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.