Tropical theta functions and log Calabi-Yau surfaces

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dc.contributor.advisorKeel, Seán
dc.creatorMandel, Travis Glennen
dc.date.accessioned2014-07-01T17:57:08Zen
dc.date.issued2014-05en
dc.date.submittedMay 2014en
dc.date.updated2014-07-01T17:57:08Zen
dc.descriptiontexten
dc.description.abstractWe describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties.en
dc.description.departmentMathematicsen
dc.format.mimetypeapplication/pdfen
dc.identifier.urihttp://hdl.handle.net/2152/24935en
dc.language.isoenen
dc.subjectTheta functionen
dc.subjectLog Calabi-Yauen
dc.subjectSurfacesen
dc.subjectTropicalen
dc.subjectToricen
dc.subjectClusteren
dc.subjectMirror symmetryen
dc.subjectPolytopeen
dc.titleTropical theta functions and log Calabi-Yau surfacesen
dc.typeThesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorThe University of Texas at Austinen
thesis.degree.levelDoctoralen
thesis.degree.nameDoctor of Philosophyen

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