The question of determining the ways in which different-dimensional spheres can wrap around each other, formally known as the homotopy groups of spheres, is the most difficult and fundamental question in homotopy theory. In the last 20-30 years, relationships between these groups and modular forms have become apparent, giving a tight connection to number theory.
A second-cousin of the homotopy groups of spheres is an object called the spectrum Q(l), which is built out of close relatives of modular forms. The goal of this IGL project is to explicitly compute the homotopy groups of Q(l).
While the theory behind this project includes extensive amount of homotopy theory, the practice of it is algebraic in nature. Students will have the opportunity to choose the mix of theory and computation that suits their interests.
Somme abstract algebra (eg. Math 417) or at least linear algebra (eg. 416) are necessary for success. Optional: Math 418, 453
We will be using Magma and/or Sage software, but no prior experience with them is necessary.