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2019 Houston Summer School on Dynamical Systems

Houston Summer School on Dynamical Systems

May 30 - June 6, 2019

The Department of Mathematics at University of Houston will host the seventh annual Houston Summer School on Dynamical Systems from May 30 - June 6, 2019.

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.

School Poster

Undergraduate participants

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:
  1. 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.
  2. 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.
  3. 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 March 1, 2019.

Funding for this event is provided by the NSF grant DMS-1900964.
NSF logo
[most of these open in the same, separate, tab]

Buildings and restaurants

Schedule of lectures

Videos of lectures

Possible evening and weekend activities

Notes for the undergraduate prep sessions

List of project papers

More about decay of correlations via the transfer operator

Webmaster   University of Houston    ---    Last modified:  November 05 2018 - 20:46:43

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