The Work of the 2014 Fields Medalists

The Work of the 2014 Fields Medalists Welington de Melo, Bjorn Poonen, Jeremy Quastel, and Anton Zorich The Notices solicited the following articles about the works of the four individuals to whom Fields Medals were awarded at the International Congress of Mathematicians in Seoul, South Korea, in August 2014. This was a historic occasion, as it marked the first time since the medal was established in 1936 that a woman was among the recipients. The International Mathematical Union also issued news releases describing the medalists' work, and these appeared in the October 2014 issue of the Notices. --Allyn Jackso

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature

Large deviations of multichordal SLE0+_{0+}, real rational functions, and zeta-regularized determinants of Laplacians

We prove a strong large deviation principle (LDP) for multiple chordal SLE$_{0+}$ curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, that satisfies PDEs arising as a semiclassical limit of the Belavin-Polyakov-Zamolodchikov equations of level two in conformal field theory with central charge $c \to -\infty$. Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the $\kappa \to 0+$ limit of the multiple SLE$_\kappa$. As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition by a M\"obius transformation.Comment: 66 pages, 4 figures. Mostly minor edits, updates, and clarifications: addressed subtleties in the definition of curve spaces and hulls. To appear in JEM