UC San Diego Group Actions Seminar

Thursdays 10:00 - 10:50 AM

In-person meetings are held in AP&M 7218 and Zoom meetings are held here. Please send an email to one of the organizers to get the the password.

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

Fall 2023 Speakers Past Speakers

Spring 2023

April 13: Zhongkai Tao (UC Berkeley)

Location: AP&M 7218 and Zoom

Title: Fractal uncertainty principle via Dolgopyat's method in higher dimensions

Abstract: The fractal uncertainty principle (FUP) was introduced by Dyatlov and Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in spectral gaps and quantum chaos on hyperbolic manifolds and has recently been an active area of research in harmonic analysis. I will talk about the history of the fractal uncertainty principle and explain its applications to spectral gaps. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a general fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov--Jin using Dolgopyat's method. As an application, we give effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups.
April 27: Srivatsav Kunnawalkam Elayavalli (Institute for Pure and Applied Mathematics, UCLA)

Location: AP&M 7218 and Zoom

Title: Conjugacy for almost homomorphisms of sofic groups

Abstract: I will discuss recent joint work with Hayes wherein we show that any sofic group G that is initially sub-amenable (a limit of amenable groups in Grigorchuk's space of marked groups) admits two embeddings into the universal sofic group S that are not conjugate by any automorphism of S. Time permintting, I will also characterise precisely when two almost homomorphisms of an amenable group G are conjugate, in terms of certain IRS's associated to the two actions of G. One of the applications of this is to recover the result of Becker-Lubotzky-Thom around permutation stability for amenable groups. The main novelty of our work is the usage of von Neumann algebraic techniques in a crucial way to obtain group theoretic consequences.
May 4: Pratyush Sarkar (UCSD)

Location: AP&M 7218 and Zoom

Title: Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Abstract: Let $\Gamma < G = \operatorname{SO}(n, 1)^\circ$ be a Zariski dense torsion-free discrete subgroup for $n \geq 2$. Then the frame bundle of the hyperbolic manifold $X = \Gamma \backslash \mathbb{H}^n$ is the homogeneous space $\Gamma \backslash G$ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup of $G$. Suppose $X$ is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. Endow $\Gamma \backslash G$ with the unique probability measure of maximal entropy called the Bowen-Margulis-Sullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing. The proof uses a countably infinite coding of the flow and the latest version of Dolgopyat's method. To overcome the difficulty in applying Dolgopyat's method due to the cusps of non-maximal rank, we prove a large deviation property for symbolic recurrence to certain large subsets of the limit set of $\Gamma$.

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May 11 Sam Mellick (McGill University)

Location: AP&M 7218 and Zoom

Title: Vanishing of rank gradient for lattices in higher rank Lie groups via cost

Abstract: In 2016 Abért, Gelander, and Nikolov made what they called a provocative conjecture: for lattices in higher rank simple Lie groups, the minimum size of a generating set (rank) is sublinear in the volume. I will discuss our solution to this conjecture. It is a corollary of our main result, where we establish "fixed price one" for a more general class of "higher rank" groups. No familiarity of fixed price or cost is required for the talk. Joint work with Mikołaj Frączyk and Amanda Wilkens.
May 18: Cancelled
May 25: Timothée Bénard (Centre for Mathematical Sciences, Cambridge UK)

Title: Random walks with bounded first moment on finite volume spaces

Abstract: We consider a finite volume homogeneous space endowed with a random walk whose driving measure is Zariski-dense. In the case where jumps have finite exponential moment, Eskin-Margulis and Benoist-Quint established recurrence properties for such a walk. I will explain how their results can be extended to walks with finite first moment. The key is to make sense of the following claim: "the walk in a cusp goes down faster that some iid Markov chain on R with negative mean". Joint work with N. de Saxcé.

