Geometry of spanning lattice polytopes

Project description

Polytopes are one of the most fundamental shapes in geometry including polygons and the Platonic solids (tetrahedron, cube, octahedron, etc.). They can be characterised as shapes in Euclidean space enclosed by planes and accept notions such as vertices and edges. Lattice polytopes are polytopes whose vertices have integral coordinates. They provide a fascinating bridge between geometry and number theory that has been studied since the ancient Greeks. They are central in modern mathematics and their study has been amazingly exciting leading to key discoveries and recent breakthroughs in areas such as algebraic geometry, optimisation theory, representation theory, and statistics. New aspects are constantly being explored, such as the ‘spanning property’ of lattice polytopes. Initial work with my collaborators shows that the implications of this property are extremely powerful.

In this project, we will study fundamental geometric and combinatorial features of spanning lattice polytopes of ‘bounded combinatorial complexity’. Due to its links to algebraic geometry and combinatorics, this topic can either lead to a hands-on thesis crunching millions of examples with a computer, or a theoretical piece of work, exploring the underlying theory, depending on the student’s preference.