### Model theory and the distribution of orders in number fields

## Abstract

Forthcoming events in this series

Thu, 12 Feb 2015

17:30 -
18:30

L6

Jamshid Derakhshan

(Oxford University)

Recently Kaplan, Marcinek, and Takloo-Bighash have proved an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field. An essential ingredient in their poof is a p-adic volume formula. I will present joint results with Ramin Takloo Bighash on model-theoretic generalizations of the volume formulas and discuss connections to number theory.

Thu, 05 Feb 2015

17:30 -
18:30

L6

Nicolai Vorobjov

(University of Bath)

Let $K\subset {\mathbb R}$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g. a semi-algebraic or a real analytic set. A definable family $\{S_\delta\ | 0<\delta\in{\mathbb R}\}$ of compact subsets of $K$, is called a *monotone family* if $S_\delta\subset S_\eta$ for all sufficiently small $\delta>\eta>0$. The main result in the talk is that when $\dim K=2$ or $\dim K=n=3$ there exists a definable triangulation of $K$ such that for each (open) simplex $\Lambda$ of the triangulation and each small enough $\delta>0$, the intersections $S_\delta\cap\Lambda$ is *equivalent* to one of five (respectively, nine) standard families in the standard simplex (the equivalence relation and a standard family will be formally defined). As a consequence, we prove the two-dimensional case of the topological conjecture on approximation of definable sets by compact families.

This is joint work with Andrei Gabrielov (Purdue).

Thu, 29 Jan 2015

17:30 -
18:30

L6

Harry Schmidt

(University of Basel)

Let ${\cal E}$ be a family of elliptic curves over a base variety defined over $\mathbb C$. An additive extension ${\cal G}$ of ${\cal E}$ is a family of algebraic groups which fits into an exact sequence of group schemes $0\rightarrow {\mathbb G}_{\rm a}\rightarrow {\cal G}\rightarrow {\cal E}\rightarrow 0$. We can define the special subvarieties of ${\cal G}$ to be families of algebraic groups over the same base contained in ${\cal G}$. The relative Manin-Mumford conjecture suggests that the intersection of a curve in ${\cal G}$ with the special subvarieties of dimension 0 is contained in a finite union of special subvarieties.

To prove this we can assume that the family ${\cal E}$ is the Legendre family and then follow the strategy employed by Masser-Zannier for their proof of the relative Manin-Mumford conjecture for the fibred product of two legendre families. This has applications to classical problems such as the theory of elementary integration and Pell's equation in polynomials.

Thu, 04 Dec 2014

17:30 -
18:30

L5

Adam Harris

(UEA)

I will outline some recent work with Jonathan Kirby regarding the first stage in the construction of the pseudo j-function. In particular, I will go through the construction of the analogue of the canonical countable pseudo exponential field as the "Fraisse limit" of a category of "partial j-fields". Although I will be talking about the j-function throughout the talk, it is not necessary to know anything about the j-function to get something from the talk. In particular, even if you don't know what the j-function is, you will still hopefully have an understanding of how to construct the countable pseudo-exp by the end of the talk.

Thu, 20 Nov 2014

16:30 -
17:30

Boris Zilber

(Oxford)

Note: joint with Philosophy of Physics.

Venue: Lecture Room, Radcliffe Humanities, ROQ.

Thu, 13 Nov 2014

17:30 -
18:30

L6

Robert Henderson

(UEA)

Little is known about C_exp, the complex field with the exponential function. Model theoretically it is difficult due to the definability of the integers (so its theory is not stable), and a lack of clear algebraic structure; for instance, it is not known whether or not pi+e is irrational. In order to study C_exp, Boris Zilber constructed a class of pseudo-exponential fields which satisfy all the properties we desire of C_exp. This class is categorical for every uncountable cardinal, and other more general classes have been defined. I shall define the three main classes of exponential fields that I study, one of which being Zilber's class, and show that they exhibit "stable-like" behaviour modulo the integers by defining a notion of independence for each class. I shall also explicitly apply one of these independence relations to show that in the class of exponential fields ECF, types that are orthogonal to the kernel are exactly the generically stable types.

