(September 9) Rich Schwartz:
Billiards Obtuse and Irrational.
Amazingly, it is not known if every triangle
admits a periodic billiard path. The case of acute
triangles was settled affirmatively in the 1770s; the
case of right triangles was settled affirmatively in
the 1990s; the case of rational triangles was settled
affirmatively in the 1980s. This leaves obtuse
triangles in which the angles are not rational
multiples of Pi. I will demonstrate a computer
program I wrote which searches for periodic billiard
paths in triangles - particularly obtuse, irrational
triangles. Using the program I can see (and prove)
that a triangle has a periodic billiard path provided
all its angles are less than 100 degrees. In this
talk I will present some of the experimental evidence
for the 100 degree result, and explain how the
evidence is converted into a proof.
(September 16) Mike Boyle:
Equivariant flow equivalence and positive K-theory
for G- shifts of finite type.
(joint work with Mike Sullivan)
Here G is a finite group acting freely on an irreducible
shift of finite type (SFT). There is a "positive
K-theory" classification framework for conjugacy
and flow equivalence of G-SFTs. (For flow equivalence,
we can equivalently think of free G actions on
mapping tori of SFTs, and orientation preserving
homeomorphisms respecting G-actions.)
In this framework, we reduce the flow equivalence of
SFTs to the problem of determining whether given
two square matrices A,B over ZG there are matrices
U,V which are products of basic elementary matrices
over ZG such that UAV = B.
For ZG=Z, this requires only the calculation of the
Smith normal form. For nontrivial G, simple examples
and K-theoretic invariants indicate the the problem
is much more interesting and difficult.
(September 23) Yuri Kifer :
Advances in fully coupled averaging for dynamical
systems
abstract in PS ,
abstract in pdf
(October 7) Dmitry Treschev :
Classical approach to quantum mechanics
We propose a new language for quantum mechanics.
By using this language, one can see that quantum mechanics
is a direct generalization of classical one and that basic
classical quantities are projections of the corresponding
quantum analogs. This language gives us a possibility to
use analogs of classical methods in quantum situation
(changes of variables, normal forms, etc).
(October 21) Todd Fisher :
Non-Locally Maximal Hyperbolic Sets and Markov Partitions
I give new examples of hyperbolic sets that are not contained in
locally maximal ones. These examples are robust and can be built on any
compact
manifold of dimension greater than one. Further, I will show that any
hyperbolic set is included in a hyperbolic set with a Markov
partition.
(November 4) Leonid Koralov :
An Inverse Problem for Gibbs Fields
Given a potential of pair interaction and a value of activity,
one
can consider the Gibbs distribution in a finite domain $\Lambda
\subset \mathbb{Z}^d$. It is a classical result that for small values of
activity there exist the infinite volume ($\Lambda \rightarrow
\mathbb{Z}^d$) limiting Gibbs distribution and the infinite volume
correlation functions. In this talk we consider the converse
problem - we show that given $\rho_1$ and $\rho_2(x)$, where
$\rho_1$ is a constant and $\rho_2(x)$ is a function on
$\mathbb{Z}^d$, which are sufficiently small, there exist a pair
potential and a value of activity, for which $\rho_1$ is the
density and $\rho_2(x)$ is the pair correlation function.
(November 18) James Propp :
When algebraic entropy vanishes
If one studies dynamics in the setting of algebraic
geometry (using birational maps between projective varieties),
a very natural invariant is algebraic entropy, as defined by
Bellon and Viallet. It can be thought of as a generalization
of the topological entropy of a torus endomorphism, and it is
related to Arnold's notion of intersection complexity. When
a map has algebraic entropy zero, nice things can arise,
including intimate links with more familiar forms of dynamics;
for instance, the birational map (x:y:z)->(xy:y^2+z^2:xz) from
CP^2 to itself is related to the golden-mean shift, and a more
complicated version of this map (best thought as an "algebraic
cellular automaton") is related to the dimer model in the plane.
It appears that algebraic entropy is always the logarithm of an
algebraic integer, but it can be shown that the degree-sequence
itself need not satisfy a linear recurrence.
(December 9) Konstantin Khanin :
Rigidity of Critical Circle Maps
We shall discuss recent results on rigidity for circle maps
and universal behaviour of renormalizations.
It turns out that in the presence of singularities the rigidity
is more robust and it holds for all irrational rotation numbers.
We prove that two analytical critical circle maps with the same order
of the critical points and the same irrational rotation numbers are
C^1-smoothly conjugated.
