Barcelona Summer School on Floer Theory

Floer homology was originally constructed to establish a deep connection between Hamiltonian dynamics and the topology of the underlying symplectic manifold. Since then, the theory has developed and expanded into many subfields of mathematical physics as well as pure mathematics (low-dimensional topology, string theory, algebraic geometry...). The aim of the summer school is to equip attendees with a basic understanding of the theory so that they can access the literature and apply it to various fields.

The summer school will consist of 10 hours of introductory lectures on Floer homology and 5 hours of more specialized talks presenting the different facets and applications of Floer theory. The school is aimed at Ph.D. students, postdocs and senior researchers, who would like to expand their symplectic horizons. The course assumes no prior knowledge of Floer theory, but a basic knowledge of symplectic geometry, functional analysis (variational methods) and algebraic topology (homology theory) is required.

Specialized Talks:

Maciej Borodzik

Title: Knot Floer homology and applications

Abstract: In this introductory talk, I will introduce Heegaard Floer theory for 3-manifolds and for links. I will give some explanations and hints, why the theory is so powerful in solving open problems in 3-manifolds.

Federica Pasquotto

Title: Exact contact embeddings and Rabinowitz-Floer Homology 

Abstract: Rabinowitz Floer Homology (RFH) was originally introduced by Cieliebak and Frauenfelder as an extension of Floer theory to the case of a special Lagrange multiplier functional (the Rabinowitz action functional). It can detect closed Reeb orbits of arbitrary period on fixed hypersurfaces of contact type, and it is closely related to the concept of displaceability. In this talk, I will outline the construction of RFH, illustrate its relationship to Hamiltonian/Reeb dynamics and symplectic homology, and discuss some of its applications. If time permits, I will also talk about the case of non-compact hypersurfaces, in particular those which arise as level sets of so called tentacular Hamiltonians (joint work with Jagna Wiśniewska).

Oliver Fabert

Title: Topology, symplectic topology, … what’s next?

Abstract: In topology one uses Morse theory to prove lower bounds for the number of 0-dimensional objects, namely critical points of smooth functions on smooth manifolds. In symplectic topology one uses Floer theory to prove lower bounds for the number of 1-dimensional objects, namely solutions to Hamiltonian ODEs. In this talk I will outline how the framework of symplectic topology can be generalized to study 2-dimensional or even higher-dimensional objects, leading to a class of first-order (Hamiltonian) PDEs sharing similar rigidity properties. In the same way as the Hamiltonian ODEs provide a generalized framework for classical mechanics, our class of Hamiltonian PDEs shall provide a generalized framework for studying equilibrium states of reaction-diffusion systems. This is joint work with my PhD student Ronen Brilleslijper.

Jagna Wiśniewska

Title: Lagrangian Rabinowitz Floer homology in celestial mechanics

Abstract: The origins of the field of symplectic geometry and Hamiltonian dynamics go back to early nineteenth century when Lagrange introduced the notion of a symplectic manifold to study the three body problem. The three body problem considers the motions of three heavenly bodies under the influence of their gravitational field. Our project is motivated by the following practical question: Can we send a rocket between any two points in the gravitational field of the Moon and the Earth, using the engines only at the beginning and at the end of the journey? In my talk I will introduce the Lagrangian Rabinowitz Floer homology as the perfect tool to answer this question. Based on joint work with Kai Cieliebak, Urs Frauenfelder and Eva Miranda.

 Organizing Committee

• Josep Fontana McNally - josep.fontana () estudiantat.upc.edu

• Jagna Wiśniewska - jagna.janina.wisniewska () upc.edu