As in past years, the school is designed for
graduate students; however, 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.
Participants should plan to arrive on May 29; the
summer school will begin around 9:30am on Thursday,
May 30 and end around 4pm on Thursday, June 6, with
a free day on Sunday, June 2.
We anticipate being able to provide lodging for all
participants, and to reimburse travel
expenses.
A number of participant spots are reserved for
undergraduate students, who will first attend
several preliminary lectures on May 28-29
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 participants
should plan to arrive on May 27.
Undergraduate students interested in participating
in the event should apply following the instructions below.
Descriptions
of short courses
The following short courses are planned:
Basics of ergodic theory
(Alan Haynes, Joanna Furno - University
of Houston) These lectures are intended to bring all
participants up to speed with basic
definitions, results, and proofs from the
theory of measure preserving dynamical
systems, including: recurrence, the Birkhoff
ergodic theorem, spaces of measures, Koopman
operators and spectral theory, basic entropy
theory, and techniques for studying dynamics
on compact groups.
Uniformly hyperbolic systems
(Vaughn Climenhaga -
University of Houston) Hyperbolicity, stable and unstable
manifolds, hyperbolic basic sets, structural
stability, Markov partitions and symbolic
dynamics will be covered. This course will
provide the background and framework for the
course on statistical properties of hyperbolic
systems and Jana Rodriguez Hertz's course on
Pesin theory and non-uniformly hyperbolic
systems.
Statistical properties in hyperbolic
dynamics
(Matthew Nicol, William 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.
Elements of Pesin Theory
(Jana Rodriguez Hertz - Southern
University of Science and Technology,
Shenzhen, China) This course will introduce criteria to
identify hyperbolic ergodic components of a
smooth invariant measure. Topics to be covered
will include the foundational work of Pesin on
non-uniformly hyperbolic systems (Pesin
Theory), as well as Lyapunov exponents,
Oseledets theorem, Local manifold theorem,
important examples of nonuniformly hyperbolic
systems and absolute continuity.
Introduction to Quantum walks
(Jake Fillman - Virginia
Tech)
These lectures will introduce quantum walks as
quantum mechanical versions of random walks. The
course will discuss the relation of quantum walks
to classical random walks, and some of their
fundamental properties. There will be a review of
some background: Hilbert spaces, tensor products,
spectral theory of unitary operators, connections
to spectral theory and orthogonal polynomials.
The dominant theme will be the connections
between spectral theory, dynamics, and other
areas of mathematics.
Bratteli diagrams, flat surfaces and the
hierarchical structure of minimal systems
(Rodrigo
Treviño - University of Maryland) These lectures will serve as an introduction
to two families of concepts: the first concept
is that of a flat surface and the dynamics
which are studied on these types of surfaces,
which go under the name of translation flows.
The study of translation flows is one of the
main focus areas of Teichmüller dynamics,
which is an active area with connections to
other fields of mathematics. The second
concept is that of a Bratteli diagram.
Bratteli diagrams were introduced in the study
of a particular class of C*
algebras, but they have also turned out to be
extremely useful in revealing the hierarchical
and asymptotic structure of minimal systems.
Both of these concepts are easy to define but
have a very rich structure. The course will
connect these two concepts and prove a version
of Masur's criterion for unique ergodicity for
translation flows.
Prerequisites and
application process
Topics graduate students participating in the school
should be familiar with include measure theory and
basic functional analysis. 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 complete
this form -- you will need
following information:
Your name, current institution, program and
year of study, and 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, as well as your
background in the prerequisite topics of measure
theory and functional analysis.
A brief description of your mathematical
interests, particularly as they relate to the
topic of the summer school.
Undergraduate studentsinterested 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.
To contact the organizers, please use uh.summer.school@gmail.com.
The deadline for applications to be guaranteed
full consideration is
March 1,
2019.
Funding for this event is provided by the NSF grant
DMS-1900964.