(February 3) Sam Lightwood :
Bounded Orbit Injections and
Suspension Equivalence for Minimal Z^2 Actions on Cantor Sets
(joint work with Nic Ormes)
We define a weakened form of flow equivalence which we call
suspension equivalence. In the setting of minimal Z actions on
Cantor sets by homeomorphisms, previous results by Boyle and by Parry and
Sullivan may be combined to allow suspension equivalence to be identified
with the existence of bounded orbit injections into a
common system. We have extended this to the setting of Z^2 minimal
actions on Cantor sets by commuting homeomorphisms. We will describe
the result, the ingredients of the proof and indicate what we see as
the obstruction to dimensions greater than 2.
(February 10) Michal Misiurewicz :
Rotation theory for billiards
Rotation theory deals with the limits of ergodic averages (everywhere,
not almost everywhere) for a given observable. This observable is
usually some kind of displacement in a covering space. For billiards,
there are various cases: billiard inside a table with or without an
obstacle, toral billiard with obstacles, etc., and various approaches:
one can look at the flow or at a discrete system in different ways. In
certain cases with the right approach one can prove interesting
properties of the rotation sets. This is a work in progress, joint
with Alexander Blokh and Nandor Simanyi.
(March 3) Alexander Kechris :
Group actions, equivalence relations and set-theoretic rigidity
phenomena
I will give an introduction to a theory of complexity of
classification problems in mathematics and discuss its connections with
set-theoretic versions of rigidity phenomena for measure preserving actions
of discrete groups.
(March 31) Joe Auslander :
Recurrence in zero dimensional flows
Let (X,G) be a flow, where X is a compact metric space and
G is a finitely generated group. Recurrence is defined in terms
of certain subsets of G called cones. We say that x in X is
recurrent if for every cone C and every neighborhood U of x,
the intersection of U and Cx is nonempty.
Now suppose X is zero dimensional. Then (X,G) is pointwise
recurrent iff (X,G) is pointwise almost periodic iff the orbit
closure relation is closed. A corollary is that a distal action
of G on a zero dimensional space is equicontinuous.
This is joint work with Eli Glasner and Benjy Weiss.
(April 7) Stephane Sabourau :
Systolic volume and minimal entropy on aspherical manifolds
The systole of a nonsimply connected closed manifold is defined as the
length of the shortest noncontractible loop. In 1983, M. Gromov
established the first systolic inequality for manifolds of dimension at
least three. Namely, he proved that the systole of every essential
Riemannian n-manifold with unit volume is bounded from above by a
constant depending only on n.
In this talk, we will present new systolic inequalities for aspherical
manifolds of dimension greater than three. In particular, we will show
how, on such manifolds, the systolic volume is related to the minimal
entropy.
(April 14) Keith Burns :
Ergodicity for partially hyperbolic diffeomorphisms
A famous recent result of Pugh and Shub states that
a volume preseving partially hyperbolic diffeomorphism is ergodic
under certain conditions. Some of these conditions are natural and
inevitable. Others should be unnecessary. The talk will describe recent
work with Amie Wilkinson, which weakens the hopefully unnecessary
conditions and at the same time simplifies the proof.
(April 21) Todd Fisher :
Hyperbolic sets with nonempty interior
In this talk we answer the question: Is every hyperbolic set with nonempty
interior Anosov? The answer in general is no. We give two examples of
hyperbolic sets with nonempty interior that are not Anosov. Additionally, we
give sufficient conditions for a hyperbolic set with nonempty interior to be
Anosov.
(April 28) Tomasz Downarowicz :
Faces of simplices of invariant measures
By an "assignment" we mean a function
associating to extreme points of some abstract Choquet
simplex ergodic measure-preserving transformations
identified up to measure-theoretic isomorphism.
Every topological dynamical system yields naturally such
an assignment on its set of invariant measures
(we call such assignments "natural").
As a non-uniquely ergodic version of a Jewett-Krieger type
statement, it is interesting to chatracterize natural assignments,
specifically assignment occuring in minimal systems (call them "minimal").
We will shed some light on this problem. Two facts are crucial:
(May 5) Stanley Eigen :
An Aperiodic Tiling of The Plane from a
Dynamical System via Beatty Sequences
Wang tiles are unit squares with colored edges. When used to tile the
plane, they are placed edge-to-edge with matching colors. By a piecewise,
multiplicative function f(x) on an interval [a,b) we mean there is a partition
of [a,b) ={a = x_0 < x_1 < \cdots < x_n = b} where f(x) is multiplicative
--- that is,
f(x) = q_i x on each subinterval, with the q_i rational --- and the resulting
map is surjective. Such a function can be used to create a set of Wang tiles
by employing the Beatty sequence for real numbers. Aperiodicity of the
function will result in a set of tiles that tile the plane aperiodically.
these ideas will be presented through a detailed exposition of an example
of Culik (which is a slight modification of an example of Kari).