# Ehud Hrushovski: Expanding polynomials and stability

Logic Seminar (HUJI)
The logic seminar  takes place on Wednesdays at 16:00, in room 209.
The speaker on October 30 is Ehud Hrushovski.

Title: Expanding polynomials and stability.Abstract: Let $F(X,Y)$ be a polynomial, say over $\mathbb Z$.    In the paper by Tao, he showed that – unless $F$ is conjugate to $+$ or $\cdot$ in an appropriate sense – $F(X,Y)$ is   ‘moderately expanding’, meaning  that for some $\epsilon>0$,  for any large prime $p$ and any $Z \subset GF(p)$ with $|Z|> p^{1-\epsilon}$, $F(Z,Z)$ contains a nonzero proportion of the points of $GF(p)$.  Along the way he proved a strong version of Szemeredi’s regularity lemma for graphs definable over finite fields.

This paper has a number of strong connections to model theory and stability, and I will discuss some of them.    In particular, the Szermeredi type lemma is closely related to the fundamental theorem of stability, Shelah‘s ‘uniqueness of non-forking extensions’.

The model theory incidentally facilitates some generalizations (e.g. for prime powers, and for rational functions rather than polynomials.)