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This seminar talk from the "Spectral Geometry in the clouds" series features Jaume de Dios Pont from ETH Zurich exploring the fascinating Hot Spots conjecture in thermal dynamics. Discover how a homogeneous, insulated object with non-uniform initial temperature evolves toward thermal equilibrium, and examine the long-standing question of which points take the longest to reach this equilibrium state. Learn about Rauch's initial conjecture that maximum temperature points would approach the boundary over time, and how this was disproved by Burdzy and Werner for planar domains with holes. Follow the evolution of the conjecture through Kawohl and Bañuelos-Burdzy's work suggesting it might still hold for convex sets across all dimensions. Explore the speaker's innovative approach drawing from convex analysis, particularly how dimension-free results often have natural log-concave extensions. Understand the construction of a log-concave analog to the Hot Spots conjecture and see how this construction ultimately disproves the conjecture for convex sets in high dimensions.
