Summer Graduate School

Geometric Measure Theory (GMT) has proved to be a key tool in the study of Geometric Variational Problems and of the fine structures of sets and measures. In the last decade we have seen a series of new exciting developments in geometric measure theory and its applications, ranging from regularity theory to harmonic analysis to geometric analysis and free boundary problems.

A recurrent theme in GMT is the following. One often starts with a problem formulated in the smooth category (for instance finding the minimal surface spanning a given boundary or finding a closed minimal submanifold in given Riemannian manifold). In solving the problem, one is often forced to enlarge the class of competitors so to make it easier to find solutions. Once these solutions are found in the larger class, in order to solve the original problem one has to develop a suitable regularity theory to prove that the found solutions belong to the initial category of smooth objects or that, if this is not the case, that they are “as regular as possible”.

The scope of the school is to make the students acquainted with the basic techniques needed to solve the above program. This is also a great entry point for student to learn a series of techniques that are indeed ubiquitous in analysis, such as blow-up analysis, dimension reduction, monotonicity formulae, harmonic approximations, . . . .

The school will consist of three interrelated courses, aimed to introduce the main concepts in Geometric Measure Theory.

(1) Sets and measure in the Euclidean space, Guido De Philippis (Courant Institute of Mathematical Sciences)
(2) Theory of currents, Annalisa Massaccesi (University of Padua)
(3) Allard regularity theory, Camillo De Lellis (Institute of Advanced Study)

School Structure

There will be two lectures each day, and substantial time devoted to problem sessions.

Prerequisites

Real Analysis: We expect the students to be familiar with the Lebesgue theory of integration: definition of Lebesgue measure, measurable sets, integrals, Fatou Lemma, Monotone and Dominated Convergence theorems, L p -spaces, Fubini Theorem. Some knowledge of abstract measure theory could be useful, but the main concepts will be reviewed. Good reference for this material is Royden and Fitzpatrick, “Real Analysis”, Chapters 1-4, 7. The material in Chapters 17-22 can also be useful, but will be quickly revised.

Functional Analysis: We expect the students to be familiar with some basic concepts in functional analysis, in particular: Hilbert and Banach Spaces, weak and weak* topologies and Banach-Alaoglu Theorem. Good references for this material are Brezis, “Functional Analysis”, Chapter 3, and Royden and Fitzpatrick, “Real Analysis”, Chapters 14-15.

Sobolev Spaces and Partial Differential equations: We would like the students to be familiar with the basic properties of harmonic functions as in Evans, “Partial Differential Equations”, Chapter 2. Knowledge of some basic Sobolev spaces and distribution theory is also advised, for instance Evans, “Partial Differential Equations”, Chapter 5, and Folland, “Real Analysis”, Chapter 9.

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.