Year 4 project – Tropical curves

This project is about the emerging field of tropical geometry, a piecewise-linear shadow of classical algebraic geometry whose surprisingly rich structure reveals deep connections with combinatorics, algebraic geometry, and number theory.

Background and motivation

The fundamental idea of tropical geometry is to replace the classical operations of addition and multiplication with the tropical operations \(a \oplus b = \min(a, b)\) and \(a \odot b = a + b\). Under this substitution, polynomial equations become piecewise-linear functions, and classical algebraic curves — defined by polynomial equations — become tropical curves: combinatorial objects made up of line segments and rays meeting at vertices, living in \(\mathbb{R}^n\). Despite their simpler structure, tropical curves retain a surprising amount of geometric information about their classical counterparts.

One way to make this precise is through the notion of amoebas: the image of a classical algebraic curve under the coordinatewise logarithm map, which converges to a tropical curve in a well-defined limit. This limiting process, known as Maslov dequantization, is one of the conceptual cornerstones of the subject.

A central theme of the project is understanding the correspondence between tropical and classical curves. Key tools here include the Newton polygon of a polynomial, which encodes the combinatorial structure of a tropical curve, and Viro’s patchworking theorem, which shows how tropical and classical curves can be related in a controlled and explicit way. These tools lead to striking applications in enumerative geometry, solving questions such as: how many algebraic curves of a given degree and genus pass through a prescribed set of points? Such counts can often be computed purely combinatorially via tropical methods.

Reading list

A great friendly introduction to tropical curves is in the paper [BS] by Brugallé and Shaw. To learn the theory more in details, we will use [B] as a first reference for the foundations of the subject, and [BIMS] for a longer one on advanced topics. To situate the theory in the broader context of tropical geometry, we will use the standard tropical reference book [MS], by Maclagan and Sturmfels. Potential advanced topics for dissertation include:

• Studying the combinatorial structure of tropical plane curves and their dual Newton subdivisions.
• Investigating tropical intersection theory and the tropical Bézout theorem.
• Exploring Viro’s patchworking theorem and its applications to the topology of real algebraic curves.
• Applying floor diagrams to classical and tropical curve counting problems in enumerative geometry.

Mode of operation & evidence of learning

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites and Co-requisites

It is essential that students have taken Algebra II, Topology II and Geometry III. Helpful modules are Galois Theory III and Codes & Cryptography III, but they are not essential for the success of the project.

References

• [B] E. Brugallé - “Some aspects of tropical geometry” Link
• [BS] E. Brugallé and K. Shaw - “A Bit of Tropical Geometry” Link
• [BIMS] E. Brugallé, I. Itenberg, G. Mikhalkin, and K. Shaw - “Brief introduction to tropical geometry” Link
• [MS] D. Maclagan and B. Sturmfels - “Introduction to tropical geometry” Link