Speaker: Dr. Yingying Cai (Universitat Autònoma de Barcelona)

Title: Quantitative estimates for the dimension drop of harmonic measure on Ahlfors regular boundaries, part I: Background, Main Results, and Planar Domains & part II: Higher Dimensional Estimates and Proof Techniques

Time: 10:00-11:00 July 27, 2026 (Monday) & July 28, 2026 (Tuesday)

Place: MCM110

Abstract: In the first part of this two-hour series, I will introduce the fundamental concepts and recent developments regarding the dimension drop of harmonic measure. We consider domains of the form $\Omega = \mathbb{R}^{n+1} \setminus E$, where the boundary $E$ is an $s$-Ahlfors regular compact set. A central question in geometric measure theory is understanding how the geometric properties of the boundary, specifically non-flatness, influence the dimension of the associated harmonic measure. I will begin by reviewing classical results and providing the necessary background. Then, based on joint work with Xavier Tolsa, I will state our main quantitative estimates for the dimension drop. In the remainder of this first talk, we will focus on the case of planar domains. I will explain how we establish a quantitative threshold $s_0 = 1 - c\delta_0^2$ under Azzam's uniform non-flatness condition ($\beta_\infty + \beta_{\text{hole}} \ge \delta_0$), highlighting the key geometric intuitions in $\mathbb{R}^2$.

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.