{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/117315"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/117315","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Mayer-Vietoris property for relative symplectic cohomology","abstract":"In this thesis, I construct and investigate the properties of a Floer theoretic invariant called relative symplectic cohomology. The construction is based on Hamiltonian Floer theory. It assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. I show the existence of restriction maps, and prove that they satisfy the Hamiltonian isotropy invariance property, discuss a Kunneth formula, and do some example computations. Relative symplectic cohomology is then used to establish a general criterion for displaceability of subsets. Finally, moving on to the main contribution of my thesis, I identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 co isotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.","abstract_html":"In this thesis, I construct and investigate the properties of a Floer theoretic invariant called relative symplectic cohomology. The construction is based on Hamiltonian Floer theory. It assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. I show the existence of restriction maps, and prove that they satisfy the Hamiltonian isotropy invariance property, discuss a Kunneth formula, and do some example computations. Relative symplectic cohomology is then used to establish a general criterion for displaceability of subsets. Finally, moving on to the main contribution of my thesis, I identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 co isotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.","abstract_has_math":false,"creators":["Varolgunes, Umut"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/117315"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 115-118)."]},{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, I construct and investigate the properties of a Floer theoretic invariant called relative symplectic cohomology. The construction is based on Hamiltonian Floer theory. It assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. I show the existence of restriction maps, and prove that they satisfy the Hamiltonian isotropy invariance property, discuss a Kunneth formula, and do some example computations. Relative symplectic cohomology is then used to establish a general criterion for displaceability of subsets. Finally, moving on to the main contribution of my thesis, I identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 co isotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Mayer-Vietoris property for relative symplectic cohomology"]}]}],"canonical_facts":{"dc:contributor.advisor":["Paul Seidel."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:creator":["Varolgunes, Umut"],"dc:date.accessioned":["2018-08-08T19:48:57Z"],"dc:date.available":["2018-08-08T19:48:57Z"],"dc:date.issued":["2018"],"dc:description":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 115-118)."],"dc:description.abstract":["In this thesis, I construct and investigate the properties of a Floer theoretic invariant called relative symplectic cohomology. The construction is based on Hamiltonian Floer theory. It assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. I show the existence of restriction maps, and prove that they satisfy the Hamiltonian isotropy invariance property, discuss a Kunneth formula, and do some example computations. Relative symplectic cohomology is then used to establish a general criterion for displaceability of subsets. Finally, moving on to the main contribution of my thesis, I identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 co isotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors."],"dc:description.degree":["Ph. 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