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Kuranishi atlases and genus zero GromovWitten invariants
http://academiccommons.columbia.edu/catalog/ac:197454
Castellano, Robert
http://dx.doi.org/10.7916/D89W0FF0
Mon, 11 Apr 2016 18:24:16 +0000
Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of Jholomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero GromovWitten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that GromovWitten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero GromovWitten invariants, we show that they satisfy the GromovWitten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.
Mathematics
rtc2119
Mathematics, Mathematics (Barnard College)
Dissertations

A geometric construction of a Calabi quasimorphism on projective space
http://academiccommons.columbia.edu/catalog/ac:188472
Carneiro, Andre R.
http://dx.doi.org/10.7916/D8N29W9T
Fri, 17 May 2013 00:00:00 +0000
We use the rotation numbers defined by ThÃ©ret in [T] to construct a quasimorphism on the universal cover of the Hamiltonian group of CP^n. We also show that this quasimorphism agrees with the Calabi invariant for isotopies that are supported in displaceable subsets of CP^n.
Mathematics
arc2142
Mathematics (Barnard College), Mathematics
Dissertations

Lattice Subdivisions and Tropical Oriented Matroids, Featuring Products of Simplices
http://academiccommons.columbia.edu/catalog/ac:131450
Piechnik, Lindsay C.
http://hdl.handle.net/10022/AC:P:10240
Fri, 29 Apr 2011 00:00:00 +0000
Subdivisions of products of simplices, and their applications, appear across mathematics. In this thesis, they are the tie between two branches of my research: polytopal lattice subdivisions and tropical oriented matroid theory. The first chapter describes desirable combinatorial properties of subdivisions of lattice polytopes, and how they can be used to address algebraic questions. Chapter two discusses tropical hyperplane arrangements and the tropical oriented matroid theory they inspire, paying particular attention to the previously uninvestigated distinction between the generic and nongeneric cases. The focus of chapter three is products of simplices, and their connections and applications to ideas covered in the first two chapters.
Mathematics
lp2149
Mathematics (Barnard College), Mathematics
Dissertations