Past Logic Seminar

Zilber constructed an exponential field B, which is conjecturally isomorphic to the complex exponential field. He did so by giving axioms in an infinitary logic, and showing there is exactly one model of those axioms. Following a suggestion of Zilber, I will give a different list of axioms satisfied by B which, under a number-theoretic conjecture known as CIT, describe its complete first-order theory

It is known that the expansion of the real field by some quasianalytic algebras of functions are o-minimal and polynomially bounded. We prove that, for these structures, the preparation theorem for definable functions proved by L. van den Dries and P. Speissegger has an explicit form, from which it is easy to deduce a quantifier elimination result.

We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely algebraic geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a (small upgrade of a) theorem of Abramovich and Karu about weak toroidalization of morphisms. Our method also yields a new alternative approach to the model theory of henselian valued fields (including the Ax-Kochen-Ersov transfer principle and quantifier elimination).

Games are ubiquitous in set theory and in particular can be used to build models (often using some large cardinal property to justify the existence of strategies). As a reversal one can define large cardinal properties in terms of such games.

We look at some such that build models through indiscernibles, and that have recently had some effect on structures at aleph_2.

A $C${\em -relation} is the ternary relation induced by an ultrametric distance, in particular a valuation on a field, when we only remember the relation:

$C(x;y,z)$

iff $d(x,y)

   I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.

   The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.

We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
1

Consider the valued field $\mathbb{R}((\Gamma))$ of generalised series, with real coefficients and

monomials in a totally ordered multiplicative group $\Gamma$ . In a series of papers,

we investigated how to endow this formal algebraic object with the analogous

of classical analytic structures, such as exponential and logarithmic maps,

derivation, integration and difference operators. In this talk, we shall discuss

series derivations and series logarithms on $\mathbb{R}((\Gamma))$ (that is, derivations that

commute with infinite sums and satisfy an infinite version of Leibniz rule, and

logarithms that commute with infinite products of monomials), and investigate

compatibility conditions between the logarithm and the derivation, i.e. when

the logarithmic derivative is the derivative of the logarithm.

Vopenka's Principle is a very strong large cardinal axiom which can be used to extend ZFC set theory. It was used quite recently to resolve an important open question in algebraic topology: assuming Vopenka's Principle, localisation functors exist for all generalised cohomology theories. After describing the axiom and sketching this application, I will talk about some recent results showing that Vopenka's Principle is relatively consistent with a wide range of other statements known to be independent of ZFC. The proof is by showing that forcing over a universe satisfying Vopenka's Principle will frequently give an extension universe also satisfying Vopenka's Principle.

Motivic exponential integrals are an abstract version of p-adic exponential integrals for big p. The latter in itself is a flexible tool to describe (families of) finite expontial sums. In this talk we will only need the more concrete view of "uniform in p p-adic integrals"

instead of the abstract view on motivic integrals. With F. Loeser, we obtained a first transfer principle for these integrals, which allows one to change the characteristic of the local field when one studies equalities of integrals, which appeared in Ann. of Math (2010). This transfer principle in particular applies to the Fundamental Lemma of the Langlands program (see arxiv). In work in progress with Halupczok and Gordon, we obtain a second transfer principle which allows one to change the characteristic of the local field when one studies integrability conditions of motivic exponential functions. This in particular solves an open problem about the local integrability of Harish-Chandra characters in (large enough) positive characteristic.

In 1974 Haim Gaifman conjectured that if a first-order theory T is relatively categorical over T(P) (the theory of the elements satisfying P), then every model of T(P) expands to one of T.

The conjecture has long been known to be true in some special cases, but nothing general is known. I prove it in the case of abelian groups with distinguished subgroups. This is some way outside the previously known cases, but the proof depends so heavily on the Kaplansky-Mackey proof of Ulm's theorem that the jury is out on its generality.

The n-amalgamation property has recently been explored in connection with generalised imaginaries (groupoid imaginaries) by Hrushovski. This property is useful when studying models of a stable theory together with a generic automorphism, e.g.

elimination of imaginaries (e.i.) in ACFA may be seen as a consequence of 4-amalgamation (and e.i.) in ACF.

The talk is centered around 4-amalgamation of stably dominated types in algebraically closed valued fields. I will show that 4-amalgamation holds in equicharacteristic 0, even for systems with 1 vertex non stably dominated. This is proved using a reduction to the stable part, where 4-amalgamation holds by a result of Hrushovski. On the other hand, I will exhibit an NIP (even metastable) theory with 4-amalgamation for stable types but in which stably dominated types may not be 4-amalgamated.

Simple groups of Lie type have a purely group theoretic characterization in terms of subgroup configurations. We here show that for certain Fraisse limits, the automorphism group is a simple group.

Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.

One may answer a question of Macintyre by showing that there are recursive existentially closed dimension groups.  One may build such groups having most of the currently known special properties of finitely generic dimension groups, though no finitely generic dimension group is arithmetic.

tba

As everyone knows, one popular notion of a (Scott-Ersov) domain is defined as a bounded complete algebraic cpo. These are closely related to algebraic lattices: (i) A domain becomes an algebraic lattice with the adjunction of an (isolated) top element. (ii) Every non-empty Scott-closed subset of an algebraic lattice is a domain. Moreover, the isolated (= compact) elements of an algebraic lattice form a semilattice (under join). This semilattice has a zero element, and, provided the top element is isolated, it also has a unit element. The algebraic lattice itself may be regarded as the ideal completion of the semilattice of isolated elements. This is all well known. What is not so clear is that there is an easy-to-construct domain of countable semilattices giving isomorphic copies of all countably based domains. This approach seems to have advantages over both the so-called "information systems" or more abstract lattice formulations, and it makes definitions of solutions to domain equations very elementary to justify. The "domain of domains" also has a natural computable structure

Jointly with Number Theory

Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety.