Thu, 06 Nov 2014

17:30 -
18:30

L6

Luca Spada

(Salerno and Amsterdam)

The aim of this talk is to provide a general setting in which a number of important dualities in mathematics can be framed uniformly. The setting comes about as a natural generalisation of the Galois connection between ideals of polynomials with coefficients in a field K and affine varieties in K^n. The general picture that comes into sight is that the topological representations of Stone, Priestley, Baker-Beynon, Gel’fand, or Pontryagin are to their respective classes of structures just as affine varieties are to K-algebras.

Tue, 28 Oct 2014

17:00 -
18:00

C2

Mike Prest

(Manchester)

Note: joint with Algebra seminar.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them. But not all. To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:

G. Puninski and M. Prest, Ringel's conjecture for domestic string algebras, arXiv:1407.7470;

R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

Thu, 23 Oct 2014

17:30 -
18:30

L6

Volker Halbach

(Oxford)

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

Thu, 16 Oct 2014

17:30 -
18:30

L6

Mario Edmundo

(Universidade de Lisboa)

The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimals sheaves, which might be useful on it own.

Thu, 19 Jun 2014

17:15 -
18:15

L6

Jamshid Derakhshan

(Oxford)

This is joint work with Angus Macintyre.

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

To prove this we prove universal definability of the valuation rings of such fields using work of Cluckers-Derakhshan-Leenknegt-Macintyre on existential

definability, quantifier elimination of Basarab-Kuhlmann for valued fields in a many-sorted language involving higher residue rings and groups,

a model completeness theorem for certain pre-ordered abelian groups which generalize Presburger arithmetic (we call finite-by-Presburger groups),

and an interpretation of higher residue rings of such fields in the higher residue groups.

Thu, 12 Jun 2014

17:15 -
18:15

L6

Katrin Tent

(Muenster)

Finite sharply 2-transitive groups were classified by Zassenhaus in the 1930's. It has been an open question whether infinite sharply 2-transitive group always contain a regular normal subgroup. In joint work with Rips and Segev we show that this is not the case.

Thu, 05 Jun 2014

17:15 -
18:15

L6

Charlotte Kestner

(Central Lancashire)

I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. This is joint work with William Anscombe.

Thu, 29 May 2014

17:15 -
18:15

L6

Andrew Brooke-Taylor

(Bristol)

Cardinal characteristics of the continuum are (definitions for) cardinals that are provably uncountable and at most the cardinality c of the reals, but which (if the continuum hypothesis fails) may be strictly less than c. Cichon's diagram is a standard diagram laying out all of the ZFC-provable inequalities between the most familiar cardinal characteristics of the continuum. There is a natural analogy that can be drawn between these cardinal characteristics and highness properties of Turing oracles in computability theory, with implications taking the place of inequalities. The diagram in this context is mostly the same with a few extra equivalences: many of the implications were trivial or already known, but there remained gaps, which in joint work with Brendle, Ng and Nies we have filled in.

Thu, 22 May 2014

17:15 -
18:15

L5

Will Anscombe

(Leeds)

A 1-dimensional asymptotic class (Macpherson-Steinhorn) is a class of finite structures which satisfies the theorem of Chatzidakis-van den Dries-Macintyre about finite fields: definable sets are assigned a measure and dimension which gives the cardinality of the set asymptotically, and there are only finitely many dimensions and measures in any definable family. There are many examples of these classes, and they all have reasonably tame theories. Non-principal ultraproducts of these classes are supersimple of finite rank.

Recently this definition has been generalised to `Multidimensional Asymptotic Class' (joint work with Macpherson-Steinhorn-Wood). This is a much more flexible framework, suitable for multi-sorted structures. Examples are not necessarily simple. I will give conditions which imply simplicity/supersimplicity of non-principal ultraproducts.

An interesting example is the family of vector spaces over finite fields with a non-degenerate bilinear form (either alternating or symmetric). If there's time, I will explain some joint work with Kestner in which we look in detail at this class.

Thu, 13 Mar 2014

17:15 -
18:15

L6

Colin McLarty

(Case Western Reserve)

Several number theorists have stressed that the proofs of FLT focus on small concrete arithmetically defined groups rings and modules, so the steps can be checked by direct calculation in any given case. The talk looks at this in relation both to Hilbert's idea of contentual (inhaltlich) mathematics, and to formal provability in Peano arithmetic and other stronger and weaker axioms.

