Project Overview
Flow polytopes: a marriage between graph theory and polyhedral geometry
Department(s)
Mathematics
Abstract
What is a polytope? If you've ever played a game with dice, then you've encountered polytopes: cubes and pyramids are two of the most well-known examples, but there's also the dodecahedron (where twelve regular pentagons fit together to make a 3D shape), the octahedron (made up of eight regular triangles), and the icosahedron (made up of 20 regular triangles). These five make up the Platonic solids and have been studied since antiquity.
Polytopes appear in all sorts of applications, including linear and integer programming, statistics, physics, and even the arts. In this project, we will be thinking about polytopes that arise from other mathematical structures: graphs. A graph is a set of points, or nodes, with edges between some pairs of those points. Again, graphs are incredibly useful for modeling, but we will be focusing on the purely mathematical nature of them.
Graphs and polytopes can interact in numerous fascinating ways, but this project will focus on one: something called the flow polytope of a graph. Although the Platonic solids live in 3D space, most polytopes -- flow polytopes included -- can only be constructed in four, five, six, or higher dimensions. This presents an obvious problem: how can we hope to understand a polytope if we can't visualize it? It turns out that properties of the corresponding graph can tell us a great deal about the polytope even though we can't see it! Since graphs can always be drawn on a page, we now have a way of analyzing complicated high-dimensional objects using easy-to-draw pictures.
Despite the correspondence, there are some very basic questions about flow polytopes that remain unanswered. Perhaps the most basic question is: What is its volume? There are nice and simple formulas for cubes and pyramids, but the formluas get much more challenging to write down in higher dimensions. In this project, we will investigate this problem and others from multiple directions. There will be frequent use of programming to make and test conjectures. With a bit of luck, this project will become an article we submit for publication.
Student Qualifications
MATH 214 and (MATH 250 or MATH 310 or MATH 360) with grades of B or higher. Programming experience desirable but not required.
Project Length
8 weeks
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