexity
  
  Abstract: We will give an introduction to local entropy theory and we will trace the descriptive complexity of different families of topological dynamical systems with completely positive entropy (CPE) and uniform positive entropy (UPE). Joint work with Udayan B. Darji.
  
  video
  
  
• September 30: Forte Shinko (California Institute of Technology)
  
  Title: Realizations of equivalence relations and subshifts
  
  Abstract:Every continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results for variants of this problem, and we investigate the connection to subshifts.
  
  video, slides
  
  
• October 7: Riley Thornton (UCLA)
  
  Title: Cayley Diagrams and Factors of IID Processes
  
  Abstract:A Cayley diagram is a labeling of a graph $G$ that encodes an action of a group which induces $G$.  For instance, a $d$-edge coloring of a $d$-regular tree is a Cayley diagram for the group $(\mathbb{Z}/2\mathbb{Z})^{*d}$.  In this talk, we will investigate when a Cayley graph $G=(\Gamma, E)$ admits an $\operatorname{Aut}(G)$-f.i.i.d. Cayley diagram and show that $\Gamma$-f.i.i.d. solutions to local labeling problems for such graphs lift to $\operatorname{Aut}(G)$-f.i.i.d. solutions.
  
  video
  
  
• October 14: Anton Bernshteyn (Georgia Institute of Technology)
  
  Title: Equivariant maps to free and almost free subshifts
  
  Abstract: Let $\Gamma$ be a countably infinite group. Seward and Tucker-Drob proved that every free Borel action of $\Gamma$ on a Polish space $X$ admits a Borel equivariant map $\pi$ to the free part of the Bernoulli shift $k^\Gamma$, for any $k \geq 2$. Our goal in this talk is to generalize this result by putting extra restrictions on the image of $\pi$. For instance, can we ensure that $\pi(x)$ is a proper coloring of the Cayley graph of $\Gamma$ for all $x \in X$? More generally, can we guarantee that the image of $\pi$ is contained in a given subshift of finite type?
  
  The main result of this talk is a positive answer to this question in a very broad (and, in some sense, optimal) setting. The main tool used in the proof of our result is a probabilistic technique for constructing continuous functions with desirable properties, namely a continuous version of the Lovȧsz Local Lemma.
  
  video
  
  
• October 21: Pratyush Sarkar (Yale University)
  
  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.
  
  video
  
  
• October 28: Wooyeon Kim (ETH Zurich)
  
  Title: Effective equidistribution of expanding translates in $ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$
  
  Abstract: In this talk, we discuss effective versions of Ratner’s theorems in the space of affine lattices. For $d \geq 2$, let $Y=ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$, $H$ be a minimal horospherical group of $SL_d(\mathbb{R})$ embedded in $ASL_d(\mathbb{R})$, and $a_t$ be the corresponding diagonal flow. Then $(a_t)$-push-forwards of a piece of $H$-orbit become equidistributed with a polynomial error rate under certain Diophantine condition of the initial point of the orbit. This generalizes the previous results of Strömbergsson for $d = 2$ and of Prinyasart for $d = 3$.
  
  video
  
  
• November 4: Aaron Calderon (Yale University)
  
  Title: Random hyperbolic surfaces via random flat surfaces
  
  Abstract: What does it mean to pick a “random” hyperbolic surface, and how does one even go about “picking” one? Mirzakhani gave an inductive answer to this question by gluing together smaller random surfaces along long curves; this is equivalent to studying the equidistribution of certain sets inside the moduli space of hyperbolic surfaces. Starting from first concepts, in this talk I’ll explain a new method for building random hyperbolic surfaces by building random flat ones. As time permits, we will also discuss the application of this technique to Mirzakhani’s “twist torus conjecture.” This is joint work (in progress) with James Farre.
  
  video
  
  
• November 18: Sunrose Shrestha (Wesleyan University)
  
  Title: Periodic straight-line flows on the Mucube
  
  Abstract: The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to- edge via translations) has been widely studied due to connections to polygonal billiards and Teichmüller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on the Mucube – an infinite $\mathbb{Z}^3$ periodic half-translation square-tiled surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions for the straight-line flow on the Mucube – in terms of a subgroup of $\mathrm{SL}_2 \mathbb{Z}$. We will use the latter characterization to obtain the group of derivatives of affine diffeomorphisms of the Mucube.  This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.
  
  video
  
  
• December 2: Josh Southerland (University of Washington)
  
  Title: Towards a shrinking target property for primitive square-tiled surfaces
  
  Abstract: In this talk, I will discuss ongoing work to develop a method for proving a shrinking target property on primitive square-tiled surfaces that comes from the action of a subgroup $G$ of its Veech group. Our main tool is the construction of a Fourier-like transform which we can use to relate the $L^2$-norm of the Koopman operator induced by $G$ to the $L^2$-norm of a Markov operator related to a random walk on $G$.