Forthcoming events in this series
Linear equations over multiplicative groups in positive characteristic, sums of recurrences, and ergodic mixing
Abstract
Dependent Pairs
Abstract
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.
Models of quantum phenomena
Abstract
[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).
Defining Z in Q
Abstract
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.
Model completeness results for certain Pfaffian structures
Abstract
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.
Definably compact, connected groups are elementarily equivalent to compact real Lie groups
Abstract
(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).
17:00
On Intersection with Tori
Abstract
Isomorphism Types of Maximal Cofinitary Groups
Abstract
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.
15:15
Representations of positive real polynomials
Abstract
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.
16:00
Characterizing Z in Q with a universal-existential formula
Abstract
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 and Inductive Definitions
Abstract
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.
The real field with an irrational power function and a dense multiplicative subgroup
15:15
Schanuel’s Conjecture and free E-rings in o-minimal structures
Abstract
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.
15:15
Definability in differential Hasse fields and related geometric questions
Abstract
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.
14:15
Strong theories, weight, and the independence property
Abstract
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.
14:15
Arithmetic in groups of piece-wise affine permutations of an interval
Abstract
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.
10:00
Zariski reducts of o-minimal structures
Abstract
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.
14:15
Non Archimedian Geometry and Model Theory
Abstract
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
14:15
Small subgroups of the circle group
Abstract
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.