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June 1: Etienne BONNAFOUX (École Polytechnique)

Title: Counting of pairs of saddle connections for a typical flat surface of a Sl(2,R)-invariant measure

Abstract: Asymptotic counting of geometric objects has a long history. In the case of flat surfaces, various works by Masur and Veech showed the quadratic asymptotic growth of the number of saddle connections of bounded length. In this spirit, Athreya, Fairchild and Masur showed that, for almost any flat surface, the number of pairs of saddle connection of bounded length and bounded virtual area increases quadratically with the constraint on length. In this case the << almost all >> is with respect to the so-called Masur-Veech measure.

To demonstrate this, they use tools of ergodic theory (hence the result is true almost everywhere). This result can be extended in several ways, giving an error term or extending it to almost any SL(2,R)-invariant measure. We will present several useful tools for tackling these questions.

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June 8: Anthony Sanchez (UCSD)

Location: AP&M 7218 and Zoom

Title: Effective equidistribution of large dimensional measures on affine invariant submanifolds

Abstract: The unstable foliation, that changes the horizontal components of period coordinates, plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle invariant measures.

In this talk we show that measures of large dimension equidistribute in affine invariant manifolds and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang. Background knowledge on translation surfaces and homogenous dynamics will be explained.

Winter 2023

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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Fall 2022

October 6: Andrei Alpeev (Euler International Mathematical Institute)

Title: Amenabilty and random orders

Abstract: An invariant random order is a shift-invariant measure on the space of all orders on a group. It is easy to show that on an amenable group, any invariant random order could be invariantly extended to an invariant random total order. Recently, Glaner, Lin and Meyerovitch showed that this is no longer true for SL_3(Z). I will explain, how starting from their construction, one can show that this order extension property does not hold for non-amenable groups, and discuss an analog of this result for measure preserving equivalence relations.

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October 13: Konrad Wrobel (McGill University)

Title: Orbit equivalence and wreath products

Abstract: We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let F be a nonabelian free group. In particular, we show that the wreath products A ≀ F and B ≀ F are orbit equivalent for any pair of nontrivial amenable groups A, B. This is joint work with Robin Tucker-Drob.

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October 20: Florent Ygouf (Tel Aviv University)

Title: Horospherical measures in the moduli space of abelian differentials

Abstract: The classification of horocycle invariant measures on finite volume hyperbolic surfaces with negative curvature is known since the work of Furstenberg and Dani in the seventies: they are either the Haar measure or are supported on periodic orbits. This problem fits inside the more general problem of the classification of horospherical measures in finite volume homogenous spaces.

In this talk, I will explain how similar questions arise in the moduli space of abelian differentials (and more generally in any affine invariant manifolds) and will discuss a notion of horospherical measures in that context. I will then report on progress toward a classification of those horospherical measures and related topological results. This is a joint work with J. Smillie, P. Smillie and B. Weiss.

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October 27: Elad Sayag (Tel Aviv University)

Title: Entropy, ultralimits and Poisson boundaries

Abstract: In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of G-spaces, and give a new description of the Poisson-Furstenberg boundary of (G,k) as an ultra-limit of G action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

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November 3: Nachi Avraham-Re'em (Hebrew University of Jerusalem)

Title: Symmetric Stable Processes Indexed by Amenable Groups - Ergodicity, Mixing and Spectral Representation

Abstract: Stationary symmetric \alpha-stable (S\alpha S) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and LÃ©vy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary S\alpha S process is characterized by its spectral representation. While this result is known when the process is indexed by \mathbb{Z} or \mathbb{R}, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary S\alpha S processes indexed by amenable groups admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

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November 10: Rogelio Niño (National Autonomous University of Mexico, Morelia)

Title: Arithmetic Kontsevich-Zorich monodromies of origamis

Abstract: We present families of origamis of genus 3 that have arithmetic Kontsevich-Zorich monodromy in the sense of Sarnak. It is known this is true for origamis of genus 2, however the techniques for higher genera should be different. We present an outline of the proof for the existence of these families.

