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.
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 14-15
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:
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.
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.
Statistical mechanics and thermodynamic
formalism
(Vaughn Climenhaga - University
of Houston)
These lectures will introduce basic models for
statistical mechanics of lattice gases and
explain how the mathematical techniques used in
their analysis can be translated to the setting
of dynamical systems, leading to the theory of
thermodynamic formalism, equilibrium states, and
SRB measures.
Dynamics of quantum spin systems
(Anna Vershynina -
University of Houston)
These lectures will present the analysis of
dynamics in quantum systems. The main question of
consideration here is how fast does information
spread in quantum systems? The direct answer to
this question could be "instantaneously", due to
a quantum phenomena called entanglement.
Entanglement allows far away quantum particles to
"feel" each instantaneously in a way that is
stronger than any classical correlation.
Fortunately for the future of the quantum
computer, this answer is not the end of the
story. The ground-breaking discovery is that, up
to exponentially-small error, the information
spreads with finite speed even in quantum
systems, making them essentially robust against
small perturbations. This is known as
Lieb-Robinson bounds, or locality estimates.
Dynamical approaches to the spectral theory of
operators
(David Damanik - Rice
University)
The goal of these lectures is to give an overview
of foundational techniques and results underlying
dynamical approaches to the study of spectra of
operators associated to physical systems, such as
Schrödinger operators associated to graphs.
Topics to be covered include cocycles, Lyapunov
exponents, rotation numbers, uniform
hyperbolicity, and reducibility. A major theme
will be the study of systems with potentials
displaying "aperiodic order," i.e. between random
and periodic, and a number of examples, such as
the Fibonacci Hamiltonian, will be considered in
detail.
Dynamics on homogeneous spaces, with applications
to number theory
(Seonhee Lim
- Seoul National University)
The lectures will begin with the introduction of
closed linear groups, lattices, and natural
dynamical systems on their quotients. Motivating
examples will be actions defined by
multiplication by diagonal matrices on quotients
of \(\mathrm{PSL}_n(\mathbb{R})\). After
explaining Hopf's proof of the ergodicity of the
geodesic flow, these lectures will aim to provide
an overview of the proof by Margulis of the
Oppenheim conjecture. If time permits they will
also include a discussion of Eskin, Margulis, and
Mozes's proof(s) of quantitative versions of the
Oppenheim conjecture.
Prerequisites and
application process
Graduate students participating in the school should be
familiar with the following prerequisite material:
measure theory, basic functional analysis, and algebra
(mainly concerning groups and linear algebra). 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
with 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 algebra
(mainly concerning groups and linear algebra).
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.
To contact the organizers, please use
uh.summer.school@gmail.com.
The deadline for applications to be guaranteed full
consideration is February 28,
2018.
Funding for this event is provided by the NSF grant
DMS-1800669.
