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 $i\neq j$, 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.