Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021

• Exponential mixing of 3D Anosov flows

  Speaker: Zhiyuan Zhang (CNRS, Universite Paris 13) - https://sites.google.com/site/homepageofzhiyuanzhang/home
  

  When: Thu, September 3, 2020 - 2:00pm
  Where: Zoom
  

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  Abstract: We show that a topologically mixing $C^\infty$ Anosov flow on a 3-dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential. This is a joint work with Masato Tsujii.
  
• Spectral gaps for a class of random products of bounded linear operators

  Speaker: Joseph Horan (Department of Mathematics and Statistics, University of Victoria) -
  

  When: Thu, September 17, 2020 - 2:00pm
  Where: Zoom
  

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  Abstract: For a primitive Markov chain (represented by a non-negative matrix where a power has all positive entries), we may use the classical Perron-Frobenius theorem to conclude that the Markov chain has a unique stationary distribution and all initial states converge to that distribution with some exponential decay rate determined by the largest and second largest eigenvalues of the matrix. What happens when we allow random products (cocycles) of matrices, or of bounded linear operators? In the first part of this talk, we set up and present a new cocycle Perron-Frobenius theorem, and give a couple of interesting applications at a high level, including the situation of a cocycle of ”paired tent maps”. In the second part of this talk (after the break), we give an idea of the proof of the theorem and elaborate on the paired tent maps example.
  
  The video for the talk can be found here: https://umd.zoom.us/rec/share/eo4LSKlEdHSIaYrvB8ZkIz3hp-hsusuIxxMGOYwS7Nd80qdhDH-pvGsBV9-mR_-z.9_MONl1MX3enVk9k
  
• Effective equidistribution of horospherical flows in infinite volume

  Speaker: Nattalie Tamam (UC San Diego) - https://www.math.ucsd.edu/~natamam/
  

  When: Thu, September 24, 2020 - 2:00pm
  Where: Zoom
  

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  Abstract: We want to provide effective information about averages of orbits of the horospherical subgroup acting on a hyperbolic manifold of infinite volume. We start by presenting the setting and results for manifolds with finite volume. Then, discuss the difficulties that arise when studying the infinite volume setting, and the measures that play a crucial role in it. This is joint work with Jacqueline Warren.
  The video for the talk can be found here: https://umd.zoom.us/rec/share/XZao71i2C4E_HIcJXPZdHdl4_FUaJZCAkMCo87E6SaKRA_wbjM6w-t5Sf0hVxAze.T2Rc998HpDJsUqOZ
  
• Stabilizers in group Cantor actions and measures

  Speaker: Olga Lukina (University of Vienna) - https://sites.google.com/view/olgalukina
  

  When: Thu, October 1, 2020 - 2:00pm
  Where: Zoom
  

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  Abstract: Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has full measure. On the other hand, there are many examples of group actions, where every point has a non-trivial stabilizer. In this talk, we introduce an analog of the notion of an essentially free action for such highly non-free actions, using the concept of holonomy. For equicontinuous actions of countable groups on Cantor sets, we answer the following question: under what conditions there exists a subgroup H of G, such that the stabilizers of almost all points in X are conjugate to H? These conditions are that the action is locally quasi-analytic and locally non-degenerate. We discuss applications of this work to the study of the properties of the invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Groeger.
  The video for the talk can be found here: https://umd.zoom.us/rec/share/HnhA7kV6wAR8qJHb36ohsQbia_LQKZJZWTG3jgekMuletS-5zMkI3NszXaBvbwHA.IIfOLN8Itmqo6JFb
  
• Classification and statistics of cut-and-project sets

  Speaker: Barak Weiss (Tel Aviv University) - http://www.math.tau.ac.il/~barakw/
  

  When: Thu, October 8, 2020 - 2:00pm
  Where: Online
  

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  Abstract: Cut and project sets are a well-studied model of aperiodic order, with connections to diverse topics in crystallography, mathematical physics, ergodic theory, and geometry of discrete sets.We introduce a class of so-called Ratner-Marklof-Strombergsson measures. These are probability measures supported on parameter spaces of cut-and-project sets in $R^d$ which are invariant and ergodic for the action of the groups $ASL(d,R)$ or $SL(d,R)$. We classify the measures that can arise in terms of data of algebraic groups. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. With this in hand we derive results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.
  
