### 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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