In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance.

We discuss some new results on the distribution of such "special points"

on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions.

I will give a brief introduction to the theory of heights in the talk.

We consider valued fields with a distinguished isometry or contractive derivation, as valued modules over the Ore ring of difference operators. This amounts to study linear difference/differential

equations with respect to the distinguished isometry/derivation.

Under certain assumptions on the residue field, but in all characteristics, we obtain quantifier elimination in natural languages, and the absence of the independence property.

We will consider other operators of interest.

Elimination of wild ramification is used in the structure theory of valued function fields, with applications in areas such as local uniformization (i.e., local resolution of singularities) and the model theory of valued fields. I will give a survey on the role that Artin-Schreier extensions play in the elimination of wild ramification, and corresponding main theorems on the structure of valued function fields. I will show what these results tell us about local uniformization. I have shown that local uniformization is always possible after a separable extension of the function field of the algebraic variety (separable "alteration"). This was extended to the arithmetic case in joint work with Hagen Knaf. Recently, Michael Temkin has proved local uniformization by purely inseparable alteration.

Further, I will describe a classification of Artin-Schreier extensions with non-trivial defect. It can be used to improve one of the above mentioned main theorems ("Henselian Rationality"). This could be a key for a purely valuation theoretical proof of Temkin's result. On the other hand, the classification shows that separable alteration and purely inseparable alteration are just two ways to eliminate the critical defects. So the existence of these two seamingly "orthogonal" local uniformization results does not necessarily indicate that local uniformization without alteration is possible.

(IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber

( IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber

Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.

We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals. We suggest a new construction for approximating a large class of definable sets, including the sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets.

Those compact sets bound from above the homotopies and homologies of the approximated sets.

The construction is applicable to images under definable maps.

Based on this construction we refine the previously known upper bounds on Betti numbers of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae, and prove similar new upper bounds, individual for different Betti numbers, for their images under arbitrary continuous definable maps.

Joint work with A. Gabrielov.

The talks will discuss relations between two major conjectures in the theory of groups of finite Morley rank, a modern chapter of model theoretic algebra. One conjecture, the famous the Cherlin-Zilber Algebraicity Conjecture formulated in 1970-s states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture, due to Hrushovski and more recent, states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points.

Hrushovski showed that the Cherlin-Zilber Conjecture implies his conjecture. Proving Hrushovski's Conjecture and reversing the implication would provide a new efficient approach to proof of Cherlin-Zilber Conjecture.

Meanwhile, the machinery that is already available for the work at pseudofinite/finite Morley rank interface already yields an interesting

result: an alternative proof of the Larsen-Pink Theorem (the latter says, roughly speaking, that "large" finite simple groups of matrices are Chevalley groups over finite fields).

I will discuss the special properties of dimension groups obtained by model-theoretic forcing

Let R be a number field (or a recursive subring of anumber field) and consider the polynomial ring R[T].

We show that the set of polynomials with integercoefficients is diophantine (existentially definable) over R[T].

Applying a result by Denef, this implies that everyrecursively enumerable subset of R[T]^k is diophantine over R[T].

I'll give a brief survey of what is known about the density of rational points on definable sets in o-minimal expansions of the real field, then discuss improving these results in certain cases.

After a short introduction to the section conjecture, I plan to present a "minimalistic" form of the birational p-adic section conjecture. The result is related to both: Koenigsmann's proof of the birational p-adic section conjecture, and a "minimalistic" Galois characterisation of formally p-adic valuations.

VC dimension and density are properties of a collection of sets which come from probability theory.  It was observed by Laskowski that there is a close tie between these notions and the model-theoretic property called NIP. This tie results in many examples of collections of sets that have finite VC dimension. In general, it is difficult to find upper bounds for the VC dimension, and known bounds are mostly very large. However, the VC density seems to be more accessible. In this talk, I will explain all of the above acronyms, and present a theorem which gives an upper bound (in some cases optimal) on the VC density of formulae in some examples of NIP theories. This represents joint work of myself with M. Aschenbrenner, A. Dolich, D. Macpherson and S.

Starchenko.

I will speak about the CBP (canonical base property) for types of finite SU-rank. This property first appears in a paper by Pillay and Ziegler, who show that it holds for types of finite rank in differentially closed fields of characteristic 0, as well as in existentially closed difference fields. It is unknown whether this property holds for all finite rank types in supersimple theories. I will first recall the definition of a canonical base, and give some natural examples. I will then  talk about a reduction of the problem (which allows one to extend the Pillay-Ziegler result to existentially closed fields of any characteristic), and finally derive some consequences of the CBP, in particular the UCBP, thus answering a question of Moosa and Pillay.  If time permits, I will show an application of these results to difference

fields.

I will talk about two pieces of work on finite covers:

(i) Work of Hrushovski which, for a stable theory, links splitting of certain finite covers with higher amalgamation properties;

(ii) Joint work of myself and Elisabetta Pastori which uses group cohomology to investigate some non-split finite covers of the set of k-sets from a disintegrated set.