Toric and Tropical Geometry: Positivity and Completion
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Cai, May
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Abstract
This thesis is divided into the introductory chapter and then three main chapters. The the introductory first chapter is devoted to providing some preliminary background on polytopes, tropical geometry, and tropical linear algebra.
The second chapter consists of a result on the probability completion of log-linear models. In particular, given partial entry-wise information about a point in some toric variety intersected with the set of probability vectors, we describe the number of completions of that partial point into an actual point in the semi-algebraic set, as well as necessary conditions to be a valid partial point. A preprint of the content of this chapter can be found online at https://arxiv.org/abs/2312.15154, and was joint work with Cecilie Olesen Recke and Thomas Yahl.
The third chapter is concerned with the tropical variety of symmetric tropical rank at most 2. We discuss a refinement of the fan of the variety that gives a tree characterization of the variety, as in Markwig and Yu, and from this we deduce the shellability for the tropical variety as well as a condition to verify independence in the algebraic matroid of this variety. This chapter was joint work with Kisun Lee and Josephine Yu, and a preprint of the content can be found online at https://arxiv.org/abs/2404.08121.
The fourth chapter focuses on applying notions of tropical positivity developed by Brandenburg, Loho, and Sinn to tropical symmetric determinantal varieties. It describes the ``positive'' and ``really positive'' parts of the tropical varieties of rank 2 symmetric matrices, using the tree characterization established in the third chapter, as well as of the symmetric tropical determinantal hypersurface. We also re-prove certain results about the really positive part of the tropical varieties of rank 2 usual tropical matrix and of the usual tropical determinantal hypersurface. This chapter was joint work with Abeer al Ahmadieh and Josephine Yu.
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2024-04-28
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Dissertation