This lecture, part of the Zygmund Calderón Lectures in Analysis (2025) series at the University of Chicago Department of Mathematics, features MIT professor David Jerison exploring the geometry of level sets of solutions to elliptic PDE. Discover how curved level sets of semilinear elliptic equations are through an analysis inspired by Hamilton and Perelman's work on mean curvature flow and Ricci flow. Learn about fundamental tools of algebraic and differential geometry, including Jacobi functions and the Levi-Civita affine connection. While the lecture discusses potential applications to level sets of eigenfunctions, it primarily focuses on establishing foundations with harmonic functions. This one-hour lecture represents the third installment in the distinguished lecture series.