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Orienting Moduli Spaces of Flow Trees for Symplectic Field TheoryPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, 2016. , 37 p.
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 92
##### Keyword [en]

Contact manifolds, Legendrian submanifolds, Lagrangian immersions, Legendrian contact homology, Morse flow trees, Determinant line bundles, Orientation of moduli spaces, Exact Lagrangian cobordisms
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-269551ISBN: 978-91-506-2523-3OAI: oai:DiVA.org:uu-269551DiVA: diva2:885069
##### Public defence

2016-02-19, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2016-01-28 Created: 2015-12-17 Last updated: 2016-01-28
##### List of papers

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology.

In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm.

In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically. This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees of the Lagrangian cobordisms.

In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.

1. Orientations of Morse flow trees in Legendrian contact homology$(function(){PrimeFaces.cw("OverlayPanel","overlay883365",{id:"formSmash:j_idt432:0:j_idt436",widgetVar:"overlay883365",target:"formSmash:j_idt432:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. A note on orientations of exact Lagrangian cobordisms with cylindrical Legendrian ends$(function(){PrimeFaces.cw("OverlayPanel","overlay883366",{id:"formSmash:j_idt432:1:j_idt436",widgetVar:"overlay883366",target:"formSmash:j_idt432:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Area-preserving isotopies of self-transverse immersions of S (1) in a"e(2)$(function(){PrimeFaces.cw("OverlayPanel","overlay615396",{id:"formSmash:j_idt432:2:j_idt436",widgetVar:"overlay615396",target:"formSmash:j_idt432:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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