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Graduate Student Analysis Seminar: This lecture contains a lot of overlap (but not too much): A friendly introduction to the Kakeya problem

Ryan Bushling, University of Washington
Thursday, January 12, 2023 - 12:30pm to 1:30pm
PDL C-401

The Kakeya problem of geometric measure theory asks the following: for \$n \geq 2\$, does there exist a Borel set \$B \subset \mathbb{R}^n\$ such that (1) \$B\$ contains a line segment of unit length in every direction and (2) \$\mathcal{L}^n(B) = 0\$? Mathematicians have constructed many examples of such Besicovitch sets since the question was first posed, but related problems remain wide open, including the validity of the famous Kakeya conjecture contending that every Besicovitch set in \$\mathbb{R}^n\$ has Hausdorff dimension \$n\$. In this expository talk, we survey the history of the Kakeya problem, prove some rudimentary results in support of the Kakeya conjecture, and discuss some of its applications to harmonic analysis.