The Focus Period 2020 will be entirely held online via Zoom. The link will be provided close to the date of the event to people who have registered (use the link below). During registration you will be asked to upload an updated CV and to indicate if you are interested in the either one or both weeks of the Focus Period.Registration Form
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Abstract: We will study the Laplace eigenfunctions on Riemannian manifolds and introduce some tools that are specific to the two-dimensional case, including a Carleman estimate and a weak subharmonicity of some auxiliary function. Among the main results are the bounds of the order of vanishing of eigenfunctions, the estimates of their zero sets, and Bernstein’s type inequalities. In the first two lectures we plan to go over results from the 1990s which were obtained by Donnelly and Fefferman and by Dong. The last two lectures will provide an overview of more recent ideas and results on Laplace eigenfunctions on surfaces. Despite recent progress, there are many open problems in the field, some of which will be discussed in the final lectures.
Absract: In this short course I will present the general theory of branched transport. This is a variant of the classical Monge-Kantorovich optimal mass transport, characterized by the fact that the efficiency of a network which achieves the transportation is measured by a functional which favors large flows of masses and penalizes diffusion. Part of the presentation will require specific tools from Geometric Measure Theory and in particular from the Theory of Currents, which will be the content of the first part of the course. After discussing some well-known features of the most popular models of branched transport, including the equivalence between their Eulerian and Lagrangian formulation, and some qualitative properties of the minimizers such as the absence of cycles, I will focus on recent results concerning the well-posedness of the problem.
Abstract: We will introduce and study two classical free boundary problems: the one-phase (Bernoulli) problem, and the obstacle problem. For each of them, we will present first some basic properties, such as optimal regularity of solutions and non-degeneracy, and then turn our attention to the regularity of free boundaries. Finally, at the end of the lectures we will discuss some recent results on singular points, as well as some natural questions that remain open.