CANCELED: Möbius Inversion: From Number Theory to Topological Data Analysis

The classical Möbius inversion formula was introduced to number theory in 1832 by August Ferdinand Möbius. It relates two arithmetic functions (e.g., Euler’s Phi function) in terms of sums over divisors of a given integer. In 1962, Gian-Carlo Rota introduced a vast generalization of this idea for functions defined over partially ordered sets (posets). Rota applied his generalized Möbius inversion formula to numerous problems in combinatorics. Since then, Möbius inversion has found applications beyond combinatorics and number theory to fields such as physics, group theory, algebraic topology, and more recently, topological data analysis.

In this talk, I will introduce Rota’s formulation of Möbius inversion with concrete examples, including Möbius’s original application to number theory. I’ll then describe how Möbius inversion has become a foundational tool in topological data analysis. Finally, time permitting, I’ll describe a recent categorification of Möbius inversion that provides a richer description of certain functions on posets. This latter work is joint with Amit Patel.

Presented by Alex Elchesen, Postdoctoral Fellow, Colorado State University