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.

In this dissertation, I prove a number of stability theorems for volume forms and symplectic forms in the noncompact setting, as well as a semiglobal classification result of finite dimensional integrable Hamiltonian systems. Volume forms and symplectic forms are, roughly, structures on smooth manifolds that measure volumes and $2$-dimensional areas. A work of Darboux in 1882 ruled out any local invariants of symplectic forms. Moser proved in 1965 that on a compact manifold, we can not get non-diffeomorphic volume forms without changing the volume or non-diffeomorphic symplectic forms with a smooth deformation inside a cohomology class. Moser's result on volume forms was generalized to noncompact manifolds by Greene and Shiohama. I develop these stability results in two directions. For volume forms, I find the extra conditions for Greene--Shiohama theorem to hold for smooth families of volume forms. The case of smooth families fits into the more general framework of fiber bundles with compact base and noncompact fiber. I define the concept of an exhausted fiber bundle which is exhausted by a smooth function compatible with the fiber bundle structure. On an exhausted fiber bundle, two fiberwise defined volume forms are fiberwise diffeomorphic under similar conditions as the smooth family case. For symplectic forms, the notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser stability theorem in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and completeness assumptions on the symplectic forms at infinity I show that the stability holds for a cohomologuous smooth family of symplectic forms subject to a growth condition at the infinity, which I call having bounded log-variation.

Integrable systems are, roughly, dynamical systems with the maximal amount of conserved quantities. The symplectic theory of integrable systems started from the action-angle theorem of Minuer in 1937 and Liouville--Arnold in 1963, which was extended to a global version by Duistermaat in 1980. These results clarified the symplectic structures near regular points and compact regular fibers of the momentum map. Eliasson in 1984 (complemented by V{\~u} Ng{\d{o}}c--Wacheux in 2013) proved that near a nondegenerate singular point the integrable system is symplectomorphic to its linear model, called the Eliasson local normal form. The neighborhoods of a compact connected fiber with only one focus-focus point and without other singular points in a $4$-dimensional integrable system is classified by San V{\~u} Ng{\d{o}}c in 2002, by a formal power series. I prove that a compact connected fiber with multiple focus-focus points and without other singular points in a $4$-dimensional integrable system is classified by a tuple formal power series as many as singular points.