Logic List Mailing Archive

Isaac Newton Institute Program "Model Theory and Applications to Algebra and Analysis"; January to July 2005

Isaac Newton Institute for Mathematical Sciences
Model Theory and Applications to Algebra and Analysis

17 Jan--15 Jul 2005

http://www.newton.cam.ac.uk/programs/MAA/maa.html

Organisers: Professor Z Chatzidakis (CNRS), Professor A Pillay (Illinois),
Professor A Wilkie (Oxford)

Programme theme

Model theory is a branch of mathematical logic dealing with abstract
structures (models), historically with connections to other areas of
mathematics. In the past decade, model theory has reached a new maturity,
allowing for a strengthening of these connections and striking
applications to diophantine geometry, analytic geometry and Lie theory, as
well as strong interactions with group theory, representation theory of
finite-dimensional algebras, and the study of the p-adics. The main
objective of the semester will be to consolidate these advances by
providing the required interdisciplinary collaborations.

Model theory is traditionally divided into two parts pure and applied.
Pure model theory studies abstract properties of first order theories, and
derives structure theorems for their models. Applied model theory on the
other hand studies concrete algebraic structures from a model-theoretic
point of view, and uses results from pure model theory to get a better
understanding of the structures in question, of the lattice of definable
sets, and of various functorialities and uniformities of definition. By
its very nature, applied model theory has strong connections to other
branches of mathematics, and its results often have non-model-theoretic
implications. A substantial knowledge of algebra, and nowadays of
algebraic and analytic geometry, is required.


The programme will concentrate on the following areas:

Pure model theory. We expect further developments in the use of stability
theory techniques in unstable contexts (simple theories, algebraically
closed valued fields) and in non-elementary classes.

Model theory of fields with operators, and connections with arithmetic
geometry. The model theory of differentially closed fields and of other
fields with operators has been at the centre of model-theoretic proofs of
results in arithmetic geometry. The Zil'ber programme of pseudo-analytic
functions is also expected to have some interesting consequences.

O-minimality and related topics. O-minimality is a property of ordered
structures, yielding results akin to traditional real analytic results,
such as the classical Finiteness Theorems for subanalytic sets (cell
decompositions, Whitney stratifications, etc.). Mathematically central,
new examples of o-minimal structures have emerged, and the logical theory
has had applications to Lie theory, to asymptotics, and to neural
networks.

Henselian fields. Model theory of Henselian fields, and in particular of
p-adic fields and Arc spaces. Connections with algebraic and analytic
geometry. Study of cohomology theories and motives, aiming at uniformity
results. Study of compact complex manifolds, and uses of stability.

Model theory of groups. We plan to have a workshop on groups of Finite
Morley rank, a topic connected to the Classification of finite simple
groups via its techniques and its aims. The recent (and very exciting)
developments in the model theory of non-abelian free groups should also be
studied, depending on its degree of maturity.