Construction of flows without periodic orbits on 3-manifolds
Ana Rechtman, Université de Strasbourg
The already disproved Seifert’s conjecture stated the existence of a periodic orbit for every non-singular flow on S3. P. Schweitzer constructed the first example of a C1 flow without periodic orbits, and later K. Kuperberg constructed smooth, and even real analytic, examples. We can still talk about Seifert’s conjecture for restrictive families of flows, as for example flows that preserve a volume.
On the other hand, there are some families of flows that always have periodic orbits. I will present Schweitzer and Kuperberg’s constructions, and discuss the results on the existence of periodic orbits.
From foliations to contact structures (and back?)
Thomas Vogel, Max-Planck-Institut für Mathematik
In this course we will study the relationship between foliations and contact structures approximating them (in dimension 3). In particular, I will outline the proof of a theorem of Eliashberg and Thurston stating that all foliations without spherical leaves can be approximated by contact structures. Natural questions in this context are: Is the apporximating contact structure unique? Which contact structures are obtained by such an approximation.
Moduli spaces of vector bundles over a real curve
Thomas Baird, Memorial University
Moduli spaces of holomorphic bundles over a complex projective curve have been a important object of study in mathematics for more than 50 years. In a highly influential paper from the ’80s, Atiyah and Bott used Morse theory and the Yang-Mills functional to compute the rational Betti numbers of these moduli spaces. More recently, the moduli space of vector bundles over a real curve has garnered a great deal of interest. I will define these moduli spaces, and explain how to adapt the Atiyah-Bott method to compute their Betti numbers in characteristic 2.
Contact Structures and Open Book Decomposition of Moment-Angle Manifold
Yadira Lizeth Barreto Felipe
The purpose of the talk is to describe contact structures on certain manifolds, called moment – angle manifolds, using open book decompositions as a tool. These manifolds appears in the construction of a family of complex, compact, non symplectic manifolds given by Santiago López de Medrano and Alberto Verjovsky.