Virtual Talks

Daniel Brake ,   Colorado State University
PhD student, advised by Vakhtang Putkaradze
Irreversibility of fluid suspended beads in Couette flow
      At the Stokes limit, fluid equations become time-independent, meaning that the flow is reversible. In a Couette cell, one drives fluid motion by turning concentric cylinders. If the fluid is driven, withReynolds number Re<<1,for some time and then the driving cylinder is reversed for an equal period of time, the fluid will return to its original position. In a sense, we can ``unstir'' the fluid.
      Experimentally, the addition of a single plastic bead of size substantially smaller than the apparatus preserves this reversibility, in that the beadas well as the fluidreturns to its original position. But with the presence of only twosuspendedbeads, the system becomes ``irreversible'': the beads return to macroscopically noticeably different positions. With three, the irreversibility is even more pronounced.
      Further, the surface features of the beads contributes to the irreversibility. A pair (or triplet) of very smooth beads will return more closely to their initial positions, whereas a pair of relatively rough, although macroscopically spherical, beads will exaggerate the difference.
      Computationally elaborate and comprehensive models, including lubrication theory and boundary layer finite element methods, have been developed to investigate this phenomenon. However, they are as time-intensive to run as the original experiments, making it difficult to gather data by running a large number of experiments.
      To overcome the CPU/experiment time barrier, a simpler method has been developed, using only the Stokes flow Couette fluid velocity field and an adjustment for contact, and it approximates the system and its irreversibility decently well. I will present results of these simpler simulations, specifically relating to approximation of the experiment by a 2D system of circles.

Aaron Brown ,   Tufts University
PhD student, advised by Boris Hasselblatt
Non-expanding attractors
      Let X be a non-expanding hyperbolic attractor with 1-dimensional unstable manifolds. We show that the dynamics on X is conjugate to a solenoidal automorphism, which can be explicitly described in terms of the induced action on the fundamental group of the basin of X. As a corollary, we show that no examples of hyperbolic attractors exist in 3 dimensions beyond those already studied and classified in the literature. Our proof relies heavily on the method of proof for the Franks-Newhouse theorem on codimension-1 Anosov diffeomorphisms.

Josh Clemons,   University of North Carolina, Chapel Hill
PhD student, advised by Jane Hawkins
Mandelbrot sets and elliptic functions on square lattices
      In this talk we show that the parameter space of Weierstrass elliptic functions on square lattices contains infinitely many Mandelbrot sets. We also discuss the implications of this result. For example, we show the existence of Siegel disks and of so called "contained" Misiurewicz points.
      For the talk itself, click here.

Gernot Greschonig ,   University of Vienna
Real extensions of distal minimal flows and continuous topological ergodic decompositions
      The talk presents a structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows with a compactly generated Abelian transformation group (e.g. Z^d flows and R^d flows). The main result states that every such extension can be represented by a perturbation of a Rokhlin skew product, and as a corollary follows that the topological ergodic decomposition of the skew product is continuous with respect to the Fell topology. The proof of the result uses of course Furstenberg's structure theorem for distal flows.

Andrew Parrish,   University of Memphis
Pointwise Convergence of Ergodic Averages in Orlicz Spaces
      You can click HERE for the Beamer virtual talk described in the abstract below.
      In this virtual talk we will explore the role of the function space in the question of pointwise convergence of ergodic averages along subsequences. We argue that the behavior of these averages in the Orlicz spaces contained in L^1 but containing all L^p, p>1, is of particular interest, especially when viewed in the context of the recent startling resolutions of some long-standing questions in subsequence ergodic theory. We will also discuss how results in Orlicz spaces can run counter to intuition gained from well-known results in harmonic analysis.

Mrinal Roychowdhury, University of Texas Pan-American
Quantization dimension for some Moran measures
For the abstract, click here.

Mrinal Roychowdhury, University of Texas Pan-American
Quantization dimensions and Markov measure associated with recurrent self-similar set
For the abstract, click here.

Ilya Vinogradov,   Princeton University
PhD student, advised by Yakov Sinai
Limiting distribution of trajectories of several rotations in shrinking intervals
      Let d and n be a positive integers; consider d rotations acting on the 1-torus. We show that in the Poisson scaling regime the numbers of points of the form m_1\alpha_1+...+m_d\alpha_d, m_i < N in n intervals with fixed centers have a joint limiting distribution as N\to\infty. This distribution has finitely many moments. If we then let d tend to infinity, we typically get independent Poisson vectors, but for some centers of the n intervals components of the vector stay entangled even after taking d to infinity.