RésumésJohn Ball: Function spaces for liquid crystals
In models of liquid crystals, the description of defects depends on the choice of order parameter. For a lowdimensional order parameter such as the director (describing the mean orientation of the constituent rodlike molecules) defects may be described by mathematical singularities in the order parameter; however, in the corresponding OseenFrank theory some of these singularities have infinite energy. For a higherdimensional order parameter such as the de Gennes Qtensor, defects can have internal structure and may not be actual singularities. The course will discuss some of the issues surrounding the choice of order parameters and corresponding function spaces, and how they influence the description of defects, drawing on lessons from solid mechanics. Usually only point and line defects are considered, but the course will also consider the possibility of surface defects and their potential relevance for nematic elastomers, order reconstruction and smectic thin films.
Fabrice Bethuel : Sobolev maps between manifolds and branched transportation
Robert Hardt: Spaces of Flat and Normal Chains and Cochains
Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, Whitney introduced the geometric "flat norm" on polyhedral chains in Euclidean space and then realized flat chains as the flat norm completion of polyhedral chains. Flat cochains were members of the topological dual space. Federer and Fleming also considered these in the sixties in connection with massminimization among chains with a given boundary or cycles in a given homology class. Their work used a regularity property of the spaces, that they be Euclidean Lipschitz neighborhood retracts. These spaces include smooth manifolds and polyhedra, but not algebraic varieties or subspaces of some Banach spaces. In works with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate
Augusto Ponce: Sobolev spaces into manifolds: a toolbox
Sobolev maps with values into manifolds can be defined in two nonequivalent ways: as Sobolev functions whose target is a manifold or by completion of smooth maps under the Sobolev norm. We explain why these approaches may lead to different objects, depending on the topology of the target.
Tristan Rivière: The variations of YangMills Lagrangian
YangMills theory is growing at the interface between high energy physics and mathematics. It is well known that YangMills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential topology, Hitchin Kobayashi conjecture relating holomorphic bundles over Kähler manifolds and Mumford stability in complex geometry or also Gromov Witten invariants in symplectic geometry...etc. While the influence of Gauge theory in geometry is quite notorious, one tends sometimes to forget that YangMills theory has been also at the origin of fundamental progresses in the nonlinear analysis of Partial Differential Equations in the last decades.
The purpose of this minicourse is to present the variations of this important lagrangian. We shall raise analysis questions such as existence and regularity of YangMills minimizers or critical point of Yangmills lagrangian in general. We will first describe during the first half of the course the progresses which have been made in these directions for the critical dimension 4 and below. At this occasion we will mostly insist on the important contributions by K.Uhlenbeck from the late seventies  early eighties. The second part of the minicourse will be devoted to the study of YangMills fields in dimension larger than 4. We will first describe the LinTian concentration compactness method in super critical dimension as well as the epsilonregularity results obtained by TaoTian and MeyerRivière.
In the last part of the course we shall present very recent results obtained in collaboration with M. Petrache regarding the adhoc framework for producing YangMills minimizers in dimension larger than 4.
The following notes issued from a minicourse given by the author on the subject at the 9th summer school in Differential Geometry at the Korean Institute for Advanced Studies between june 23rd and june 27th 2014 can serve as a guide for audience: lecture notes
