Past Logic Seminar

Ax's theorem on the dimension of the intersection of an algebraic subvariety and a formal subgroup (Theorem 1F in "Some topics in differential algebraic geometry I...") implies Schanuel type transcendence results for a vast class of formal maps (including exp on a semi-abelian variety). Ax stated and proved this theorem in the characteristic 0 case, but the statement is meaningful for arbitrary characteristic and still implies positive characteristic transcendence results. I will discuss my work on positive characteristic version of Ax's theorem.

The transfer principle of Ax-Kochen-Ershov says that every first order sentence φ in the language of valued fields is, for p sufficiently big, true in ℚ_p iff it is true in \F_p((t)). Motivic integration allowed to generalize this to certain kinds of non-first order sentences speaking about functions from the valued field to ℂ. I will present some new transfer principles of this kind and explain how they are useful in representation theory. In particular, local integrability of Harish-Chandra characters, which previously was known only in ℚ_p, can be transferred to \F_p((t)) for p >> 1. (I will explain what this means.)

This is joint work with Raf Cluckers and Julia Gordon.

Sela showed that the theory of the non abelian free groups is stable. In a joint work with Sklinos, we give some characterization of the forking independence relation between elements of the free group F over a set of parameters A in terms of the Grushko and cyclic JSJ decomposition of F relative to A. The cyclic JSJ decomposition of F relative to A is a geometric group theory tool that encodes all the splittings of F as an amalgamated product (or HNN extension) over cyclic subgroups in which A lies in one of the factors.

An externally definable set of a first order structure $M$ is a set of the form $X\cap M^n$ for a set $X$ that is parametrically definable in some elementary extension of $M$. By a theorem of Shelah, these sets form again a first order structure if $M$ is NIP. If $M$ is a real closed field, externally definable sets can be described as some sort of limit sets (to be explained in the talk), in the best case as Hausdorff limits of definable families. It is conjectured that the Shelah structure on a real closed field is generated by expanding the field with convex subsets of the line. This is known to be true in the archimedean case by van den Dries (generalised by Marker and Steinhorn). I will report on recent progress around this question, mainly its confirmation on real closed fields that are close to being maximally valued with archimedean residue field. The main tool is an algebraic characterisation of definable types in real closed valued fields. I also intend to give counterexamples to a localized version of the conjecture. This is joint work with Francoise Delon.

Non-trivial henselian valuations are often so closely related to the arithmetic of the underlying field that they are encoded in it, i.e., that their valuation ring is first-order definable in the language of rings. In this talk, we will give a complete classification of all henselian valued fields of residue characteristic 0 that allow a (0-)definable henselian valuation. This requires new tools from the model theory of ordered abelian groups (joint work with Franziska Jahnke).

The Schelling segregation model has been extensively studied, by researchers in fields as diverse as economics, physics and computer science. While the explicit concern when the model was first introduced back in 1969, was to model the kind for racial segregation observed in large American cities, the model is sufficiently abstract to apply to almost situation in which agents or nodes arrange themselves geographically according to a preference not to be of a minority type within their own neighbourhhood. Kirman and Vinkovik have established, for example, that Schelling's model is a finite difference version of a differential equation describing interparticle forces (and applied in the modelling of cluster formation). Despite the large literature relating to the model, however, it has largely resisted rigorous analysis -- it has not been possible to prove the segregation behaviour easily observed when running simulations. For the first time we have now been able to rigorously analyse the model, and have also established some rather surprising threshold behaviour.

This talk will require no specialist background knowledge.

It has long been a challenge to synthesize the complementary insights offered by model theory and category theory. A small fragment of that challenge is to understand ultraproducts categorically. I will show that, granted some general categorical machinery, the notions of ultrafilter and ultraproduct follow inexorably from the notion of finiteness of a set. The machine in question, known as the codensity monad, has existed in an underexploited state for nearly fifty years. To emphasize that it was not constructed specifically for this purpose, I will mention some of its other applications. This talk represents joint work with an anonymous referee. Little knowledge of category theory will be assumed.

I shall give a non-technical survey of Pure Inductive Logic, a branch of Carnap's Inductive Logic which was

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.

Masser recently proved a bound on the number of rational points of bounded height on the graph of the zeta function restricted to the interval [2,3]. Masser's bound substantially improves on bounds obtained by Bombieri-Pila-Wilkie. I'll discuss some results obtained in joint work with Gareth Boxall in which we prove bounds only slightly weaker than Masser's for several more natural analytic functions.

The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to

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 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.

(Joint work with Nikolaos Galatos.) Proof-theoretic methods provide useful tools for tackling problems for many classes of algebras. In particular, Gentzen systems admitting cut-elimination may be used to establish decidability, complexity, amalgamation, admissibility, and generation results for classes of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing a family resemblance to groups, such methods have so far met only with limited success. The main aim of this talk will be to explain how proof-theoretic methods can be used to obtain new syntactic proofs of two core theorems for the class of lattice-ordered groups: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

(Joint work with Artem Chernikov.) In the talk, we will first recall some basic results on valued difference fields, both from an algebraic and from a model-theoretic point of view. In particular, we will give a description, due to Hrushovski, of the theory VFA of the non-standard Frobenius acting on an algebraically closed valued field of residue characteristic 0, as well as an Ax-Kochen-Ershov type result for certain valued difference fields which was proved by Durhan. We will then present a recent work where it is shown that VFA does not have the tree property of the second kind (i.e., is NTP2); more generally, in the context of the Ax-Kochen-Ershov principle mentioned above, the valued difference field is NTP2 iff both the residue difference field and the value difference group are NTP2. The property NTP2 had already been introduced by Shelah in 1980, but only recently it has been shown to provide a fruitful ‘tameness’ assumption, e.g. when dealing with independence notions in unstable NIP theories (work of Chernikov-Kaplan).

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

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.

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