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November 17: Jayadev Athreya (University of Washington)

Location: AP&M 6402 and on Zoom

Title: Variance bounds for geometric counting functions

Abstract: Inspired by work of Rogers in the classical geometry of numbers, we'll describe how to obtain variance bounds for classical geometric counting problems in the settings of translation surfaces and hyperbolic surfaces, and give some applications to understanding correlations between special trajectories on these types of surfaces. Parts of this will be joint work with Y. Cheung and H. Masur; S. Fairchild and H. Masur; and F. Arana-Herrera, and all of this has been inspired by joint work with G. Margulis.
November 22 (Tuesday) at 4 PM: Ruixiang Zhang (University of California Berkeley)

(Joint with the Combinatorics Seminar)

Location: AP&M 5829

Title: A nonabelian Brunn-Minkowski inequality

Abstract: The celebrated Brunn-Minkowski inequality states that for compact subsets X and Y of R^d, m(X + Y)^(1/d) >= m(X)^(1/d) + m(Y)^(1/d) where m is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Yifan Jing and Chieu-Minh Tran, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of SL_2(R)), we also obtain partial results. In this talk I will discuss this inequality and explain important ingredients, old and new, in our proof.
November 29 (Tuesday) at 2 PM: Camille Horbez (CNRS, Laboratoire de MathÃ©matiques d'Orsay)

(Joint with the Functional Analysis Seminar)

Location: AP&M 7321

Title: Measure equivalence rigidity among the Higman groups

Abstract: The Higman groups were introduced in 1951 (by Higman) as the first examples of infinite finitely presented groups with no nontrivial finite quotient. They have a simple presentation, with k >= 4 generators, where two consecutive generators (considered cyclically) generate a Baumslag-Solitar subgroup. Higman groups have received a lot of attention and remain mysterious in many ways. We study them from the viewoint of measured group theory, and prove that many of them are superrigid for measure equivalence (a notion introduced by Gromov as a measure-theoretic analogue of quasi-isometry). I will explain the motivation and context behind this theorem, some consequences, both geometric (e.g. regarding the automorphisms of their Cayley graphs) and for associated von Neumann algebras. I will also present some of the tools arising in the proof. This is joint work with Jingyin Huang.

Spring 2022

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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Winter 2022

January 6: Gil Goffer (Weizmann Institute of Science)

Title: Is invariable generation hereditary?

Abstract: We will discuss the notion of invariably generated groups, with various motivating examples. We will then see how hyperbolic groups and small cancellation theory are used in answering the question in the title, which was asked by Wiegold and by Kantor-Lubotzky-Shalev. This is a joint work with Nir Lazarovich.

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January 13: Siyuan Tang (Indiana University)

Title: Nontrivial time-changes of unipotent flows on quotients of Lorentz groups

Abstract: The theory of unipotent flows plays a central role in homogeneous dynamics. Time-changes are a simple perturbation of a given flow. In this talk, we shall discuss the rigidity of time-changes of unipotent flows. More precisely, we shall see how to utilize the branching theory of the complementary series, combining it with the works of Ratner and Flaminio-Forni to get to our purpose.

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January 27: Sebastián Barbieri Lemp (Universidad de Santiago de Chile)

Title: Self-simulable groups

Abstract: We say that a finitely generated group is self-simulable if every action of the group on a zero-dimensional space which is effectively closed (this means it can be described by a Turing machine in a specific way) is the topological factor of a subshift of finite type on said group. Even though this seems like a property which is very hard to satisfy, we will show that these groups do exist and that their class is stable under commensurability and quasi-isometries of finitely presented groups. We shall present several examples of well-known groups which are self-simulable, such as Thompson's V and higher-dimensional general linear groups. We shall also show that Thompson's group F satisfies the property if and only if it is non-amenable, therefore giving a computability characterization of this well-known open problem. Joint work with Mathieu Sablik and Ville Salo.

