Benjamin Filippenko

Title: A polyfold proof of the Arnol'd conjecture

Abstract: Polyfold theory, based on sc-calculus, is a non-linear Fredholm theory over spaces of Sobolev class maps modulo a reparameterization action and with varying domains and targets, developed by Hofer-Wysocki-Zehnder to resolve transversality issues in moduli problems in symplectic topology. I will explain a polyfold proof, joint with Katrin Wehrheim, of the Arnol'd conjecture: the number of 1-periodic orbits of a nondegenerate 1-periodic Hamiltonian on a closed symplectic manifold is at least the sum of the Betti numbers. Our proof is a polyfold construction of the Piunikhin-Salamon-Schwarz morphisms between Morse and Floer theory for general closed symplectic manifolds, which relies on the Symplectic Field Theory polyfolds constructed by Joel Fish and Helmut Hofer. In the first two lectures, I will introduce Hamiltonian Floer theory, sc-calculus, polyfold Fredholm theory and its main transversality theorem, and new fiber product and implicit function theorems in polyfold theory required for the Arnol'd conjecture proof.

Joel W. Fish

Title: Feral Curves and Minimal Sets

Abstract: I will discuss some recent joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral pseudoholomorphic-curves) to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian on R⁴, then no non-empty regular energy level of H is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

Title: Properties of Feral Curves I

Abstract: I will discuss some technical details regarding properties of feral pseuodoholomorphic curves. Specifically, I will focus on the target-local Gromov compactness theorem, and an estimate for exponential area bounds.

Title: Properties of Feral Curves II

Abstract: I will discuss some technical details regarding properties of feral pseuodoholomorphic curves. Specifically, I will focus on the connected-local area bound and asymptotic curvature bound.

Daniel Peralta-Salas

Title: An introduction to Beltrami fields

Abstract:

A divergence-free vector field in a Riemannian 3-manifold whose curl is proportional to the field itself (through a not necessarily constant factor) is called a Beltrami field. From the physical viewpoint, Beltrami fields are a particularly relevant class of stationary solutions to the Euler equations. The goal of this short course is to introduce a selected kit of tools to study Beltrami fields, which are instrumental to analyze the topology of the vortex lines and tubes and to establish the existence of vortex reconnections in Navier-Stokes. The first tool that I will present is a Cauchy-Kowalewsky theorem for the curl operator. Then I will explain the local-global flexibility of the Beltrami equation by proving a Runge-type approximation theorem. Finally, I will introduce an inverse localization technique that allows one to transplant Beltrami fields from Euclidean space into the torus or the sphere. The connections with contact geometry and some selected open problems will be also discussed.

Francisco Javier Torres de Lizaur

Title: A characterization of 3D steady Euler flows using commuting zero-flux homologies

Abstract: I will show how to characterize those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. For a given vector field, the existence of such a metric depends on some approximation properties of its asymptotic cycles. This extends Sullivan’s homological characterization of geodesible flows in the volume preserving case. As an application I will prove that the steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). This is joint work with Ana Rechtman and Daniel Peralta-Salas.

Gunther Uhlmann

Title: Calderón's Inverse Problem

Abstract: We will consider some of developments over the last 40 years concerning the following inverse problem proposed by Calderón in 1980. Can one determine the conductivity of a medium by making voltage and current measurements at the boundary? This problem arises in medical imaging, geophysics and non-destructive evaluation of materials among several other applications. In mathematical terms it can be formulated as whether one can determine the coefficients of an elliptic equation, the conductivity equation, from the associated Dirichlet-to-Neumann map. We will concentrate on the study of complex geometrical optics (CGO) solutions that have played an essential role in this inverse problem and other inverse problems.