Past Logic Seminar

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

Abstract available at: http://people.maths.ox.ac.uk/~kirby/LInnocente.pdf

I will prove that certain pairs of ordered structures are dependent. There are basically two cases depending on whether the smaller structure is dense or discrete. I will discuss the proofs of two quite general theorems which construe the dividing line between these cases. Among examples are dense pairs of o-minimal structures in the first case, and tame pairs of o-minimal structures in the latter. This is joint work with P. Hieronymi.

[This is a joint seminar with OASIS]

A formulation of quantum mechanics in terms of symmetric monoidal categories

provides a logical foundation as well as a purely diagrammatic calculus for

it. This approach was initiated in 2004 in a joint paper with Samson

Abramsky (Ox). An important role is played by certain Frobenius comonoids,

abstract bases in short, which provide an abstract account both on classical

data and on quantum superposition. Dusko Pavlovic (Ox), Jamie Vicary (Ox)

and I showed that these abstract bases are indeed in 1-1 correspondence with

bases in the category of Hilbert spaces, linear maps, and the tensor

product. There is a close relation between these abstract bases and linear

logic. Joint work with Ross Duncan (Ox) shows how incompatible abstract

basis interact; the resulting structures provide a both logical and

diagrammatic account which is sufficiently expressive to describe any state

and operation of "standard" quantum theory, and solve standard problems in a

non-standard manner, either by diagrammatic rewrite or by automation.

But are there interesting non-standard models too, and what do these teach

us? In this talk we will survey the above discussed approach, present some

non-standard models, and discuss in how they provide new insights in quantum

non-locality, which arguably caused the most striking paradigm shift of any

discovery in physics during the previous century. The latter is joint work

with Bill Edwards (Ox) and Rob Spekkens (Perimeter Institute).

I will present a universal definition of the integers in the field of rational numbers, building on work discussed by Bjorn Poonen in his seminar last term. I will also give, via model theory, a geometric criterion for the non-diophantineness of Z in Q.

I show that the expansion of the real field by a total Pfaffian chain is model complete in a language with symbols for the functions in the chain, the exponential and all real constants. In particular, the expansion of the reals by all total Pfaffian functions is model complete.

(joint work with E. Hrushovski and A. Pillay)

If G is a definably compact, connected group definable in an o-minimal structure then, as is known, G/Z(G) is semisimple (no infinite normal abelian subgroup).

We show, that in every o-minimal expansion of an ordered group:

If G is a definably connected central extension of a semisimple group then it is bi-intepretable, over parameters, with the two-sorted structure (G/Z(G), Z(G)). Many corollaries follow for definably connected, definably compact G.
Here are two:

1. (G,.) is elementarily equivalent to a compact, connected real Lie group of the same dimension.

2. G can be written as an almost direct product of Z(G) and [G,G], and this last group is definable as well (note that in general [G,G] is a countable union of definable sets, thus not necessarily definable).

Cofinitary groups are subgroups of the symmetric group on the natural numbers

(elements are bijections from the natural numbers to the natural numbers, and

the operation is composition) in which all elements other than the identity

have at most finitely many fixed points. We will give a motivation for the

question of which isomorphism types are possible for maximal cofinitary

groups. And explain some of the results we achieved so far.

We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that

the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is

bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on

$S(h)$, admits a representation

$f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$

with each $\sigma_i$ being a sum of squares of real polynomials.

If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean.

There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however,

interested in an effective procedure to decide whether h is archimedean or not.

In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision

problem. Recently, S. Wagner has proved decidability for all dimensions using among others

model theoretic tools like the Ax-Kochen-Ershov Theorem.

Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points.

Fixed-point logics are a class of logics designed for formalising

recursive or inductive definitions. Being initially studied in

generalised recursion theory by Moschovakis and others, they have later

found numerous applications in computer science, in areas

such as database theory, finite model theory, and verification.

