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Square Functions and Rectifiability

Emily Casey, University of Washington
Thursday, May 11, 2023 - 12:30pm to 1:20pm

In 1989 Bishop, Carleson, Garnett, and Jones proved in Harmonic Measures Supported on Curves that the boundary of a Jordan domain having no tangent points \$\mathcal{H}^1\$-a.e. is equivalent to the harmonic measures relative to the interior and exterior of that Jordan domain being mutually singular. Following these results, it was shown that at almost every tangent point, the boundary satisfies a certain integral condition. Carleson conjectured that the converse also held: at almost every point where the boundary satisfied a certain integral condition, there is a classical tangent. This conjecture remained open until 2021, finally being resolved by Jaye, Tolsa, and Villa in their paper A proof of Carleson's \$\varepsilon^2\$-conjecture. In the first part of this talk, we will explore the key ideas in the proof given by Jaye, Tolsa, and Villa. Later, we will discuss some work towards quantitative results both with the \$\varepsilon\$ function and also a related geometric square function.