AVERAGING AND FERMI ACCELERATION

Dmitry Dolgopyat

University of Maryland

Abstract: Averaging is one of classical tools in dynamics. However, despite two centuries of work, there are still many open problems. In this course I will illustrate some recent results and open problem in a model case of Fermi acceleration where one would like to describe the evolution of energy for a particle with the environment.

NONLOCAL MINIMAL SURFACES

Serena Dipierro

University of Western Australia

Abstract: We present the theory of nonlocal minimal surfaces, with special attention to interior regularity, sheeting phenomena, maximum principle, boundary stickiness and connection with phase transitions.

RIGIDITY RESULTS AND BOCHNER TYPE TECHNIQUES IN RIEMANN GEOMETRY

Luciano Mari

Università degli Studi di Milano Statale

Abstract: This minicourse aims to introduce some techniques used to prove splitting and topological/geometric rigidity theorems in Riemannian Geometry. I will highlight how the stability/finite index of a Schrodinger operator naturally appearing in the problem forces the rigidity phenomenon. Among the applications, time permitting I will discuss some results in Schoen-Yau’s theory of harmonic maps,

Li-Tam-Wang’s theory of Harmonic functions, and minimal (hyper)surfaces.

DYNAMICAL DECOMPOSITIONS OF SURFACE DIFFEOMORPHISMS

Sylvain Crovisier

Université Paris-Saclay

Abstract: In order to describe the dynamics of a diffeomorphism, one first try to decompose the system into invariant elementary pieces, that may be studied separately. For hyperbolic dynamics, the decomposition is provided by Smale’s « spectral decomposition theorem », and the number of pieces is finite. These lectures deal with the decomposition of general smooth diffeomorphisms on surfaces.

One will mainly focus on systems which have infinitely many pieces: typical examples have been constructed by Newhouse. Several view points will be covered: topological (filtration, renormalization) or measurable (Borel homoclinic classes). In good cases, the dynamics inside each piece may be studied as for uniformly hyperbolic systems: most of their orbits equidistribute towards natural invariant measures (physical or maximizing the entropy), and their statistics are described by limit theorems. I will also discuss how the dynamics degenerates along sequences of distinct pieces, in terms of their dynamical invariants (e.g. the entropy and the Lya-punov exponents).