A Lie-theoretic trichotomy in Diophantine geometry and arithmetic dynamicsMPIM

by Robin Zhang (MIT)

Wednesday Nov 12, 2025, 12:00 PM → 1:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Extra talk

How can the finite/infinite dichotomy of the Killing–Cartan classification of simple Lie groups & algebras appear in arithmetic geometry? I will explain how this Lie-theoretic dichotomy is realized in the finiteness or infinitude of positive integer solutions to certain Diophantine equations and explore some of its implications for classical questions studied by Gauss, Mordell, Coxeter, Conway, and Schinzel in combinatorics and number theory. I will then switch gears to the arithmetic dynamics of cluster Donaldson–Thomas transformations, which refines the Diophantine realization of the finite/infinite dichotomy into a finite/affine/indefinite trichotomy that matches the Kac–Moody classification of infinite-dimensional Lie algebras. No background in arithmetic geometry or cluster algebras will be assumed.