Joaquim Brugués defended his PhD thesis Floer homology for b-symplectic manifolds, supervised by Eva Miranda and Sonja Hohloch on 20th of March 2024 within the UPC doctoral program in Applied Mathematics and the PhD program (Doctor in Science: Mathematics) at University of Antwerp.
Thesis summary
Symplectic geometry is the natural setting for the study of classical mechanics in their full generality. Symplectic manifolds provide an environment for the intersection of the fields of differential geometry, topology and analysis and for results that bridge between them. In that line, Arnold put forward in 1963 a conjecture which proposed that the lower bound to the number of periodic orbits of a Hamiltonian vector field in a compact symplectic manifold is the sum of its Betti numbers. Thus, dynamics are bounded by topology, in a way analogous to that of Morse theory. The general proof of the Arnold conjecture took several decades and inspired key developments in the field. One of the most important of such developments was presented by Floer in the 1980’s, taking inspiration from previous work by Conley and Zehnder in variational methods to tackle the Arnold conjecture, and from Gromov’s work studying pseudoholomorphic curves. Floer was able to define a homology that combined these approaches to relate the periodic orbits of a Hamiltonian vector field to the homology of the underlying manifold, thus proving the conjecture in a wide range of symplectic manifolds.
Symplectic manifolds can be seen as particular cases of Poisson manifolds, which have a foliated symplectic structure. For instance, the subject of this work is the study of bm-symplectic manifolds, having a structure that is symplectic almost everywhere but with a singularity in a hypersurface, sometimes called the “singular hypersurface”, where the structure is that of a symplectic foliation of maximal rank (namely, of corank 1). In our work we set out to explore possible generalizations of the Arnold conjecture and constructions of Floer-type homologies in bm-symplectic manifolds. We studied the dynamics of Hamiltonian vector fields in this context, focusing on the presence of periodic orbits, and developed an understanding about how could these be understood as regular symplectic Hamiltonian vector fields under a technique used to study bm-symplectic topology called desingularization.
By means of this desingularization procedure we successfully found lower bounds in the case of b2k-symplectic manifolds. Moreover, when restricting ourselves to bm-symplectic surfaces we managed to find a strict lower bound. This lower bound is of particular interest because it combines the usual understanding of the homology of a surface with the relative position of the singular hypersurface (in this case, the singular curve) within the surface. This opens an intriguing question about the nature of the Arnold conjecture in the bm-symplectic setting, in which not only is the topology of the underlying manifold relevant, but also the topology induced by the relative position of the singular hypersurface.
On a slightly different approach, studying pseudoholomorphic curves in bm-symplectic manifolds, we were able to define a chain complex (and therefore a homology) analogous to that of Floer. However, computing this homology (as in finding an isomorphism to a homology induced by topology) is still an open question.
In this work we also studied the notion of integrable systems in bm-symplectic manifolds, and more specifically that of b-semitoric systems. We studied the basic properties of these systems and proved that they cannot have singular points in the hypersurface. Then we adapt two important examples of semitoric systems, the coupled spin-oscillator and the coupled angular momenta, into b-semitoric systems by adding a singularity. This provides us with some examples which we are able to study explicitly, and which we hope will provide guidance in some future complete classification of this type of systems.
Selected publications: [1], [2].
References
[1] J. Brugués, S. Hohloch, P. Mir, and E. Miranda, Constructions of b-semitoric systems, Journal of Mathematical Physics 64 (2023) no. 7.
[2] J. Brugués, E. Miranda, and C. Oms, The Arnold conjecture for singular symplectic manifolds, Journal of Fixed Point Theory and Applications 26 (2024) no. 2 (16 pages).