Dynamics/Operator Algebra Seminar Spring 2015

The Dynamics/Operator Algebra seminar will meet on Thursday afternoons at 2:30 in Clearihue A127. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title
22/1/2015 No seminar  
29/1/2015 Jason Siefken (UVic) A Minimal Subsystem of the Kari Culik Tilings
5/2/2015 Joseph Horan (UVic) Block triangularization of matrix cocycles
12/2/2015 Reading Break  
19/2/2015 Raimundo Briceño (UBC) Gibbs measures, hard constraints and (boundary) phase transitions
26/2/2015 Ian Putnam (UVic) Some new minimal homeomorphisms, why they shouldn't exist and why it's important that they do
5/3/2015 Alexander Holroyd (Microsoft) Finitely dependent coloring
12/3/2015 Matthew Junge (Washington) The frog model on trees.
19/3/2015 Seth Chart (UVic) A Lasota-Yorke inequality for a class of expanding interval maps.
26/3/2015 Peter Dukes (UVic) A class of piecewise rotations in finite fields and graph one-factorizations
2/4/2015 Doug Lind (Washington) Algebraic actions of the discrete Heisenberg group

Date: 22/1/2015

Date: 29/1/2015
Speaker: Jason Siefken (UVic)
Title: A Minimal Subsystem of the Kari Culik Tilings
Abstract: The set of 13 Kari Culik tiles is currently the smallest know set of Wang tiles (square tiles with colored edges and the rule that two tiles may lie adjacent iff their common edges share the same color) that tile the plane only in an aperiodic way. They do so for fundamentally different reasons than previously known tilings. A subset of Kari Culik tilings have rows which may be interpreted as Sturmian sequences. This talk will show how this Sturmian-like subset can be thought of as a generalization of rotation sequences and how to get explicit waiting time bounds for n*m configurations.

Date: 5/2/2015
Speaker: Joseph Horan
Title: Block triangularization of matrix cocycles
Abstract: When dealing with matrices, putting them into a triangular form often yields meaningful information about their structure. In the case of invertible real matrix cocycles over an invertible, ergodic map, the Multiplicative Ergodic Theorem provides a subspace decomposition of $\mathbb{R}^n$, which may be interpreted as a statement about block diagonalizing the cocycle. One can then ask if cocycles may always be block triangularized, not just block diagonalized; we answer this question in the negative, and therefore show that the MET is in this sense optimal.

Date: 19/2/2015
Speaker: Raimundo Briceño (UBC)
Title: Gibbs measures, hard constraints and (boundary) phase transitions
Abstract: Inspired by topological notions coming from symbolic dynamics, we introduce a new combinatorial condition (TSSM) on the support of Z^d Markov random fields, especially useful when dealing with hard constrained systems. We establish some of its properties and different characterizations. We also show how TSSM is related with the absence of phase transitions in the context of nearest-neighbour Gibbs measures

Date: 26/2/2015
Speaker: Ian Putnam (UVic)
Title: Some new minimal homeomorphisms, why they shouldn't exist and why it's important that they do
Abstract: It is an old question in dynamical systems: which compact metric spaces admit minimal (uniquely ergodic) homeomorphisms? The circle obviously has irrational rotations, but for higher dimensional spheres, this question becomes quite interesting. One of the major positive results is by Fathi and Herman which asserts that odd dimensional spheres admit minimal, uniquely ergodic homeomorphisms. On the other hand, the most famous negative result is the Hopf-Lefschetz theorem which, in particular, shows that even dimensional spheres do not. After discussing these, I will give an example of another space which admits a minimal, uniquely ergodic homeomorphism, managing in a crafty manner to avoid the Hopf-Lefschetz theorem, which says that it shouldn't. This has interesting consequences for George Elliott's classification program for C*-algebras. I'll discuss these, without assuming any prior knowledge of C*-algebras.

