Packing capacities of semitoric polygons

Faculty Member

Joseph Palmer

Project Image

How can we study something four-dimensional? It turns out that in some special cases, four dimensional objects can be represented by two-dimensional pictures. This way, we can study the two-dimensional picture to learn about the four dimensional object.

In this project, we study certain invariants of four dimensional symplectic manifolds called equivariant packing capacities. These invariants are computed by filling the manifold with four-dimensional balls in a special way, but actually this whole question can be translated into a two dimensional problem about filling certain polygons with triangles. Packing capacities for symplectic toric manifolds can be computed by studying certain polygons called Delzant polygons, but a more interesting (and complicated) case arises when studying packing capacities for semitoric manifolds. In this case, the single Delzant polygon is replaced by an infinite family of polygons known collectively as the semitoric polygon of the system.

In this project we will learn about semitoric polygons, we will learn the rules for packing these polygons with triangles, and we will try to compute the packing capacities of some specific examples.

 

Team Meetings

Approximately once per week

Project Difficulty

Intermediate

Undergrad Prerequisites

Linear algebra, and a good understanding of abstract algebra (groups) would be useful.