**TASK II -
MODEL THEORY OF FIELDS**

AND APPLICATIONS

**
Description**

This topic is at the heart of our project, and is the one where the
connections between model theory and other branches of mathematics are
most striking. The use of techniques coming from geometric stability has
led to several applications outside of model theory. While most of the
questions listed below have a model-theoretic inspiration, their
solution will mostly involve proving algebraic results.

The topic can roughly be divided into two parts: Model-theoretic
study of various fields, in particular with operators such as
derivations, automorphisms, the
λ-functions of separably closed fields, pseudo-analytic
functions; Study of various objects from algebraic geometry and
uniformity results.

**
List of specific problems**

**II.1. Model theory of fields and
applications to algebraic geometry**

a) Investigate groups definable in
SCF: in particular, whether definable simplicity implies simplicity

b) Develop model theory (e.g., find
model companion, prove simplicity, describe imaginaries, definable groups,
minimal types, nonmodular types, induced structure on definable sets, etc.)
of fields expanded by one or several operators, e.g.: algebraically closed
fields with (commuting) derivations and/or automorphisms, separably closed
fields with stacks of Hasse derivations, derivations of the Frobenius, non-standard
Frobenius.

c) Extend model theory of Buium's D-varieties
to difference varieties (ACFA) and Hasse D-varieties (SCF).

d) Investigate D-groups in differential
Galois theory (non-linear case), making the connection with the Malgrange
construction of the Galois groupoid of a foliation. Calculate the groups
for classical equations (eg Painlevé equations). Sharpen Hrushovski's
effective determination of the Galois group of a linear differential equation
(improve the bound on the order of tensor powers). Similar questions for
difference Galois groups (e.g., of q-difference equations). Connections with
the difference Galois theory developped by Singer and Van der Put.

e) Apply model theory to the study
of special points of abelian and Shimura varieties, in the style
of the Manin-Mumford theorem and André-Oort Conjecture.

f) Continue the study of difference algebraic
geometry : difference schemes, varieties, blow-ups, (co-)homology groups
etc.

**II.2. PAC fields and finite fields**

a) Prove Zilber trichotomy for reducts
of pseudofinite fields, if necessary assuming omega-categoricity.

b) Describe imaginaries in PAC fields,
and use this to study groups interpretable in omega-free PAC fields.

** II.3. Fields with Pseudo-exponentiation, and analytic expansions of the complex numbers. **

a) Prove pseudo-exponentiation gives
an analytic Zariski structure; likewise expansions by power functions. Find
examples from quantum tori and non-commutative geometry. Examples from Hrushovski
amalgamation.

b) Refute Zilber s conjecture that complex pseudo-exponentiation is exponentiation

** II.4. Motivic integration**

Develop a theory of motivic integration
with parameters and a sensible theory of constructible motivic functions

** II.5. Weil Cohomology**

a) Examine non-standard Weil cohomologies.

b) Connections of non-standard
Frobenius with conjectured semisimplicity of action of Frobenius on Weil
cohomology.

c) Determine in which cases the p-th
cohomological dimension of a field is an elementary invariant of the field.

**Progress**

Year 1