Date: 5/3/2015
Speaker: Alexander Holroyd (Microsoft Research)
Title: Finitely dependent coloring
Abstract: A central concept of probability and ergodic theory is mixing in its various forms. The strongest and simplest mixing condition is finite dependence, which states that variables at sufficiently well separated locations are independent. A 50-year old conundrum is to understand the relationship between finitely dependent processes and block factors (a block factor is a finite-range function of an independent family). The issue takes a surprising new turn if we in addition impose a local constraint (such as proper coloring) on the process. In particular, this has led to the discovery of a beautiful yet mysterious stochastic process that seemingly has no right to exist.

Date: 12/3/2015
Speaker: Matthew Junge (University of Washington)
Title: The frog model on trees
Abstract: Fix a graph G and place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Set things in motion by having awake frogs perform independent simple random walk, waking any "sleepers" they encounter. Say the model is recurrent if the root is a.s. visited by infinitely many frogs and otherwise transient. When G is the rooted d-ary tree with one-frog-per-site we prove a phase transition from recurrence to transience as d increases. Alternatively, for fixed d with Poi(m)-frogs-per-site we prove a phase transition from transience to recurrence as m increases. The proofs use two different recursions and two different versions of stochastic domination. Several open problems will be discussed. Joint with Christopher Hoffman and Tobias Johnson.

Date: 19/3/2015
Speaker: Seth Chart (UVic)
Title: A Lasota-Yorke inequality for a class of expanding interval maps.
Abstract: In their 1973 paper, Lasota and Yorke observed that that the transfer operator associated to a piecewise expanding map of the interval acting on functions of bounded variation has a regularizing effect. This property allowed them to prove that the map possesses absolutely continuous invariant measures. The main step in this proof is the calculation of the so called Lasota-Yorke inequality.
The classical definition of variation is not easily generalized to higher dimensions. Many variants of multidimensional variation were introduced in the literature. The widely accepted modern definition of variation is due to Ennio De Giorgi and is formulated in terms of C^{1} test functions. The modern definition generalizes nicely to higher dimensions.
In this talk we will prove a Lasota-Yorke inequality for a class of expanding interval maps using the modern definition of variation. The formulation allows for the main technical issue encountered in proving the inequality to be treated in a particularly transparent and geometric manner.

Date: 26/3/2015
Speaker: Peter Dukes (UVic)
Title: A class of piecewise rotations in finite fields and graph one-factorizations
Abstract: Piecewise rotations in the complex plane divide the space into two half-planes, let's say upper and lower, sending points in the upper (lower) half-plane to a right (respectively, left) translation, followed by a common rotation. The orbits of such maps lead to some beautiful pictures and itineraries of points lead to some interesting symbolic dynamics. Early researchers include Boshernitzan, Cheung, Goetz, and Quas. Curiously, I had been interested in a (superficially, at least) similar map over finite fields. The motivation comes from a problem in combinatorics and extremal graph theory. A one-factorization of a graph is a colouring or partition of its edges into perfect matchings (also called one-factors). The union of two perfect matchings is a disjoint union of even-length cycles, and it is of interest to study the possible cycle lengths arising from such unions in a one-factorization. For instance, a one-factorization is called "perfect" if the union of any two colour classes is a Hamiltonian cycle. At the other extreme, if the union of any two colour classes always shatters into small components, this answers a question of Haggkvist.
I will discuss how a class of one-factorizations of complete graphs arising from finite fields can give some neat results on cycle structures. The idea is essentially a study of the dynamics of my finite analog of piecewise rotations. This is based on joint work with Jeff Dinitz.
As fair warning, I admit that the connection with dynamics is a bit of a stretch; however, the parallels with the planar case may be amusing.

Date: 2/4/2015
Speaker: Doug Lind (Washington)
Title: Algebraic actions of the discrete Heisenberg group
Abstract: Actions of groups like Z or Z^d using automorphisms of compact abelian groups have served as a source of inspiration and rich examples in dynamics for over fifty years. More recently, the study of similar actions of groups such as the discrete Heisenberg group have revealed a host of new phenomena and connections to areas such as von Neumann algebras. Using concrete examples that anyone can understand, I will sketch some of the previous theory, and indicate some of the many fascinating open problems remaining. This is necessarily just a sampler, in the spirit of Halmos's dictum for math talks "to attract and inform”.

For previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010
Spring 2011
Spring 2012
Spring 2013
Spring 2014