  
  Joint work with Rene Ruehr and Yotam Smilansky.
  
  The video for the talk can be found here: https://umd.zoom.us/rec/share/7Xp1Vaxv8BqDSNBa9ohAV_to41vq-ojww4o02CXSwKQzWQMLd4ub6aZl3Kwu_X63.uQ3uImM3h9amliA3
  
• Thermodynamics of smooth models of pseudo-Anosov homeomorphisms

  Speaker: Dominic Veconi (Penn State) - http://dominic.veconi.com/
  

  When: Thu, October 15, 2020 - 2:00pm
  Where: Online
  

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  Abstract: We discuss the thermodynamic and ergodic properties of a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the pseudo-Anosov map is uniformly hyperbolic outside of a neighborhood of a set of singularities, and the trajectories are slowed down so the differential is the identity at the singularities. Using Young towers, we prove existence and uniqueness of equilibrium measures for geometric $t$-potentials. This family of equilibrium measures includes a unique smooth SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.
  
  
  The video for the talk can be found here: https://umd.zoom.us/rec/share/Hg5XzKKFbyIntuH_z2V_UWvqPsICaTBJRS-xqOoT-YNDu4pwC5oMwpOQTiWtrS6N.qYoDBloWbmYkllor
  
  
• Large intersection classes for pointwise emergence

  Speaker: Agnieszka Zelerowicz (UMD) -
  

  When: Thu, October 22, 2020 - 2:00pm
  Where: Online
  

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  Abstract: Recently, P. Berger introduced a concept of metric emergence to quantitatively study the non-uniqueness of statistics (such as Newhouse or KAM phenomena). Later, S. Kiriki, Y. Nakano, and T. Soma introduced a concept of pointwise emergence to quantitatively study non-existence of averages, and constructed a residual subset of the full shift with high pointwise emergence. In this talk I will consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. I will present my joint work with Yushi Nakano, where we show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property.
  
  The video for the talk can be found here: https://umd.zoom.us/rec/share/l_9KxEJQyjaY37_KKuatfQbn4rFyJXq89TjtBOvEkiBm8tujkQntI5GJrLshPwfU.VP31-rdQqsBurokM
  
• Quasi-periodic attractors for dissipative systems

  Speaker: Alessandra Celletti (University of Rome Tor Vergata) - http://www.mat.uniroma2.it/celletti/
  

  When: Thu, October 29, 2020 - 2:00pm
  Where: Online
  

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  Abstract: We consider the existence of invariant attractors for the specific case of dissipative systems known as conformally symplectic systems, which are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter. We provide a KAM theorem in an a-posteriori format: assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby. The theorem does not assume that the system is close to integrable.
  
  This method can be also used to get different results: (i) a breakdown criterion for invariant attractors; (ii) an efficient algorithm to generate the solution, which can be implemented successfully in model problems; (iii) the existence of whiskered tori for conformally symplectic systems, (iv) the analyticity domains of the quasi-periodic attractors in
  the symplectic limit.
  
  
  The content of this talk refers to works in collaboration with R. Calleja and R. de la Llave.
  
  The recorded video for the talk can be found here: https://umd.zoom.us/rec/share/kKQ2GSxJLFOt5X9caTpAlimdg8NFpkeDxUhbwg3_LhSar0aOLahkXcCniX4fNx02.bQDTym7jft59hb-u
  
• Multiscale substitution tilings

  Speaker: Yotam Smilansky (Rutgers University) - https://sites.math.rutgers.edu/~ys755/
  

  When: Thu, November 5, 2020 - 2:00pm
  Where: Zoom
  

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  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.
  
• Entropy rate of product of independent processes

  Speaker: Joanna Kulaga-Przymus (Nicolaus Copernicus University in Toruń) - https://www-users.mat.umk.pl/~joasiak/
  

  When: Thu, November 12, 2020 - 2:00pm
  Where: Online
  

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  Abstract:
  
  Let $\mb{X}$ and $\mb{Y}$ be two-sided stationary processes taking values in $\{0,1\}$. Let $\mb{M}$ be their coordinatewise product: $M_i=X_i \cdot Y_i$. During my talk I will discuss the entropy rate of $\mb{M}$.
  
