McMaster University, December 5 - 8, 2014
I will give a brief overview of the geometric ideas behind the proof. If time permits I will also discuss some diophantine applications.
If $V \subseteq \mathbb C^n$ is a non-weakly-special variety, then $V \cap Iso (\bar a)$ is not Zariski dense in $V$. We will discuss how to use differential algebra to give an effective upper bound on the degree of the Zariski closure of $V \cap Iso (\bar a)$. In the case that one knows that $V \cap Iso (\bar a)$ is zero-dimensional (e.g. $V$ is a curve or $V$ contains no weakly special varieties), this gives an effective bound on the number of points in the intersection.
The proofs use intersection theory in jet spaces and various notions from geometric model theory.