Aim of the workshop
To give a series of lectures explaining the proof by Ehud Hrushovski of
the twisted Lang-Weil estimates for difference
varieties. Link to
the paper "The elementary theory of the Frobenius Automorphism" (preprint).
Here are some slides of some survey lectures given at the Orsay Workshop Model
theory of difference fields and applications, in December 2011:
Non-standard
Frobenius: Introduction, by M. Hils.
Plan détaillé de la preuve de Hrushovski, by G. Giabicani
and Y. Laszlo.
Lectures will begin around 9:00 am on Monday 25 March and
probably end around 4:00 pm on Friday 29 March.
Modus Operandi
Participants are supposed to have a working knowledge of the
prerequisites (see below). We hope that most of them will participate
actively in the programme.
A set of lecture notes, in a preliminary form, is expected to be
available before the start of the workshop.
Prerequisites
All participants are supposed to have a working knowledge of the basics
both in model theory and in algebraic geometry.
For example, in model theory, be comfortable with at least the first six sections
of Dave Marker's Orsay lecture notes:
Sections 1-3: Language, Structures and Theories, The Compactness Theorem,
Ultraproducts and Compactness
Sections 4-6: Complete Theories, Quantifier Elimination, Algebraically
Closed Fields
In Algebraic Geometry
The basic definitions for example:
Robin Hartshorne, Algebraic Geometry, Chap. I-II
Some further suggestions
Valued fields:
- O. Zariski et P. Samuel: "Commutative algebra II" (Ch VI), GTM, Springer.
-
Notes of a course given by Z. Chatzidakis on valued fields (in
French), chapter 1, and maybe parts of chapters 2 and 3.
Difference algebra:
- Richard Cohn, Difference algebra, (Ch 1-3).
-
Lecture Notes "Algebraic difference equations" by Michael Wibmer.
Intersection theory: Basic definitions and results will be
recalled but it could be useful to look at (or browse through) one of the
following:
- W. Fulton: "Introduction to intersection theory in algebraic geometry"
(first 3 sections), Cbms Regional Conference Series in Mathematics,
A.M.S. 1984.
- I. R. Shafarevich: "Basic algebraic geometry 1" (Ch IV, sections 1 and
2)
Advanced algebraic geometry (Around the Weil conjectures): Again, this is not required, but here is a possible
reference:
- B. Mazur: "Eigenvalues of Frobenius acting on algebraic varieties over
finite fields", in: "Algebraic Geometry - Arcata 1974", Proc. of Symposia in Pure Mathematics, Vol XXIX, pp 231-262.
Contact
If you wish to be put on the mailing list, please write an e-mail to
zoe_at_math.univ-paris-diderot.fr
Organisation Committee: Z. Chatzidakis, F. Loeser, T. Scanlon.
Scientific Committee: J.-B. Bost, E. Bouscaren, Z. Chatzidakis, M. Hils,
Y. Laszlo, F. Loeser, T. Scanlon.
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