# Potential 4th year projects 2021

This page is just to give some ideas of potential suitable projects. Variants on this topic may be possible. I will accept at most two students per year over all projects.

## Algebraic geometry

• Tropical geometry. Tropical Geometry is an emerging area at the intersection of algebraic geometry and polyhedral combinatorics with applications to other areas. At its most basic, it is geometry where addition and multiplication are replaced by minimum and addition respectively. This turns familiar geometric objects, such as circles, into piecewise linear objects, which can be studied using combinatorial methods. This project would be to investigate recovering invariants of plane curves using these ideas.
• Moduli spaces of genus zero curves. The moduli space of genus zero curves is the first element in an important family of varieties in algebraic geometry. The goal of this project is to understand recent advances in the birational geometry of this space.

## Commutative algebra.

• In characteristic zero it has long been known (longer than the definition of algebraically closure!) that the field of Puiseux series cup_{n \geq } K((t^{1/n})) is an algebraically closed field that contain the algebraic closure of K(t). However it contains elements that are not algebraic over K(t). In characteristic p, or if we replace K(t) by K(t_1,...,t_n), there are also algebraically closed fields containing K(t_1,..,t_n) known, but the details of the precise algebraic closure are not completely understood. The project would be to understand the current status, and work towards extensions.
• The Hilbert function of a homogeneous ideal I in a polynomial ring R records the dimension of each graded piece of R/I. Macaulay's theorem characterises which Hilbert functions are possible when the grading is the standard one (the degree of every variable is one). The project is to expand this to some more general grading. This is secretly an algebraic geometry topic, but the techniques needed are only commutative algebra.

## Combinatorics

Matroids are a combinatorial abstraction of linear algebra, and also of parts of graph theory. The project is to understand realization spaces of matroids. This could have connections to algebraic geometry, depending on the taste of the student. This project would suit someone who liked the combinatorics or combinatorial optimisation modules, but these are not prerequisites.