Tropical Geometry

Jeffrey Giansiracusa - email: jeffrey.giansiracusa@durham.ac.uk

Description

Tropical geometry is a relatively new area of mathematics that touches algebraic geometry (the geometry of shapes defined by polynomial equations), combinatorics and polyhedral geometry, optimisation, computer science, and bioinformatics. It came to prominence when Mikhalkin used it to give a method for computing a certain sequence of numbers called the Gromov-Witten invariants of the project plane. It also provided a new proof of a conjecture from Ed Witten in quantum gravity.

One of the entry points to tropical geometry is the idea of replacing the arithmetic operations of multiplication and addition on the real line with the operations of max and +. This gives a semiring, which is just like a ring, except that there is no subtraction; amazingly, there is a lot of interesting algebra that can be done without subtraction. Another point of entry is the combinatorial theory of matroids (abstracting notions from linear algebra line linear dependence).

Tropical geometry is an excellent way to start to learn about the ideas of algebraic geometry, which is one of the pillars of pure mathematics. One of the things tropical geometry is good at is computing how various shapes intersect, leading to the famous theorem of Bézout for counting the number of intersection points of projective curves and hypersurfaces. It also offers scope to explore computational methods and combinatorial ideas. This project offers a number of possible directions to suit the interests of individual students.

Prerequisites and corequisites

This project will build heavily on concepts from Algebra II (MATH2581)

Taking Geometry III (MATH3201) might be beneficial, but it is not a requirement.

Resources

An excellent introductory survey paper: Ralph Morrison, Tropical Geometry. https://arxiv.org/abs/1908.07012

Some good lecture notes by Diane Maclagan: https://homepages.warwick.ac.uk/staff/D.Maclagan/papers/ManchesterAllLectures.pdf