Past Logic Seminar

Van den Dries has proved the decidability of the ring of algebraic integers, the integral closure of the ring of integers in
the algebraic closure of the rationals.  A well-established analogy leads to ask the same question for the ring of complex polynomials.
This turns out to go the other way, interpreting the rational field.    An interesting structure on the
limit of Jacobians of all complex curves is encountered along the way. 

The concept of pseudofinite dimension for ultraproducts of finite structures was introduced by Hrushovski and Wagner. In this talk, I will present joint work with D. Macpherson and C. Steinhorn in which we explored conditions on the (fine) pseudofinite dimension that guarantee simplicity or supersimplicity of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of droping of the pseudofinite dimension. Also, under a suitable assumption, it can be shown that a measure-theoretic condition is equivalent to loc

*/ /*-->*/ A {\em monoid} is a semigroup with identity. A {\em finitary property for monoids} is a property guaranteed to be satisfied by any finite monoid. A good example is the maximal condition on the lattice of right ideals: if a monoid satisfies this condition we say it is {\em weakly right noetherian}. A monoid $S$ may be represented via mappings of sets or, equivalently and more concretely, by {\em (right) $S$-acts}. Here an $S$-act is a set $A$ together with a map $A\times S\rightarrow A$ where $(a,s)\mapsto as$, such that

for all $a\in A$ and $s,t\in S$ we have $a1=a$ and $(as)t=a(st)$. I will be speaking about finitary properties for monoids arising from model theoretic considerations for $S$-acts.

Let $S$ be a monoid and let $L_S$ be the first-order language of $S$-acts, so that $L_S$ has no constant or relational symbols (other than $=$) and a unary function symbol $\rho_s$ for each $s\in S$. Clearly $\Sigma_S$ axiomatises the class of $S$-acts, where

\[\Sigma_S=\big\{ (\forall x)(x\rho_s \rho_t=x\rho_{st}):s,t\in S\big\}\cup\{ (\forall x)(x\rho_1=x)

\}.\]

Model theory tells us that $\Sigma_S$

has a model companion $\Sigma_S^*$ precisely when the class

${\mathcal E}$ of existentially closed $S$-acts is axiomatisable and

in this case, $\Sigma_S^*$ axiomatises ${\mathcal E}$. An old result of Wheeler tells us that $\Sigma_S^*$ exists if and only if for every finitely generated right congruence $\mu$ on $S$, every finitely generated $S$-subact of $S/\mu$ is finitely presented, that is, $S$ is {\em right coherent}. Interest in right coherency also arises from other considerations such as {\em purity} for $S$-acts.

Until recently, little was known about right coherent monoids and, in particular, whether free monoids are (right) coherent.

I will present some work of Gould, Hartmann and Ru\v{s}kuc in this direction: specifically we answer positively the question for free monoids.

Where $\Sigma_S^*$ exists, it is known to be

stable, and is superstable if and only if $S$ is weakly right noetherian.

By using an algebraic description of types over $\Sigma_S^*$ developed in the 1980s by Fountain and Gould,

we can show that $\Sigma_S^*$ is totally

transcendental if and only if $S$ is weakly right noetherian and $S$ is {\em ranked}. The latter condition says that every right congruence possesses a finite Cantor-Bendixon rank with respect to the {\em finite type topology}.

Our results show that there is a totally transcendental theory of $S$-acts for which Morley rank of types does not coincide with $U$-rank, contrasting with the corresponding situation for modules over a ring.

I will consider automorphism groups of countable structures acting continuously on compact spaces: the viewpoint of topological dynamics. A beautiful paper of Kechris, Pestov and Todorcevic makes a connection between this and the ‘structural Ramsey theory’ of Nesetril, Rodl and others in finite combinatorics. I will describe some results and questions in the area and say how the Hrushovski predimension constructions provide answers to some of these questions (but then raise more questions). This is joint work with Hubicka and Nesetril.

An abelian l-group G is essentially a partially ordered subgroup of functions from a set to a totally ordered abelian group such

that G is closed under taking finite infima and suprema. For example, G could be the continuous semi-linear functions defined on the open
unit square, or, G could be the continuous semi-algebraic functions defined in the plane with values in (0,\infty), where the group
operation is multiplication. I will show how G, under natural geometric assumptions, can be interpreted (in a weak sense) in its lattice of
zero sets. This will then be applied to the model theory of natural divisible abelian l-groups. For example we will see that the
aforementioned examples are elementary equivalent. (Parts of the results have been announced in a preliminary report from 1987 by F. Shen
and V. Weispfenning.)

