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This lecture by Alessio Figalli explores the mathematical intricacies of free boundary regularity in obstacle problems, a field that has fascinated mathematicians for over 60 years. Learn about the fundamental theory behind obstacle problems, which naturally emerge when studying elastic membranes constrained by contact with objects. Discover recent advancements in the regularity theory of free boundaries as Figalli guides you through this complex mathematical landscape. The talk is part of the Simons Collaboration on the Localization of Waves Annual Meeting 2025.
