Toric and semitoric packing capacities

Faculty Member

Joseph Palmer
A 4 sided irregular polygon is shown with dotted lines breaking it into 4 pieces, leaving a large area in the center shaded in gray, and three areas along the edges shaded in purple.

In physics and mathematics, it is typical to run into four dimensional objects, but of course it can be difficult to study the geometry of objects with dimensions higher than three! So how can we do it? In some highly symmetric cases, high dimensional objects can be represented by something less complicated.

In this project, we will be studying certain two dimensional polygons which actually represent four dimensional objects. This will allow us to use linear algebra and simple analysis to understand the properties of so-called symplectic toric and symplectic semitoric 4-manifolds. We will be performing some computations by hand, and also working on computer programs (mostly using python) to compute invariants of the polygons, which in turn gives information about the associated 4-manifolds.

Team Meetings

Weekly

Project Difficulty

Intermediate

Undergrad Prerequisites

Some linear algebra would help a lot. Abstract algebra would be nice, but is not necessary. Some students will probably use python (or C++), but knowledge of these languages is not strictly necessary.