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Explore the intricate relationship between geometry and partial differential equations in this illuminating lecture. Delve into how optimal geometric structures and shape evolution are governed by PDEs, and discover their recurring presence across diverse scientific, engineering, and mathematical fields. Examine the geometric invariance that makes these equations canonical and their ability to describe phenomena seemingly unrelated to geometry. Learn how geometry can unlock the structure of equations, leading to fundamental PDE tools, while also understanding the pivotal role of analysis in geometric development. Investigate this principle through several fundamental equations, starting with a long-standing geometric problem that leads to optimal regularity for viscosity solutions of a degenerate elliptic PDE. Progress to using PDEs to comprehend optimal shapes and geometric evolution, gaining simultaneous insight into both analysis and geometry and their interplay.
