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

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Abstract
The problems of symplectic embeddings have found applications in as farreaching 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 3manifold. 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 oddindexed 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)  

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Degree programme:  Mathematics  
Thesis type:  Bachelor's Thesis  
Language:  English  
Date Deposited:  11 Jul 2019  
Last Modified:  16 Jul 2019 09:33  
URI:  http://fse.studenttheses.ub.rug.nl/id/eprint/20121 
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