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2017 Houston Summer School on Dynamical Systems

Houston Summer School on Dynamical Systems

May 17-25, 2017



The Department of Mathematics at University of Houston will host the fifth annual Houston Summer School on Dynamical Systems from May 17-25, 2017.

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.
     

Rough outline of schedule

The detailed schedule for May 17-25 is here.

Monday, May 15
Morning (or Sunday evening): Undergraduate participants arrive
Afternoon: Undergraduate event begins; lecture, problem session, Q&A
Tuesday, May 16
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:
  1. 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.
  2. 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.
  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.
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.
NSF logo
Schedule of lectures

Videos of lectures

Possible evening and weekend activities

Exercises

List of project papers

Notes for the undergraduate prep sessions



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