Jessica Purcell: Research Pages

RESEARCH PAGES:

  • Introduction

  • Papers

     

     

     

     

     

     

     

     

     

     

     

    • Jessica Purcell's Mathematical Interests

    I am interested in three-dimensional manifolds, hyperbolic geometry, and knot theory, and especially in problems in which all three meet.

    What is a three-dimensional manifold?

    Let's start by describing a two-dimensional manifold. A two-dimensional manifold is an object that looks flat - like a two-dimensional plane - no matter where you're standing on it, but usually isn't flat if you can stand back and look at it. For example, the surface of the earth is a two-dimensional manifold. No matter where you're standing on the earth, it looks (basically) flat. In the middle ages, a lot of smart people thought the earth was flat. However, most of us now know that the earth is round - a large sphere.

    A sphere is an example of a two-dimensional manifold. When you stand on it, it looks flat, or two-dimensional, no matter where you're standing, but all those flat-looking parts usually glue up into something that doesn't fit in two dimensions. For example, you need three dimensions to see the whole sphere.

    Now you can probably guess what a three-dimensional manifold looks like. No matter where you're standing on it, it looks like it's three-dimensional, like space. However, almost all of these manifolds don't really fit into three-dimensional space.

    This makes studying three-dimensional manifolds difficult. We'd like to be able to classify all of them.