Ever since the compilers of Euclid's Elements gave the "definitions" that "a point is that which has no part" and "a line is breadthless length", philosophers and mathematicians have worried that the basic concepts of geometry are too abstract and too idealized.  In the 20th century writers such as Husserl, Lesniewski, Whitehead, Tarski, Blumenthal, and von Neumann have proposed "pointless" approaches.  A problem more recent authors have emphasized it that there are difficulties in having a rich theory of a part-whole relationship without atoms and providing both size and geometric dimension as part of the theory.  A possible solution is proposed using the Boolean algebra of measurable sets modulo null sets along with relations derived from the group of rigid motions in Euclidean n-space. 

Given a collection F of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from F. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from F in the neighbourhood of a generic point. We prove that this description is not complete anymore in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which each suggest that at least three new definable operations need to be added to Wilkie's description in order to capture local definability in its entirety. The construction illustrates the interaction between resolution of singularities and definability in the o-minimal setting. Joint work with O. Le Gal, G. Jones, J. Kirby.

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.

References:
[1] D'Aquino, P. - Kuhlmann, S. - Lange, K. : A valuation theoretic characterization ofrecursively saturated real closed fields ,
Journal of Symbolic Logic, Volume 80, Issue 01, 194-206 (2015)
[2] Carl, M. - D'Aquino, P. - Kuhlmann, S. : Value groups of real closed fields and
fragments of Peano Arithmetic, arXiv: 1205.2254, submitted
[3] D'Aquino, P. - Kuhlmann, S : Saturated o-minimal expansions of real closed fields, to appear in Algebra and Logic (2016)
[4] Kuhlmann, S. :Ordered Exponential Fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000)
 

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on
his notion of ``extremal valued fields''. He proved that algebraically
complete discretely valued fields are extremal. However, the proof
contained a mistake, and it turned out in 2009 through an observation by
Sergej Starchenko that Ershov's original definition leads to all
extremal fields being algebraically closed. In joint work with Salih
Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate
definition and then characterized extremal valued fields in several
important cases.

We call a valued field (K,v) extremal if for all natural numbers n and
all polynomials f in K[X_1,...,X_n], the set of values {vf(a_1,...,a_n)
| a_1,...,a_n in the valuation ring} has a maximum (which is allowed to
be infinity, attained if f has a zero in the valuation ring). This is
such a natural property of valued fields that it is in fact surprising
that it has apparently not been studied much earlier. It is also an
important property because Ershov's original statement is true under the
revised definition, which implies that in particular all Laurent Series
Fields over finite fields are extremal. As it is a deep open problem
whether these fields have a decidable elementary theory and as we are
therefore looking for complete recursive axiomatizations, it is
important to know the elementary properties of them well. That these
fields are extremal could be an important ingredient in the
determination of their structure theory, which in turn is an essential
tool in the proof of model theoretic properties.

The notion of "tame valued field" and their model theoretic properties
play a crucial role in the characterization of extremal fields. A valued
field K with separable-algebraic closure K^sep is tame if it is
henselian and the ramification field of the extension K^sep|K coincides
with the algebraic closure. Open problems in the classification of
extremal fields have recently led to new insights about elementary
equivalence of tame fields in the unequal characteristic case. This led
to a follow-up paper. Major suggestions from the referee were worked out
jointly with Sylvy Anscombe and led to stunning insights about the role
of extremal fields as ``atoms'' from which all aleph_1-saturated valued
fields are pieced together.

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

I will report on joint work with Arno Fehm in which we apply
our previous `existential transfer' results to the problem of
determining which fields admit diophantine nontrivial henselian
valuation rings and ideals. Using our characterization we are able to
re-derive all the results in the literature. Also, I will explain a
connection with Pop's large fields.

Abstract: (Joint work with Sylvy Anscombe) We consider four properties 
of a field K related to the existence of (definable) henselian 
valuations on K and on elementarily equivalent fields and study the 
implications between them. Surprisingly, the full pictures look very 
different in equicharacteristic and mixed characteristic.

 If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame.  We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases.  Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing:  in this case, we are looking at the so-called "generic multiverse".

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations. 
 

Let M be an algebraic curve over an algebraically closed field and let
$(M, ...)$ be a strongly minimal non-locally modular structure with
basic relations definable in the full Zariski language on $M$. In this
talk I will present the proof of the fact that $(M, ...)$ interprets
an algebraically closed field.

We consider the decision problem of determining whether an exponential
polynomial has a real zero.  This is motivated by reachability questions
for continuous-time linear dynamical systems, where exponential
polynomials naturally arise as solutions of linear differential equations.

The decidability of the Zero Problem is open in general and our results
concern restricted versions.  We show decidability of a bounded
variant---asking for a zero in a given bounded interval---subject to
Schanuel's conjecture.  In the unbounded case, we obtain partial
decidability results, using Baker's Theorem on linear forms in logarithms
as a key tool.  We show also that decidability of the Zero Problem in full
generality would entail powerful new effectiveness results concerning
Diophantine approximation of algebraic numbers.

This is joint work with Ventsislav Chonev and Joel Ouaknine.

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.