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are the three main results presented in the thesis abstract?","Question",{"text":63,"@type":64},"The thesis gives three correspondences linking q-refined tropical curve counting and Kontsevich–Soibelman scattering diagrams to higher genus log Gromov–Witten theory, and it constructs a q-deformation of the Gross–Hacking–Keel mirror producing a deformation quantization with a canonical basis.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which mathematical areas does the work connect according to the abstract?",{"text":68,"@type":64},"The abstract states that the results lie at the intersection of tropical geometry, enumerative geometry, mirror symmetry, and non-commutative algebra.",{"name":70,"@type":61,"acceptedAnswer":71},"How are the three results logically related?",{"text":72,"@type":64},"They are logically dependent: the proof of the third result relies on the second, and the second relies on the first, while each result is independently 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the degree of  \nDoctor of Philosophy of Imperial College London  \nby  \nPierrick Bousseau  \nDepartment of Mathematics  \nImperial College  \n180 Queen's Gate, London SW7 2BZ  \nI certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline.  \nCopyright  \nThe copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.  \nThesis advisor: Professor Richard Thomas Pierrick Bousseau  \nQuantum mirrors of log Calabi-Yau surfaces and higher genus  \ncurve counting  \nAbstract  \nWe present three results, at the intersection of tropical geometry, enumerative geometry, mirror symmetry and non-commutative algebra.  \n1. A correspondence between Block-G􀁿ottsche q-re􀀌ned tropical curve counting and higher genus log Gromov-Witten theory of toric surfaces.  \n2. A correspondence between q-re􀀌ned two-dimensional Kontsevich-Soibelman scattering diagrams and higher genus log Gromov-Witten theory of log Calabi-Yau surfaces.  \n3. A q-deformation of the Gross-Hacking-Keel mirror construction, producing a deformation quantization with canonical basis for the Gross-Hacking-Keel families of log Calabi-Yau surfaces.  \nThese results are logically dependent: the proof of the third result relies on the second, whose proof itself relies on the 􀀌rst. Nevertheless, each of them is of independent interest.  \nTo my parents.  \n6  \nAcknowledgments  \nI would like to thank my supervisor Richard Thomas for continuous support and innumerous discussions, suggestions and corrections.  \nI thank the examiners, Mark Gross and Johannes Nicaise, for careful reading and corrections.  \nI thank Tom Bridgeland, Michel van Garrel, Lothar G􀁿ottsche, Mark Gross, Liana Heuberger, Ilia Itenberg, Yanki Lekili, Rahul Pandharipande, Bernd Siebert, Jacopo Stoppa for various invitations to conferences and seminars, where some results of this thesis have been presented, and for useful comments and discussions.  \nI thank Navid Nabijou, Dan Pomerleano and Vivek Shende for speci􀀌c discussions directly related to the content of this thesis.  \nThis work would not have been possible without the London mathematical environment and they are too many people to mention. This includes members and graduate students of the Departments of Mathematics of Imperial College London, UCL, KCL, and in particular my fellows of the LSGNT program.  \nFinally, thanks to all the people met around the world during conferences and having contributed to my mathematical education, and who are again too numerous to list.  \nThis work has been supported by the EPSRC award 1513338, Counting curves in algebraic geometry, Imperial College London, and has bene􀀌ted from the EPRSC [EP/L015234/1], EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.  \nContents  \nIntroduction 13  \nIntroduction to Chapter 1 ................................... 13  \nIntroduction to Chapter 2 ................................... 18  \nIntroduction to Chapter 3 ................................... 23  \n1 Tropical refined curve counting from higher genera 27  \n1.1 Precise statement of the main result ......................... 27  \n1.1.1 Toric geometry .................................. 27  \n1.1.2 Log Gromov-Witten invariants ........................ 28  \n1.1.3 Tropical curve","cbCaip7TMQq1hJ2m","https://ap.wps.com/l/cbCaip7TMQq1hJ2m","pdf",1806908,213,"English","# Introduction\n## Introduction to Chapter 1\n## Introduction to Chapter 2\n## Introduction to Chapter 3\n# 1 Tropical refined curve counting from higher genera\n## Precise statement of the main result\n## Gluing and vanishing properties of lambda classes\n## Toric degeneration and decomposition formula\n## Non-torically transverse stable log maps in X\n## Statement of the gluing formula\n## Proof of the gluing formula\n## Vertex contribution","[{\"question\":\"What are the three main results presented in the thesis abstract?\",\"answer\":\"The thesis gives three correspondences linking q-refined tropical curve counting and Kontsevich–Soibelman scattering diagrams to higher genus log Gromov–Witten theory, and it constructs a q-deformation of the Gross–Hacking–Keel mirror producing a deformation quantization with a canonical basis.\"},{\"question\":\"Which mathematical areas does the work connect according to the abstract?\",\"answer\":\"The abstract states that the results lie at the intersection of tropical geometry, enumerative geometry, mirror symmetry, and non-commutative algebra.\"},{\"question\":\"How are the three results logically related?\",\"answer\":\"They are logically dependent: the proof of the third result relies on the second, and the second relies on the first, while each result is independently interesting.\"}]","Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting - Doctoral Thesis Abstract | PDF",537]