MODNET Research Workshop, Humboldt-Universität Berlin,
September 10-14 Berlin
Lecture Courses
- Model Theory of Fields (Françoise Delon,
Université Paris 7)
Notes
Abstract: The following themes will be covered:
- Algebraically closed fields: Category of definable groups.
- Hasse derivations with emphasis on positive characteristic. Separably
closed fields of finite imperfection degree. Types, thin and very thin
types. D-algebraic geometry. Prolongations. Category of type-definable
groups. Minimal groups.
- Overview of the characteristic zero case.
-
O-minimality, Part II. On the construction of o-minimal structures (Alex Wilkie, The
University of Manchester)
Notes by participants
Abstract: My talks will be somewhat different from those that Kobi Peterzil gave at
the Camerino meeting in June in that I shall focus on examples of o-minimal
structures and how one goes about proving that they are so. An
acquaintance with the basic general theory, as presented by Kobi, would
still be desirable however, but not essential.
I shall begin with an introduction to the local theory of real analytic
functions and of semi- and sub-analytic sets and then present the
elimination theory for them in the style of van den Dries and Denef. This
leads to the o-minimality of the expansion of the real field by the
so-called globally subanalytic sets. I shall then briefly discuss the
situation for quasi-analytic classes of functions.
Some general theory follows, namely the valuation inequality for o-minimal
polynomial bounded expansions of real closed fields and I hope to be able
to explain how it leads to model completeness and o-minimality
results for expansions of such structures by the exponential function and,
if time permits, to certain tameness results for expansions of the complex
field by certain holomorphic functions.
-
Applications of Model Theory of Fields. The Zariski dichotomy and Mordell-Lang (Rahim Moosa,
University of Waterloo)
Notes by participants
Abstract: The aim of my talks will be to explain the meaning of the Zilber
dichotomy in differentially closed fields of characteristic zero and
separably closed fields of positive characteristic, and to describe, in as
much detail as time permits, how the Mordell-Lang conjecture for function
fields (i.e., Hrushovski's theorem) follows from the dichotomy. In
addition, I intend to prove the dichotomy for differentially closed fields
of characteristic zero using the method of Pillay-Ziegler. Most of the
necessary background material on differentially closed fields and
separably closed fields will be covered in Francoise Delon's lectures.
30. Juli 2007