220524 Jack Davies

Modular forms are classical objects in number theory, and Hecke operators on these modular forms are a useful organisational tool. In this talk, I will define modular forms over the complex numbers and the classical Hecke operators which act upon them. Then I will discuss a generalisation of this theory from the complex numbers to a theory for general rings, and eventually for derived rings. Finally, I would like to show some little games we can play with these kinds of derived Hecke operators, as a sample of their utility. Only basic complex analysis will be assumed for the first half, and in the second half we will use some language from modern algebraic geometry.