Pseudoholomorphic quilts with figure eight singularity
Name
941789819-MIT.pdf
Description
Full printable version
Size
11.89 MB
Format
Adobe PDF
Checksum (MD5)
c31393aeba00f197df5c2baab4ac919d
Author(s)
Bottman, Nathaniel Sandsmark
Advisor(s)
Katrin Wehrheim.
Date Issued
2015
Publisher
Massachusetts Institute of Technology
Abstract
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.
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 107-109).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
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.
Persistent DSpace Link
