Guy David (Université Paris-Saclay)
Title: Harmonic and elliptic measures on domains with rough boundaries
We'll present recent results on the support of the harmonic/elliptic measure on domains with a rough boundary. In the case of the harmonic measure, i.e., when the elliptic operator is the Laplacian, this means finding out the size of the set in the boundary where Brownian trajectories starting at a central point in the domain eventually land. Here the rough domains we consider may have a boundary of dimension other than n-1.
We'll try to insist in the presentation on issues that are connected to the geometric measure theory of the boundary, or the rough analysis on the domain (Poincaré inequalities and the such). But some standard techniques of PDE's are used in the proofs.
Topics that will be presented will include an introduction to the ``classical case'' of rough boundaries of co-dimension 1, where the main ingredients for the mutual absolute continuity of harmonic measure and surface measure are now known to be quantitative connectedness (such as NTA) and the rectifiability of the boundary.
We will then discuss extensions to higher co-dimensional boundaries, where we need to replace the Laplacian with another interesting (degenerate) elliptic operator.
We should also mention some Cantor set examples where the elliptic measure is, surprisingly, absolutely continuous.
Finally we should describe recent results where the Dirichlet boundary condition implicit in the above is replaced by a Robin boundary condition. One of our motivations there is to understand part of the story for the exchange of oxygen in the lung. We try to explain why the lung is roughly fractal.
Based on work of/with Stefano Decio, Max Engelstein, Marcel Filoche, Joseph Feneuil, Cole Jeznach, Antoine Julia, Linhan Li, Svitlana Mayboroda, Marco Michetti, and others.
De-Jun Feng (The Chinese University of Hong Kong)
Title: Dimensions of orthogonal projections of typical self-affine sets
The aim of this course is to present some dimensional results on orthogonal projections of typical self-affine sets that I have recently obtained in collaboration with Yuhao Xie. Before that, I will introduce the classical dimension theory of typical self-affine sets and measures developed by Falconer, Solomyak, Kaenmaki, Jordan-Pollicott-Simon, and others. Then I will explain how to use the ergodic theory and thermodynamic formalism to further study the dimensions of orthogonal projections of typical self-affine sets and measures along specific directions.
Olga Maleva (University of Birmingham)
Title: Typical differentiability of Lipschitz mappings
The classical Rademacher Theorem guarantees that every set P of positive Lebesgue measure in a finite-dimensional Euclidean space X contains points of differentiability of every Lipschitz mapping from X to another finite-dimensional space. In fact, almost every point of P is a point of differentiability.
A major direction in geometric measure theory has been to explore to what extent the above is true for Lebesgue null subsets of finite-dimensional spaces. First, Zahorski showed in the 1940s that for any null subset N of the set of real numbers R there is a Lipschitz function defined on R and nowhere differentiable in N. In contrast, Preiss proved in 1990 that every finite-dimensional space of dimension at least 2 has Lebesgue null subsets S such that every Lipschitz function on the whole space has points of differentiability in S. In joint papers with Dore and with Dymond we contructed examples of such S with additional important features (such as being closed of Minkowski dimension 1).
But even if there exists a Lipschitz function nowhere differentiable on a given Lebesgue null set N, one naturally wonders what can be said about differentiability of a typical (in the sense of Baire category) Lipschitz function on N. In 2020 in a joint work with Dymond we gave a complete characterisation of the subsets S of a finite-dimensional Euclidean space such that a typical 1-Lipschitz real-valued function has points of differentiability in S. It turns out that such S cannot be covered by an F-sigma purely unrectifiable set. We also showed that for all sets which can be so covered, a typical 1-Lipschitz function is nowhere differentiable, even directionally, at each point.
In a recent result, also joint with Dymond, we prove a much stronger statement. We show that in the class of 1-Lipschitz mappings between finite-dimensional normed spaces, for any given set N coverable by an F-sigma purely unrectifiable set, a typical mapping is non-differentiable at every point of N, in a very strong sense: the derivative ratio approximates every linear operator of norm at most 1. We prove this alongside a surprising result which says that for any pair of Banach spaces X and Y, no matter how good a subset P of X is, a typical 1-Lipschitz mapping between X and Y is non-differentiable at a typical point of P in the above strong sense; this result holds even for subsets P of positive Lebesgue measure of finite-dimensional spaces.
In these lectures, I will introduce the notion of Baire category and explain its connection with the topological Banach-Mazur game, used in the proof of the results about differentiability. I will discuss ideas behind, and details of, some of the proofs of the above results. I will also give an overview of questions still open and future directions of research.
Pertti Mattila (University of Helsinki)
Title:Applications of the Fourier transform to Hausdorff dimension
Starting from Kaufman's work in 1968 and Falconer's in 1982 on orthogonal projections of general Borel sets, there have been many applications of the Fourier transform to geometric problems involving Hausdorff dimension. In addition to orthogonal projections, such topics include Hausdorff dimension of plane sections, of intersections of Borel sets in general positions, and of distance sets. During roughly the last ten years related questions on radial projections have been studied intensively by Orponen, Shmerkin and others. This work is connected with the Furstenberg set problem and has been a partial ingredient in its recent solution by Ren and Wang. There are also connections to the Kakeya problem. The lectures will aim to give an overview of some of the results, methods and ideas in this scenario.
Herve Pajot (Grenoble Alpes University)
Title: Geometric analysis on singular spaces and discrete graphs
Various approaches to generalising the notion of manifolds with nonnegative Ricci curvature to the case of metric spaces have been proposed in connection with optimal transportation (Lott-Villani, Sturm, Ollivier) or with the geometry of diffusions (Bakry-Emery). One of the aims is to obtain versions in this general framework of the classical theorems of Riemannian geometry (Bonnet-Myers, Bishop-Gromov, etc.) and their applications in analysis (Poincaré or Brunn-Minkowski inequalities, for example) or in topology/geometry (Mostow-type rigidity theorems). In this course, we will discuss what happens in the case of discrete graphs, for example Cayley graphs of finitely generated groups.
These lectures are based on recent work with Emmanuel Russ.
