For almost four decades, PROMYS has been synonymous with deep exploratory mathematical learning for talented secondary school students and their teachers. In this presentation we will discuss the history of PROMYS and its underlying principles as well as strategies for developing mathematical habits of mind that encourage creativity and innovation. We will also share ideas for new outreach efforts, currently in development, designed to serve local students from underserved populations in the Boston area.

This will be an expository lecture intended for a general mathematical audience to illustrate, through examples, the theme of p-adic variation in the classical theory of modular forms. Classically, modular forms are complex analytic objects, but because their fourier coefficients are typically integral, it is possible to also do elementary arithmetic with them. Early examples arose already in the work of Ramanujan. Today one knows that modular forms encode deep arithmetic information about elliptic curves and Galois representations. Our main goal will be to illustrate these ideas through simple concrete examples.