Forthcoming events in this series

### Pure Inductive Logic

## Abstract

anticipated early on in that subject but has only recently begun to be developed as an area of Mathematical Logic. My intention

is to cover its origins and aims, and to pick out some of the key concepts which have emerged in the last decade or so.

### Rational values of certain analytic functions

## Abstract

### Multiplicity in difference geometry

## Abstract

count the number of solutions to difference polynomial equations over fields with powers of Frobenius.

We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle

of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.

### The Outer Model Programme

## Abstract

The Outer Model Programme investigates L-like forcing extensions of the universe, where we say that a model of Set Theory is L-like if it satisfies properties of Goedel's constructible universe of sets L. I will introduce the Outer Model Programme, talk about its history, motivations, recent results and applications. I will be presenting joint work with Sy Friedman and Philipp Luecke.

### An application of proof theory to lattice-ordered groups

## Abstract

### Valued difference fields and NTP2

## Abstract

### A non-desarguesian projective plane of analytic origin

## Abstract

avoiding a direct use of Hrushovski's construction. Instead we make use of the field of complex numbers with a holomorphic function (Liouville function) which is an omega-stable structure by results of A.Wilkie and P.Koiran. We first find a pseudo-plane interpretable in the above analytic structure and then "collapse" the pseudo-plane to a projective plane applying a modification of Hrushovski's mu-function.

### Topological dynamics and model theory of SL(2,R)

## Abstract

### Embeddings of the spaces of the form C(K)

## Abstract

We discuss the question of the existence of the smallest size of a family of Banach spaces of a given density which embeds all Banach spaces of that same density. We shall consider two kinds of embeddings, isometric and isomorphic. This type of question is well studied in the context of separable spaces, for example a classical result by Banach states that C([0,1]) embeds all separable Banach spaces. However, the nonseparable case involves a lot of set theory and the answer is independent of ZFC.

### Plus ultra

## Abstract

I shall present a very general class of virtual elements in a structure, ultraimaginaries, and analyse their model-theoretic properties.

### Algebraic closure in pseudofinite fields

## Abstract

has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can

be realised as ultraproducts of finite fields. A group $G$ is geometrically

represented in a theory $T$ if there are modles $M_0\prec M$ of $T$,

substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$

and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of

pseudofinite fields. We show that, geometric representation of a group whose order

is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity

in models of $T$. As a consequence of this, we show that, for almost all

completions of the theory of pseudofinite fields, over a substructure $A$, algebraic

closure agrees with definable closure, if $A$ contains the relative algebraic closure

of the prime field. This is joint work with Ehud Hrushovski.

### S-independence in NIP theories

## Abstract

theories, which is an attempt to capture the "stable part" of types.

### Uniformly defining valuation rings in Henselian valued fields with finite and pseudo-finite residue field

## Abstract

We give a first-order definition, in the ring language, of the ring of p-adic integers inside the field p-adic numbers which works uniformly for all p and for valuation rings of all finite field extensions and of all local fields of positive characteristic p, and in many other Henselian valued fields as well. The formula canbe taken existential-universal in the ring language. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula. For any fixed general p-adic field we give an existential formula in the ring language which defines the valuation ring.

We also state some connections to some open problems.