Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1308

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

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.

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

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

Abstract:

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1308

Speaker: Jianlu Zhang (UMD) -

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 $\mu \rightarrow 0$.

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

Speaker: Fumihiko Nakamura (Hokkaido University ) -

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

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

Speaker: Adam Kanigowski (Penn State) -

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.

Where: Kirwan Hall 1311

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

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.

Where: Kirwan Hall 1311

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

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

Where: Kirwan Hall 1311

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.

Where: Kirwan Hall 1311

Speaker: Scott Schmieding (Northwestern) - https://www.scholars.northwestern.edu/en/persons/scott-edward-schmieding

Abstract: For a shift of finite type $(X,\sigma)$, the automorphism group $Aut(\sigma)$ consists of all homeomorphisms from $X$ to $X$ which commute with the shift map $\sigma$. The group $Aut(\sigma)$ is known to contain a rich structure, and has been heavily studied over the years. To analyze a particular automorphism, one may consider its action on the dimension group, an ordered abelian group associated to the system $(X,\sigma)$. We will describe what this dimension group is, and discuss relationships between various dynamical properties of an automorphism and its action on the associated dimension group. We'll focus on connections between the topological entropy of an automorphism and spectral data coming from its action on the dimension group, and how this relates to an entropy conjecture for shifts of finite type in the spirit of Shub's classical entropy conjecture.

Where: Kirwan Hall 1311

Speaker: Mads Bisgaard (ETH Zurich, Switzerland) - https://sites.google.com/view/madsbisgaard

Abstract: Aubry-Mather theory studies the flow associated to a convex Hamiltonian H on a cotangent bundle, by finding so-called Mather measures. Symplectic topology studies properties imposed on the flow by the geometry of the cotangent bundle. It is interesting to understand how these theories can play together. In this talk I will show how different techniques from symplectic topology give rise to incarnations of Mather’s $\alpha$-function and that its relation to invariant measures continues to hold: Mather measures exist. I will discuss applications to Hamiltonian system on closed symplectic manifolds, $\mathbb R^{2n}$ and twisted cotangent bundles.

Where: Kirwan Hall 1311

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

Abstract: The N-body problem can be rephrased as theproblem of finding geodesics for a certain one-parameter family of metrics,the Jacobi-Maupertuis [JM] metrics. Marchal's lemma is the basic tool for eliminating collisionswhen using variational methods to establish existence of various "designer" orbits in the N-body problem. We rephrase Marchal's lemma as a lemma in metric geometry. The essential idea can be seen in the standard planar Kepler problem where we show that he zero-energy metric is isometric to the cone of radiius one-half which is flat everywhere except at the cone point which corresponds to collision. Inextendibility of geodesics through the cone point corresponds to the conclusion of Marchal's lemma: action minimizers cannot have collision points in their interior. This work has significant overlap with recent work of Barutello, Terracini, and Verzini. Time permitting I will also discuss a metric perspective on McGehee blow-up.

Where: Kirwan Hall 1311

Speaker: Mitsuru Shibayama (Kyoto University, Japan) - http://yang.amp.i.kyoto-u.ac.jp/~shibayama/top_e.html

Abstract: We consider the existence of solution in the planar Sitnikov problem, realizing given symbolic sequences by based on variational method. We also prove the existence of various periodic solutions and connecting orbits between them.

Where: Kirwan Hall 1311

Speaker: Michael Jakobson (UMD) - http://www.math.umd.edu/~jakobson/

Abstract: We review several results about ergodic and statistical properties of hyperbolic attractors and present new results obtained in a joint work with Lucia Simonelli (ICTP, Trieste).

Where: Kirwan Hall 1311

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

Abstract: A quasicrystal is, by one definition, a solid with non-classical diffraction pattern, e.g, 5-fold rotational symmetry. The first quasicrystal was discovered by D. Schectman at NIST in around 1984, for which he ultimately won the 2011 Chemistry Nobel prize. At the time, several physicists suggested the vertex set of a Penrose tiling (or its 3-dimensional analogue) as a model for the placement of atoms in a quasicrystal. The diffraction theory of vertex sets of Penrose-like tilings is closely tied to the dynamical spectrum (especially the point spectrum) of a corresponding type of dynamical system. In this talk, we will start with an overview of this type of dynamical system and the corresponding diffraction theory. Then we will describe some recent new results in this area, most of which concern the spectral and mixing properties of different types of substitution dynamical systems.

Where: Kirwan Hall 1311

Speaker: Alex Grigo (U of Oklahoma) -

Abstract: In this talk I will present some basic examples of transport phenomena

in classical physics and provide an overview of certain mechanical

models in which one can rigorously investigate these properties.

In particular I will focus on illustrating how transport phenomena

are related to results in the theory of hyperbolic dynamical systems

such as central limit theorems, linear response theory, and averaging theory. Then I will describe in detail some of my recent works in this area, which are in parts joint work with Leonid Bunimovich.

Where: Kirwan Hall 1311

Speaker: Molei Tao (Georgia Tech) - http://people.math.gatech.edu/~mtao8/index.html

Abstract: We consider dynamical system perturbed by small Gaussian noises, with a goal of quantifying how noises can affect the dynamics. More precisely, most likely noise-induced metastable transitions are understood by maximizing transition rate provided by Freidlin-Wentzell large deviation theory. Such transitions in gradient systems were understood and known to cross separatrices at saddle points. We instead investigate nongradient systems (possibly irreversible) using a developed tool of (generalized) orthogonal decomposition. Two examples will be described: (1) A different type of transitions, which cross hyperbolic periodic orbits, will be discussed. Corresponding numerical tools for both identifying such periodic orbits and computing transition paths will be developed. (2) The Langevin model of stochastic mechanical systems will be investigated and extended, with emphasis on how its metastable transition can differ from the overdamped (reversible) case. In addition, numerical approaches for general nongradient systems will be presented. If time permits, I will also mention how these results can help design control strategies.