John Ball: Function spaces for liquid crystals


In models of liquid crystals, the description of defects depends on the choice of order parameter. For a low-dimensional order parameter such as the director (describing the mean orientation of the constituent rod-like molecules) defects may be described by mathematical singularities in the order parameter; however, in the corresponding Oseen-Frank theory some of these singularities have infinite energy. For a higher-dimensional order parameter such as the de Gennes Q-tensor, 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 mass-minimization 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 
variational topologies for classes of flat chains and cochains in general metric spaces. With these, we homologically characterize Lipschitz path connectedness and obtain several facts about and examples of singular metric spaces that satisfy local linear isoperimetric inequalities. For example there is a topological duality between normal chain homology (involving chains of 
finite mass and boundary mass) and "charge" cohomology. We may compare this with the flat duality of Federer for general flat chains and cochains. In Euclidean space, charges are, by De Pauw, Pfeffer, Moonens, cochains that may be represented as the sum of a continuous form and the exterior derivative of a continuous form. This contrasts with the older Wolfe's theorem that a general flat cochain corresponds to bounded measurable forms or recent  theorems of Snipes for partial forms or of Petit, Rajala, Wenger for weakly differentiable cochains and Sobolev forms. 


Augusto Ponce: Sobolev spaces into manifolds: a toolbox


Sobolev maps with values into manifolds can be defined in two non-equivalent 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.
In some cases involving spheres, the obstruction can be detected using an elegant tool: the distributional Jacobian that quantifies the strength of topological singularities of a Sobolev map. The course will be devoted to explaining some classical tools to investigate these questions.


Tristan Rivière: The variations of Yang-Mills Lagrangian
Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills 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 Yang-Mills theory has been also at the origin of fundamental progresses in the non-linear analysis of Partial Differential Equations in the last decades.
The purpose of this mini-course is to present  the variations of this important lagrangian. We shall raise analysis questions such as existence and regularity of Yang-Mills minimizers or critical point of Yang-mills 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 mini-course will be devoted to the study of Yang-Mills fields in dimension larger than 4. We will first describe the Lin-Tian concentration compactness method in super critical dimension as well as the epsilon-regularity results obtained by Tao-Tian and Meyer-Rivière.
In the last part of the course we shall present very recent results obtained in collaboration with M. Petrache regarding the ad-hoc framework for producing Yang-Mills minimizers in dimension larger than 4.
The following notes issued from a mini-course 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


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