Forthcoming events in this series
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
(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
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
Fixed-Point Logics and Inductive Definitions
Abstract
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 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
14:15
Strong theories, weight, and the independence property
Abstract
14:15
Arithmetic in groups of piece-wise affine permutations of an interval
Abstract
$F$ 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
14:15
Non Archimedian Geometry and Model Theory
Abstract
14:15
Small subgroups of the circle group
Abstract
14:15
Randomised structures and theories
Abstract
Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).
10:00
Minimal definable sets in difference fields.
Abstract
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.