Tropical Positivity and Symmetric Low Rank Matrices

In 2003, Speyer and Williams introduced two definitions for positive tropicalizations of algebraic varieties over complex Puiseux series and proved their equivalence for Grassmannians. The two corresponding notions of positive tropical generators were studied by Brandenburg, Loho, and Sinn in 2022. In this talk, I will provide a brief introduction to tropical geometry and then present a combinatorial description of the tropicalization of the space of symmetric rank two matrices, along with their real and positive parts. We will compare the two notions of tropical positivity for rank 2 or corank 1 symmetric matrices. The results are based on joint works with Abeer Al Ahmadieh, May Cai, and Kisun Lee.