Exposés / Vorträge

Elise Bonhomme (Saclay)

Discrete approximation of the Griffith functional.

This joint work with Jean-François Babadjian is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for thisfunctional. It is proved to be approximated in the sense of Γ-convergence by a sequence of integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies.I will start with an introduction of the Griffith energy, then I will present our main result and try tomake you intuit some of the main ideas of its proof.

Penelope Gehring (Potsdam)

Initial boundary value problems for Lorentzian Dirac operators.

Second order differential operators, such as wave operators, are some of the most studied objects in
mathematics as well as in physics. On the other hand, the properties of first order operators are still not as well-understood, specially on spacetimes with boundary.
In this talk, I will give an introduction to non-local boundary conditions for elliptic first order differential operators on Riemannian manifolds with boundary and will point out the difference to hyperbolic operators on Lorentzian manifolds.
Then, I will discuss initial boundary value problems of Lorentzian Dirac operators on spacetimes with timelike boundary with non-local boundary conditions. More precisely, I will analyze how to choose non-local boundary conditions such that the associated initial boundary value problems are well-posed.

Marius Müller (Leipzig)

Elliptic equations with surface measures.

In 1812, S. D. Poisson has examined how electric charges generate an electric field E⃗ . This is how the famous Poisson equation −∆u = f came into mathematics. The electric field is computed via E⃗ = ∇u.Poisson assumed that the electric charges lie on a 3D-object (a charge carrier) and the field lines spread into a 3D-domain. If the charge carrier is however very thin, one might as well want to look at it as a 2-dimensional surface Γ ⊂ R3.While 3D-charge carriers can be modeled by a function f, the 2D-setting gives rise to (Hausdorff) measures, leading to the equation −∆u = H2 Γ, which we will explain.The questions we examine is whether in this setting the electric field E⃗ = ∇u is bounded/continuous close to the charge carrier Γ. The answers depend highly on the geometry of Γ. Mathematically, this question is thought of as a question of regularity for a PDE involving measures.

Title and abstract in PDF.

Philipp Reiser (Fribourg, CH)

Surgery on Riemannian manifolds of positive Ricci curvature.

Ricci curvature is a notion of curvature that appears in many contexts in geometry and also has connections to other branches of mathematics, such as analysis, topology and mathematical physics. It is therefore natural to ask, which manifolds admit a Riemannian metric whose Ricci curvature satisfies certain given conditions, such as being positive or negative. On the other hand, surgery is a topological concept: it is a procedure to modify manifolds by certain cut and paste operations that is widely used in topology, for example in the classification of manifolds. In this talk I will introduce the notions of both Ricci curvature and surgery and then show how surgery can be used to construct examples of Riemannian manifolds with positive Ricci curvature.

Tommaso Rossi (Bonn)

The heat content in sub-Riemannian geometry.

Sub-Riemannian geometry is a vast generalization of Riemannian geometry, where a smoothly varying metric is defined only on a subset of preferred directions. In this setting, we study the small-time asymptotics of the heat content associated with smooth non-characteristic domains. The heat content associated with an open and bounded set is the total amount of heat contained in the set at time t, assuming that its boundary is perfectly insulated. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the complete asymptotic series and we compute explicitly its coefficients in terms of sub-Riemannian geometrical invariants of the domain. This is a joint work with Luca Rizzi.

David Tewodrose (Nantes)

How much singular a limit of manifolds with uniform curvature bounds may be?

The formation of singularities along a sequence of smooth Riemannian manifolds of same dimension is a frequent situation in eometric analysis. For instance, a Ricci flow may develop singularities in finite time, as the Ricci flow equation is a non-linear PDE which may admit divergent maximal solutions. In presence of such singularities, it is natural to look for theorems which classify them and provide information on the geometry they encode or the structure of the set they form. A classical setting studied in the literature is when the sequence of Riemannian manifolds satisfies a uniform curvature assumption. For instance, in a celebrated series of papers from the late nineties, Jeff Cheeger and Tobias Colding developed a structure theory for limits of smooth Riemannian manifolds satisfying a uniform lower bound. In this talk, I will present recent work with Gilles Carron and Ilaria Mondello where we extend the Cheeger-Colding theory to a larger class of limit spaces, namely those obtained from smooth Riemannian manifolds whose optimal lower bound on the Ricci curvature has negative part is a suitable Kato class. Our framework allows the Ricci curvature to degenerate towards - infinity, but in a way that is controlled by the heat kernel.

Mélanie Theillière (Luxembourg)

Why use corrugations?

The word "corrugation" is a synonym of "wave" or "ondulation". In maths, it appears in the Theory of corrugations of Thurston that he uses to remove singular points and, in particular, to evert a sphere. A more general theory, the Convex Integration of Gromov is also based on a deformation formula and allows to solve a large family of differential problem. We'll see the main idea of Convex Integration, from the problem to remove singular point to the problem of build isometric map. At the end, we'll see an explicit C^1 isometric embedding of the hyperbolic plane. This last construction is a joint work with the Hevea team.

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