Lefschetz Fibrations, Mapping Tori and Dynamics on Moduli Spaces of Objects

Dynamical systems (mathematical models of systems changing in time) describe many processes that affect us, and have given rise to some of the hardest questions in science, such as the multi-body problem in celestial mechanics. The first part of this project aims to explore an entirely new kind of dynamics, in which the states of the system (such as positions of particles) move around in time, without a global motion of the entire system. This seems paradoxical, and indeed one expects it to happen mostly in situations that are far from applications. Nevertheless, it has been shown that the phenomenon is mathematically possible, and because dynamical systems thinking provides such a powerful intuition, it makes sense to want to stretch its limits as far as possible. Any evidence of additional complexity in classical mechanical systems, even if it directly affects only a few cases, ultimately changes how we think of the complexity of such systems in general. The second part of this project deals with a phenomenon in mathematics which arises from its current close exchange of ideas with string theory: namely, the appearance of complicated explicit functions (typically, of one variable). From the viewpoint of topology, which is more qualitative, one hopes to minimize the amount of information that needs to be encoded inside such functions. For instance, if the functions themselves solve a differential equation, they can be recovered from a finite amount of information. String theory has been very effective in providing such a characterization, but this project aims (in a special case) for a more direct and simpler description. It should be viewed as an exercise in "noncommutative geometry", which is the mathematicians' way to prepare ourselves for thinking beyond conventional notions of space (which is one of the big challenges in contemporary mathematics and physics).

Symplectic manifolds have a rich internal structure. This can be approached from a variational viewpoint (capacities, Hofer norms, spectral invariants), or from string theory and mirror symmetry. Nevertheless, the basic known invariants are a collection of numbers, or homology classes (Gromov-Witten invariants). There is information beyond that (Lagrangian submanifolds, Fukaya categories), but it is not directly amenable to being used as a classification tool. The project intends to attack this situation by looking at dynamical systems acting on the Fukaya category. The idea is start with geometric considerations such as flux, and export them to other situations. The approach is designed to be applied to a specific class of symplectic manifolds, related to mapping tori. The other major topic is a way of computing Fukaya categories, using Lefschetz pencils. While there is a body of previous work in this direction, it is restricted to the exact (or monotone) situation, and does not address the challenge of understanding the infinite series that arise in the Calabi-Yau case. The PI's aim is to describe those series in a more direct way than is provided by the standard framework of mirror symmetry (Gauss-Manin connections, mirror maps); this description would then also be more general, since it is ultimately independent of mirror symmetry considerations.