An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota

TL;DR

This paper explores the historical and mathematical development of Grassmann's ideas, tracing their influence through Rota's work and culminating in a new understanding of Grassmann's regressive product, with ongoing research into its algebraic structure.

Contribution

It provides a historical and mathematical analysis connecting Grassmann's original concepts to modern algebraic structures like the Whitney algebra of a matroid and advances understanding of Grassmann's regressive product.

Findings

Connection between Grassmann's ideas and Rota's algebraic frameworks

Development of the Whitney algebra of a matroid

Ongoing work on the nature of Grassmann's regressive product

Abstract

We sketch the outlines of Gian Carlo Rota's interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney algebra of a matroid, and finally to a resolution of the question "What, really, was Grassmann's regressive product?". This final question is the subject of ongoing joint work with Andrea Brini, Francesco Regonati, and William Schmitt. The present paper was presented at the conference "The Digital Footprint of Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It will appear in proceedings of that conference, to be published by Springer Verlag.