Irrational ellipsoid embeddings and a canonical grading of embedded contact homology

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The main content of this thesis is two papers. The first paper defines a canonical $\Q$-grading of the embedded contact homology chain complex under certain conditions. ECH is an invariant of 3-manifolds isomorphic to the Seiberg-Witten Floer cohomology as defined by Kronheimer and Mrowka. The underlying chain complex comes with a relative $\Z/p$-grading, or, an absolute $\Z/p$-grading after fixing a reference generator. However, when $p=0$, we show that one can actually specify an absolute grading of the chain complex without having to choose a reference generator. Explicitly, that grading set is a subset of $\Q$ isomorphic to $\Z$. As a consequence, $ECH(Y,\lambda)$ is canonically $\Q$-graded whenever $H_1(Y)$ is torsion.

The second paper is an application of ECH capacities to the problem of embedding ellipsoids into irrational ellipsoids. More specifically, we analyze lattice point counting functions to deduce that the existence of an accumulation point of an infinite staircase is impossible for all irrational targets minus a set of irrationals in bijection with $\N$, for which the question has yet to be resolved.