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.
In this project, we'll explore different kinds of modular forms and connections between them coming from various operators, as well as implications for homotopy theory. While the inspiration behind this project includes extensive amount of homotopy theory, the practice of it is algebraic and number theoretic in nature.
Students will have the opportunity to choose a mix of theory and computation that suits their interests.
Some abstract algebra (eg. Math 417) or at least linear algebra (eg. 416) are necessary for success. Optional: Math 418, 453.
Interested students may use Magma and/or Sage software, but no prior experience with them is necessary.