Pure Maths

This will be a talk aimed at a wide audience. In the first part, I will recall the notion of an elliptic curve over the complex numbers and I will review the beautiful theory that allows us to classify them. Following this, I will recall what it means for an elliptic curve to have complex multiplication (CM) and I will discuss major advances over the last 30-40 years that describe how tuples of CM elliptic curves are distributed. Finally, I will venture into the realm of "unlikely intersections", which aims to study the distribution of other forms of "additional structure" on elliptic curves and their generalizations.

In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions, including the notion of irretractable solutions, and explore how these solutions correspond with matched pairs of groups. Finally, I will show how each irretractable solution lifts to provide the full classification of all bijective solutions.