Dynamics Archives for Academic Year 2018

C*-algebras, topological dynamics, and noncommutative dimensions

When: Thu, August 30, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Jianchao Wu (Penn State University) - http://www.personal.psu.edu/jxw710/
Abstract: C*-algebras are a kind of operator algebras tailored to describe noncommutative (i.e., quantum) topological spaces via functional analytical means. A major source of examples throughout the history of C*-algebra theory lies in the construction of crossed products from topological dynamical systems. This bridge between operator algebras and dynamics, valid also in the measure-theoretical setting, has proven immensely fruitful. On the other hand, the dimension theory of C*-algebras, which studies analogs of classical dimensions for topological spaces, is young but has been gaining momentum lately thanks to the pivotal role played by the notion of finite nuclear dimension in the classification program of simple separable nuclear C*-algebras. The confluence of these two themes leads to the question: What type of topological dynamical systems give rise to crossed product C*-algebras with finite nuclear dimension? I will present some recent work on this problem.

Homogeneous Structures from Topological Flows and Applications in Dynamics

When: Thu, September 6, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Kurt Vinhage (University of Chicago) - http://home.uchicago.edu/~kvinhage/
Abstract: A standard way of building a homogeneous structure on a smooth manifold X is to find a family of finite-dimensional subspace of vector fields which span the tangent space of X at every point and are closed under taking Lie brackets. We will describe a more topological approach to building a homogeneous structure, which has applications to smooth dynamical systems which have associated structures which are not a priori smooth.
In the first hour, I will discuss basic definitions and sketch an application of these tools to local rigidity of abelian group actions. In the second hour, I will discuss the details of the proof and possible future directions. Joint with Zhenqi Wang.

Regular and semi regular minimal flows

When: Thu, September 13, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Joe Auslander (UMD) -
Abstract: Let (X,T) be a minimal flow and let x,y in X. An obvious necessary condition for there to be an automorphism f of (X,T) with f(x)=y is that (x,y) be an almost periodic point of the product flow The flow (X,T) is said to be regular if this is always the case. Regular minimal flows are the minimal left ideals of the enveloping semigroup of a flow.

A further necessary condition for the existence of an automorphism is given in terms of the automorphism group of the universal minimal flow, namely that if (m,n) projects to (x,y) and g(m)=n, then g must be in the normalizer of the Ellis group of (X,T). When this always occurs (X,T) is said to be semi regular. Every minimal flow has a semi regular one in the same proximal class.

Dynamics of slow-fast Hamiltonian Systems

When: Thu, September 20, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Thibaut Castan (UMD) -
Abstract: The presence of gaps in the distribution of asteroids in the asteroid belt between Mars and Jupiter is far from being well understood. A common physical explanation is the presence of resonance in a system Sun-Jupiter-Asteroid, however this phenomenon is not fully understood yet. In 1983, this problem was restated as the study of a slow-fast Hamiltonian system, and allowed to understand the chaotic motion observed in simulations at the 3:1 resonance.
More precisely, when the fast system has a hyperbolic fixed point, the motion for initial condition close to the separatrices has yet to be understood. When close to the separatrix, the usual averaging of the fast system does not hold anymore, leading to some chaotic behavior for the slow system.
We will study how to understand the seemingly chaotic behavior of this Hamiltonian system.

Stochastic diffusive behavior at Kirkwood gaps

When: Thu, September 27, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Vadim Kaloshin (UMD) - https://www.math.umd.edu/~vkaloshi/
Abstract: One of well known indications of instability in the Solar system is presence of Kirkwood gaps in the Asteroid belt. They are caused by interaction of Asteroids with Jupiter. We study the restricted Sun-Jupiter-Asteroid problem. One mechanism of creation for the 3:1 Kirkwood gap was discovered by Wisdom and indepently by Neishtadt. We propose another mechanism, based on apriori chaotic underlying dynamical structure, and exhibit chaotic behavior. As an indication of chaos we show that eccentricity behaves like a stochastic diffusion process. More exactly, we establish probability measures such that the distributions of the projection onto eccentricity under the push forward in a proper scale weakly converges to a stochastic diffusion process on the line. This is a joint work with M. Guardia, P. Martin, and P. Roldan.

Cohomological equations and Ruelle resonances

When: Thu, October 11, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Giovanni Forni (UMD) -
Abstract: We summarize results on the cohomological equation for translation flows on translation surfaces (myself, Marmi, Moussa and Yoccoz, Marmi and Yoccoz) and apply these results to the asymptotic of correlations for pseudo-Anosov maps, which were recently obtained by a direct method by Faure, Gouezel and Lanneau. In this vein, we consider the generalization of this asymptotic to generic Teichmueller orbits (pseudo-Anosov maps correspond to periodic Teichmueller orbits) and to (partially hyperbolic) automorphisms of Heisenberg nilmanifolds (from results on the cohomological equation due to L. Flaminio and myself).