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February 3: Julien Melleray (Université Lyon 1)

Title: From invariant measures to orbit equivalence, via locally finite groups

Abstract: A famous theorem of Giordano, Putnam and Skau (1995) states that two minimal homeomorphisms of a Cantor space X are orbit equivalent (i.e, the equivalence relations induced by the two associated actions are isomorphic) as soon as they have the same invariant Borel probability measures. I will explain a new "elementary" approach to prove this theorem, based on a strengthening of a result of Krieger (1980). I will not assume prior familiarity with Cantor dynamics. This is joint work with S. Robert (Lyon).

video
February 10: Lauren Wickman (University of Florida)

Title: Knaster Continua and Projective Fraïssé Theory

Abstract: The Knaster continuum, also known as the buckethandle, or the Brouwer–Janiszewski–Knaster continuum can be viewed as an inverse limit of 2-tent maps on the interval. However, there is a whole class (with continuum many non-homeomorphic members) of Knaster continua, each viewed as an inverse limit of p-tent maps, where p is a sequence of primes. In this talk, for each Knaster continuum K, we will give a projective Fraïssé class of finite objects that approximate K (up to homeomorphism) and examine the combinatorial properties of that the class (namely whether the class is Ramsey or if it has a Ramsey extension). We will give an extremely amenable subgroup of the homeomorphism group of the universal Knaster continuum.

video
February 17: Social Hour
February 24 at 10:00 AM: J. Moritz Petschick (Heinrich Heine University Düsseldorf)

Title: Groups of small period growth

Abstract: The concept of period growth was defined by Grigorchuk in the 80s, but still there are only a few examples of groups where we can estimate this invariant. We will sketch a connection to the Burnside problems and introduce a family of groups with very small period growth, answering a question by Bradford.

video
March 3 (joint with Probability Seminar) at 3:00 PM: Tom Hutchcroft (California Institute of Technology)

Location: AP&M 6402 and on Zoom

Title: The Ising model on nonamenable groups

Abstract: I will outline a proof that the Ising model has a continuous phase transition on any nonamenable Cayley graph. This will involve some neat probabilistic applications of ergodic-theoretic machinery such as factors of IID and the spectral theory of group actions. I will aim to make the talk accessible to a broad community.

video
March 10: Yan Mary He (University of Oklahoma)

Location: AP&M 7218 and on Zoom

Title: A quantitative equidistribution of angles of multipliers of hyperbolic rational maps

Abstract: In this talk, we will consider the angular component of multipliers of repelling cycles of a hyperbolic rational map in one complex variable. Oh-Winter have shown that these angles of multipliers uniformly distribute in the circle (-\pi, \pi]. Motivated by the sector problem in number theory, we show that for a fixed $K \geq 1$, almost all intervals of length 2\pi/K in (-\pi, \pi] contains a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in K. This is joint work with Hongming Nie.

video

Fall 2021
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 11: Veteran's Day Holiday
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
November 25: Thanksgiving Holiday
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 $G$.

Spring 2021
June 1: Osama Khalil (University of Utah)

Title: On the Mozes-Shah phenomenon for horocycle flows on moduli spaces
Abstract: The Mozes-Shah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) SL(2,ℝ). We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.

video
May 25: Joshua Frisch (Caltech)

Title: Which Linear Groups have bounded harmonic functions?
Abstract: The Poisson boundary of a group is a measure space which serves a dual purpose. From the perspective of random walks it represents the range of distinct asymptotic possibilities that a random walk on the group might take. From the perspective of harmonic analysis it classifies the space of bounded harmonic functions on that group. Understanding the Poisson boundary of a group is intimately related to the algebraic and geometric properties of that group. The most basic question one can ask about the Poisson boundary is whether it is trivial (equivalently whether there are any non-constant bounded harmonic functions on that group). In this talk I will survey some core ideas around the Poisson boundary and then focus on the case of linear groups. In particular I will give a complete characterization of when linear groups over positive characteristic fields admit any non-constant bounded harmonic functions. This is joint work with Anna Erschler.

video
May 18: Philipp Kunde (Anatole Katok Center for Dynamical Systems and Geometry)

Title: Anti-classification results for the Kakutani equivalence relation
Abstract: Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the \emph{isomorphism problem} to classify invertible measure-preserving transformations (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classification is impossible. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classification problem can be considered. In joint work with Marlies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable.

video
May 11: Cagri Sert (UZH)

