Symplectic invariants and moduli spaces of integrable systems

TL;DR: Palmer et al. as mentioned in this paper introduced the notion of a semitoric helix, which is the natural combinatorial invariant of integrable systems, and constructed a meaningful metric on general integrably Hamiltonian systems by defining a metric on such a collection.

Abstract: Author(s): Palmer, Joseph | Advisor(s): Pelayo, Alvaro | Abstract: In this dissertation I prove a number of results about the symplectic geometry of finite dimensional integrable Hamiltonian systems, especially those of semitoric type. Integrable systems are, roughly, dynamical systems with the maximal amount of conserved quantities. Though the study of integrable systems goes back hundreds of years, the earliest general result in this field is the action-angle theorem of Arnold in 1963, which was later extended to a global version by Duistermaat. The results of Atiyah, Guillemin-Sternberg, and Delzant in the 1980s classified toric integrable systems, which are those produced by effective Hamiltonian torus actions. Recently, Pelayo-Vu Ngoc classified semitoric integrable systems, which generalize toric systems in dimension four, in terms of five symplectic invariants. Using this classification, I construct a metric on the space of semitoric integrable systems. To study continuous paths in this space produced via symplectic semitoric blowups, I introduce an algebraic technique to study such systems by lifting matrix equations from the special linear group SL(2,Z) to its preimage in the universal cover of SL(2,R). With this method I determine the connected components of the space of semitoric integrable systems. Motivated by this algebraic technique, I introduce the notion of a semitoric helix; the natural combinatorial invariant of semitoric systems. By applying a refined version of the algebraic method to semitoric helixes I classify all possible minimal semitoric integrable systems, which are those that do not admit a symplectic semitoric blowdown. I also produce invariants of integrable systems designed to respect the natural symmetries of such systems, especially toric and semitoric ones. For any Lie group G, I construct a G-equivariant analogue of the Ekeland-Hofer symplectic capacities. I give examples when the capacity is an invariant of integrable systems, and I study the continuity of these capacities using the metric I defined on semitoric systems. Finally, as a first step towards constructing a meaningful metric on general integrable systems, I provide a framework to study convergence properties of families of maps between manifolds which have distinct domains by defining a metric on such a collection.