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Explore advanced spectral geometry through this mathematical seminar examining the fundamental gap conjecture and its geometric implications. Delve into the groundbreaking 2011 proof by Andrews and Clutterbuck that established sharp lower bounds for the difference between the first two Dirichlet Laplacian eigenvalues in convex sets, and discover how recent research has strengthened this inequality by quantifying the geometric excess of the gap in terms of flatness. Learn about the innovative localized variational interpretation of the fundamental gap that enables dimension reduction through convex partitions using Payne-Weinberger techniques. Understand how the proof combines new sharp results for one-dimensional Schrödinger eigenvalues with measure potentials alongside thorough geometric analysis of convex cell partitions. Examine the quantitative form of the Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of convex sets in ℝᴺ, which provides a stronger version of the 2007 Hang-Wang conjecture. Gain insights into cutting-edge research that addresses Yau's 1990 question about rigidity while advancing our understanding of spectral geometry through collaborative work between Vincenzo Amato, Ilaria Fragalà, and the presenter from Université de Savoie.
