2022 MAA Distinguished Lecture Series featuring Ken Ono

"Arithmetic and Geometric Means form Jellyfish Swarms of Curves"

Classical work of Euler and Gauss on "Arithmetic and Geometric Means” (AGM) produces wonderful rapidly convergent sequences with common limit. Euler’s famous formula for pi is an extraordinary example. Here the speaker will introduce the AGM for finite fields, where finite directed graphs are spawned in lieu of infinite sequences. The collection of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e., tentacles connected to a bell head). These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves, and this interpretation gives a striking description of Gauss's class numbers as counts of spots on these jellyfish.