By Henri Berestycki, EHESS, Paris

I will present a system of equations describing the effect of including a line (the "road") with a specific diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. One can determine the asymptotic shape of propagation in every direction influenced by the road. I will discuss several further effects such as transport, reaction on the road or the influence of various parameters. I will also describe some variants of this system. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

By Yoshihiro Tonegawa, Tokyo Institute of Technology

By Guido De Philippis, SISSA, Trieste

I will show a general structure theorem for the singular part of A-free Radon measures, where A is a linear PDE operator. By applying the theorem to suitably chosen dierential operators A ,one can obtain a simple proof of Alberti's rank-one theorem and its extensions to functions of bounded defor-mation (BD). I will also show some consequences concerning the sharpness of Rademacher Theorem and the structure of Ambrosio{Kirchheim top-dimensional metric current in Rd.

By Jun Zhang, New York University / NYU Shanghai

Advancing interfaces are ubiquitous in nature, can be found over many length scales and in many systems. In addition to the numerical and analytical studies in this field, experimentalists have also tried to create simple situations when an interface advances in the presence of uncorrelated, stochastic noise. In this talk, I will present a few examples that have emerged from physics laboratories and the results are discussed in some detail.

By Tristan Rivière, ETH Zurich

How much does it cost...to knot a closed simple curve? To cover the sphere twice? To realize such or such homotopy class ? ...etc.
All these questions consisting of assigning a "canonical" number and possibly an optimal "shape" to a given topological operation are known to be mathematically very rich and to bring together notions and techniques from topology, geometry and analysis.
In this talk we will concentrate on the operation consisting of everting the 2 sphere in the 3 dimensional space. Since Smale's proof in 1959 of the existence of such an operation the search for effective realizations of such eversions has triggered a lot of fascination and works in the math community. The absence in nature of matter that can interpenetrate and the quasi impossibility, up to the advent of virtual imaging, to experience this deformation is maybe the reason for the difficulty to develop an intuitive approach on the problem.
We will present the optimization of Sophie Germain conformally invariant elastic energy for the eversion. Our efforts will finally bring us to consider more closely an integer number together with a mysterious minimal surface.