UC Santa Cruz

This dissertation consists of a mixture of two different topics from two separate areas of geometry. The first part of the thesis deals with the classification of a particular type of geometric distribution from subriemannian geometry. The second half of this dissertation originates from a problem in symplectic geometry concerning the degenerate case of the Arnold conjecture.

The first part of the dissertation is motivated by the work of A. Castro, R. Montgomery, and M. Zhitomirskii and investigates the classification, up to local diffeomorphism, of a certain type of geometric distribution known as a Goursat multi-flag. Montgomery and Zhitomirskii approached this classification problem by working with a structure called the Monster Tower, which is comprised of a sequence of manifolds. Each level of this tower is constructed through a process called Cartan prolongation. Montgomery and Zhitomirskii pointed out that the problem of classifying the points within each level of the tower is equivalent to the problem of classifying Goursat multi-flags. This work is an extension of the classification work with the

R³ Monster Tower that was initiated by Castro and Montgomery. We will present two different methods for classifying Goursat 2-flags of small critical length within the Monster Tower. In short, we classify the orbits within the first four levels of the Monster Tower and show that there is a total of 34 orbits in the fourth level of the tower.

The second part of this thesis is concerned with a problem from symplectic geometry that was posed by V. Ginzburg. This research looks at the size of the fixed point set of a Hamiltonian diffeomorphism on a closed symplectic manifold which is both rational and weakly monotone. We show that there exists a non-trivial cycle of fixed points whenever the action spectrum is smaller, in a certain sense, than required by the Ljusternik-Schirelman theory. For instance, in the aspherical case, we prove that when the number of points in the action spectrum is less than or equal to the cup length of the manifold, then the cohomology of the fixed point set must be non-trivial. This is a consequence of a more general result which applies to all weakly monotone manifolds and asserts that the same is true when the action selectors are related by an equality of the Ljusternik-Schirelman theory.