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The Minimal Obstruction Problem of Ellipsoids into Balls

Heaney, Robert Jack (2019) The Minimal Obstruction Problem of Ellipsoids into Balls. Bachelor's Thesis, Mathematics.


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The problems of symplectic embeddings have found applications in as far-reaching areas as combinatorics. One aspect of this paper will be to describe the tools used to solve symplectic embedding problems. Those tools are Embedded Contact Homology (ECH) and ECH capacities. The ECH is defined via counts of $J$-holomorphic curves inside of a symplectization of a contact 3-manifold. The capacities are then derived from this homology. The main result is to reverify a minimal obstruction problem. Namely, finding those $\mu$ such that $E(1,\mu^2)$ symplectically embeds into $B^4(\mu)$. The solution to which appear as $\mu = g_{n+1}/g_n$, $n\in \mathbb{Z}_{\geq 0}$ where the $g_n$'s are odd-indexed Fibonacci numbers. Through combinatorial methods, a Diophantine equation will be derived, with a solution set directly associated with solving the minimal obstruction problem. The Diophantine equation is solved via graph theory. The graph theory approach is very likely applicable to other minimal obstruction problems.

Item Type: Thesis (Bachelor's Thesis)
Supervisor nameSupervisor E mail
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 11 Jul 2019
Last Modified: 16 Jul 2019 09:33

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