RUI: Volumes in tropical geometry

Throughout the last several decades, tropical geometry has emerged as an influential bridge between the disparate subjects of algebraic and discrete geometry. In essence, tropical geometry replaces geometric spaces modeled by nonlinear equations (a parabola or a sphere, for example) with geometric spaces modeled by linear equations (a line or a plane, for example). Crucially, this bridge runs in both directions, allowing one to study the rich structure of nonlinear spaces using linear and combinatorial techniques while also allowing one to import the deep geometric framework of algebraic geometry into the study of combinatorics. This project will build a new lane in this bridge that is centered around the classical concept of volume, with applications in both combinatorics and algebraic geometry. In addition to the intellectual and mathematical outcomes of this project, the principal investigator will use the line of research problems in this project as an avenue to train and support a diverse community of student researchers at his home institution of San Francisco State University, preparing them to succeed in PhD programs and research careers in the sciences. One of the most important ways in which volumes arise in algebraic geometry is through the study of divisors on algebraic varieties, which are fundamental objects for studying the defining equations of a variety. Given a divisor on a projective variety, there are at least two volume-theoretic interpretations for the degree of its top power: it is the volume of the associated compact Riemannian manifold, and it is the volume of the Newton-Okounkov body associated to the divisor. This project will develop parallels of these notions in tropical geometry by introducing volume-theoretic tools for studying divisors and intersection numbers on tropical varieties. Applications of the volume-theoretic tools introduced in this project include a new geometric understanding of recent influential results concerning log-concavity of characteristic polynomials of matroids, allowing one to generalize these log-concavity results to intersection numbers on a much larger class of tropical varieties than was accessible by previous approaches, as well as the development of new tropical methods for studying cones of divisors on tropical compactifications of algebraic varieties. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.