Description: The standard format is a talk in two halves separated by a break, with the first half aimed at nonspecialists.
Organizers: Peter NandoriJoe Auslander
When: Thursdays @ 2:00pm - 4:00pm
Where:  Math 1311
Dynamics Website: www-math.umd.edu/research/research-groups/dynamics.html
Pre-2012 Archives: http://dynamics.math.umd.edu/seminar/

Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018

  • Infinite mixing for one-dimensional maps with an indifferent fixed point

    Speaker: Marco Lenci (Universita' di Bologna) - http://www.dm.unibo.it/~lenci/

    When: Thu, August 31, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: In the first part of the talk, I will give some background on the
    question of mixing for dynamical systems preserving an infinite
    measure (a.k.a. 'infinite mixing'). Then I will recall and discuss the
    definitions of 'infinite-volume mixing' that I have introduced in
    recent years, with a survey on some examples of dynamical systems
    which verify or do not verify such definitions. Among these examples
    there will be one-dimensional intermittent maps, the subject of recent
    work with C. Bonanno and P. Giulietti.
    In the second part of the talk, I will better state the results for
    the intermittent maps: they comprise a class of expanding maps of
    [0,1] with a 'strongly neutral' fixed point in 0 and a class of
    expanding maps of the real line with strongly neutral fixed point at
    infinity. I will give a sketch of how some of the definitions of
    infinite-volume mixing are proved or disproved. Finally I will show
    how one property, called global-local mixing, entails certain limit
    theorems for our intermittent maps.
  • Ground states for Frenkel-Kontorova models on quasicrystals

    Speaker: Rodrigo Trevino (UMD) - http://trevino.cat

    When: Thu, September 7, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: The Frenkel-Kontorova model was first proposed in the 1930's to describe the structure and dynamics of a crystal lattice in the vicinity of a dislocation core, and by now has found many uses outside of solid state physics. Viewed from a dynamical systems point of view, it exhibits a lot of rich behavior tied to all sorts of great theories (e.g. KAM theory and Aubry-Mather theory) and fundamental open questions (e.g. Lyapunov exponents for the standard map).

    I will talk about this model in the setting where the crystal is aperiodic. In this setting, most of the dynamics are no longer available, but some tools developed to study the (periodic) classical model are still useful. I will talk about how one of them in particular, the so-called anti-integrable limit, is useful to find ground states (also known as equilibrium configurations). No background on the model will be assumed.
  • Mixing and local limit theorem for some hyperbolic flows

    Speaker: Peter Nandori (UMD) - http://math.umd.edu/~pnandori/

    When: Thu, September 28, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: We consider a special flow over a mixing map with some hyperbolicity.
    In case the roof function is square integrable, we find a set of conditions, under which the flow is mixing and also satisfies the local limit theorem. In case the roof function is non-integrable, we identify another set of conditions that imply Krickeberg mixing. The most important condition is the local limit theorem for the underlying map. We check that the conditions are satisfied for some examples, such as Axiom A flows, Sinai billiards, geometric Lorenz attractors (finite measure case) and suspensions over Pomeau-Manneville maps (finite and infinite measure cases). The talk is based on joint work with Dmitry Dolgopyat.
  • Fourier decay and spectral gaps on hyperbolic surfaces - UNUSUAL TIME

    Speaker: Semyon Dyatlov (MIT) - http://math.mit.edu/~dyatlov/

    When: Tue, October 17, 2017 - 2:00pm
    Where: Kirwan Hall 1308

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    Abstract: Let $M$ be a nonelementary convex co-compact hyperbolic surface. It is well-known that the Selberg zeta function $Z_M(s)$ has a zero at $s=\delta$ and no zeroes to the right of it, where $0 \delta-\varepsilon\}$. An application is an asymptotic counting formula for lengths of closed geodesics with remainder of relative size $O(e^{-\varepsilon t})$.

    The key ingredient of the proof is a Fourier decay bound for the Patterson-Sullivan measure on the limit set. This bound relies on the nonlinearity of the transformations generating the corresponding group as well as bounds on exponential sums which follow from the discretized sum-product theorem. The Fourier decay bound implies a fractal uncertainty principle for the limit set, which in turn gives the gap. This talk will include an introduction to transfer operators on Schottky groups, which are used throughout the proofs.

    This talk is based on joint works with Jean Bourgain and Maciej Zworski.
  • Can you hear the shape of a drum and deformational spectral rigidity of convex axis-symmetric planar domains.

    Speaker: Vadim Kaloshin (UMD) - https://www.math.umd.edu/~vkaloshi/

    When: Thu, October 19, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: M. Kac popularized the question 'Can you hear the shape of a drum?'. Mathematically, consider a bounded planar domain $\Omega$ and
    the associated Dirichlet problem
    $$
    \Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.
    $$
    The set of $\lambda$’s such that this equation has a solution, denoted
    $\mathcal L(\Omega)$ is called the Laplace spectrum of $\Omega$.
    Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

    Consider the billiard problem inside $\Omega$. Call the length spectrum the closure
    of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of
    the wave trace function, generically, the Laplace spectrum determines the length
    spectrum. In the space of strictly convex axis-symmetric domains we shall discuss
    Sarnak's question whether one can deform such a domain without changing
    its spectrum. This is based joint works with J. De Simoi, A. Figalli and Q. Wei.
  • POSTPONED TO NOV 2

    Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/

    When: Thu, October 26, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract:
  • Statistical properties of the Standard map with increasing coefficient

    Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/

    When: Thu, November 2, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: The Chirikov standard map $F_L$ is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Analysis in this direction is notoriously difficult, and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any value of L.

