Holly Miller, Temple University
This talk will focus on a few notions of differentiation for modular forms. The space of modular forms of positive weight k and level a congruence subgroup is not closed under differentiation with respect to its complex variable. The derivative of a modular form is, however, "close" to being a modular form, and this allows the deficiency to be corrected in several ways. Our first avenue for correcting this will be through the Serre derivative, and our exploration of this will allow us to conclude that no modular form satisfies a non-linear second-order differential equation, while every (even quasi-)modular form satisfies a non-linear third order differential equation. We will use this to show that $E_2$, $E_4$, and $E_6$ are algebraically independent. Our second route will be through the Rankin-Cohen bracket, which we will use to sketch the proof that every modular form satisfies a linear ODE. As an application of this, we will also sketch a proof that $\zeta(3)$ is irrational.