Fundamental Theorems of Calculus and Zinbiel Algebras

Abstract: Many concepts from calculus can be formalized in algebra. Derivation formalize the differential operator and are linear maps which satisfy the Leibniz rule. Integrations instead formalize the integral operator and are linear maps which satisfy the Rota-Baxter rule, which is an integral only integration by parts. On the other hand, Zinbiel algebras are a special kind of commutative non-unital associative algebras that capture  the notion of riffle shuffle permutations. In this talk, I will explain how giving a Zinbiel algebra is equivalent to giving a derivation and an integration which together satisfy the two fundamental theorems of calculus. This talk is based on my paper https://arxiv.org/abs/2401.08223 and is a fun talk that should be accessible to all, even undergraduate students.