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

We sketch the outlines of Gian Carlo Rota’s interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre[13, 15] of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Gian Carlo variously called Grassmann-Cayley algebra and 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. 1. Almost ten years later We are gathered today in order to renew and deepen our recollection of the ways in which our paths intersected that of Gian Carlo Rota. We do this in poignant sadness, but with a bitter-sweet touch: we are pleased to have this opportunity to meet and to discuss his life and work, since we know how Gian Carlo transformed us through his friendship and his love of mathematics. We will deal only with the most elementary of geometric questions; how to represent pieces of space of various dimensions, in their relation to one another. It’s a simple story, but one that extends over a period of some 160 years. We’ll start and finish with Hermann Grassmann’s project, but the trail will lead us by Giuseppe Peano, Hassler Whitney, to Gian Carlo Rota and his colleagues. Before I start, let me pause for a moment to recall a late afternoon at the Accademia Nazionale dei Lincei, in 1973, on the eve of another talk I was petrified to give, when Gian Carlo decided to teach me how to talk, so I wouldn’t make a fool of myself the following day. The procedure was for me to start my talk, with an audience of one, and he would interrupt whenever there was a problem. We were in that otherwise empty conference hall for over two hours, and I never got past my first paragraph. It was terrifying, but it at least got me through the first battle with my fears and apprehensions, disguised as they usually are by timidity, self-effacement, and other forms of apologetic behavior. 2. Synthetic Projective Geometry Grassmann’s plan was to develop a purely formal algebra to model natural (synthetic) operations on geometric objects: flat, or linear pieces of space of all possible dimensions. His approach was to be synthetic, so that the symbols in his algebra 1 ar X iv :0 90 4. 33 49 v1 [ m at h. H O ] 2 1 A pr 2 00 9