Collaborative Research: Toric Geometry, Tropical Geometry, and Combinatorial Buildings

This project involves research at the intersection of algebraic geometry and combinatorics. Algebraic geometry is the study of solution sets of polynomial equations called algebraic varieties. It has applications in many fields as diverse as high energy physics, coding, cryptography, and mathematical biology.

Understanding how the shape of the solution set changes as the coefficients are varied is one of the oldest and central questions in the field. Such continuous deformations, which appear in all branches of algebraic geometry and its applications, are called algebraic families. An important example is the geometric Langlands program, which is concerned with understanding principal bundles on curves, a very special class of families.

Bundles are also main players in gauge theory in high energy physics. Algebraic families are the central focus of this project. The research aims to introduce new methods to classify and compute with algebraic families. The project will provide research training opportunities for graduate students.

Toric varieties are a large class of varieties whose geometry is intimately connected with combinatorics of convex lattice polytopes. They play a central role in contemporary algebraic geometry. Tropical geometry, a relatively recent area of research, concerns study of piecewise linear geometry and has roots in convex optimization.

Tropical geometry translates numerous questions in algebra and geometry into combinatorial and convex geometric questions that are often more tractable. The theory of buildings is an area of combinatorial geometry that has deep connections with topology and differential geometry. It aims to unravel hidden combinatorial geometric structures in matrix groups and related spaces.

The topics of this research revolve around the common theme of studying families of algebraic varieties over a toric variety, or a toric family for short. The main insight is that the combinatorics needed to understand toric families comes from both tropical geometry and the theory of buildings. The approach followed in this project will lead to the development of new techniques in algebraic geometry and related fields.

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.