**Title: **Elliptic Curves, Isogenies, and Volcanoes

**Abstract:** Elliptic curves have long been of interest in mathematics. Over the last twenty-five years, they have been used in the proof of Fermat’s last theorem, factoring algorithms, and cryptography. One reason for their ubiquity is that the points of an elliptic curve can be made into an abelian group, using simple geometry. In this talk, I will discuss isogenies of elliptic curves, and some of their applications in number theory and cryptography. Isogenies are maps between elliptic curves, which preserve the group structure. In particular, I will give some explicit new formulas to evaluate isogenies, which are the fastest known. I will also talk about “isogeny volcanoes” and some tricks to compute them more efficiently.