Homological Mirror Symmetry and the Thomas-Yau Conjecture

Brongers, Bram (2023) Homological Mirror Symmetry and the Thomas-Yau Conjecture. Master's Thesis / Essay, Mathematics.

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Abstract

This thesis consists of two parts. The first part reviews homological mirror symmetry (HMS), and the second part discusses the Thomas-Yau conjecture, which is inspired by HMS. The first part starts by discussing relevant differential geometry, primarily Calabi-Yau manifolds and symplectic geometry. This is so that we can define the Fukaya category and the bounded derived category of coherent sheaves, related through HMS. We explain the string theoretic origins of mirror symmetry, and subsequently HMS, as it pertains to the two aforementioned categories. We discuss multiple examples along the way, placing an emphasis on complex tori. In the second part of the text, we turn our eyes to the Thomas-Yau conjecture, which is motivated by homological mirror symmetry and the notion of "stability", which was not necessarily precise at the time the conjecture was formulated. We explain the reasoning that went into the conjecture, going into more detail than the source material and making the analogy with the Kobayashi-Hitchin correspondence evident. We work out some examples, again focusing on tori, some of which do not appear to be available in the literature. Finally, we explain the updated version of the Thomas-Yau conjecture, taking into account a precise notion of "stability", called a Bridgeland stability condition. This leads to the Thomas-Yau-Joyce conjecture, which has some implications for the enumerative geometry of Calabi-Yau threefolds that we will conclude with.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Martynchuk, N.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	31 Jul 2023 10:09
Last Modified:	09 Aug 2023 12:28
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/30506

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