A common feature of most fixed-point logics is that they extend a basic

logical formalism such as first-order or modal logic by explicit

constructs to form fixed points of definable operators. The type of

fixed points that can be formed as well as the underlying logic

determine the expressive power and complexity of the resulting logics.

In this talk we will give a brief introduction to the various extensions

of first-order logic by fixed-point constructs and give some examples

for properties definable in the different logics. In the main part of

the talk we will concentrate on extensions of first-order

logic by least and inflationary fixed points. In particular, we

compare the expressive power and complexity of the resulting logics.

The main result will be to show that while the two logics have rather

different properties, they are equivalent in expressive power on the

class of all structures.

In recent years Schanuel’s Conjecture (SC) has played a fundamental role

in the Theory of Transcendental Numbers and in decidability issues.

Macintyre and Wilkie proved the decidability of the real exponential field,

modulo (SC), solving in this way a problem left open by A. Tarski.

Moreover, Macintyre proved that the exponential subring of R generated

by 1 is free on no generators. In this line of research we obtained that in

the exponential ring $(\mathbb{C}, ex)$, there are no further relations except $i^2 = −1$

and $e^{i\pi} = −1$ modulo SC. Assuming Schanuel’s Conjecture we proved that

the E-subring of $\mathbb{R}$ generated by $\pi$ is isomorphic to the free E-ring on $\pi$.

These results have consequences in decidability issues both on $(\mathbb{C}, ex)$ and

$(\mathbb{R}, ex)$. Moreover, we generalize the previous results obtaining, without

assuming Schanuel’s conjecture, that the E-subring generated by a real

number not definable in the real exponential field is freely generated. We

also obtain a similar result for the complex exponential field.

I will give a few model theoretic properties for fields with a Hasse derivation which are existentially closed. I will explain how some type-definable sets allow us to understand properties of some algebraic varieties, mainly concerning their field of definition.

I will explain the connection between Shelah's recent notion of strongly dependent theories and finite weight in simple theories. The connecting notion of a strong theory is new, but implicit in Shelah's book. It is related to absence of the tree property of the second kind in a similar way as supersimplicity is related to simplicity and strong dependence to NIP.

Bardakov and Tolstykh have recently shown that Richard Thompson's group

$F$ interprets the Arithmetic $(\mathbb Z,+,\times)$ with parameters. We

consider a class of infinite groups of piecewise affine permutations of

an interval which contains all the three groups of Thompson and some

classical families of finitely presented infinite simple groups. We have

interpreted the Arithmetic in all the groups of this class. In particular

we have obtained that the elementary theories of all these groups are

undecidable. Additionally, we have interpreted the Arithmetic in $F$ and

some of its generalizations without parameters.

This is a joint work with Tuna Altınel.

This is joint work with Assaf Hasson. We consider non-locally modular strongly minimal reducts of o-minimal expansions of reals. Under additional assumptions we show they have a Zariski structure.

We shall present work in progress in collaboration with E. Hrushovski on the geometry of spaces of stably dominated types in connection with non archimedean geometry \`a la Berkovich

There is a well-behaving class of dense ordered abelian groups called "regularly dense ordered abelian groups". This first order property of ordered abelian groups is introduced by Robinson and Zakon as a generalization of being an archimedean ordered group. Every dense subgroup of the additive group of reals is regularly dense. In this talk we consider subgroups of the multiplicative group, S, of all complex numbers of modulus 1. Such groups are not ordered, however they have an "orientation" on them: this is a certain ternary relation on them that is invariant under multiplication. We have a natural correspondence between oriented abelian groups, on one side, and ordered abelian groups satisfying a cofinality condition with respect to a distinguished positive element 1, on the other side. This correspondence preserves model-theoretic relations like elementary equivalence. Then we shall introduce a first-order notion of "regularly dense" oriented abelian group; all infinite subgroups of S are regularly dense in their induced orientation. Finally we shall consider the model theoretic structure (R,Gamma), where R is the field of real numbers, and Gamma is dense subgroup of S satisfying the Mann property, interpreted as a subset of R^2. We shall determine the elementary theory of this structure.

