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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• March 31:Andy Zucker (University of California San Diego)
  
  Location: AP&M 6402 and on Zoom
  
  Title: Minimal subdynamics and minimal flows without characteristic measures
  
  Abstract: Given a countable group G and a G-flow X, a probability measure on X is called characteristic if it is Aut(X, G)-invariant. Frisch and Tamuz asked about the existence, for any countable group G, of a minimal G-flow without a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection F of infinite subgroups of G, when is there a faithful G-flow for which the action restricted to any member of F is minimal? Joint with Joshua Frisch and Brandon Seward.
  
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• April 7:Frank Lin (Texas A&M University)
  
  Title: Entropy for actions of free groups under bounded orbit equivalence
  
  Abstract: Joint work with Lewis Bowen.  The f-invariant is a notion of entropy for probability measure preserving (pmp) actions of free groups.  It is invariant under measure conjugacy and has some similarities to Kolmogorov-Sinal entropy.  Two pmp actions are orbit equivalent if their orbits can be matched almost everywhere in a measurable fashion.  Although entropy in general is not invariant under orbit equivalence, we show that the f-invariant is invariant under the stronger notion of bounded orbit equivalence.
  
  
• April 14:Srivatsa Srinivas (University of California San Diego)
  
  Location: AP&M 6402 and on Zoom
  
  Title: An Escaping Lemma and its implications
  
  Abstract: Let $\mu$ be a measure on a finite group $G$. We define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z}) \times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. Further suppose that the support of $\mu$ generates $G$. Then we show that there are constants $L, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq \frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its implications. This talk is based on joint work with Professor Alireza Salehi-Golsefidy.
  
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• April 21:Seonhee Lim (Seoul National University)
  
  Location: AP&M 6402 and on Zoom
  
  Title: Complex continued fractions and central limit theorem for rational trajectories
  
  Abstract: In this talk, we will first introduce the complex continued fraction maps associated with some imaginary quadratic fields (d=1,2,3,7,11) and their dynamical properties. Baladi-Vallee analyzed (real) Euclidean algorithms and proved the central limit theorem for rational trajectories and a wide class of cost functions measuring algorithmic complexity. They used spectral properties of an appropriate bivariate transfer operator and a generating function for certain Dirichlet series whose coefficients are essentially the moment generating function of the cost on the set of rationals. We extend the work of Baladi-Vallee for complex continued fraction maps mentioned above. (This is joint work with Dohyeong Kim and Jungwon Lee.)
  
  
• April 28: Osama Khalil (University of Utah)
  
  Location: Zoom
  
  Title: Mixing, Resonances, and Spectral Gaps on Geometrically Finite Manifolds
  
  Abstract: I will report on work in progress showing that the geodesic flow on any geometrically finite, rank one, locally symmetric space is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure of maximal entropy. The method is coding-free and is instead based on a spectral study of transfer operators on suitably constructed anisotropic Banach spaces, ala Gouezel-Liverani, to take advantage of the smoothness of the flow. As a consequence, we obtain more precise information on the size of the essential spectral gap as well as the meromorphic continuation properties of Laplace transforms of correlation functions.
  
  
• May 5: Matthew Welsh (University of Bristol)
  
  Location: AP&M 6402 and on Zoom
  
  Title: Bounds for theta sums in higher rank
  
  Abstract: In joint work with Jens Marklof, we prove new upper bounds for theta sums -- finite exponential sums with a quadratic form in the oscillatory phase -- in the case of smooth and box truncations. This generalizes results of Fiedler, Jurkat and Körner (1977) and Fedotov and Klopp (2012) for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio (2015). Key inputs in our approach include the geometry of Sp(n, Z) \ Sp(n, R), the automorphic representation of theta functions and their growth in the cusp, and the action of the diagonal subgroup of Sp(n, R).
  
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• May 12: Yair Hartman (Ben-Gurion University)
  
  Location: Zoom
  
  Title: Tight inclusions
  
  Abstract: We discuss the notion of "tight inclusions" of dynamical systems which is meant to capture a certain tension between topological and measurable rigidity of boundary actions, and its relevance to Zimmer-amenable actions. Joint work with Mehrdad Kalantar.
  
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• May 19: Robin Tucker-Drob (University of Florida)
  
  Location: AP&M 6402 and on Zoom
  
  Title: Amenable subrelations of treed equivalence relations and the Paddle-ball lemma
  
  Abstract: We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.
  
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• May 26: Dami Lee (University of Washington)
  
  Location: AP&M 6402 and on Zoom
  
  Title: Computation of the Kontsevich--Zorich cocycle over the Teichmüller flow
  
  Abstract: In this talk, we will discuss the dynamics on Teichmüller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the SL(2,R)-orbit on the moduli space. To study the dynamics of Teichmüller flow of the SL(2,R)-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the KZ monodromy. This is part of an ongoing work with Anthony Sanchez.
  
  
• June 2: Israel Morales Jimenez (Universidad Nacional Autónoma de México)
  
  Location: Zoom
  
  Title: Big mapping class groups and their conjugacy classes
  
  Abstract: The mapping class group, Map(S), of a surface S, is the group of all isotopy classes of homeomorphisms of S to itself. A mapping class group is a topological group with the quotient topology inherited from the quotient map of Homeo(S) with the compact-open topology. For surfaces of finite type, Map(S) is countable and discrete. Surprisingly, the topology of Map(S) is more interesting if S is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a Polish group. In recent literature, this last class of groups is called “big mapping class groups”. In this talk, I will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. This was a joint work with Jesús Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.
  
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