Dynamics Seminar Autumn 2020

The Dynamics/Operator Algebra seminar will meet on Monday afternoons at 2:30 in Zoom. All are welcome. Please contact Chris Bose or Anthony Quas for the zoom link. The list of seminars is as follows. For further information, please contact one of us at cbose(a)uvic.ca or aquas(a)uvic.ca

Date Speaker Title
21/9/2020 Alex Deane (UVic) The Lightning Model
28/9/2020 Gourab Ray (UVic) Proper planar 3-colorings as local functions of iid (Part II)
5/10/2020 Thomas Hughes (McGill) Hit and miss with the (α,β)-superprocess
12/10/2020   No seminar (Thanksgiving)
19/10/2020 at 11 a.m. Jairo Bochi (Universidad Catolica de Chile) Diversity of statistical behavior in dynamical systems
26/10/2020 Anthony Quas (UVic) Flexibility of the pressure function
2/11/2020 Matt Nicol (Houston) Large deviations and central limit theorems for sequential and random dynamical systems
9/11/2020   No seminar: reading break
16/11/2020 Jacob Richey (UBC) Finding the source of a random process
11 a.m. on 23/11/2020 Benoît Saussol Mikado of geodesics on negatively curved manifolds
30/11/2020 Amie Wilkinson (Chicago) Topological and measure rigidity for the strong unstable foliation of an Anosov diffeomorphism


Date: 21/9/2020
Speaker: Alex Deane
Title: The Lightning Model
Abstract: We explore a novel percolation-type model that was proposed as a crude model of lightning propagation. In short, each cell in a network is randomly assigned a potential (a uniform [0,1] random variable). The lightning then propagates from a cell i to a neighbouring cell j if the potential at j is less than i's potential plus a transfer threshold, t. We found that for small values of t, there is no long-range propagation of the "lightning", while for larger values of t, the lightning almost certainly propagates. This model is interesting for a number of reasons: the directionality of the potential bonds (for positive values of t); and the fact that conditioned on the existence of a long path to a node j, the conditional distribution of its potential is strongly affected
Date: 28/9/2020
Speaker: Gourab Ray
Title: Proper planar 3-colorings as local functions of iid (Part II)
Abstract: We consider the unique measure of maximal entropy for proper 3-colorings of Z2, or equivalently, the so-called zero-slope Gibbs measure. We will prove that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation- equivariant function of independent and identically distributed random variables placed on Z2.​ Joint with Y. Spinka (UBC). This is part II of a pair of presentations. Part I that was given in pre-covid era. I will recall all the relevant information, and everything will be self-contained.
Date: 5/10/2020
Speaker: Thomas Hughes
Title: Hit and miss with the (α,β)-superprocess
Abstract: Superprocesses are measure-valued Markov processes modelling spatial branching behaviour in continuous space and time. In this talk I will consider the superprocess associated to an α-stable spatial motion and a (1+β)-stable branching mechanism in the parameter regime in which it has a density. After introducing this process and some classical results, I will discuss some newly proven path properties of the density. These include (i) strict positivity of the density at a fixed time (for certain parameters) and (ii) a classification of the measures which the density charges almost surely when conditioned on survival. The duality between the superprocess and a fractional parabolic PDE is central to our method, and I will discuss how the probabilistic statements above correspond to new results about singular solutions to the PDE.
Date: 19/10/2020
Speaker: Jairo Bochi (Universidad Catolica de Chile)
Title: Diversity of statistical behavior in dynamical systems
Abstract: For chaotic dynamical systems, it is unfeasible to compute long-term orbits precisely. Nevertheless, we may be able to describe the statistics of orbits, that is, to compute how often an orbit will visit a prescribed region of the phase space. Different orbits may or may not follow different statistics. I will explain how to measure the statistical diversity of a dynamical system. This diversity is called emergence, is independent of the traditional notions of chaos.
I will begin by discussing classic problems of discretization of metric spaces and measures. Then I will apply these ideas to dynamics and define two forms of emergence. I will present several examples, culminating with new dynamical systems for which emergence is as large as we could possibly hope for. This talk is based on joint work with Pierre Berger (Paris).
Date: 26/10/2020
Speaker: Anthony Quas
Title: Flexibility of the pressure function
Abstract: Pressure plays a critical role in the thermodynamic formalism, giving a unifying description of many invariant measures of dynamical importance, as well as giving rise to a procedure for computing the Hausdorff dimension of some dynamically-defined sets. In particular, for a dynamical system T:X->X and for a potential function f:X->R, one studies the function g(t)=Pressure(t*f). One can show that provided T has finite topological entropy, the function g(t) is convex, and the set of intercepts with the y-axis forms a bounded sub-interval of [0,infty).
We will recall the definition and basic properties of pressure, and show that the constraint above is essentially the only restriction on g. (Joint work with Tamara Kucherenko)
Date: 2/11/2020
Speaker: Matt Nicol (Houston)
Title: Large deviations and central limit theorems for sequential and random dynamical systems
Abstract: We obtain large and moderate deviations for both sequential and random compositions of slowly mixing intermittent type maps. We also address the question of whether or not centering is needed for quenched central limit theorems (joint with Felipe Perez Pereira and Andrew Torok)
Date: 16/11/2020
Speaker: Jacob Richey (UBC)
Title: Finding the source of a random process
Abstract: Consider a diffusion spreading through a network. Given a snapshot of the history, can the starting point be determined? I will discuss the ideas and problems surrounding this question in two contexts: for simple random walk / Brownian motion, and rumour spread in social networks. For random walks, I will sketch how the theory of self-intersections lends a hand; for rumour spread, I will present the state-of-the-art, an algorithm called adaptive diffusion, discuss its shortfalls, and suggest a path forward.​
Date: 11am on 23/11/2020
Speaker: Benoît Saussol (Brest)
Title: Mikado of geodesics on negatively curved manifolds
Abstract: Recently Athreya, Lalley, Sapir and Wroten have been interested in the tangle of geodesics in a compact riemaniann surface of negative curvature. One question is to understand locally the picture of a geodesic segment of length T, in a vicinity of any point given on the surface. With Françoise Pène we recover their main result, applying our work on spatio-temporal point processes for visits to small sets.
Date: 30/11/2020
Speaker: Amie Wilkinson (Chicago)
Title: Topological and measure rigidity for the strong unstable foliation of an Anosov diffeomorphism
Abstract:
For previous semesters, see
Spring 2006
Fall 2006
Spring 2007
Spring 2008
Spring 2009
Spring 2010
Spring 2011
Spring 2012
Spring 2013
Spring 2014
Spring 2015