Mathematical logic seminar - April 30, 2003

Abstract:

In 2001, Anand Pillay and Martin Ziegler proposed an alternate model-theoretic proof to the Mordell-Lang Conjecture in characteristic zero. Their proof followed many of the steps of Hrushovski's construction, however the proofs diverged at one key step. Whereas Hrushovski used deep model-theoretic results about Zariski geometries, Pillay and Ziegler utilized a natural generalization of differential tangent spaces, called differential jet spaces. In this talk, I will transfer the statement of the Mordell-Lang conjecture given in my previous lecture to a second "relative form" which is amenable to model-theoretic methods. I will then provide an outline of the key steps in the jet spaces proof of the relative form of the theorem. In particular, I will develop the idea of a jet space and discuss how Pillay and Ziegler employed them to obtain key model-theoretic information.