Mathematics Department

# Operator Algebras and Dynamics Seminar

## Spring 2018

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Apr
30
• Understanding the flow of large dimensional oscillators via the centre manifold
Irina Popovici
USNA
Time: 03:45 PM

#### View Abstract

Many of the phenomena modeled by dynamical systems have large dimensionality and therefore have complicated dynamics (flocking, synchronization, consensus forming). They often exhibit oscillations in multiple planes, so understanding their behavior begins with studying the flow on their large dimensional centre manifolds. I will talk about the subtleties associated with parametrizing manifolds that are non-unique, and with approximating flows for nonlinear systems that are not stable to perturbations. The applications range from (classic) low-dimensional systems to recent results regarding the stability of some structures in a swarming model.
• Apr
23
• On amenability of topological groups
Christian Rosendal
University of Illinois at Chicago
Time: 03:45 PM

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I will discuss the Folner type conditions that come up in characterizing amenability of general topological groups, in particular in recent work of Schneider and Thom, and compare this with the Ramsey type characterizations offered by Moore for the particular case of non-Archimedean Polish groups.
• Apr
16
• Bratteli diagrams and the ergodic theory of random substitution tilings of R^d
Rodrigo Trevino
The University of Maryland
Time: 03:45 PM

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We study tilings obtained by performing random substitutions on a given set of prototiles in R^d. These tilings can be seen as given by random applications of a collection of graph iterated function systems obtained by extending the blowup construction of M. Barnsley and A. Vince. We use Bratteli diagrams to organize the hierarchical structure of the tilings and introduce renormalization dynamics through the shift of the Bratteli diagrams. The Lyapunov spectrum of the induced cocycle on the cohomology bundle describes growth properties of ergodic averages of the R^d action on the tiling space. The associated invariant distributions help explain questions physical nature such as behavior of diffraction measures as well as the spectral properties of Schrodinger operators associated to these tilings. This generalizes several works focused on self-similar tilings. This is joint work with Scott Schmieding.
• Apr
09
• Stability of Nonlinear Swarms on Flat and Curved Surfaces
MIDN Carl Kolon
USNA
Time: 03:45 PM

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This talk concerns the stability of a nonlinear parabolic potential model for a swarm. We will discuss the global behavior of a simplified decoupled system using Lyapunov stability theory and Lienard's theorem, and the local behavior of a system of more agents. We will then present a generalization of this model to Riemannian manifolds, and present numerical results displaying their behavior.
• Apr
02
• Infinite measure mixing in mechanical systems
Peter Nandori
The University of Maryland
Time: 03:45 PM

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We consider certain physical systems that preserve a natural absolutely continuous infinite measure. For such systems, we study three notions of mixing: (1) Krickeberg mixing, i.e. mixing of nice, compactly supported functions, called local functions, (2) local global mixing, i.e. when one function is local and the other one has convergent averages over large boxes (called global function) and (3) global global mixing, when both functions are global. The last two definitions are due to Marco Lenci. We prove some abstract theorems in both discrete and continuous time. Then we apply these theorems for specific examples such as suspensions over Pomeau-Maneville maps, billiard flows and ping-pong models. Joint work with Dmitry Dolgopyat.
• Mar
26
• Essential Freeness of Stationary Actions of Lattices
Darren Creutz
USNA
Time: 03:45 PM

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Generalizing a specific case of the Margulis Normal Subgroup Theorem to actions, Stuck and Zimmer proved that every ergodic measure-preserving action of an irreducible lattice is a semisimple real Lie group, all of whose factors is higher-rank, on a nonatomic probability space is essentially free. More recently, partly in joint work with J. Peterson, we generalized this to lattices in all semisimple groups having at least one higher-rank factor. However, since such lattices are nonamenable, the existence of invariant measures for their actions is not guaranteed. A more natural question to consider is that of stationary measures (which always exist). For such a lattice, the natural stationarity condition to consider is that the measure be stationary under the measure that makes the Poisson boundary of the lattice be that of the semisimple group. We prove that every stationary nonatomic action of a lattice is essentially free.
• Mar
19
• Dynamical properties of maps on quasi-graphs
Piotr Oprocha
AGH University of Science and Technology (Krakow, Poland)
Time: 03:45 PM

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Quasi-graphs are natural generalizations of topological graphs. The simplest example of such space is the Warsaw circle. Even from this simplest example it is clear that the structure of ω-limit set can be richer than is possible in graph maps. On the other hand, some similarities still exist. In this talk we will focus mainly on topological entropy and properties of invariant measures.
• Mar
02
• A (potential) topological/C*-algebra approach to rigidity for lattices (joint w/ M. Kalantar)
Darren Creutz
Time: 03:45 PM

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Rigidity results for lattices have almost exclusively come from studying measurable actions and the corresponding noncommutative generalization to von Neumann algebras, the central ingredient being the Poisson boundary. A relatively unknown topological equivalent of the Poisson boundary, called the Furstenberg boundary, appears to capture even more information about the group at the expense of being quite difficult to work with in comparison. The corresponding noncommutative C*-algebras arising from it, and their inclusions in one another, likewise seem to capture more information but are harder to understand. I will present a potential approach to a new proof of Margulis' Normal Subgroup Theorem based on these ideas and outline why myself and M. Kalantar believe they have the potential to prove rigidity results currently unreachable.
• Feb
26
• On the rigidity of uniform Roe algebras of coarse spaces
Bruno de Mendonca Braga
York University
Time: 03:45 PM

#### View Abstract

(joint with Ilijas Farah) Given a coarse space $(X,\mathcal{E})$, one can define a $C^*$-algebra $C^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willet that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this talk, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.