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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• January 12: David Aulicino (Brooklyn College and the CUNY Graduate Center)
  
  Title: Siegel-Veech Constants of Cyclic Covers of Generic Translation Surfaces
  
  Abstract: We consider generic translation surfaces of genus g>0 with n>1 marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.
  
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• January 19: Karl Winsor (Fields Institute)
  
  Title: Uniqueness of the Veech 14-gon
  
  Abstract: Teichmüller curves are algebraic curves in the moduli space of genus g Riemann surfaces that are isometrically immersed for the Teichmüller metric. These curves arise from SL(2,R)-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. A Teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree g. In genus 2, Calta and McMullen independently discovered an infinite family of algebraically primitive Teichmüller curves. However, in higher genus, such curves seem to be much rarer. We will discuss a result that shows that the regular 14-gon yields the unique algebraically primitive Teichmüller curve in genus 3 of a particular combinatorial type. All relevant notions will be explained during the talk.
  
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• January 26: Samantha Fairchild (Max Planck Institute)
  
  Title: Shrinking rates of horizontal gaps for generic translation surfaces
  
  Abstract: A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.
  
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• February 2: Tina Torkaman (Harvard University)
  
  Title: Intersection number and intersection points of closed geodesics on hyperbolic surfaces
  
  Abstract: In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.
  
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• February 7 (Tuesday) at 11 AM: Jingyin Huang (Ohio State University)
  
  (Joint with Functional Analysis Seminar)
  
  Location: AP&M 6402
  
  Title: Integral measure equivalence versus quasi-isometry for some right-angled Artin groups
  
  Abstract: Recall that two finitely generated groups G and H are quasi-isometric, if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We study the relation between these two notions in the class of right-angled Artin groups. In this talk, we show if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. Interestly, this class of groups are rigid for a reason which is quite different from other cases of measure equivalence rigidity. We will also do a quick survey of relevant measure equivalence rigidity and quasi-isometric rigidity results of other classes of groups, motivating our choice of right-angled Artin groups as a playground. This is joint work with Camille Horbez.
  
  
• February 9: Gil Goffer (UCSD)
  
  Location: AP&M 7218 and Zoom
  
  Title: compact URS and compact IRS
  
  Abstract: I will discuss compact uniformly recurrent subgroups and compact invariant random subgroups in locally compact groups, and present results from ongoing projects with Pierre-Emanuel Caprace and Waltraud Lederle, and with Tal Cohen.
  
  
• February 16: Or Landesberg (Yale University)
  
  Location: AP&M 7218 and Zoom
  
  Title: Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces
  
  Abstract: Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner's theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.
  
  
• February 23: Homin Lee (Northwestern University)
  
  Location: AP&M 7218 and Zoom
  
  Title: Higher rank lattice actions with positive entropy
  
  Abstract: We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in SLnR (n is at least 3). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most (n-1), the action is either isometric or projective. Both cases, we don't have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed n-manifold by a lattice in SLnR (n is at least 3) then the lattice should be commensurable with SLnZ. This is the work in progress with Aaron Brown.
  
  
• March 2: Félix Lequen (Cergy-Pontoise University)
  
  Title: Bourgain's construction of finitely supported measures with regular Furstenberg measure
  
  Abstract: The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalisations of the Cantor middle third set, and then the case of random iterations of matrices in SL(2, R) acting on the real projective line, where the stationary measure is unique under certain conditions, and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has a finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. I will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups.
  
  
• March 9: Zvi Shem-Tov (Institute for Advanced Study)
  
  Title: Arithmetic Quantum Unique Ergodicity for 3-dimensional hyperbolic manifolds
  
  Abstract: The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk I will discuss a recent work on this problem for arithmetic quotients of the three dimensional hyperbolic space. I will discuss our key result that Hecke eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman.
  
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• March 16: Emilio Corso (University of British Columbia, Vancouver)
  
  Title: Asymptotic behaviour of expanding circles on compact hyperbolic surfaces
  
  Abstract: Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis' thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underlying this linkage allows to transfer mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behaviour of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem; predictably, the latter fail to improve upon the currently best known bound, achieved via finer methods by Selberg more than half a century ago.
  
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