Explore the intriguing world of partial differential equations and convexity in this 56-minute conference talk by Benjamin Weinkove at ICBS2024. Delve into the fascinating interplay between elliptic partial differential equations and convexity properties, examining how convexity is often preserved under evolution equations. Discover surprising instances where convexity can be broken, including examples from the porous medium equation, the Stefan problem, and the quasi-static droplet model. Gain valuable insights into these complex mathematical concepts and their applications in this thought-provoking presentation.