Speaker: Dr. Yingying Cai，Universitat Autònoma de Barcelona

Title: Quantitative estimates for the dimension drop of harmonic measure on Ahlfors regular boundaries

Language: Chinese

Time & Venue: 2026.07.28 10:00-11:00 MCM110

Abstract: In the second part of the talk, we will dive into the higher-dimensional setting, where the geometry of the boundary and the behavior of the harmonic measure become significantly more complex. We will focus on boundaries satisfying a uniform $L^2$-based non-flatness condition ($\beta_2 \ge \delta_0$). I will detail our result showing that under this $\beta_2$ condition, the dimension of the harmonic measure drops strictly below $s$, provided $s$ is sufficiently close to $n$ (specifically, for $s \in (n - c\delta_0^2, n]$). The majority of this session will be devoted to the technical core of the proofs. I will outline the main machinery and techniques developed in our joint work with Xavier Tolsa. In particular, we will discuss the delicate relationship between $\beta$-numbers and Riesz transforms—an approach in the broader study of rectifiability—and demonstrate how we use this interplay to get the result in higher dimensions.