We study the Pauli operator in a two-dimensional, connected
domain with Neumann or Robin boundary condition. We prove a sharp
lower bound on the number of negative eigenvalues reminiscent of the
Aharonov-Casher formula. We apply this lower bound to obtain a new
formula on the number of eigenvalues of the magnetic Neumann Laplacian
in the semi-classical limit. Our approach relies on reduction to a
boundary Dirac operator. We analyze this boundary operator in two
different ways. The first approach uses Atiyah-Patodi-Singer index
theory. The second approach relies on a conservation law for the
Benjamin-Ono equation. This is a joint work with S. Fournais, R. L.
Frank, M. Goffeng, M. Sundqvist.

For general initial data without any structural assumption, the Prandtl equations are usually ill-posed in the Sobolev space because the loss of tangential derivatives occurs in a non-local term. We will study the well-posedness property of the Prandtl equations in the critical Gevrey space of index 2. The proof combines a new cancellation mechanism with the abstract Cauchy-Kovalevskaya theorem, to overcome the difficulty of the loss of derivatives in the system.

Nous prouvons un théorème d'interpolation pour des fonctionnelles nonlinéaires définies sur des échelles d'espaces de Banach qui généralisent les espaces de Besov. La démonstration est auto-contenue et indépendante de tout résultat antérieur concernant la théorie de l'interpolation. Comme corollaire, nous en déduisons un théorème de continuité automatique pour le flot d’une équation quasi-linéaire. Précisément, nous montrons que la continuité de celui-ci découle automatiquement des estimations qui sont habituellement prouvées pour prouver l’existence et l’unicité des solutions.

A compact invariant set of a flow is called locally maximal when it is the largest invariant set in some neighborhood. In this talk, based on joint work with Erman Cineli, Viktor Ginzburg, and Basak Gürel, I will present a "forced existence" result for the closed orbits of certain Reeb flows on spheres of arbitrary odd dimension:

• If the contact form is non-degenerate and dynamically convex, the presence of a locally maximal closed orbit implies the existence of infinitely many closed orbits.

• If the locally maximal closed orbit is hyperbolic, the assertion of the previous point also holds without the non-degeneracy and with a milder dynamically convexity assumption.

These statements extend to the Reeb setting earlier results of Le Calvez-Yoccoz for surface diffeomorphisms, and of Ginzburg-Gürel for Hamiltonian diffeomorphisms of certain closed symplectic manifolds.