Almgren's Center Manifold in a Simple Setting

Explore a specialized 4-lecture course analyzing Almgren's center manifold theory through a simplified toy case of a single sheet minimal surface. Learn to prove direct C^{3,α} regularity for minimal surfaces using elementary considerations without relying on Schauder's estimates, following the approach developed by De Lellis and Spadaro. Master the fundamental concepts through detailed examination of a case study that demonstrates center manifold techniques in geometric measure theory. Gain insights into the broader center manifold framework essential to Almgren's theory and its modern applications in minimal surface regularity. Access comprehensive lecture notes and benefit from the expertise of Camillo De Lellis from the Institute for Advanced Study. Prerequisites include basic knowledge of elliptic partial differential equations (harmonic functions and properties), harmonic analysis (maximal functions and estimates), and elementary geometric measure theory (Hausdorff measures, rectifiable sets, area formula). The course was presented at the 28th annual PCMI Summer Session in 2018 as part of the harmonic analysis research theme, providing advanced graduate-level instruction in this specialized area of mathematical analysis.