Master Course: Geometric Analysis on Rⁿ

Abstract

During the spring semesters of three consecutive years (2023-2025) I have been teaching at University Complutense of Madrid a Master Course called Técnicas de Análisis Geométrico. Part of this course considers advanced analysis tools and explores geometric and measure related properties of the Euclidean space Rⁿ. Fundamental results in the area are considered, like, for instance, the Whitney extension theorem, Rademacher’s theorem, the Morse-Sard theorem, or Aleksandroff theorem among others. As an effort of making all this content available and enyoable to undergraduate students, and trying to provide all needed details in the proofs, I have developed the following notes. One big premise that I tried to keep when making these notes was that they are as self-contained as possible and that no big results could be used without being proved before. Moreover, the reader may find at the end of each chapter a large number of exercises that try to explore some of the key facts and subtleties behind all this theory. It must be mentioned that there exist similar works considering this very same topic, or similar ones. Indeed, Hajłasz, and Kinunen’s notes were extremely useful during the development of this course. Still, I wanted to add my own flavor in some of the main results. Lastly, but not least, I thank my students for important suggestions and comments that helped clari fying technical steps.