CeCANT - Dr Felipe Rincón

Tropical ideals

Supervisor: Dr Felipe Rincón

Project description:

Tropical algebraic geometry is a young but rapidly developing field that translates intricate algebraic and geometric problems into combinatorial ones. In this setting, algebraic varieties (that is, geometric objects defined by polynomial equations) are transformed into simpler, discrete geometric structures called tropical varieties. These tropical counterparts retain essential asymptotic information about the original varieties and offer powerful combinatorial tools for understanding their properties.

In tropical geometry, arithmetic takes place in the “tropical semifield”, where addition is replaced by taking the maximum of two numbers, and multiplication becomes their usual sum. This simple change gives rise to a remarkably rich mathematical world where algebraic geometry meets combinatorics, polyhedral geometry, and matroid theory.

Over the past two decades, tropical geometry has led to profound results across mathematics. Highlights include Mikhalkin’s computation of Gromov-Witten invariants of the complex projective plane, tropical proofs of the Brill-Noether theorem, new approaches to mirror symmetry, and the Adiprasito-Huh-Katz proof of the Heron-Rota-Welsh conjecture for matroids; a milestone that contributed to June Huh’s 2022 Fields Medal.

Most existing work in tropical geometry has focussed on the geometric properties of tropical varieties, while the deeper algebraic framework underlying them has remained less developed. The emerging theory of tropical schemes seeks to fill this gap, much as Grothendieck’s theory of schemes underpins modern algebraic geometry. In this direction, Maclagan and Rincón introduced tropical ideals: combinatorial structures built from matroids that capture the behaviour of ideals in a polynomial ring. This construction provides a broad connection between tropical geometry, commutative algebra, and matroid theory.

Several research directions can be pursued within this project, depending on the student’s background and interests. A central problem is the realisability question: which polyhedral complexes arise as tropicalisations of algebraic varieties? For tropical ideals, the analogous problem asks which polyhedral complexes can occur as their varieties. Since every tropicalisation of a classical ideal yields a tropical ideal, the class of varieties of tropical ideals already includes all tropicalisations of algebraic varieties. However, some tropical ideals may define non-realisable tropical varieties, that is, polyhedral complexes that do not correspond to any algebraic variety. This project aims to investigate this fundamental open question in the field.

Clarifying this boundary will shed light on how rigid or flexible tropical ideals are. If all their varieties are realisable, tropical ideals form a highly constrained algebraic class; if not, their flexibility could provide new algebraic tools for studying more general polyhedral complexes. This mirrors recent developments in matroid theory, where algebraically inspired techniques have driven major advances on long-standing problems.

A very concrete case arises from the work of Rincón and Draisma, who showed that no tropical ideal can have a variety equal to the Bergman fan of the Vámos matroid plus the rank-2 uniform matroid on 3 elements. However, it remains unknown whether the same holds for the Bergman fan of the Vámos matroid itself. Exploring possible tropical scheme structures on this fan represents a natural and promising approach that will form the core of this PhD research.

References:

• Diane Maclagan and Felipe Rincón, Tropical ideals, Compositio Mathematica, 154(3), 640-670, 2018.
• Jan Draisma and Felipe Rincón, Tropical ideals do not realise all Bergman fans, Research in the Mathematical Sciences, 8(3), 44, 2021.
• Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry (Vol. 161), American Mathematical Society, 2015.

Funding Notes: