Dynamical Systems - Summer School  26 - 30 August 2019

Part of the Semester : Dynamics with Structures

08:00 - 17:00
Room : GA 3 21

Organiser

Jörg Schmeling, Lund University

The summer school will provide courses from experts in the areas of the scope of the program ''Dynamics with Structures''. It is intended to provide surveys of modern directions in the theory of dynamical systems on the level of graduate students and junior researchers. The goal is to propagate modern directions of research to younger scientists and show the variety of methods and results obtained by imposing different structures on a dynamical system. It will be a unique possibility to give an insight on all three main events of the semester. The topics will reach from Conservative Dynamics and Hamiltonian Systems over Algebraic and Number Theoretic Systems and Methods, Analytic Dynamical Systems and Thermodynamic Formalism. Besides the lectures the audience is encouraged to have active discussions with the lecturers and other experts.


School programme:

Dynamical approaches to the spectral theory of operators.
Lecturer: David Damanik (Rice University)
Abstract: The goal of these lectures is to give an overview of fundamental techniques and results underlying dynamical approaches to the study of spectra of operators associated to physical systems, with the primary examples given by Schrödinger operators on the line. Topics to be covered include linear cocycles, Lyapunov exponents, rotation numbers, uniform hyperbolicity, and reducibility. A major theme will be the study of systems displaying either almost periodicity or randomness.

Dynamics on homogeneous spaces and interactions with number theory.
Lecturer: Anish Ghosh (Tata Institute of Fundamental Research)
Abstract: I will talk about the ergodic theory of Lie group actions on homogeneous spaces. This subject has seen tremendous activity in the last few years, much of it motivated by its connections to number theory. In this lecture series, we will discuss mixing for group actions and its consequences as well as measure rigidity (especially Ratner's theorems) and its consequences for Diophantine approximation.

Periodic orbits of Hamiltonian systems: the Conley conjecture, pseudo-rotations and holomorphic curves.
Lecturer: Viktor Ginzburg (University of California Santa Cruz)
Abstract: One distinguishing feature of Hamiltonian dynamical systems is that such systems, with few notable exceptions, tend to have numerous periodic orbits. For instance, for many symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many periodic orbits unconditionally. This fact, usually referred to as the Conley conjecture, has by now been established for a broad class of manifolds. However, the Conley conjecture obviously fails for some, even very simple, manifolds such as the sphere. These spaces admit Hamiltonian diffeomorphisms with few periodic orbits -- the so-called pseudo-rotations -- which are of particular interest and occupy a very special place in dynamics. Symplectic topological methods and, in particular, Floer theory turn out to be the right tools to study pseudo-rotations in all dimensions and recently a connection between the existence of pseudo-rotations and the Gromov-Witten invariants has been discovered. 
We will start these lectures with the background results on the Conley conjecture and then focus on the dynamics of Hamiltonian pseudo-rotations and the connection between pseudo-rotations and quantum homology.

Dynamics of singular Riemann surface foliations.
Lecturer: Nessim Sibony (Université Paris-Sud, Orsay)
Abstract: Consider a polynomial differential equation in two complex variables. The time is complex. In order to study the global behavior of the solutions, it is convenient to consider the extension as a foliation in the projective plane. This is an example of a singular foliation by Riemann surfaces.
I will discuss some recent results around the following questions. What is the ergodic theory of such systems, in a general compact Kahler manifold? How do the leaves distribute in a generic case?
The main tool is the potential theory of positive ddc-closed currents and their geometry, which I will introduce. They provide the analogue of "invariant measures" and permit to prove unique ergodicity theorems when all the singularities are hyperbolic. The averaging used is inspired by Nevanlinna's theory. I will also give examples of the use of pluri-potential theory in discrete dynamics. I will also describe analogies with the theory of polynomial automorphisms of in two complex variables, which exhibit similar rigidity phenomena.


