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oai:www.repository.cam.ac.uk:1810/3715272024-07-24T00:43:37Zcom_1810_213747com_1810_256064col_1810_213748

Floer theory and spectral networks https://doi.org/10.17863/CAM.110686 Nho, Yoon Jae Keating, ailsa This thesis consists of two papers. In the first paper, we study the relationship between Gaiotto-Moore-Neitzke's non-abelianization map and Floer theory. Given a complete GMN quadratic differential $\phi$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the complement of the poles of $\phi$. In the case where the spectral curve $\Sigma_{\phi}$ is exact with respect to the canonical Liouville form on $T^{\ast}\tilde{C}$, we show that an ``almost flat" $GL(1;\mathbb{C})$-local system $\mathcal{L}$ on $\Sigma_{\phi}$ defines a Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< \epsilon\leq 1$. Then we show that for small enough $\epsilon$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$. In the second paper, we extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\RR^n\cap (\RR^{k}\times \sqrt{-1}\RR^{n-k})$ inside $\CC^n$. Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach. University of Cambridge 2024-06-30 Thesis Doctoral eng https://www.repository.cam.ac.uk/handle/1810/371527 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/8b4a505d-8edf-4d99-87fe-008763ec0186/download a5024db01c87235a8a71fd14d7bdcae8 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/9bd05024-5745-4459-a316-a591acfbd5b1/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ symplectic spectral curves quadratic differentials