Dynamics/Applied Seminar: Spring 2009

The Dynamics/Applied seminar will meet on Friday afternoons (unless otherwise stated) at 2:30 in COR A128. All are welcome. The list of seminars is as follows. For further information, please contact me at  

Date Speaker Title:
27/2 Amites Sarkar (Western Washington) Partitioning Random Geometric Covers
6/3   No seminar
13/3 Chris Bose Strange Eigenmodes for Invertible, Measure Preserving Systems
17/3 Ethan Coven (Wesleyan) Every odometer is a cellular automaton
20/3 Vipul Periwal (NIH) A model of liver regeneration 
27/3 Mike Hochman (Princeton) Local entropy and projections of dynamically defined fractals
3/4 Bob Burton (Oregon State) A Brief Illustrated Story About Square Ice in The Random Snow


Date: 27th February
Speaker: Amites Sarkar (Western Washington)
Title: Partitioning Random Geometric Covers
Abstract: I'll present some new results on partitioning both random and non-random geometric covers. For the random results, let P be a Poisson process of intensity one in the infinite plane R^2, and surround each point x of P by the open disc of radius r centered at x. Now let S_n be a fixed disc of area n>>r^2, and let C_r(n) be the set of discs which intersect S_n. Write E_r^k for the event that C_r(n) is a k-cover of S_n, and F_r^k for the event that C_r(n) may be partitioned into k disjoint single covers of S_n. I'll sketch a proof of the inequality Prob(E^k_r\F^k_r)<c_k/log n, which is best possible up to a constant. The non-random result is a classification theorem for covers of R^2 with half-planes that cannot be partitioned into two single covers. It was motivated by a desire to understand the obstructions to k-partitionability in the original random context. This is all joint work with Paul Balister, Bela Bollobas and Mark Walters.


Date: 13th Match
Speaker: Chris Bose (UVic)
Title: Strange Eigenmodes for Invertible, Measure Preserving Systems
Abstract: The term strange eigenmode in dynamical systems refers to a situation where some large-scale structure persists for an uncharacteristically long time under action by an otherwise mixing flow. This curious and not-very-well-defined concept has been observed in nature (ocean flows, for example) and in some mathematical models. For one-dimensional uniformly expanding dynamics a reasonable explanation for the phenomenon can be given through the presence of large, non-peripheral eigenvalues for the associated transfer operator (acting on a suitable space of regular functions) For mulidimensional uniformly hyperbolic systems, this theory breaks down and one must instead look at an expanded space of 'functions' to serve as the eigenmode. We present a preliminary report. This is joint work with Gary Froyland, School of Mathematics and Statistics, University of New South Wales, AUSTRALIA


Date: 17th March
Speaker: Ethan Coven (Wesleyan)
Title: Every odometer is a cellular automaton
Abstract: An odometer is the +1 map on a countable product of finite cyclic groups, addition with "carrying." Think of an old-fashioned car odometer with spinning wheels. A one-dimensional cellular automaton is a map given by a local rule on a space of sequences with entries from a finite alphabet. In two dimensions, think of the Game of Life. I will show that every odometer is topologically conjugate to a "gliders with reflecting walls" cellular automaton, and I will characterize those odometers that are topologically conjugate to cellular automata with the simplest, non-trivial, algebraic local rules: $x_i \mapsto x_i + x_{i+1} \mod n$.


Date: 20th March
Speaker: Vipul Periwal (NIH)
Title: A model of liver regeneration
Abstract: The network of interactions underlying liver regeneration is robust and precise with liver resections resulting in controlled hyperplasia (cell proliferation) that terminates when the liver regains its lost mass. The interplay of cytokines and growth factors responsible for the inception and termination of this hyperplasia is not well understood. A model is developed for this network of interactions based on the known data of liver resections. This model reproduces the relevant published data on liver regeneration and provides geometric insights into the experimental observations. The predictions of this model are used to suggest two novel strategies for speeding up liver mass recovery and a strategy for enabling liver mass recovery in cases where a resection leaves less than 20$\%$ of the liver that would otherwise result in complete loss of liver mass.


Date: 27th March
Speaker: Mike Hochman (Princeton)
Title: Local entropy and projections of dynamically defined fractals
Abstract: If a closed subset X of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of X (since the projection is Lipschitz), and also no larger than 1 (since it is a subset of a line). A classical theorem of Marstrand says that for any such X, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to min(1,dim(X)). In general, there can be uncountably many exceptional directions. An old conjecture of Furstenberg is that if A,B are subsets of [0,1] invariant respectively under x2 and x3 mod 1, then for their product, X=AxB, the only exceptional directions in Marstrand's theorem are the two trivial ones, namely the projectsion onto the x and y axes. Recently, Y. Peres and P. Shmerkin proved that this is true for certain self-similar fractals, such as regular Cantor sets. I will discuss the proof of the general case, which relies on a method for computing dimension using local entropy estimates. I will also describe some other applications. This is joint work with Pablo Shmerkin.


Date: 3rd April
Speaker: Bob Burton (Oregon State)
Title: A Brief Illustrated Story About Square Ice in The Random Snow
Abstract: Square Ice is a simple model for a kind of frozen water in semi-crystalline form. Much of its interest arises from from its nearly unbounded mutability and the continuing connections being made with other parts of mathematics, especially dynamical systems and number theory. A simple way to think of square ice is as a tiling system with 1 by 1 monominoes which can be of three colors: medium orange, pale blue, and poppy purple. The are used to tile the plane according to the lattice as randomly as possible with the restriction that no two adjacent tiles can have the same color. This model belongs to only two nontrivial classes of subshifts of finite type (SOFTs) whose entropy is exactly computable (4/3)^(3/2) = (8/9) sqrt(3). We will show pictures and puzzles, constructions and points of contact and connections with symbolic subshifts, substitution systems, Farey trees, representations of general Z^2 indexed dynamical systems., and so on, taking care to protect our time for general discussion so all will have ample opportunity to provide insights.



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