The Zilber trichotomy principle gives a precise sense in which the structure on a sufficiently well behaved one-dimensional set must have one of only three possible kinds: disintegrated (meaning that there may be some isolated correspondences, but nothing else), linear (basically coming from an abelian group with no extra structure), or algebro-geometric (essentially coming from an algebraically closed field). This principle is true in differentially closed fields when “one dimensional” is understood as “strongly minimal” (proven by Hrushovski and Sokolovic using the theory of Zariski geometries and then by Pillay and Ziegler using jet spaces).

When working with differentially closed fields with finitely many, but more than one, distinguished commuting derivations, there are sets which from a certain model theoretic point of view (having to do with the notion of a regular type) are one dimensional even though they are infinite dimensional from the point of view of differential dimension. Moosa, Pillay and Scanlon showed that a weakening of the trichotomy principle is true for these sets: if there is a counter example to the trichotomy principle, then one can be found for a set defined by linear PDEs.

In this lecture, I will explain in detail what the trichotomy principle means in differential algebra, how the reduction to the linear case works, and then how one might approach the open problems.

This is a joint event of the CUNY Logic Workshop and the Kolchin Seminar in Differential Algebra, as part of a KSDA weekend workshop.