Pseudoholomorphic quilts with figure eight singularity
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
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| Other Authors: | |
| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2016
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| Online Access: | http://hdl.handle.net/1721.1/101823 |
| _version_ | 1826215159302979584 |
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| author | Bottman, Nathaniel Sandsmark |
| author2 | Katrin Wehrheim. |
| author_facet | Katrin Wehrheim. Bottman, Nathaniel Sandsmark |
| author_sort | Bottman, Nathaniel Sandsmark |
| collection | MIT |
| description | Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. |
| first_indexed | 2024-09-23T16:17:31Z |
| format | Thesis |
| id | mit-1721.1/101823 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T16:17:31Z |
| publishDate | 2016 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1018232019-04-10T17:38:01Z Pseudoholomorphic quilts with figure eight singularity Bottman, Nathaniel Sandsmark Katrin Wehrheim. Massachusetts Institute of Technology. Department of Mathematics. Massachusetts Institute of Technology. Department of Mathematics. Mathematics. Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. Cataloged from PDF version of thesis. Includes bibliographical references (pages 107-109). In this thesis, I prove several results toward constructing a machine that turns Lagrangian correspondences into A[infinity],-functors between Fukaya categories. The core of this construction is pseudoholomorphic quilts with figure eight singularity. In the first part, I propose a blueprint for constructing an algebraic object that binds together the Fukaya categories of many different symplectic manifolds. I call this object the "symplectic A[infinity]-2-category Symp". The key to defining the structure maps of Symp is the figure eight bubble. In the second part, I establish a collection of strip-width-independent elliptic estimates. The key is function spaces which augment the Sobolev norm with another term, so that the norm of a product can be bounded by the product of the norms in a manner which is independent of the strip-width. Next, I prove a removable singularity theorem for the figure eight singularity. Using the Gromov compactness theorem mentioned in the following paragraph, I adapt an argument of Abbas-Hofer to uniformly bound the norm of the gradient of the maps in cylindrical coordinates centered at the singularity. I conclude by proving a "quilted" isoperimetric inequality. In the third part, which is joint with Katrin Wehrheim, I use my collection of estimates to prove a Gromov compactness theorem for quilts with a strip of (possibly non-constant) width shrinking to zero. This features local C[infinity]-convergence away from the points where energy concentrates. At such points, we produce a nonconstant quilted sphere. by Nathaniel Sandsmark Bottman. Ph. D. 2016-03-25T13:38:08Z 2016-03-25T13:38:08Z 2015 2015 Thesis http://hdl.handle.net/1721.1/101823 941789819 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 109 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Bottman, Nathaniel Sandsmark Pseudoholomorphic quilts with figure eight singularity |
| title | Pseudoholomorphic quilts with figure eight singularity |
| title_full | Pseudoholomorphic quilts with figure eight singularity |
| title_fullStr | Pseudoholomorphic quilts with figure eight singularity |
| title_full_unstemmed | Pseudoholomorphic quilts with figure eight singularity |
| title_short | Pseudoholomorphic quilts with figure eight singularity |
| title_sort | pseudoholomorphic quilts with figure eight singularity |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/101823 |
| work_keys_str_mv | AT bottmannathanielsandsmark pseudoholomorphicquiltswithfigureeightsingularity |