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Analysis Seminar

Thursday, February 22, 2024 - 3:30pm

Kevin Ren

Princeton University


University of Pennsylvania


Fix a real number 0 < s <= 1. A set E in the plane is a s-Furstenberg set if there exists a line in every direction that intersects E in a set with Hausdorff dimension s. For example, a planar Kakeya set is a special case of a 1-Furstenberg set, and indeed we know that 1-Furstenberg sets have Hausdorff dimension 2. However, obtaining a sharp lower bound for the Hausdorff dimension of s-Furstenberg sets for any 0 < s < 1 has been a challenging open problem for half a century. In this talk, I will explain the recent resolution of the Furstenberg set conjecture using tools from Fourier analysis and additive combinatorics. Joint works with Yuqiu Fu and Hong Wang.