Potential 4th year projects 2018

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.

Commutative algebra.

It was recently shown that there is a bound B(r,d) on the degrees of a Grobner basis of an ideal generated by r polynomials of degree at most d (not depending on of the number of variables), but the value of this bound is not know in most cases. This project would be to understand some of the proofs of this fact, and investigate some special cases of B(r,d).

Combinatorics

Matroids are a combinatorial abstraction of linear algebra, and also of parts of graph theory. I have two potential 4th year projects in this area. One is to understand the theory of valuated matroids, which is a further generalization. The other is to understand realization spaces of matroids. Both could have connections to algebraic geometry, depending on the taste of the student. These project would suit someone who liked the combinatorics or combinatorial optimisation modules, but these are not prerequisites.