Lecture Course: Introduction to Convex Integration

Start: June 10, 2027, 14:00 - 16:00 h
Further dates:
June 11, 14, 16 and 18, 2027, 14:00 - 16:00 h

Abstract: The flexibility of isometric immersions is a recurring theme in geometric analysis. Under sufficient regularity, rigidity is expected: existence and properties of an isometric immersion are constrained by the geometry of the underlying manifold. On the other hand, the celebrated Nash–Kuiper theorem reveals a strikingly different picture at lower regularity, raising a fundamental question about the transition between rigidity and flexibility. In this course, I will introduce the main ideas and techniques of convex integration, using the isometric immersion problem and the closely related Monge–Ampère system as guiding examples. I will survey recent developments in the quest for the Hölder regularity thresholds between rigidity and flexibility, with particular emphasis on how these thresholds depend on dimension and codimension, and on the mechanisms that make increasingly higher regularity accessible. Beyond these geometric problems, convex integration has emerged as a remarkably versatile approach to constructing solutions of partial differential equations, with applications across a broad range of problems in mathematical analysis.

Aims and methods of the course:
The aim of the course is to introduce the main ideas and techniques of convex integration and to explain the current state of the art. We will discuss the step and stage construction, the Nash–Kuiper iteration, Källen's iteration trick, and the regularity results known today, with particular emphasis on the isometric immersion and Monge–Ampère problems. As time permits, we may also explore connections to nonlinear elasticity, the scaling of non-Euclidean elastic energies in prestrained thin films, and the quantitative isometric immersion problem.

Lecture Course Announcement (pdf) - will follow

Link to the course directory - will follow