  The motivation is twofold. Furstenberg in his seminal paper asked when $\mb{X}$ can be recovered from $\mb{X}+\mb{Y}$ and showed that this is the case whenever the corresponding dynamical systems are disjoint. I will discuss an analogous problem for $\mb{X}+\mb{Y}$. As we admit zero as a value, we cannot use logarithm to reduce our problem to Furstenberg's setting. We show that if $\mb{Y}$ is of zero entropy rate than for a large class of positive entropy rate processes $\mb{X}$, the entropy rate of $\mb{M}$ is strictly lower than that of $\mb{X}$ and thus $\mb{X}$ cannot be recovered from $\mb{M}$.
  
  Another reason we are interested in the entropy rate of $\bm{M}$ comes from dynamics. More precisely, Mariusz Lemanczyk, Benjamin Weiss and me left some open problems related to invariant measures for $\mathscr{B}$-free systems. The most prominent example of such a system is the subshift generated by the square of the Moebius function.
  
  During my talk I will show how these two settings are related to each other and I will discuss the following questions:
  a) is there a general formula for the entropy rate of $\mb{M}$?
  b) when is $\mb{M}$ of positive entropy?
  c) what is the relation between the entropy rates of $\mb{M}$ and $\mb{X}$?
  
  The talk is based on joint work with Michał Lemańczyk https://arxiv.org/pdf/2004.07648.pdf
  
• Linear repetitive Delone sets beyond Abelian groups

  Speaker: Felix Pogorzelski (Universität Leipzig) - http://www.math.uni-leipzig.de/~pogorzelski/
  

  When: Thu, November 19, 2020 - 2:00pm
  Where: Zoom
  

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  Abstract: There is no rigorous mathematical definition of a quasicrystal. In spaces with some group translation the latter term usually refers to well-scattered point sets (Delone sets) that are not periodic but display long range symmetries. Classes of these sets such as model sets have already been studied by Meyer in the 70's, i.e. some time before Shechtman's discovery of physical alloys with non-periodic molecular structure in 1982 (Nobel prize for chemistry in 2011). In this talk we focus on non-periodic point sets in (possibly non-Abelian) lcsc groups that are not too far from crystals in a dynamical sense. The main focus will be on the generalization of the concept of linearly repetitive Delone sets known from Euclidean space. Although they are not found easily, non-periodic, linearly repetitive Delone sets exist in many non-Abelian groups as well. Going beyond a result from Lagarias and Pleasants for R^d, we explain how to prove unique ergodicity for the associated dynamical systems for a class of Lie groups of polynomial volume growth. Joint work with Siegfried Beckus and Tobias Hartnick.
  
• On the dimension drop conjecture for diagonal flows on the space of lattices

  Speaker: Shahriar Mirzadeh (Michigan State) - Abstract:
  

  When: Thu, December 3, 2020 - 2:00pm
  Where: Online
  

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  Abstract:
  Consider the set of points in a homogeneous space $X = G/\Gamma$ whose ${g_t}$-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when has real rank 1. \\
  In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: $G=\SL_{m+n}(\mathbb{R})$, $\Gamma= \SL_{m+n}(\mathbb{Z})$, and $g_t=\diag (e^{t/m}, \cdots, e^{t/m},e^{-t/n}, \cdots, e^{-t/n})$. We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.
  
• The nature of equations and the equations of nature

  Speaker: Jim Yorke (UMD) - http://yorke.umd.edu
  

  When: Thu, February 4, 2021 - 2:00pm
  Where: Probably online
  
• TBA

  Speaker: Tushar Das (Department of Mathematics & Statistics, University of Wisconsin-La Crosse) - https://www.uwlax.edu/profile/tdas/
  

  When: Thu, February 11, 2021 - 2:00pm
  Where: Online
  
• TBA

  Speaker: Matthew Foreman (UC Irvine) -
  

  When: Thu, February 18, 2021 - 2:00pm
  Where: Online
  
• TBA

  Speaker: David Kerr (University of Münster) - https://www.math.tamu.edu/~kerr/
  

  When: Thu, February 25, 2021 - 2:00pm
  Where: Online