# Model Theory Days May 21-22, 2015

All talks are in lecture room C124

#### Thursday May 21st

10.15-11.25 Andres Villaveces: Model theory for modular invariants.

13.00-14.10 Jonathan Kirby: Examples of quasiminimal classes.

Abstract: I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and some open questions. Then I will explain some joint work with Martin Bays in which we have constructed variants of the pseudo-exponential fields (originally due to Boris Zilber) which are quasimininal and discuss progress towards the problem of showing that complex exponentiation is quasiminimal.

14.20-15.30 Boris Zilber: Geometric dualities and model theory.

Abstract: Geometries can be given in a direct semantic way, say as a complex or real manifold, or more abstractly, by corresponding 'co-ordinate' algebras. A duality of this kind becomes highly non-trivial in cases of schemes of arithmetic type and for non-commutative co-ordinate algebras and C*-algebras. I will discuss these issues from model-theoretic perspective. A detailed analysis has been worked out for the canonical commutation relation(s) underlying quantum mechanics. Some applications will be mentioned.

#### Friday May 22nd

10.15-11.25 Boris Zilber: Geometric dualities and model theory.

13.00-14.10 Misha Gavrilovich: The lifting property: a common pattern for definitions in topology and finite group theory.

Abstract: We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of sujectivity and injectivity of maps; as well as of being connected, being compact and being sequentially compact; the separation axioms T0 and T1 in topology, having dense image, induced (pullback) topology; the properties of a finite group being abelian, perfect, soluble or being of order prime to p. This lets us express the statement of Feit-Thompson theorem (that a finite group with odd number of elements is necessarily soluble) as an implication between lifting properties.

We also offer a brief speculation that these observations may lead to a weaker than usual logic (deductive calculus) which expresses these properties.