Area-preserving maps of surfaces: complex and PL versions

When: Thu, October 18, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Simion Filip (Institute for Advanced Study and Harvard) - http://www.math.ias.edu/~sfilip/
Abstract: I will start by introducing K3 surfaces, a class of (compact, algebraic) surfaces with automorphisms which preserve a natural area form. Cantat & Dupont showed that if the volume form is also the measure of maximal entropy, then the examples must come from linear maps of tori. I will explain a different and much shorter proof of this result, joint with Valentino Tosatti, using Ricci-flat metrics. After that, I will discuss what happens in a 1-parameter family of such automorphisms, as the surface degenerates. One gets PL maps of real 2-dimensional spheres which inherit many of the structures available in complex geometry.

Preperiodic points of polynomials

When: Tue, October 23, 2018 - 2:00pm
Where: MTH 0104
Speaker: Laura DeMarco (Northwestern) - http://www.math.northwestern.edu/~demarco/
Abstract: I will discuss results about the geometry of the periodic (or pre-periodic) points for rational maps on P^1. In new work with Holly Krieger and Hexi Ye, we give a uniform bound on the number of shared preperiodic points for two distinct complex polynomials of the form f(z) = z^2 + c. The question was inspired by analogous results/questions in arithmetic geometry (and the topic of my Number Theory seminar), and its proof involves p-adic analysis and heights -- combined with features of the Julia sets of the quadratic polynomials.

Simple Low-dimensional Paradigms for higher-dimensional chaos

When: Thu, October 25, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: James Yorke (UMD) - http://www.chaos.umd.edu/~yorke/
Abstract: The most frequently studied dynamical systems are low dimensional and all the periodic orbits in a chaotic set have the same number of unstable dimensions, but this property seems to fail in high dimensional systems.

In this talk I continue this theme. Here we define a property called ``hetero-chaos'', in which, along with the usual properties of chaos, there is a dense set of k-dimensionally unstable periodic orbits, and this holds for more than one k. We provide examples including a piecewise linear generalized baker map. I will explain how to prove ergodicity for our baker-like skew-product maps in 3D. This talk summarizes joint work with Shuddho Das, Yoshi Saiki, Miguel Sanjuan, and Hiroki Takahashi.

Multiscale substitution schemes and Kakutani sequences of partitions.

When: Tue, October 30, 2018 - 2:00pm
Where: MTH 0104
Speaker: Yotam Smilansky (Hebrew University of Jerusalem) -
Abstract: Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.

Zimmer's conjecture for co-compact lattices in simple complex Lie groups

When: Thu, November 1, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Zhiyuan Zhang (Institute of Advanced Study, Princeton) - https://sites.google.com/site/homepageofzhiyuanzhang/
Abstract: We will discuss Zimmer's conjecture on the actions of lattices in higher rank Lie groups on manifolds of small dimension. This conjecture is recently proved by Aaron Brown, David Fisher and Sebastian Hurtado for split simple real Lie groups. Then we discuss a recent generalisation of their result to simple complex Lie groups. This is a joint work with Jinpeng An and Aaron Brown.

Dynamics on abelian covers

When: Thu, November 8, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Wenyu Pan (Penn State) - https://wenyupanblog.wordpress.com
Abstract: In the first part, I will present a measure rigidity result on abelian covers. A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. I will present a joining classification result of horocycle flows on hyperbolic surfaces of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface. In the second part, I will talk further about ergodic properties of the geodesic flow/ frame flow on a general abelian cover. In particular, I will present the local mixing property of the geodesic flow/ frame flow, which is introduced to substitute the well-known strong mixing property in infinite volume setting. Part of the talk is based on the joint work with Hee Oh.

Diffraction from mathematical quasicrystals

When: Thu, November 15, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Alan Haynes (University of Houston) - https://www.math.uh.edu/~haynes/
Abstract: In the first part of this talk we will discuss the mathematics behind diffraction and explain some basic examples, such as diffraction from lattices. Next we will introduce cut and project sets, which are dynamically defined models for physical materials called quasicrystals. We will explain the classical approach to calculating the diffraction patterns seen from these objects, using the Poisson summation formula. In the second part of the talk, which is joint work with Michael Baake, we will take a more practical point of view and attempt to quantify how much the diffraction patterns observed from finite patches of cut and project sets deviate from the infinite models. Our methods, which are explicit and geared towards numerical computation, demonstrate the importance of Diophantine approximation to accurately determining complex phases and amplitudes of diffraction patterns produced by finite patches.

Coding geodesic flows and various continued fractions

When: Tue, November 27, 2018 - 2:00pm
Where: Kirwan Hall 1308
Speaker: Claire Merriman (UIUC) - https://faculty.math.illinois.edu/~emerrim2/
Abstract: I will connect continued fractions with even or odd partial quotients to geodesic flows on modular surfaces. The connection between geodesics on the modular surface PSL(2,Z)\H and regular continued fractions was established by Series, and we extend this to the odd and grotesque continued fractions and even continued fractions. This is joint work with Florin Boca.

Dynamics on Clebsch's Diagonal Cubic Surface.

When: Thu, November 29, 2018 - 2:00pm
Where: Kirwan Hall 1311
Speaker: William Goldman (UMD) - https://www.math.umd.edu/~wmg/
Abstract: TBA


When: Thu, February 7, 2019 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Tushar Das (University of Wisconsin) - https://sites.google.com/a/uwlax.edu/tdas/research
Abstract: TBA