Abstracts
Matthias Aschenbrenner, University
of Illinois at Chicago
Faithfully flat Lefschetz extensions
We call a commutative ring of characteristic zero a Lefschetz ring if
it is (isomorphic to) an ultraproduct of rings of prime characteristics.
Every uncountable algebraically closed field of characteristic zero
is Lefschetz. In this talk I will discuss the following question: which
Noetherian rings of characteristic zero admit a faithfully flat embedding
into a Lefschetz ring? It is well-known that any polynomial ring over
an uncountable algebraically closed field of characteristic zero admits
such an embedding (van den Dries and Schmidt). I will show that each
Noetherian local ring of equicharacteristic zero admits a faithfully
flat Lefschetz extension, which is even functorial in a way. I will
then briefly indicate some applications to commutative algebra.
Rahim Moosa, Masachussetts Institute of
Technology
Differential Jet Spaces
Jet spaces in algebraic and complex analytic geometry are useful tools
in studying the universal family of subvarieties of a given variety.
Pillay and Ziegler have developed the notion of differential jet spaces
for differential varieties of finite-rank over differentially closed
fields of characteristic 0. This provides a direct argument for the
Zilber dichotomy in DCF_0, avoiding the deep machinery of Zariski geometries.
I will describe on going work with Anand Pillay and Thomas Scanlon in
which we introduce and study differential jet spaces of arbitrary (that
is, possibly infinite-rank) differential varieties in the context of
several commuting derivations.
Kobi Peterzil, University of Haifa
Three examples of nonstandard analytic sets, in o-minimal structures.
Let M be an o-minimal expansion of a real closed field R, and let K
be its algebraic closure. K-manifolds and K-analytic sets are defined
in direct analogy to complex manifolds and complex analytic sets. I
will discuss recent work (still in progress) in o-minimal K-analytic
geometry, through 3 examples of nonstandard K-manifolds in o-minimal
structures:
1. The field K, expanded by all K-analytic subsets of K^n, is just
a pure algebraically closed field.
2. The standard (compact) Hopf manifold can be expanded by a new K-analytic
automorphism, not definable in CCM (Compact Complex Manifolds).
3. The nonstandard 1-dimensional torus E, equipped by all K-analytic
subsets is a locally modular, 1-dimensional, Zariski structure (when
he underlying o-minimal structure expands T_{an,exp}). Thus E is not
isomorphic to any strongly minimal set in CCM.
Along the way to proving (1) and (3) we prove (strong) analogues of
Remmert's Proper Mapping Theorem and Chow's Theorem for K-analytic sets.
James Tyne, Ohio State University
Towards an exponential Wilkie inequality
Valuation theory has been successfully used to produce several
results concerning o-minimal expansions of fields, but some of these
results require the additional assumption that the expansion is power
bounded. This is because the proofs of these results depend upon a version
of the Wilkie inequality which is only valid if no exponential function
is definable. Thus, the formulation of a version of the Wilkie inequality
that holds in the exponential case would presumably allow these results
to be generalized. One recently formulated version of a general Wilkie
inequality is far from ideal, but has yielded some results and applications
that I will survey.
Matt Valeriote, McMaster University
Varieties with few finite members
The /G/-spectrum of a variety /V/ is the function /g/_/V/ , where /g/_/V/
(/n/) is equal to the number of /n/-generated members of /V/, up to
isomorphism. The study of /G/-spectra was initiated by Berman and Idziak
in the 1990s.
In my talk I will discuss a result with Idziak and McKenzie on finitely
generated varieties whose /G/-spectra are polynomially bounded. It turns
out that these varieties have very nice structure. I'll relate this
to earlier results on the spectrum function for arbitrary varieties.
Yoav Yaffe, McMaster University
Lie Differential Fields - Model Completion and a version of Hensel's
Lemma
A {\em Lie differential field} (LDF) is a field given with a finite
number of derivations satisfying some given commutation relations. As
an example take the field $F$ of rational functions on some $n$-dimensional
algebraic variety, with $n$ $F$-linearly-independent vector fields acting
as derivations on $F$.
I will give a short account of the model completion of LDFs, based
on extending to the PDE context the concept (due to L. Blum) of a {\em
generic} solution to (formal) ODEs.
I will then describe a more geometric version of the above model completion,
using {\em torsor fields} on {\em pro-algebraic varieties} (all concepts
will hopefully be defined).
Finally I will discuss valued LDFs with the same interaction as in
Scanlon, i.e. every ideal of the value ring is a differential ideal.
I will give a version of Hensel's Lemma suitable for this context, where
the condition of Scanlon's DHL is replaced by the assumption that the
residue of the given approximate solution is a regular point of a suitable
pro-algebraic variety over the residue field.
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