Talks

• Camilla Brizzi (München); On the Entropic Optimal Transport in $L^{\infty }$
  
  The Optimal Transport problem in $L^{\infty }$, i.e. the problem of minimizing the $L^{\infty }$-norm of the cost function among the transport plans, is a nonconvex and thus a presumably more difficult problem. Due to the success of entropic approximation and of Sinkhorn's algorithm, seeking an analogue for the infinity case seems quite natural. I will show the $\mathrm{\Gamma }$-convergence of the regularized functionals to the one related to the OT problem in $L^{\infty }$ and that every cluster point of the minimizers is a $\infty$-cyclically monotone transport plan which is for some cost functions a solution of the Monge problem and thus a map. Some numerical simulation will be also provided in order to give a better understanding of the results.
  
  
• Yann Chaubet (Paris); The narrow capture problem for geodesic Levy flights
  
  
  
  
• Annina Iseli (Fribourg CH); Thurston obstructions and the geometry of snowspheres
  
  We consider a specific family of Thurston maps that arises from Schwarz reflections on flapped pillows. Combining a counting argument with Thurston's characterization of rational maps, we establish anecessary and sufficient condition for a map in this family to be realized by a rational map. This resultgeneralizes to the case of all Thurston maps with four postcritical points. This is joint work with M. Bonk and M. Hlushchanka. By a theorem of Bonk-Meyer our results are related to the study of the geometry offractal snowspheres.
  
  
• Camille Labourie (Erlangen); Epsilon-regularity theorems for Griffith minimizers
  
  The Griffith energy is a functional introduced by Francfort and Marigo to model the equilibrium state of a brittle fracture in linear elasticity. The formulaGgiven below is a bit simplified.Let $\mathrm{\Omega }$ be a bounded open set of ${ℝ}^N$, which stands for the reference configuration (without crack) of a homogeneous, isotropic, brittle body. We apply a deformation (constant over time) at the boundary of the solid and we assume that the solid only undergoes infinitesimal transformations. Francfort and Marigo formulate the problem as the minimization of $G(u,K)={\int }_{\mathrm{\Omega }⁄K}|e(u){|}^{2}dx+{\mathscr{H}}^{n-1}(K)$ among pairs $(u,K)$ such that K is a (N-1)-dimensional subset of $\mathrm{\Omega }$, $u:\mathrm{\Omega }\to {ℝ}^N$ is a smooth function which satisfies a Dirichlet condition at the boundary $\partial \mathrm{\Omega }$, and the matrix $e(u):=\frac{1}{2}(Du+D{u}^T)$ is the symmetric part of the gradient of u. We interpretuas a displacement field (the deformation of the solid is $x\mapsto x+u(x)$), the matrix e(u) as its linear strain tensor (it describes the local deformation of $\mathrm{\Omega }$), K as a crack.The energy G(u,K) puts in competition the elastic energy stored outside of the crack and the surface energy required to create the crack. This formulation does not say anything about the topology and the regularity of the crack a priori and as of yet, we know very few regularity results about minimizers.Although it looks like the Mumford-Shah functional, the Griffith energy provides in fact a lot of suprising new difficulties as one works with the symmetrized gradient instead of the full gradient. The goal of the talk is to present a partialε-regularity theorem for Griffith minimizers in any dimension N (joint work with Antoine Lemenant) and its optimal variant in dimension 2 (joint work with Manuel Friedrich and Kerrek Stinson)
  
  
• Florian Litzinger (Magdeburg); Singularities of low entropy high codimension curve shortening flow
  
  The evolution of closed curves in the plane by their curvature is a well-studied geometric PDE. The behaviour of curves in higher codimension under this flow, however, is much less understood. In this talk, I will give an introduction to some of the classical results in the area. Moreover, I will present some new results concerning the flow of curves in any codimension subject to a bound on a suitably defined entropy functional.
  
  
• Christian Scharrer (Bonn) ; Short closed geodesics and the Willmore energy
  
  The Willmore bending energy of a closed surface is given by the integrated squared mean curvature. It is minimised by the unit sphere, thus quantifies the defect of a surface to be round. On the unit sphere each geodesic is closed and has the same length as the unit circle. Indeed, any spherical surface contains a closed geodesic of minimal length. I will show that the length of this shortest closed geodesic can be bounded from below in terms of the Willmore energy.
  
  
• Timo Schultz (Bielefeld) ; On absolute continuity and bounded variation of curves of probability measures
  
  Given a curve of probability measures in p-Wasserstein space for p>1, it is known that the curve is absolutely continuous with finite kinetic energy if and only if, on one hand, it is obtained as a superposition of absolutely continuous curves, and on the other, if and only if it solves the continuity equation driven by a p-integrable vector field. Both characterisations fail in the case p=1, which can be easily seen by looking at a linear interpolation of two dirac masses. In this talk, I will discuss analogous results in the case p=1 obtained by relaxing the absolute continuity in the first characterisation, and the notion of continuity equation in the second. A brief introduction to optimal transport and Wasserstein-spaces is provided.