### Mini-Courses

Guy David (Université Paris-Sud)

*Harmonic and elliptic measures associated to sets of codimensions larger than 1*

Abstract: We intend to describe a recent attempt, with Max Engelstein, Joseph Feneuil, and Svitlana Mayboroda, to study an analogue of harmonic measure for domains of **R**^{n} with an Ahlfors regular boundary of dimension d < n-1. For such domains, the usual harmonic measure (associated to the Laplacian) does not always make sense, so we use some degenerate elliptic operators, with coefficients tending to ∞ at the boundary in a controlled way. We define an elliptic measure associated to these operators, prove good general properties (such as doubling), and under some strong rectifiability assumptions on the boundary (small Lipschitz graph, probably uniformly rectifiability) prove its absolute continuity with respect to the Hausdorff measure. We are interested in converse results, but they seem to be harder to get. The lectures will probably focus more on the geometric aspects of the proofs.

Thierry De Pauw (East China Normal University)

*Rectifiable and flat G-chains*

Abstract: I will give a modern presentation of the theory of rectifiable and flat chains initially due to H. Federer and W.H. Fleming in 1960. The course will not focus on the context of currents but rather that of a completion of the group of chains, following H. Whitney. This permits to treat at once all coefficient groups. The ambient spaces allowed will be Euclidean, Riemannian, normed finite dimensional and Finsler. After setting up the group of polyhedral chains and their basic operations I will introduce the group of flat chains together with their basic tools : boundary, push-forward and slicing. The two main theorems of the subject will be discussed in some detail: deformation and rectifiability. The course will close on some applications: isoperimetric inequality and mass minimizing representatives of integral homology classes of compact Riemannian manifolds.

De-Jun Feng (The Chinese University of Hong Kong)

*Dimension theory of fractal measures*

Abstract: Iterated function system (IFS) is a broad scheme for generating fractal sets and measures. In this short course, we discuss the ergodic and dimensional properties of certain fractal measures associated with conformal and affine IFSs. We prove the exact dimensionality of these fractal measures and give its applications to the dimensions of fractal sets, as well as their projections and slices.

Antoine Lemenant (Université Paris 7)

*Regularity theory for a class of free-discontinuity problems arising in physical mechanics, from Mumford-Shah to Griffith*

Abstract: The aim of this course is to expose some recent developments about the minimizers of the Griffith energy coming from the variational model of crack propagation. In particular, we aim to describe in detail the proof of a C^1 regularity result in 2D that I have recently obtained in collaboration with J-F Babadjian and F. Iurlano (preprint 2019). Before that, I will describe how to prove some C^1 regularity for an almost minimal 1-dimensional connected set in the simple 2D framework. Then, I will explain how to adapt the proof to the classical Mumford-Shah functional, and finally to the more complicated Griffith functional. At the end I will discuss the issue of non connected minimizers, which is a delicate question related to the so-called Mumford-Shah conjecture. Depending on the remaining time, I will perhaps mention the so-called optimal compliance problem for which a full C^1 regularity result has been recently obtained using similar technics [Chambolle-Lamboley-Lemenant-Stepanov (SIAM JMA 2017)].

Ulrich Menne (National Taiwan Normal University) & Mario Santilli (University of Augsburg)

*Relating curvatures of geometric measure and convex geometry*

Abstract: Curvatures in convex geometry are given as functions on the normal bundle induced by the Steiner formula. Appropriately adapting the notion of normal bundle, the same holds for arbitrary closed subsets ofEuclidean space. The canonical second-order rectifiable stratification of such sets on the other hand induces curvatures in the sense of geometric measure theory. The latter curvatures account precisely for the "absolutely continuous" part of the curvatures of convex geometry. The class of sets for which both curvatures agree promises to be a versatile tool in the study of geometric variational problems involving measure theoretic surfaces (varifolds) of bounded mean curvature.

The course shall survey this content. It requires a thorough knowledge of general measure theory but is self-contained with respect to those parts of descriptive set theory, multilinear algebra, geometric measure theory, and convex geometry that occur.

### Talks

Julien Barral (Université Paris 13)

*Dimensions of Statistically Self-affine Sierpinski Sponges in ***R**^{d}

Abstract: I will explain how one can extend in any Euclidean space the result by Gatzouras and Lalley on the Hausdorff and Minkowski dimensions of statistically self-affine Sierpinski carpets. While Gatzouras and Lalley exploit Bedford’s approach to the deterministic case, our approach (in a joint work with D.-J. Feng) combines Bedford’s one and the variational one used by McMullen and Kenyon-Peres for the deterministic case, as well as our recent study of projections of Mandelbrot measures.

Chang-Xing Miao (Institute of Applied Physics and Computational Mathematics)

*函数谱几何的分解与指数求和*

Abstract: 主要讨论函数Fourier变换的支集的几何如何影响函数在物理空间所发生的结构性干涉，着重讨论函数的频率支在Gauss曲率非零的光滑超曲面所导致了slab-型的square-平方函数估计、decoupling估计与指数和估计. 这些估计不仅限制性猜想、光滑紧流形上自伴微分算子特征函数的 估计、丢番图不等式整数解估计、周期色散方程的Strichartz估计、Bochner-Riesz平均等发挥重用作用，同时在观念上实现了调和分析、偏微分方程、解析数论、几何测度论等研究领域融合. 作为应用，介绍Bourgain-Demeter-Guth一举解决了沉睡近一个世纪的Vinogradov猜想；以及我们在流形上波动方程解的局部光滑性猜想上所取得的研究进展. (Talk in Chinese)

Zhi-Ying Wen (Tsinghua University)

*自相似集的一些问题*

Abstract: 自相似集是生成方式最简单、最基本、最重要的分形集. 本报告介绍自相似集研究中一些尚未解决的基本问题. (Talk in Chinese)

Da-Chun Yang (Beijing Normal University)

*Ball Average Characterizations of Function Spaces*

Abstract: It is well known that function spaces play an important role in the study on various problems from analysis. In this talk, we present pointwise and ball average characterizations of function spaces including Sobolev spaces, Besov spaces and Triebel-Lizorkin spaces on the Euclidean spaces. These characterizations have the advantages so that they can be used as the definitions of these function spaces on metric measure spaces. Some open questions are also presented in this talk.

Xiao-Ping Yang (Nanjing University)

*Measure Estimates for Nodal Sets of Solutions to Some Elliptic Equations*

Abstract: In this talk, we will discuss measure estimates for nodal sets of solutions to some elliptic equations. After introducing frequency functions, doubling indexes and establishing some estimates, the (almost) monotonicity formulae and the relationship between Frequency functions and doubling indexes are established. We develop an iteration procedure to investigate the measure bounds for nodal sets of these solutions. This is a joint work with Tian Long.

Ping Zhang (Academy of Mathematics and Systems Science, CAS)

*Global well-posedness of 3-D anisotropic Navier-Stokes system with small unidirectional derivative*

Abstract: In [15], the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier-Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier-Stokes system (*ANS*) with only horizontal dissipation. More precisely, given initial data u_{0}=(u_{0}^{h}, u_{0}^{3}) ∈ ℬ^{0,½}, (*ANS*) has a unique global solution provided that |D_{h}|^{-1}∂_{3}u_{0} is sufficiently small in the scaling invariant space ℬ^{0,½}.