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.
Lecturer: Viktor Ginzburg (University of California Santa Cruz)
Title: Periodic orbits of Hamiltonian systems: the Conley conjecture, pseudo-rotations and holomorphic curves (in 4 parts).
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.