Questions from the audience
Name University Dates of visit
Magnus Aspenberg Lund University 20/08/2019 - 30/08/2019
Cengiz Aydin Université de Neuchâtel 19/08/2019 - 30/08/2019
Polina Baron National Research University "Higher School of Economics" 25/08/2019 - 31/08/2019
David Bechara Ruhr Universität Bochum 18/08/2019 - 31/08/2019
Michael Benedicks Uppsala University 18/08/2019 - 31/08/2019
Joé Brendel Université de Neuchâtel 19/08/2019 - 30/08/2019
Mats Bylund Lund University 25/08/2019 - 01/09/2019
Solly Coles University of Warwick 25/08/2019 - 01/09/2019
Lucas Dahinden Universität Heidelberg 18/08/2019 - 31/08/2019
David Damanik Rice University 25/08/2019 - 31/08/2019
Henrik Ekström Lund University 26/08/2019 - 30/08/2019
Gerard Farré Puiggalí KTH 25/08/2019 - 01/09/2019
Mohamadreza Farshidfar University of Neuchâtel 18/08/2019 - 30/08/2019
Yaniv Ganor Tel Aviv University 18/08/2019 - 31/08/2019
Anish Ghosh Tata Institute of Fundamental Research 25/08/2019 - 01/09/2019
Viktor Ginzburg University of California Santa Cruz 18/08/2019 - 29/08/2019
Vasilii Goriachkin Lund University 18/08/2019 - 30/08/2019
David Jarossay University of Geneva 26/08/2019 - 30/08/2019
Oliver Katsikas Université de Fribourg 18/08/2019 - 31/08/2019
Seongchan Kim Université de Neuchâtel 19/08/2019 - 30/08/2019
Philipp Kunde Hamburg 25/08/2019 - 30/08/2019
Joaquín Lema Universidad de la República 17/08/2019 - 31/08/2019
Bing Li South China University of Technology 18/08/2019 - 31/08/2019
Yaoqiang Li Sorbonne Université - Campus Pierre et Marie Curie (Paris 6) 25/08/2019 - 30/08/2019
Bowen Liu Universität Augsburg 26/08/2019 - 30/08/2019
Ezequiel Maderna Universidad de la República 19/08/2019 - 30/08/2019
Khudoyor Mamayusupov National Research University Higher School of Economics 25/08/2019 - 30/08/2019
Rodica Georgiana Marineac Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest 25/08/2019 - 30/08/2019
Benjamin Meco Uppsala Universitet 25/08/2019 - 31/08/2019
Robert Nicholls University of Augsburg 19/08/2019 - 30/08/2019
Ryuma Orita Tokyo Metropolitan University 18/08/2019 - 31/08/2019
Ashraf Owis Cairo 25/08/2019 - 31/08/2019
Yakov Pesin Pennsylvania State University 19/08/2019 - 31/08/2019
Boris Petkovic KTH Royal Institute of Technology 25/08/2019 - 01/09/2019
Abror Pirnapasov Ruhr University Bochum 18/08/2019 - 31/08/2019
Mark Pollicott University of Warwick 26/08/2019 - 30/08/2019
Pedram Safaee UNINE 26/08/2019 - 30/08/2019
Felix Schlenk Université de Neuchâtel 19/08/2019 - 30/08/2019
Jörg Schmeling Lund University 18/08/2019 - 01/09/2019
Samuel Senti Universidade Federal do Rio de Janeiro 19/08/2019 - 30/08/2019
Nessim Sibony Université Paris-Sud (Orsay) 25/08/2019 - 31/08/2019
Iryna Sivak EPFL 26/08/2019 - 30/08/2019
Anastasios Stylianou University of Warwick 26/08/2019 - 30/08/2019
Polina Vytnova University of Warwick 16/08/2019 - 30/08/2019
Vijay Kumar Yadav Indian Institute of Technology (Banaras Hindu University) 26/08/2019 - 30/08/2019
Lixuan Zheng Universite Paris-Est Creteil Val de Marne 25/08/2019 - 30/08/2019
Total Guests : 46
Name University Dates of visit
David Damanik Rice University 25/08/2019 - 31/08/2019
Anish Ghosh Tata Institute of Fundamental Research 25/08/2019 - 01/09/2019
Viktor Ginzburg University of California Santa Cruz 18/08/2019 - 29/08/2019
Nessim Sibony Université Paris-Sud (Orsay) 25/08/2019 - 31/08/2019
Total Guests : 4

Monday 26 August
08:30-09:30 CIB front desk registration and Welcome Coffee
09:30-10:30 Viktor Ginzburg - Part I
10:30-11:00 Coffee Break
11:00-12:00 Nessim Sibony - Part I
12:00-14:00 Lunch
14:00-15:00 Anish Ghosh - Part I
15:00-15:30 Coffee Break
15:30-16:30 David Damanik - Part I

Tuesday 27 August
09:30-10:30 David Damanik - Part II
10:30-11:00 Coffee Break
11:00-12:00 Viktor Ginzburg - Part II
12:00-14:00 Lunch
14:00-15:00 Nessim Sibony - Part II
15:00-15:30 Coffee Break
15:30-16:30 Anish Ghosh - Part II

Wednesday 28 August
09:30-10:30 Nessim Sibony - Part III
10:30-11:00 Coffee Break
11:00-12:00 Viktor Ginzburg - Part III
12:00-14:00 Lunch
14:00-15:00 Viktor Ginzburg - Part IV
Social Dinner

Thursday 29 August
09:30-10:30 Nessim Sibony - Part IV
10:30-11:00 Coffee Break
11:00-12:00 Anish Ghosh - Part III
12:00-14:00 Lunch
14:00-15:00 David Damanik - Part III
15:00-15:30 Coffee Break

Problems/Questions

Friday 30 August
09:30-10:30 David Damanik - Part IV
10:30-11:00 Coffee Break
11:00-12:00 Anish Ghosh - Part IV