### Topics in Toric and Tropical Geometry: Positivity and Completion

- Series
- Dissertation Defense
- Time
- Monday, April 1, 2024 - 11:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 114
- Speaker
- May Cai – Georgia Institute of Technology – mcai@gatech.edu

This defense will also be on zoom at: https://gatech.zoom.us/j/99428720697

In this defense we describe three topics in tropical and toric positivity and completion. In the first part, we describe the finite completability of a partial point to a log-linear statistical model: a toric variety restricted to the probability simplex. We show when a generic point in some projection of a log-linear model has finite preimage, and the exact number of preimages in such a case. In the second part, we describe the tropical variety of symmetric tropical rank 2 matrices. We give a description of the tropical variety as a coarsening of the simplicial complex of a type of bicolored trees, and show that the tropical variety is shellable. Finally, we discuss two tropical notions of positivity, and give results on the positive part of certain tropical determinantal varieties.

Committee:

Josephine Yu, Georgia Institute of Technology (Advisor)

Matt Baker, Georgia Institute of Technology

Greg Blekherman, Georgia Institute of Technology,

Kaie Kubjas, Aalto University

Anton Leykin, Georgia Institute of Technology

Thesis draft:

Link