Algebra Seminar, Cameron Hill (Wes): 'The Lang-Weil bounds and the geometry of pseudo-finite fields.'

Friday, April 17, 2015
1:10 PM - 2:00 PM (ET)
ESC 638
Event Type
David Pollack,

Abstract: The Lang-Weil bounds are a venerable fact of algebraic geometry that provide reasonably precise estimates of the cardinalities of varieties in finite fields in terms of their dimensions (as calculated in algebraic closures). One model-theoretic corollary of this theorem is the fact that every pseudo-finite field admits a very fine-grained geometry analogous to (but distinct from) the geometry of its algebraic closure, and which also accommodates a larger family of definable sets than varieties alone. In this talk, I will discuss other routes to this geometry on a pseudo-finite field that do not use the Lang-Weil bounds as a starting point. This approach also supplies a novel proof of a relaxed version the Lang-Weil bounds themselves, and I will try to point out how this proof is really different from the classical argument.

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