This course will offer an introduction to some selected current topics in geometric analysis. It should be of interest to students of Geometry, Analysis, and Mathematical Physics. The course is composed by two parts

-First part (Prof. Francesca Da Lio): "An Excursion into Conformally Invariant Variational Problems". Since the early 50’s the analysis of critical points to conformally invariant Lagrangians has raised a special interest, due to the important role they play in physics and geometry. This mini-course will focus on the description of some fundamental tools to show regularity and compactness results of critical points to some local and nonlocal conformally invariant variational problems. -Second part (Prof. Tristan Rivière): "Minmax Methods in the Calculus of Variations of Curves and Surfaces". The study of the variations of curvature functionals takes its origins in the works of Euler and Bernoulli from the eighteenth century on the Elastica. Since these very early times, special curves and surfaces such as geodesics, minimal surfaces, elastica, Willmore surfaces, etc. have become central objects in mathematics much beyond the field of geometry stricto sensu with applications in analysis, in applied mathematics, in theoretical physics and natural sciences in general. Despite its venerable age the calculus of variations of length, area or curvature functionals for curves and surfaces is still a very active field of research with important developments that took place in the last decades. This mini-course will concentrate on the various minmax constructions of these critical curves and surfaces in euclidean space or closed manifolds. It will start by recalling the origins of minmax methods for the length functional and present in particular the “curve shortening process” of Birkhoff. It will follow an explanation of the generalization of Birkhoff’s approach to surfaces and the ”harmonic map replacement” method by Colding and Minicozzi. Then some fundamental notions of Palais-Smale deformation theory in infinite dimensional spaces will be recalled and applied to the construction of closed geodesics and Elastica. In the second part of the mini-course a new method based on smoothing arguments combined with Palais-Smale deformation theory for performing successful minmax procedures for surfaces will be presented.

In presenza

Presentazione orale

P.D.2-2 - Giugno