Potential 4th year projects 2021

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