LEAPS-MPS: Commutative Algebra of Semirings Through the Lens of Algebraic Geometry and Combinatorics

A semiring is an algebraic structure that satisfies the same axioms as a ring, except that subtraction is not required. Although this idea may have seemed unusual when first introduced by Vandiver in 1934, many familiar examples exist, such as the natural numbers and the nonnegative real numbers. Semirings have since become central in areas of theoretical computer science, including automata theory and optimization, and linear algebra over semirings has been widely applied in various scientific contexts. In recent years, semirings have attracted growing interest in pure mathematics due to their deep connections to tropical geometry and geometry over the field with one element. This project will develop new concepts and tools in the theory of semirings and apply them to problems in algebraic geometry and combinatorics. As part of the broader impacts, the PI will organize an undergraduate research symposium twice during the grant period and lead a research seminar at SUNY New Paltz. The PI also plans to reinstate the department’s mathematics seminar and mentor undergraduate research projects, providing students with meaningful opportunities to engage in mathematics and present their work at conferences.

This project investigates the commutative algebra of semirings and its applications to algebraic geometry and combinatorics. Semirings lie at the intersection of several foundational approaches to tropical geometry, while simultaneously providing a framework for introducing positivity into algebraic geometry by considering structures like the natural numbers or the nonnegative real numbers. The research consists of two interconnected programs. The first aims to build the foundation for algebraic geometry over semirings by studying basic geometric objects such as (equivariant) vector bundles and moduli spaces like Picard schemes or stacks, with particular attention to tropical schemes. The second develops a representation-theoretic framework in which vector spaces are replaced by valuated matroids—combinatorial structures that serve as tropical analogues of linear spaces. This framework, known as tropical representation theory, is closely connected to the study of group actions on matroids. The project will use algebraic constructions, combinatorial models, and computational methods to develop new tools and insights that may inform and extend classical representation theory. These efforts aim to deepen the mathematical foundations of semirings and tropical geometry while fostering a research-active environment for undergraduate students.

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.

Award Number: 2532394
Principal Investigator: Jaiung Jun
Funds Obligated: $250,000
State: NY