H. Jerome Keisler suggested to associate to each classical structure M a family of "random" structures consisting of random variables with values in M . Viewing the random structures as structures in continuous logic one is able to prove preservation results of various "good" model theoretic properties e.g., stability and dependence, from the original structure to its randomisation. On the other hand, simplicity is not preserved by this construction. The work discussed is mostly due to H.

Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).

I will speak about the Zilber trichotomy for weakly minimal difference varieties, and the definable structure on them.

A difference field is a field with a distinguished automorphism $\sigma$. Solution sets of systems of polynomial difference equations like

$3 x \sigma(x) +4x +\sigma^2(x) +17 =0$ are the quantifier-free definable subsets of difference fields. These \emph{difference varieties} are similar to varieties in algebraic geometry, except uglier, both from an algebraic and from a model-theoretic point of view.

ACFA, the model-companion of the theory of difference fields, is a supersimple theory whose minimal (i.e. U-rank $1$) types satisfy the Zilber's Trichotomy Conjecture that any non-trivial definable structure on the set of realizations of a minimal type $p$ must come from a definable one-based group or from a definable field. Every minimal type $p$ in ACFA contains a (weakly) minimal quantifier-free formula $\phi_p$, and often the difference variety defined by $\phi_p$ determines which case of the Zilber Trichotomy $p$ belongs to.

I will push Schanuel's conjecture in four directions: defining a dimension

theory (pregeometry), blurred exponential functions, exponential maps of

more general groups, and converses. The goal is to explain how Zilber's

conjecture on complex exponentiation is true at least in a "geometric"

sense, and how this can be proved without solving the difficult number

theoretic conjectures. If time permits, I will explain some connections

with diophantine geometry.

We survey the classification of structures interpretable in o-minimal theories in terms of thorn-minimal types. We show that a necessary and sufficient condition for such a structure to interpret a real closed field is that it has a non-locally modular unstable type. We also show that assuming Zilber's Trichotomy for strongly minimal sets interpretable in o-minimal theories, such a structure interprets a pure algebraically closed field iff it has a global stable non-locally modular type. Finally, if time allows, we will discuss reasons to believe in Zilber's Trichotomy in the present context

To understand the definable sets of a theory, it is helpful to have some invariants, i.e. maps from the definable sets to somewhere else which are invariant under definable bijections. Denef and Loeser constructed a very strong such invariant for the theory of pseudo-finite fields (of characteristic zero): to each definable set, they associate a virtual motive. In this way one gets all the known cohomological invariants of varieties (like the Euler characteristic or the Hodge polynomial) for arbitrary definable sets.

I will first explain this, and then present a generalization to other fields, namely to perfect, pseudo-algebraically closed fields with pro-cyclic Galois group. To this end, we will construct maps between the set of definable sets of different such theories. (More precisely:

between the Grothendieck rings of these theories.) Moreover, I will show how, using these maps, one can extract additional information about definable sets of pseudo-finite fields (information which the map of Denef-Loeser loses).

By classical results of Tarski and Artin-Schreier, the elementary theory of the field of real numbers can be axiomatized in purely Galois-theoretic terms by describing the absolute Galois group of the field. Using work of Ax-Kochen/Ershov and a p-adic analogue of the Artin-Schreier theory the same can be proved for the field $\mathbb{Q}_p$ of p-adic numbers and for very few other fields.

Replacing, however, the absolute Galois group of a field K by that of the rational function field $K(t)$ over $K$, one obtains a Galois-theoretic axiomatiozation of almost arbitrary perfect fields. This gives rise to a new approach to longstanding decidability questions for fields like

$F_p((t))$ or $C(t)$.

Countable Borel equivalence relations arise naturally as orbit equivalence

relations for countable groups. For each countable Borel equivalence relation E

there is an infinitary sentence such that E is equivalent to the isomorphism

relation on countable models of that sentence. For first order theories the

question is open.