In this talk, we share a small connection between information theory, algebra, and topology—namely a correspondence between Shannon entropy and functions on topological simplices satisfying the Leibniz rule. We begin by reviewing an important algebraic property of Shannon entropy known as the chain rule and show that it is essentially the Leibniz rule when viewed from the perspective of operads. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in the 1950s and a recent variation given by Tom Leinster. This is "part 2" of a talk given at this seminar in 2018, where we shared initial results of this work.

### Deformation Theory Seminar

Monday, June 27, 2022 - 2:00pm

#### Tai-Danae Bradley

SandboxAQ and The Master's University