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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• December 15: Tsviqa Lakrec (The Hebrew University)

  
  Title:   Equidistribution of affine random walks on some nilmanifolds

  Abstract:   We consider the action of the group of affine transformations on nilmanifolds. Given a probability measure on this group and a starting point x, a random walk on the nilmanifold is defined. Consider the distribution of the point after m random steps. We show that under certain assumptions, that hold for Heisenberg nilmanifolds for example, the distribution of this point converges to the Haar measure on the nilmanifold as m goes to infinity, unless there is the obvious obstruction that the orbit closure of x by the semigroup generated by the support of the random walk measure is a finite homogeneous union of affine sub-nilmanifolds. Furthermore, this result is quantitative and gives a rate for the convergence to Haar measure (equidistribution) depending on how close the starting point and random walk measure are to such an obstruction. This talk is based on joint works with Weikun He and Elon Lindenstrauss.

  slides   video

• December 8: Yotam Smilansky (Rutgers University)

  
  Title:   Multiscale substitution tilings

  Abstract:   Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

  slides   video

• December 1 (Joint with the Functional Analysis Seminar): Remi Boutonnet (CNRS/Bordeaux University)

  
  Title: Stationary actions of higher rank lattices on non-commutative spaces

  Abstract: I will present new results about stationary actions of higher rank semi-simple lattices on compact spaces, in the spirit of Nevo and Zimmer's work. Then I will explain how these results generalize to stationary actions on C*-algebras (i.e. "non-commutative" spaces) and give consequences about unitary representations of these lattices and their characters. All these results can be seen as generalizations of Margulis normal subgroup theorem at different levels. This is based on joint works with Cyril Houdayer, Uri Bader and Jesse Peterson.

• November 24: Christopher Shriver (UC Los Angeles)

  
  Title: Sofic entropy and the (relative) f-invariant

  Abstract:  In this talk I will explain an interpretation (due to Lewis Bowen) of the f-invariant as a variant of sofic entropy: it is the exponential growth rate of the expected number of “good models” for an action over a random sofic approximation. I will then introduce the relative f-invariant and provide a similar interpretation of this quantity. This provides a formula for the growth rate of the expected number of good models over a type of stochastic block model.

  slides   video

• November 17: Brandon Seward (UC San Diego)

  
  Title:  An introduction to the f-invariant

  Abstract:  The f-invariant was introduced by Lewis Bowen in 2008 and is a real-valued isomorphism invariant that is defined for a large class of probability measure-preserving actions of finite-rank free groups. Most notably, the f-invariant provided the first classification up to isomorphism of Bernoulli shifts over finite-rank free groups. It is also quite useful for the study of finite state Markov chains with values indexed by a finite-rank free group. The f-invariant is conceptually similar to entropy, and it has a formal connection to sofic entropy. In this expository talk, I will introduce the f-invariant and discuss some of its basic properties.

• November 10: Octave Lacourte (Université Lyon 1)

  
  Title:  A signature for some subgroups of the permutation group of [0,1[.

  Abstract: For every infinite set X we define S(X) as the group of all permutations of X. On its subgroup consisting of all finitely supported permutations there exists a natural group homomorphism signature. However, thanks to an observation of Vitali in 1915, we know that this group homomorphism does not extend to S(X). In the talk we extend the signature on the subgroup of S(X) consisting of all piecewise isometric elements (strongly related to the Interval Exchange Transformation group). This allows us to list all of its normal subgroups and gives also informations about an element of the second cohomology group of some slides

  slides   video

• October 27: Jacqueline Warren (UC San Diego)

  
  Title:  Effective equidistribution of horospherical flows in infinite volume

  Abstract:  By Ratner's famous equidistribution theorem, we know that unipotent orbits in finite volume quotients of Lie groups equidistribute in their closures. Often, in applications, one needs to know more: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is joint work with Nattalie Tamam.

  slides   video

• October 20: Anthony Sanchez (University of Washington)

  
  Title:  Gaps of saddle connection directions for some branched covers of tori

  Abstract:  Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. A natural question to ask is: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.

  slides   video