Dynamics/Operator Algebra Seminar Spring 2014

The Dynamics/Operator Algebra seminar will meet on Wednesday afternoons at 3:30 in Clearihue A303. All are welcome. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title
22/1/2014 Boualem Khouider (UVic) A coarse grained stochastic multi-type particle interacting model for tropical convection: nearest neighbour interactions
29/1/2014 Eric Foxall (UVic) New results for the two-stage contact process
3/2/2014 Todd Kemp (UCSD) Heat Flow and Brownian Motion on NxN Matrix Lie Groups, and the Large-N Limit
19/2/2014 No seminar
26/2/2014 Ian Putnam (UVic) Some algebraic dynamical systems and Markov partitions for them
5/3/2014 Sarah Saeidi Gholikandi One-dimensional solenoids
12/3/2014 Máté Wierdl (U. Memphis) Random differences of arithmetic progressions in sets of positive density
19/3/2014 Terry Loring (U. New Mexico) Numerical Experiments on the Modulus of Continuity of Matrix Functions
26/3/2014 Nishant Chandgotia (UBC) Four Cycle Free Graphs and Entropy Minimality
2/4/2014 Chris Hoffman (Washington) Z2 Skew Products

Date: 22/1/2014
Speaker: Boualem Khouider (UVic)
Title: A coarse grained stochastic multi-type particle interacting model for tropical convection: nearest neighbour interactions
Abstract: Particle interacting systems on a lattice are widely used to model complex physical processes that occur on much smaller scales than the observed phenomenon one wishes to model. However, their full applicability is hindered by the curse of dimensionality so that in most practical applications a mean field equation is derived and used. Unfortunately, the mean field limit does not retain the inherent variability of the microscopic model. Recently, a systematic methodology is developed and used to derive stochastic coarse-grained birth-death processes which are intermediate between the microscopic model and the mean field limit, for the case of the one-type particle-Ising system. Here we consider a stochastic multicloud model for organized tropical convection introduced recently to improve the variability in climate models. Each lattice site is either clear sky of occupied by one of three cloud types. In earlier work, local interactions between lattice sites were ignored in order to simplify the coarse graining procedure that leads to a multi-dimensional birth-death process; Changes in probability transitions depend only on changes in the large-scale atmospheric variables. Here the coarse-graining methodology is extended to the case of multi-type particle systems with nearest neighbour interactions and the multi-dimensional birth-death process is derived for this general case. The derivation is carried under the assumption of uniform redistribution of particles within each coarse grained cell given the coarse grained values. Numerical tests show that despite the coarse graining the birth-death process preserves the variability of the microscopic model. Moreover, while the local interactions do not increase considerably the overall variability of the system, they induce a significant shift in the climatology and at the same time boost its intermittency from the build up of coherent patches of cloud clusters that induce long time excursions from the equilibrium state.
Date: 29/1/2014
Speaker: Eric Foxall (UVic)
Title: New results for the two-stage contact process
Abstract: The two-stage contact process, first introduced by Steve Krone, is a stochastic growth model in which each individual must reach a mature stage before reproducing, and is a natural generalization of the contact process. We first give some background on the contact process, outlining some important results of the 80's and 90's including complete convergence, We then introduce the two-stage process, for which the same results hold. We indicate why this should be so, by showing that the two-stage process possesses the same useful properties as the contact process. Finally, we describe an important difference in the two-stage process, namely the existence of a critical value of the maturation rate below which survival is not possible. This fact is in qualitative contrast to the corresponding two-stage branching process.

Date:3/2/2014
Speaker: Todd Kemp (UCSD)
Title: Heat Flow and Brownian Motion on NxN Matrix Lie Groups, and the Large-N Limit
Abstract: Heat flow is one of the most pervasive concepts in mathematics. Understood from antiquity as a solution to a PDE, the mid-Twentieth Century development of stochastic processes gave a new understanding, rigorously completing Einstein's picture of heat flowing through random collisions of particles with aggregate behavior described by (a mean value of) Brownian motion. This picture makes sense on Riemannian manifolds, and led to a revolution in understanding geometry through heat flow.

In this talk, I will discuss some of the standard theory of heat flow on classical Lie groups, focusing on unitary groups UN and general linear groups GLN. I will then describe my recent work on the large-N limits of the Brownian motions on these groups, their fluctuations, and applications to random matrix theory and operator algebras.

