Title: Quo Vadis


Abstract: Paraphrasing the title of Riemann’s famous lecture of 1854 I ask: What is the most rudimentary notion of a geometry? A possible answer is a path system: Consider a finite set of “points” x_1,…,x_n and provide a recipe how to walk between x_i and x_j for all i
eq j, namely decide on a path P_ij, i.e., a sequence of points that starts at x_i and ends at x_j, where P_ji is P_ij, in reverse order. The main property that we consider is consistency. A path system is called consistent if it is closed under taking subpaths. What do such systems look like? How to generate all of them? We still do not know. One way to generate a consistent path system is to associate a positive number w_ij>0 with every pair and let P_ij be the corresponding w-shortest path between x_i and x_j. Such a path system is called metrical. It turns out that the class of consistent path systems is way richer than the metrical ones.
My main emphasis in this lecture is on what we don’t know and wish to know, yet there is already a considerable body of work that we have done on the subject.
The new results that I will present are joint with my student Daniel Cizma as well as with him and with Maria Chudnovsky.

Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7f854e5f-f9bc-493b-ae87-b25f0067a6ce