As in past years, the school is designed for
graduate students; however, this year there will
also be an opportunity for undergraduate students to
participate, arriving two days early and then
staying for the main event. See below
for details.
The school will use short
lecture courses, tutorial and discussion
sessions, and student projects to explore various
topics in dynamical systems. It will be
accessible to students without a background in
dynamics, but is also intended for students who have
begun studying dynamics and wish to learn more about
this field.
We anticipate being able to
provide financial support for all participants.
Undergraduate participants
A number of participant spots are reserved for
undergraduate students, who will first attend
several preliminary lectures on May 15-16 (see the rough schedule below) to
introduce concepts needed for the short courses that
may not have appeared in the students' undergraduate
coursework so far. There will also be problem
sessions, discussion, and Q&A time on those
days, as well as continuing review sessions during
the school itself that are specifically targeted at
the undergraduate participants, with the goal of
helping them follow the graduate-level material
being presented.
Undergraduate students interested in participating
in the event should apply following the instructions below.
Descriptions
of short courses
The following short courses are planned:
Statistical properties in hyperbolic
dynamics (Vaughn Climenhaga, Will Ott,
Andrew Török - University of Houston) These lectures will introduce the notion
of decay of correlations for a dynamical
system and will describe two important methods
for establishing a rate of decay: spectral gap
(Perron--Frobenius theory) and Birkhoff cones.
Partially hyperbolic dynamics (Todd
Fisher - Brigham Young University)
Some important aspects of uniform
hyperbolicity include stable and unstable
manifolds, absolute continuity, the Hopf
argument for ergodicity, and uniqueness
results for maximizing measures. These
lectures will describe how these tools survive
(or fail to) upon passing to the partially
hyperbolic setting, and then more generally to
the class of diffeomorphisms with dominated
splittings.
Dynamical methods in Diophantine
approximation (Alan Haynes - University of
Houston)
These talks will present applications of
topological dynamics and ergodic theory to
Diophantine approximation. We plan to cover a
mixture of classical and recent result,
including: Furstenberg's Theorem and the x2x3
Conjecture, Host's and Rudolph's Theorems,
effective x2x3 results, and applications of
ergodic theory in the space of unimodular
lattices to the Mixed Littlewood Conjecture
and to higher dimensional Steinhaus problems.
Dynamics of group actions on homogeneous
spaces (Anish Ghosh - Tata Institute of
Fundamental Research)
These lectures will introduce the ergodic
theory of group actions on homogeneous spaces
of Lie groups. We will study the basics of
lattices in Lie groups with special emphasis
on the space
\(SL(n,\mathbb{R})/SL(n,\mathbb{Z})\) of
unimodular lattices, ergodicity and mixing for
group actions on these spaces and applications
to Diophantine approximation. Further topics
may include: the Howe-Moore theorem on
quantitative mixing, quantitative
nondivergence of unipotent flows and
applications, and an introduction to Ratner's
theorems.
Linearly recurrent systems (Valérie
Berthé - Univ. Paris Diderot)
A symbolic dynamical system is made of
sequences with values in a given alphabet on
which the shift acts. Such systems occur in a
natural way as codings of dynamical systems.
Among zero entropy symbolic dynamical systems,
linearly recurrent ones play a prominent role.
They are defined in terms of return times to
cylinders: return times are linear with
respect to the size of the cylinders.
Well-studied examples of linearly recurrent
systems are substitutive dynamical systems. We
investigate combinatorial, ergodic and
spectral properties of linearly recurrent
systems by introducing their description in
terms of Rohlin towers, Bratteli-Vershik
systems and S-adic systems, and by focusing on
the connections with badly approximable
numbers. We also extend the symbolic approach
to tiling dynamics and Delone point sets.
Morning: Undergraduate
lectures Afternoon: Undergraduate problem
session, Q&A All day: Graduate participants arrive Evening: Informal social gathering
Wednesday-Thursday,
May 17-18
Main summer school activities
begin Mornings: Lectures Afternoons: Lecture, then parallel
problem/review sessions
Friday, May 19
Morning: Lectures Afternoon: Introduction to projects,
formation of groups for project work
Saturday-Sunday,
May 20-21
Free days; possible social
events and informal outings
Monday-Thursday,
May 22-25
Mornings: Lectures Afternoons: Lecture, then working on
projects
Prerequisites and
application process
Graduate students participating in the school should
be familiar with the following prerequisite
material: measure theory, basic functional analysis,
basic theory of smooth manifolds. There will
be review sessions during the school to give a brief
overview of the most relevant parts of these topics.
To apply for participation in the summer school,
please send an email to uh.summer.school@gmail.com
with a short CV containing the following
information:
Your name, current institution, and program
and year of study. Please also include the
name and email address of your Ph.D. advisor or
of another mathematician who can serve as a
reference if necessary.
A list of recent mathematics courses you have
taken and the grades earned. Please
indicate your background in the prerequisite
topics of measure theory, functional analysis,
and smooth manifold theory.
A brief description of your mathematical
interests, particularly as they relate to the
topic of the summer school.
Undergraduate students interested in
participating should follow the instructions
above, and should also arrange to have a professor
send a brief letter of recommendation to Vaughn
Climenhaga at climenha@math.uh.edu.
This letter need not be long, but should attest to
the student's overall level of preparation and
ability to quickly grasp the essential parts of
advanced topics, which will be important in order to
follow the short courses at the school.
The deadline for applications to be guaranteed full
consideration is February
28, 2017.
Funding for this event is provided by NSF grants
DMS-1554794 and DMS-1700273.
