The Lang-Weil inequality and the etale fundamental group

Terence Tao
University of California, Los Angeles (UCLA)

We give an exposition of two tools of algebraic geometry of use in combinatorial geometry. The first, the Lang-Weil inequality, gives an asymptotic for the size of an algebraic variety over a large finite field. The second, the etale fundamental group, is to algebraic varieties as the topological fundamental group is to manifolds; when combined with the Lang-Weil inequality, the etale fundamental group gives information on the size of a generic fibre of a family of algebraic varieties over a finite field. If time permits, we also explain how these tools were used in the first proof of the algebraic regularity lemma. This talk is aimed at audience members with little prior exposure to algebraic geometry.

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