Title: Expanding measures: Random walks and rigidity on homogeneous spaces
Abstract: We will start by reviewing recent developments in random walks on homogeneous spaces. In a second part, we will discuss the notion of a H-expanding probability measure on a connected semisimple Lie group H. As we shall see, for a H-expanding μ with H < G, on the one hand, one can obtain a description of μ-stationary probability measures on the homogeneous space G/Λ (G Lie group, Λ lattice) using the measure classification results of Eskin-Lindenstrauss, and on the other hand, the recurrence techniques of Benoist-Quint and Eskin-Mirzakhani-Mohammadi can be adapted to this setting. With some further work, these allow us to deduce equidistribution and orbit closure description results simultaneously for a class of subgroups which contains Zariski-dense subgroups and further epimorphic subgroups of H. If time allows, we will see how, utilizing an idea of Simmons-Weiss, these also allow us to deduce Birkhoff genericity of a class of fractal measures with respect to certain diagonal flows, which, in turn, has applications in diophantine approximation problems. Joint work with Roland Prohaska and Ronggang Shi.

video
May 4, 2021: Lingfu Zhang (Princeton University)

Title: Factor of IID for the free Ising model on the d-regular tree
Abstract: It is known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations. This is a joint work with Danny Nam and Allan Sly.

video
April 27: Prasuna Bandi (Tata Institute)

Title: Density at integer points of an inhomogeneous quadratic form and linear form
Abstract: In 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form Q in n≥3 variables, Q(ℤⁿ) is dense in ℝ. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in 3 variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form. This is based on joint work with Anish Ghosh.

video
April 20: Rachel Greenfeld (UC Los Angeles)

Title: Translational tilings in lattices
Abstract: Let F be a finite subset of ℤ^d. We say that F is a translational tile of ℤ^d if it is possible to cover ℤ^d by translates of F without any overlaps. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.
video
April 6: Jenna Zomback (University of Illinois Urbana-Champaign)

Title: A backward ergodic theorem and its forward implications
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T²(x),..., Tⁿ(x)} in front of the point x. We prove a "backward" ergodic theorem for a countable-to-one pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T^-1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein. This is joint work with Anush Tserunyan.
video

Winter 2021
March 9: Pengyu Yang (ETH Zurich)

Title: Equidistribution of expanding translates of lines in SL₃(ℝ)/SL₃(ℤ)
Abstract: Let X=SL₃(ℝ)/SL₃(ℤ) and a(t)=diag(t²,t^-1,t^-1). The expanding horospherical group U⁺ is isomorphic to ℝ². A result of Shah tells us that the a(t)-translates of a non-degenerate real-analytic curve in a (U⁺)-orbit get equidistributed in X. It remains to study degenerate curves, i.e. planar lines y=ax+b. In this talk, we give a Diophantine condition on the parameter (a,b) which serves as a necessary and sufficient condition for equidistribution. Joint work with Kleinbock, Saxcé and Shah. If time permits, I will also talk about generalisations to SL_n(ℝ)/SL_n(ℤ). Joint work with Shah.
video
March 2: Sam Mellick (UMPA ens de Lyon)

Title: Point processes on groups, their cost, and fixed price for G x Z
Abstract: Invariant point processes on groups are a rich class of probability measure preserving (pmp) actions. In fact, every essentially free pmp action of a nondiscrete locally compact second countable group is isomorphic to a point process. The cost of a point process is a numerical invariant that, informally speaking, measures how hard it is to "connect up" the point process. This notion has been very profitably studied for discrete groups, but little is known for nondiscrete groups. This talk will not assume any sophisticated knowledge of probability theory. I will define point processes, their cost, and discuss why every point process on groups of the form G x Z has cost one. Joint work with Miklós Abért.
video
Febuary 23: Gianluca Basso (Université Claude Bernard Lyon 1)

Title: Topological dynamics beyond Polish groups
Abstract: When G is a Polish group, one way of knowing that it has nice dynamics is to show that M(G), the universal minimal flow of G, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of the symmetric group of a set of cardinality κ is the space of linear orders on κ-not a metrizable space, but still nice, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having nice dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like Homeo(ω₁). This is joint work with Andy Zucker.
slides video
Febuary 16: Nishant Chandgotia (TIFR Bangalore)

