Dynamics Seminar: Spring 2007

The Dynamics seminar will meet on Friday afternoons (unless otherwise stated) at 2:30 in Clearihue A204. All are welcome. The list of seminars is as follows. For further information, please contact me at  

Date Speaker Title:
12/1 Ian Putnam (UVic) Homology theories for chaotic dynamics
19/1 Ian Putnam (UVic) Homology theories for chaotic dynamics II
THURS 25/1 in Maclaurin D115 Guangyue Han (UBC) On Binary Symmetric Channels
2/2 Anthony Quas (UVic) Global properties of piecewise isometries
9/2   No Seminar
16/2 Alexander Holroyd (UBC) Random Sorting Networks
23/2   No Seminar (Reading break)
2/3   No Seminar
9/3 Yevgeniy Kovchegov (Oregon State) Mixing times via super-fast coupling
16/3 Chris Hoffman (Washington) Phase transitions for topological properties of random two dimensional simplicial complexes
23/3 Jim Campbell (Memphis) Ergodic Theorems and Helical Transforms
30/3   No Seminar

Date:12th and 19th January
Speaker: Ian Putnam
Title: Homology theories for chaotic dynamics
Abstract: The chaotic dynamics under consideration are Smale spaces: homeomorphisms of compact metric spaces possessing canonical coordinates of contracting and expanding directions. Examples include hyperbolic toral automorphisms, solenoids, substitution tiling systems and basic sets for Smale's Axiom A systems. Perhaps the most important are shifts of finite type (SFT). In the first lecture, I will describe some of these, concentrating on the SFT's. I will describe an old invariant for SFT's: the dimension group. The aim of the second talk is to use this invariant as a building block for a much more general homology theory which applies to all Smale spaces.
Date: Thursday January 25
Speaker: Guangyue Han
Title: On Binary Symmetric Channels
Abstract: A binary symmetric channel with crossover probability epsilon (BSC(epsilon)) is a widely used model of a communication channel. In a BSC(epsilon), the probability of a 1 becoming a 0 and of a 0 becoming a 1 are assumed to be equal to epsilon. We derive an asymptotic formula of entropy rate of the hidden Markov chain, observed when a Markov chain passes through a binary symmetric channel. We also derive an asymptotic formula of capacity of a binary symmetric channel with input process supported on an irreducible finite type constraint.
Date:February 2
Speaker:Anthony Quas (UVic)
Title:Global properties of Piecewise Isometries
Abstract: We consider a simple family of bijective piecewise isometries of the plane in which two halves of the plane are slid relative to each other and the entire plane is then rotated. For these maps, we prove the existence of an infinite number of `periodic islands' and use geometric properties to show that the maps satisfy a recurrence property (in spite of the fact that the underlying measure is infinite). We state a number of open questions. (joint work with Arek Goetz)
Date:February 16
Speaker:Alexander Holroyd (UBC)
Title:Random Sorting Networks
Abstract: Sorting a list of items is perhaps the most celebrated problem in computer science. If one must do this by swapping neighboring pairs, the worst initial condition is when the n items are in reverse order, in which case n choose 2 swaps are needed. A sorting network is any sequence of n choose 2 swaps which achieves this. We investigate the behavior of a uniformly random n-item sorting network as n->infinity. We prove a law of large numbers for the space-time process of swaps. Exact simulations and heuristic arguments have led us to astonishing conjectures. For example, the half-time permutation matrix appears to be circularly symmetric, while the trajectories of individual items appear to converge to a famous family of smooth curves. We prove the more modest results that, asymptotically, the support of the matrix lies within a certain octagon, while the trajectories are Holder-1/2. A key tool is a connection with Young tableaux. (joint with Omer Angel, Dan Romik and Balint Virag)

(See Ander's web page for more pictures)

Date:March 9th
Speaker:Yevgeniy Kovchegov (Oregon State)
Title:Mixing times via super-fast coupling
Abstract:We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate $n\log(n)$ with a moderate constant. This has already been shown by Diaconis and Shahshahani but no natural coupling proof has been demonstrated to date. We also enlarge the methodology of coupling to include intuitive but nonadapted coupling rules, for example, to take in account future events and to prepare for their occurrence. (Joint work with R.Burton)
Date:March 16th
Speaker:Chris Hoffman (Washington)
Title:Phase transitions for topological properties of random two dimensional simplicial complexes
Abstract:The Erdos-Renyi random graph G(n,p) is the probability measure on all graphs on the vertex set [n]={1,2,...n}, with each edge inserted independently with probability p. Usually p is defined to be a function of n, and one asks whether a graph in G(n,p) is likely to have some (monotone) property as n goes to infinity. In their seminal 1959 paper, Erdos and Renyi showed that if p << log(n)/n then G(n,p) is almost always disconnected, but if p >> log(n)/n, then it is almost always connected. We view this as a topological statement and seek two-dimensional analogues of the Erdos-Renyi Theorem. We construct a measure on two-dimensional simplicial complexes and look at two questions about phase transitions for topological properties. For which values of p is the random complex almost always simply connected, and for which values of p does the random complex almost always have trivial homology? This is based on joint work with Matt Kahle and Eric Babson.
Date:March 23rd
Speaker:Jim Campbell (Memphis)
Title:Ergodic Theorems and Helical Transforms
 Abstract
For previous semesters, see
Spring 2006
Fall 2006