Past Logic Seminar

In contrast to the Artin-Schreier Theorem, its p-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of p-adic fields.  We plan to give a characterization of p-adic p-Henselian valuations in an essentially finite way. This relates to the Z/p metabelian form of the birational p-adic Grothendieck section conjecture.

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and 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. I will also discuss some joint work with Adam Harris in which we try to build a pseudo-j-function.

In a series of recent papers David Masser and Umberto Zannier proved the relative Manin-Mumford conjecture for abelian surfaces, at least when everything is defined over the algebraic numbers. In a further paper with Daniel Bertrand and Anand Pillay they have explained what happens in the semiabelian situation, under the same restriction as above.

At present it is not clear that these results are effective. I'll discuss joint work with Philipp Habegger and Masser and with Harry Schimdt in which we show that certain very special cases can be made effective. For instance, we can effectively compute a bound on the order of a root of unity t such that the point with abscissa 2 is torsion on the Legendre curve with parameter t.

**Note change of room**

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Number Theory. Note unusual time and place**

Standard commutative algebra is based on the notions of commutative monoid, Abelian group and commutative ring. In recent years, motivations from category theory, algebraic geometry, and mathematical logic led to the development of an area that may be called commutative 2-algebra, in which the notions used in commutative algebra are replaced by their category-theoretic counterparts (e.g. commutative monoids are replaced by  symmetric monoidal categories). The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra, and to outline how this suggests counterparts of basic aspects of algebraic geometry. In particular, I will describe some joint work with Andre’ Joyal on operads and analytic functors in this context.

In joint work with Todor Tsankov, we show that the automorphism groups of countable, omega-categorical structures have Kazhdan's property (T). The proof uses Tsankov's work on the unitary representations of these groups, together with a construction of a particular free subgroup of the automorphism group.

There is a tight fit between type theories à la Martin-Löf and constructive set theories such as Constructive Zermelo-Fraenkel set theory, CZF, and its extension as well as classical Kripke-Platek set theory and extensions thereof. The technology for determining their (exact) proof-theoretic strength was developed in the 1990s. The situation is rather different when it comes to type theories (with universes) having the impredicative type of propositions Prop from the Calculus of Constructions that features in some powerful proof assistants. Aczel's sets-as-types interpretation into these type theories gives rise to  rather unusual set-theoretic axioms: negative power set and negative separation. But it is not known how to determine the proof-theoretic strengths of intuitionistic set theories with such axioms via familiar classical set theories (though it is not difficult to see that ZFC plus infinitely many inaccessibles provides an upper bound). The first part of the talk will be a survey of known results from this area. The second part will be concerned with the rather special computational and proof-theoretic behavior of such theories.

A compact space is a Rosenthal compactum if it can be embedded into the space of Baire class 1 functions on a Polish space. Those objects have been well studied in functional analysis and set theory. In this talk, I will explain the link between them and the model-theoretic notion of NIP and how they can be used to prove new results in model theory on the topology of the space of types.
 

NOTE CHANGE OF TIME AND PLACE

It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.

We present some recent work - joint with Arno Fehm - in which we give an `existential Ax-Kochen-Ershov principle' for equicharacteristic henselian valued fields. More precisely, we show that the existential theory of such a valued field depends only on the existential theory of the residue field. In residue characteristic zero, this result is well-known and follows from the classical Ax-Kochen-Ershov Theorems. In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's theory of tame fields. One corollary is an unconditional proof that the existential theory of F_q((t)) is decidable. We will explain how this relates to the earlier conditional proof of this result, due to Denef and Schoutens.
 

The field of transseries was introduced by Ecalle to give a solution to Dulac's problem, a weakening of Hilbert's 16th problem. They form an elementary extension of the real exponential field and have received the attention of model theorists. Another such elementary extension is given by Conway's surreal numbers, and various connections with the transseries have been conjectured, among which the possibility of introducing a Hardy type derivation on the surreal numbers. I will present a complete solution to these conjectures obtained in collaboration with Vincenzo Mantova.
 

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).

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.

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.
 

Note: joint with Philosophy of Physics.

Venue: Lecture Room, Radcliffe Humanities, ROQ.

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.
 

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.

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.

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.

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. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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?

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 &gt;&gt; 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.