# 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.