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.