Thu, 27 Feb 2014

17:15 -
18:15

L6

Kentaro Fujimoto

(Bristol)

Formal truth theory sits between mathematical logic and philosophy. In this talk, I will try to give a partial overview of formal truth theory, from my particular perspective and research, in connection to some areas of mathematical logic.

Thu, 13 Feb 2014

17:15 -
18:15

L6

Philip Welch

(Bristol)

It is well known that infinite perfect information two person games at low levels in the arithmetic hierarchy of sets have winning strategies for one of the players, and moreover this fact can be proven in analysis alone. This has led people to consider reverse mathematical analyses of precisely which subsystems of second order arithmetic are needed. We go over the history of these results. Recently Montalban and Shore gave a precise delineation of the amount of determinacy provable in analysis. Their arguments use concretely given levels of the Gödel constructible hierarchy. It should be possible to lift those arguments to the amount of determinacy, properly including analytic determinacy, provable in stronger theories than the standard ZFC set theory. We summarise some recent joint work with Chris Le Sueur.

Thu, 30 Jan 2014

17:15 -
18:15

L6

Dugald Macpherson

(Leeds)

A pseudofinite group is an infinite model of the theory of finite groups. I will discuss what can be said about pseudofinite groups under various tameness assumptions on the theory (e.g. NIP, supersimplicity), structural results on pseudofinite permutation groups, and connections to word maps and generalisations.

Thu, 23 Jan 2014

17:15 -
18:15

L6

Itaï Ben Yaacov

(Lyon)

In joint work with T. Tsankov we study a (yet other) point at which model theory and dynamics intersect. On the one hand, a (metric) aleph_0-categorical structure is determined, up to bi-interpretability, by its automorphism group, while on the other hand, such automorphism groups are exactly the Roelcke precompact ones. One can further identify formulae on the one hand with Roelcke-continuous functions on the other hand, and similarly stable formulae with WAP functions, providing an easy tool for proving that a group is Roelcke precompact and for calculating its Roelcke/WAP compactification. Model-theoretic techniques, transposed in this manner into the topological realm, allow one to prove further that if R(G) = W(G); then G is totally minimal.

Thu, 28 Nov 2013

17:15 -
18:15

L6

James Studd

(Oxford)

The use of tensed language and the metaphor of set "formation" found in informal descriptions of the iterative conception of set are seldom taken at all seriously. This talk offers an axiomatisation of the iterative conception in a bimodal language and presents some reasons to thus take the tense more seriously than usual (although not literally).

Thu, 21 Nov 2013

17:15 -
18:15

L6

Alex Wilkie

(Manchester)

I am interested in integer solutions to equations of the form $f(x)=0$ where $f$ is a transcendental, globally analytic function defined in a neighbourhood of $\infty$ in $\mathbb{R}^n \cup \{\infty\}$. These notions will be defined precisely, and clarified in the wider context of globally semi-analytic and globally subanalytic sets.

The case $n=1$ is trivial (the global assumption forces there to be only finitely many (real) zeros of $f$) and the case $n=2$, which I shall briefly discuss, is completely understood: the number of such integer zeros of modulus at most $H$ is of order $\log\log H$. I shall then go on to consider the situation in higher dimensions.

Thu, 14 Nov 2013

17:15 -
18:15

L6

Lee Butler

(Bristol)

A major desideratum in transcendental number theory is a simple sufficient condition for a given real number to be irrational, or better yet transcendental. In this talk we consider various forms such a criterion might take, and prove the existence or non-existence of them in various settings.

Thu, 07 Nov 2013

17:15 -
18:15

L6

Dan Isaacson

(Oxford)

In {\it Was sind und was sollen die Zahlen?} (1888), Dedekind proves the Recursion Theorem (Theorem 126), and applies it to establish the categoricity of his axioms for arithmetic (Theorem 132). It is essential to these results that mathematical induction is formulated using second-order quantification, and if the second-order quantifier ranges over all subsets of the first-order domain (full second-order quantification), the categoricity result shows that, to within isomorphism, only one structure satisfies these axioms. However, the proof of categoricity is correct for a wide class of non-full Henkin models of second-order quantification. In light of this fact, can the proof of second-order categoricity be taken to establish that the second-order axioms of arithmetic characterize a unique structure?

Thu, 31 Oct 2013

17:15 -
18:15

L6

Piotr Kowalski

(Wroclaw)

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.

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