Simon Foucart (Texas A&M University):
Semidefinite programming in approximation theory: two examples

Dario Trevisan (Universität Pisa):
A PDE approach to random matching problems

Paolo Bonicatto (Universität Basel):
Untangling of trajectories for non-smooth vector fields and Bressan’s compactness conjecture

Given d≥1, T>0 and a vector field b: [0,T] × ℝ^d → ℝ^d, we study the problem of uniqueness of weak solutions to the associated transport equation ∂_tu + b·∇u = 0 where u: [0,T] × ℝ^d → ℝ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field b. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost. In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if b is locally of class BV in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible BV vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.

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Enno Lenzmann (Universität Basel):
Energy-critical half-wave maps: solitons and Lax pair structure

The eneryg-critical half-wave maps equation arises as a classical continuum limit of Calogero-Moser and Haldane-Shastry type spin systems in one space dimension. In this talk, I will discuss some essential features such as the complete classification of traveling solitary waves with finite energy, by using a close connection to minimal surfaces with free and non-free boundary conditions. Furthermore, I will present a recently found Lax pair structure and we explain its potential applications to the dynamics of the half-wave maps equation. Finally, I will mention some open problems. This talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

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Filip Rindler (University of Warwick):
Rademacher’s Theorem, Cheeger’s conjecture and PDEs for measures

Herbert Koch (University of Bonn):
The renormalized nonlinear wave equation in 2d with additive white noise

By the classical uniformization theorem, every smooth Riemann surface is conformally diffeomorphic to a surface of constant curvature. What happens if the smooth Riemannian metric is replaced by a non-smooth distance? Does the so obtained metric surface still admit parametrizations with good geometric and analytic properties? Such questions have been widely studied in the field of Analysis on metric spaces and are important for example in view of applications to Geometric Group Theory. I will explain how one can use recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. In particular, we obtain a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres. Joint work with Alexander Lytchak.

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Daniel Cremers (TU München):
Convex relaxation methods for computer vision

Numerous problems in computer vision and image analysis can be solved by optimization methods. Yet, many classical approaches correspond to non-convex energies giving rise to suboptimal solutions and often strong dependency on appropriate initialization. In my presentation, I will show how problems like stereo or multiple view reconstruction and 3D shape matching can be tackled by means of convex relaxation methods which allow to efficiently compute globally optimal or near-optimal solutions. The arising large-scale convex problems can be solved by means of provably convergent primal-dual algorithms. They are easily parallelized on GPUs providing high-quality solutions in acceptable runtimes.

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Stefan Wenger (Université de Fribourg):
Finding good parametrizations for metric surfaces