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Multi-phase problems for harmonic measure, harmonic polynomials, and platonic solids.

Matthew Badger
Tuesday, February 12, 2019 - 2:30pm to 3:30pm
PDL C-401

Research over the last 20 years has revealed a rich landscape of results in the one and two-phase free boundary regularity problems for harmonic measure, bookended by Kenig and Toro's 1999 paper in Annals of Mathematics and Azzam, Mourgoglou, Tolsa, and Volberg's recent proof of Bishop's rectifiability conjecture. This body of work has revealed deep connections between the geometry of a domain and the harmonic measure on its boundary. In this talk, I will introduce a new class of multi-phase problems for harmonic measure on configurations of three or more domains and present initial findings on classification of blowups. An interesting feature that emerges in the multi-phase setting is that, in addition to analytic assumptions, the combinatorial arrangement of domains also influences the geometry of the boundary. This is joint work with Murat Akman.