    I will discuss two related results on a more tractable version of this problem. The first is a kind of ‘finite-time mixing estimate, indicating that for large L and on a suitable timescale, the map $F_L$ is strongly mixing. The second pertains to statistical properties of compositions of standard maps with increasing parameter L: when the parameter L increases at a sufficiently fast polynomial rate, we obtain asymptotic decay of correlations estimates, a Strong Law, and a CLT, all for Holder observables.
  • On Siegel's question and density of collisions in the Restricted Planar Circular Three Body Problem - UNUSAL DATE

    Speaker: Jianlu Zhang (UMD) -

    When: Tue, November 7, 2017 - 2:00pm
    Where: Kirwan Hall 1308

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    Abstract: For the Restricted Circular Planar 3-Body Problem, we show that there exists a full dimensional open set $U$ in phase space independent of the mass ratio $\mu$, where the set of initial points which lead to collision is $O(\mu ^{1/20} )$ dense as $\mu \rightarrow 0$.

  • Homoclinic points for geodesic flows and billiards

    Speaker: Zhihong Jeff Xia (Northwestern) - http://www.math.northwestern.edu/~xia/

    When: Thu, November 9, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: It is believed and conjectured that, for a typical Hamiltonian system, every hyperbolic periodic point has a homoclinic point. This is indeed the case in many situations. However, for geodesic flow and billiards, the usual perturbation techniques are no longer available, since there is no local perturbation. We will show that for geodesic flows on two-sphere and for convex billiards, it is still true. The proof uses prime ends and relies more on global analysis of stable and unstable manifolds, rather than perturbation techniques.
  • Asymptotic behavior of Markov operator corresponding to non-expanding piecewise linear maps

    Speaker: Fumihiko Nakamura (Hokkaido University ) -

    When: Thu, November 16, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: The non-expanding piecewise linear map, known as the Nagumo-Sato (NS)
    model, is described as $S_{\alpha,\beta}(x)=\alpha x+\beta ({\rm
    mod} 1)$, where $0
  • Poisson boundary of random walks on free semigroups

    Speaker: Behrang Forghani (University of Connecticut) - https://sites.google.com/site/behrangforghani/

    When: Thu, November 30, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: In early 60, Furstenberg employed the theory of Poisson boundary of random walks on groups to obtain several fundamental rigidity results for lattices in Lie groups. One of the main questions in the theory of random walks on groups is how to describe the Poisson boundary of a concrete random walk on a concrete group. In particular, for an arbitrary random walk on a finitely generated free group, it is conjectured that the space of infinite irreducible words equipped with the hitting measure is the Poisson boundary.

    The conjecture has been solved by Dynkin-Maljutov for a first neighborhood random walk, by Derriennic for a finitely supported random walk, and by Kaimanovich for a random walk whose both entropy and logarithmic moment are finite.

    Although the study of random walks on free semigroups is less arduous than the one of free groups, the conjecture remains unsolved for arbitrary random walks on free semigroups. In this talk, I will show the conjecture holds whenever the random walk on a free semigroup has finite entropy or finite logarithmic moment or finite w-logarithmic moment for some finite words. This talk is based on a joint work with Giulio Tiozzo from the University of Toronto.
  • Disjointness criterion for parabolic flows

    Speaker: Adam Kanigowski (Penn State) -

    When: Thu, January 25, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: We will state a general disjointness criterion for two ergodic systems. We will then show how this criterion can be used to study disjointness in the class of parabolic systems such as unipotent flows and their time changes, nil-flows and their time-changes, smooth surface flows.
  • Invariant Random Subgroups of Full Groups of Bratteli diagrams

    Speaker: Kostya Medynets (Naval Academy) - https://www.usna.edu/Users/math/medynets/index.php

    When: Thu, February 15, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: In the talk, we will classify the ergodic invariant random subgroups (IRS) of simple AF full groups. AF full groups arise as the transformation groups of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. AF full groups are complete (algebraic) invariants for the isomorphism of Bratteli diagrams. Given a simple AF full group G, we will prove that every ergodic IRS of G arises as the stabilizer distribution of a diagonal action on X^n for some n, where X is the path-space of the Bratteli diagram associated to G. This is joint work with Artem Dudko.
  • Stochastic properties of the Z^2-periodic Sinai billiard

    Speaker: (Francoise Pene) - http://lmba.math.univ-brest.fr/perso/francoise.pene/

    When: Thu, February 22, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: We study stochastic properties of the Z^2-periodic Sinai billiard (recurrence, ergodicity, mixing, decorrelation, limit theorems).
  • Families of periodic paths on the pentagon

    Speaker: Diana Davis (Swarthmore College) - http://www.swarthmore.edu/NatSci/ddavis3/Abstract: There are infinite "families" of periodic paths on the pentagon, whose members all have a similar appearance but get more and more complicated. I'll show some beautiful examples of these families, and explain how we use the group structure on the set of periodic directions to obtain them.

    When: Thu, March 1, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: There are infinite "families" of periodic paths on the pentagon, whose members all have a similar appearance but get more and more complicated. I'll show some beautiful examples of these families, and explain how we use the group structure on the set of periodic directions to obtain them.
  • TBA

    Speaker: Scott Schmieding (Northwestern) -

    When: Thu, March 8, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: TBA
  • I. Overview of my work on the N-body problem, II. Metric Approach to McGehee blow-up and Marchall's lemma.

    Speaker: Richard Montgomery (UC Santa Cruz) - https://people.ucsc.edu/~rmont/

    When: Thu, March 29, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: TBA
  • TBA

    Speaker: E. Arthur Robinson (George Washington University) - https://blogs.gwu.edu/robinson/

    When: Thu, April 26, 2018 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: TBA