What is... an elliptic curve?

In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond. They were prominently featured in Wiles' proof of Fermat's last theorem, but there are many other problems that lead to a question about elliptic curves. We will discuss some of the open problems related to elliptic curves, including conjectures that are quite controversial!
This talk was part of "What is a seminar?"
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