The computation of 3D magnetohydrostatic (MHS) equilibria is of major importance for magnetic confinement devices such as tokamaks or stellarators. In this talk I will present recent results on the existence of stepped pressure MHS equilibria in 3D toroidal domains, where the plasma current exhibits an arbitrary number of current sheets. The toroidal domains where these equilibria are shown to exist do not need to be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The proof involves three main ingredients: a Cauchy-Kovalevskaya theorem for Beltrami fields, a Hamilton-Jacobi equation on the two-dimensional torus, and a KAM theorem for 3D solenoidal fields. This is based on joint work with A. Enciso and A. Luque.

I will start by describing two "variants" of the Poisson formula that are due to Guinand and Meyer. I will then show how these two formulas can be interpreted in terms of the orthospectrum (a family of characteristic lengths) of two convex bodies. With that interpretation in mind, Guinand-Meyer’s formulas correspond to the case where the convex bodies are reduced to two points in dimension 3. I will explain how this formula can be extended to two general (strictly) convex bodies and how it encodes certain geometric quantities (mixed volumes). This was obtained in a joint work with Nguyen Viet Dang and Matthieu Léautaud.

In this talk I will give the basics of the Anderson model and then define its Landscape function (Filoche-Mayboroda 2012). I will discuss the conjectured connection between the Landscape function and the bottom of the spectrum of the Anderson Hamiltonian restricted to a large box. Finally, I will give some partial results towards such conjecture.