The course covers two separate but closely related topics. The first topic is the mean curvature flow in the framework of GMT due to Brakke. It is a flow of varifold moving by the generalized mean curvature. Starting from a quick review on the necessary tools and facts from GMT and the definition of the Brakke mean curvature flow, I will give an overview on the proof of the local regularity theorem. The second topic is the reaction-diffusion approximation of phase boundaries with key words such as the Modica-Mortola functional and the Allen-Cahn equation. Their singular perturbation problems are related to objects such as minimal surfaces and mean curvature flows in the framework of GMT.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In the original form given by Blaschke in the 1930s, the famous Principal Kinematic Formula expresses the Euler characteristic of the intersection of two sufficiently regular objects in euclidean space, integrated over the space of all possible relative positions, in terms of geometric invariants associated to each of them individually. It is natural to wonder about the precise regularity needed  for this to work. The question turns on the existence of the normal cycle  of such an object A, i.e. an integral current that stands in for its manifolds of unit normals if A is too irregular for the latter to exist in a literal sense. Despite significant recent progress, a comprehensive understanding of this construction remains maddeningly elusive. In these lectures we will discuss both of these aspects.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In the original form given by Blaschke in the 1930s, the famous Principal Kinematic Formula expresses the Euler characteristic of the intersection of two sufficiently regular objects in euclidean space, integrated over the space of all possible relative positions, in terms of geometric invariants associated to each of them individually. It is natural to wonder about the precise regularity needed  for this to work. The question turns on the existence of the normal cycle  of such an object A, i.e. an integral current that stands in for its manifolds of unit normals if A is too irregular for the latter to exist in a literal sense. Despite significant recent progress, a comprehensive understanding of this construction remains maddeningly elusive. In these lectures we will discuss both of these aspects.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In the original form given by Blaschke in the 1930s, the famous Principal Kinematic Formula expresses the Euler characteristic of the intersection of two sufficiently regular objects in euclidean space, integrated over the space of all possible relative positions, in terms of geometric invariants associated to each of them individually. It is natural to wonder about the precise regularity needed  for this to work. The question turns on the existence of the normal cycle  of such an object A, i.e. an integral current that stands in for its manifolds of unit normals if A is too irregular for the latter to exist in a literal sense. Despite significant recent progress, a comprehensive understanding of this construction remains maddeningly elusive. In these lectures we will discuss both of these aspects.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In the original form given by Blaschke in the 1930s, the famous Principal Kinematic Formula expresses the Euler characteristic of the intersection of two sufficiently regular objects in euclidean space, integrated over the space of all possible relative positions, in terms of geometric invariants associated to each of them individually. It is natural to wonder about the precise regularity needed  for this to work. The question turns on the existence of the normal cycle  of such an object A, i.e. an integral current that stands in for its manifolds of unit normals if A is too irregular for the latter to exist in a literal sense. Despite significant recent progress, a comprehensive understanding of this construction remains maddeningly elusive. In these lectures we will discuss both of these aspects.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation