Cursos
Analytic, geometric, and topological methods for Sobolev mappings to manifolds
Ponente: Antoine Detaille (Université de Lyon)Fecha: lunes 04 de noviembre de 2024 - jueves 14 de noviembre de 2024 - 12:00-13:15Lugar: Aula Gris 2, ICMAT
Resumen:
In this minicourse (4, 7, 11 & 14 November), we will explore the theory of Sobolev mappings with values into a compact manifold. The study of such mappings is motivated by applications coming from geometry, physics, computer graphics, and numerical methods, and their study per se raises many beautiful and challenging problems. The objective is to give an overview of some of these problems, and to explain a few tools that have been developed to tackle them. The ideas will systematically be illustrated on insightful model cases, avoiding too much technicality. The only prerequisite is a relative comfort with functional analysis, especially classical Sobolev spaces. All concepts of topology and geometry that will be used in the course shall be duly reminded.
Lecture 1 (4 Nov.): The main problems concerning Sobolev mappings to manifolds
Abstract: In this session, we will review the main problems that are raised by the study of Sobolev mappings with values into manifolds, such as density, extension of traces, and lifting. Other problems could be mentioned briefly if time allows. This will also be the opportunity to revise some fundamental concepts in topology that are of crucial importance to understand those problems.
Lecture 2 (7 Nov.): Analytic and topological tools for strong density
Abstract: This session will be devoted to the presentation of the strong density theorem and the building blocks of its proof, that display a fascinating interplay between analysis and topology. We shall not attempt to present the proof of the general case in full detail, but instead explain the tools on basic, and hopefully insightful, cases, in order to give the intuition about how they work.
Lecture 3 (11 Nov.): The singular set of a Sobolev mapping, or when differential geometry comes into play
Abstract: After having discovered that Sobolev mappings are not always strongly approximable by smooth mappings, we will turn to the question of detecting which mappings can nevertheless by strongly approached. This will show a beautiful connection with one more area of mathematics, namely differential geometry, through the construction of objects such as the Jacobian.
Lecture 4 (14 Nov.): The weak density problem: connections and dipoles
Abstract: In this last session, we will briefly study what happens when strong convergence is replaced by weak convergence. We will focus on a famous technique to obtain positive results of weak approximation, namely eliminating the topological singularities of a Sobolev mapping along connections between them.
