Transfer operators and anisotropic spaces for Sinai billiards by Prof. Viviane Baladi (CNRS, France)
We will present an approach to the statistical properties of
two-dimensional dispersive billiards (discrete and continuous-time)
using transfer operators acting on anisotropic Banach spaces of
distributions. We will survey results by Demers, Zhang, Liverani, etc,
and joint work with Demers and Liverani on the SRB measure, as well as
joint work with Demers on the measure of maximal entropy and more
general equilibrium states.
Dynamical spectral determination and rigidity by Prof. Jacopo de Simoi (University of Toronto, Canada)
The classical inverse problem asks to what extent it is possible to determine the shape of a domain D of the real plane (or of a surface), by the knowledge of all eigenvalues of the Laplace operator on D with assigned boundary conditions. A dynamical version of this question can be stated by replacing the set of eigenvalues of the Laplacian with the Length spectrum, that is the set of all lengths of all possible closed billiard orbits on D (or all closed geodesics in the case of manifolds). In these lectures, we will show the deep connection between the Laplace and the dynamical problem, we will present in detail some results on the dynamical side and explore the possible outcomes of the current research in this direction.
The regularity theory for Bernoulli type free boundary problems by Prof. Guido de Philippis (Courant Institute, US)
I will present an overview on the regularity theory for Bernoulli like free boundary problem, starting from the seminal work of Allt-Caffarelli in the 80’s to the recent developments obtained in collaboration with L. Spolaor and B. Velichkov.
The obstacle problem: regularity of the free boundary and analysis of singularities by Prof. Joaquim Serra (ETH Zürich, Switzerland)
The classical obstacle problem is a very paradigmatic free boundary problem with several applications in physics, probability, potential theory, finance, etc. It is equivalent, after certain transformations, to other well-known free boundary problems such as the Stefan problem.
The goal of the course will be to give an introduction to the regularity theory for the free boundary in the obstacle problem. On the first part of the course, we will revisit the classical theory from the 1970's (although not always following the original proofs). We will start discussing the existence, uniqueness, and optimal regularity of the solutions. Later we will explain some insightful examples of Levy, Kinderleherer, and Niremberg of solutions with singular free boundaries, which are constructed using complex variables. After, we shall prove the existence of blow-ups and their classification leading to the celebrated dichotomy of Caffarelli. Finally, we will discuss how to prove smoothness of the free boundary near points which have blow-ups of regular type.
On the second part of the course, we will introduce some recent developments of the theory regarding the structure of the singular set. We will introduce monotonicity formulae methods and explain why the fine analysis of singularities of the obstacle problem leads to the analysis of singular points for the so called Signorini problem.
Finally, we will prove, in the 2D case, some new results (with A. Figalli) on higher order expansions at singular points. If time allows, we will explain roughly how the content of the course is fundamental in the recent proof in dimensions 3 and 4 of a conjecture of Schaeffer on the generic regularity of the free boundary (in a joint work with A. Figalli and X. Ros-Oton).