Date: 26/2/2014
Speaker: Ian Putnam (UVic)
Title: Some algebraic dynamical systems and Markov partitions for them
Abstract: I will describe some interesting dynamical systems of an algebraic nature. In many ways, they are similar to the well-known hyperbolic automorphisms of tori, such as Arnold's cat map. The new ingredient is the use of the field of p-adic numbers in place of the real numbers. My guess here is that most people may not be very familiar with p-adic numbers so I will give a short introduction to them, their wonderful properties and what about them might appeal to people in dynamical systems. After describing the systems, I will discuss the problem of finding convenient symbolic codings for them (i.e. Markov partitions) which was done for hyperbolic toral automorphisms by Adler and Weiss. This is joint work with Nigel Burke (UVic/ Cambridge). From these codings, we are able to compute the homology, as I defined in earlier work, but this will be a brief footnote to the rest of the talk.

Date: 5/3/2014
Speaker: Sarah Saeidi Gholikandi
Title: One-dimensional solenoids
Abstract:
I will talk about one-dimensional solenoids and their homology, as defined for Smale spaces by Putnam. First I will start with definition of Smale spaces and show how shifts of finite type spaces are examples, which play an important role in this homology. Then I will go into solenoids by giving some examples. We will show how there are natural shifts of finite type which factor onto them, which is used in computing their homology. Since the process of computing this homology is complicated, I will not enter into the details of the computations and just give the results. This is joint work with M.Amini and I.F Putnam
Date: 12/3/2014
Speaker: Máté Wierdl (Memphis)
Title: Random differences of arithmetic progressions in sets of positive density
Abstract: Szemerédi's theorem on arithmetic progressions says that, for a given k, if a sequence A = a1 In this talk, we'll focus on the question if we can restrict r to be from a typical sequence. By a typical sequence, we mean a sequence which is randomly generated. For example, let us decide if a positive integer belongs to the sequence by flipping a fair coin. Is it true that, with probability 1, we can take the difference r in Szemerédi's theorem to be from this random sequence. It was Erdős and Sárközy who first examined random sequences in case k=1. In the talk, we examine the case of arbitrary k, we discuss the current state of the art and we'll see some unsolved problems. If time permits, we'll also see connections with ergodic theory.
I'll make the talk almost completely accessible to graduate students. Only some elementary probability theory will be needed.
Date: 19/3/2014
Speaker: Terry Loring (New Mexico)
Title: Numerical Experiments on the Modulus of Continuity of Matrix Functions
Abstract: If f is a real-valued function on the unit interval, we can extend its domain to matrices that are Hermitian with eigenvalues in the unit interval. Now assume f is continuous. It is uniformly continuous, on its original domain. Is f also uniformly continuous for matrices? That is, can we find eta tending to zero at zero so that, for 0<=X<=I and 0<=Y<=I, ||f(X)-f(Y)| <= eta(||X-Y||) holds for the spectral norm? Ando showed this is false for f(t)=|t-1/2| but true when f is operator monotone. A related question is uniformly bounding ||f(X)A-Af(X)|| in terms of ||XA-AX|| whenever ||A||<=1. The conjecture ||sqrt(X)A-A\sqrt(X)||<= sqrt(||XA-AX||) has been around for decades. We are able to prove this in an important special case. I will discuss the Monte Carlo methods used when computing billions of examples and present some conjectures based on the data we collected. This is joint work with my graduate student, Fredy Vides.

Date: 26/3/2014
Speaker: Nishant Chandgotia (UBC)
Title: Four Cycle Free Graphs and Entropy Minimality.
Abstract: Entropy minimality was introduced by Coven and Smítal (1992) as a property of dynamical systems which is stronger than topological transitivity and weaker than minimality. A topological dynamical system (X,T) is said to be entropy minimal if all closed T-invariant subsets of X have entropy strictly less than (X,T). In the sphere of symbolic dynamics this translates to the following: a shift-space is entropy minimal if by exclusion of any globally allowed pattern we incur a drop in the topological entropy. In one dimension it is well known that a nearest neighbour shift of finite type is entropy minimal if it is irreducible; the same is known in higher dimensions under strong irreducibility. In this talk we will discuss a class of nearest neighbour shift of finite type which appear as the space of graph homomorphisms from Z^d to graphs without four cycles; for instance, we will see why the space of 3-colourings of Z^d is entropy minimal even though it does not have any of the nice mixing properties.

Date: 2/4/2014
Speaker: Chris Hoffman (U. Washington)
Title: Z2 Skew Products
Abstract: Many constructions in ergodic theory involve skew products. The use of skew products is much more common for constructing actions of Z than for actions of other groups. In this talk I will describe how to make the Z^2 analogs of some common skew products. Along the way we will find connections to several models statistical mechanics as well as questions in computability.

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