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Orienting Moduli Spaces of Flow Trees for Symplectic Field Theory
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

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: 978-91-506-2523-3 (print)OAI: oai:DiVA.org:uu-269551DiVA: diva2:885069
Public defence
2016-02-19, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2016-01-28 Created: 2015-12-17 Last updated: 2016-01-28
List of papers
1. Orientations of Morse flow trees in Legendrian contact homology
Open this publication in new window or tab >>Orientations of Morse flow trees in Legendrian contact homology
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-269548 (URN)
Available from: 2015-12-17 Created: 2015-12-17 Last updated: 2015-12-17
2. A note on orientations of exact Lagrangian cobordisms with cylindrical Legendrian ends
Open this publication in new window or tab >>A note on orientations of exact Lagrangian cobordisms with cylindrical Legendrian ends
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-269550 (URN)
Available from: 2015-12-17 Created: 2015-12-17 Last updated: 2015-12-17
3. Area-preserving isotopies of self-transverse immersions of S (1) in a"e(2)
Open this publication in new window or tab >>Area-preserving isotopies of self-transverse immersions of S (1) in a"e(2)
2013 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 51, no 1, 85-97 p.Article in journal (Refereed) Published
Abstract [en]

Let C and C' be two smooth self-transverse immersions of S (1) into a"e(2). Both C and C' subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C' induces a one-to-one correspondence between the disks of C and C'. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C' is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C'. In this paper we show that this is also a sufficient condition.

National Category
Natural Sciences
Identifiers
urn:nbn:se:uu:diva-197954 (URN)10.1007/s11512-012-0165-9 (DOI)000316012800005 ()
Available from: 2013-04-10 Created: 2013-04-08 Last updated: 2017-12-06Bibliographically approved

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Karlsson, Cecilia

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