Title: About Borel and almost Borel embeddings for ZD actions
Abstract: Krieger's generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
video
Febuary 9: Tamara Kucherenko (City College of New York)

Title: Flexibility of the Pressure Function
Abstract: Our settings are one-dimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. This is based on joint work with Anthony Quas.
video
Febuary 2: Matthieu Joseph (ENS de Lyon)

Title: Rigidity and flexibility phenomenons in isometric orbit equivalence
Abstract: TBAIn an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.
slides video
January 19: Felix Weilacher (Carnegie Mellon University)

Title: Marked groups with isomorphic Cayley graphs but different Descriptive combinatorics.
Abstract: We discuss the relationship between the Borel/Baire measurable/measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel/Baire measurable/measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.
video
January 12: Minju Lee (Yale University)

Title: Invariant measures for horospherical actions and Anosov groups.
Abstract: Let Γ be an Anosov subgroup of a connected semisimple real linear Lie group G. For a maximal horospherical subgroup N of G, we show that the space of all non-trivial NM-invariant ergodic and A-quasi-invariant Radon measures on Γ \ G, up to proportionality, is homeomorphic to ℝ^{rank G-1}, where A is a maximal real split torus and M is a maximal compact subgroup which normalizes N. This is joint work with Hee Oh.
slides video
January 5: François Le Maître (Institut de Mathématiques de Jussieu-PRG)

Title: A decomposition for measure-preserving near-actions of ergodic full groups
Abstract: Given a measure-preserving action of a countable group on a standard probability space, one associates to it a full group which by Dye's reconstruction theorem completely remembers the associated equivalence relation whose classes are the action's orbits. A natural question is then to understand how exactly this full group encodes the properties of the associated (measure-preserving) equivalence relation. In this talk, we will see that all non-free ergodic near-actions of the full group actually come from measure-preserving actions of the equivalence relation (or of its symmetric powers), paralleling a recent result of Matte-Bon concerning actions by homeomorphisms of topological full groups. If time permits, we will explain how this can be used to show that a measure-preserving ergodic equivalence relation has property (T) if and only if all the non-free ergodic near-actions of its full group are strongly ergodic. This talk is based on an ongoing joint work with Alessandro Carderi and Alice Giraud.

Fall 2020
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.
slides video
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

Winter 2020
March 3: Amanda Wilkens (University of Kansas)

Title: Finitary isomorphisms of Poisson point processes
Abstract: As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary. This is joint work with Terry Soo.
February 25: Barak Weiss (Tel Aviv University)

Title: Spaces of cut and project quasicrystals: classification and statistics
Abstract: Cut and project sets are well-studied models of almost-periodic discrete subsets of ℝ^d. In 2014 Marklof and Strombergsson introduced a natural class of random processes which generate cut and project sets in a way which is invariant under the group ASL(d,ℝ). Using Ratner’s theorem and the theory of algebraic groups we classify all these measures. Using the classification we obtain results analogous to those of Siegel, Rogers, and Schmidt in geometry of numbers: summation formulas and counting points in large sets for typical cut and project sets. Joint work with Rene Ruehr and Yotam Smilansky.
January 14: Tushar Das (University of Wisconsin - La Crosse)

Title: A variational principle in the parametric geometry of numbers
Abstract: We describe an ongoing program to resolve certain problems at the interface of Diophantine approximation and homogenous dynamics. Highlights include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, thereby resolving a conjecture of Kadyrov-Kleinbock-Lindenstrauss-Margulis (2014) as well as answering a question of Bugeaud-Cheung-Chevallier (2016). As a corollary of the Dani correspondence principle, this implies that the set of divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions. This is joint work with David Simmons, Lior Fishman, and Mariusz Urbanski. The reduction of various problems to questions about certain combinatorial objects that we call templates along with a variant of Schmidt's game allows us to answer some of these problems, while leaving plenty that remain open. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and geometric measure theory.