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On the topology of Lagrangian submanifolds in toric symplectic manifolds
Maison d'édition
Neuchâtel
Date de parution
2022
Mots-clés
Résumé
Cette thèse consiste en quatre chapitres. Chapitre 2, <i>An introduction to the Chekanov torus</i>, est un article d’exposition introduisant tous les concepts nécessaires pour définir le tore de Chekanov. On donne une introduction d´etail´ee aux actions Hamiltoniennes d’un cercle, a` la réduction symplectique et aux déformations versales tout en montrant comment combiner ces outils pour montrer que le tore de Chekanov est exotique. Chapitre 3, <i>On the topology of real Lagrangians in toric symplectic manifolds</i>, est un article avec Joontae Kim and Jiyeon Moon qui donne une construction de sous-vari´et´es Lagrangiennes r´eelles dans les variétés toriques par rel`evement de sym´etries du polytope moment. Nous discutons également certaines propriétés des exemples construits et donnons un analogue de la construction de Delzant. Chapitre 4, <i>Real Lagrangian tori and versal deformations</i>, est un article centré autour d’obstructions à une Lagrangienne d’être réelle et, dans ce sens, Chapitre 4 est complémentaire à Chapitre 3. On donne un critère général que l’on applique par la suite aux fibres toriques et aux tores de Chekanov dans les variétés toriques. Chapitre 5, <i>Squeezing via degenerations of the complex projective plane</i>, est un appendice à l’article <i>On certain quantification of Gromov’s non-squeezing theorem</i> par Kevin Sackel, Antoine Song, Umut Varolgunes et Jonathan J. Zhu. On montre que la boule symplectique de dimension quatre privée d’un sous-ensemble de dimension de Minkowski deux peut être plongé dans un cylindre de rayon inférieur `a celui de la boule.
Abstract
This thesis consists of four main chapters. Chapter 2, <i>An introduction to the Chekanov torus</i>, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, <i>On the topology of real Lagrangians in toric symplectic manifolds</i>, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, <i>Real Lagrangian tori and versal deformations</i>, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, <i>Squeezing via degenerations of the complex projective plane</i>, is an appendix to the paper <i>On certain quantifications of Gromov’s non-squeezing theorem</i> by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.
Abstract
This thesis consists of four main chapters. Chapter 2, <i>An introduction to the Chekanov torus</i>, is an expository article introducing all the concepts necessary to define and under- stand the Chekanov torus. We give a detailed introduction to Hamiltonian circle actions, symplectic reduction, lifting techniques and versal deformations and show how those can be used to show that the Chekanov torus is exotic. Chapter 3, <i>On the topology of real Lagrangians in toric symplectic manifolds</i>, is a joint article with Joontae Kim and Jiyeon Moon focusing on constructing examples of real Lagrangian submanifolds in toric manifolds by lifting symmetries from the moment polytope. We also prove convexity and tightness for the examples we construct and give an analogue of the Delzant construction. Chapter 4, <i>Real Lagrangian tori and versal deformations</i>, is an article focusing on obstructions for a given Lagrangian submanifold to be real and is, in some sense, complementary to Chapter 3. We develop a general obstruction in terms of versal deformations and displacement energy and apply this to toric fibres and Chekanov tori in toric manifolds. Chapter 5, <i>Squeezing via degenerations of the complex projective plane</i>, is an appendix to the paper <i>On certain quantifications of Gromov’s non-squeezing theorem</i> by Kevin Sackel, Antoine Song, Umut Varolgunes and Jonathan J. Zhu. We prove that the symplectic four-ball can be squeezed after removing a subset of Minkowski dimension two.
Notes
Doctorat, Université de Neuchâtel, Institut de mathématiques
Identifiants
Type de publication
doctoral thesis
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