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\begin{document}

\begin{center}{\huge Enumerable Sets and Quasi-Reducibility}\end{center}
\begin{center}{\large Geoffrey LaForte}\end{center}
\begin{center}{\large Victoria University
of Wellington, Wellington, New Zealand}\end{center}
\begin{center}{\large (joint with R. Downey, Victoria University 
and A. Nies, University of Chicago)}\end{center}

\bigskip

Classical recursion theory first arose in order to study the
inherent difficulty of mathematical problems.   By far the most deeply studied
notion relating the difficulty of one problem to another has been
that of Turing reducibility; nevertheless, for specific problems,
particularly those arising in the study of algebraic structures,
other reducibilities are of interest.

For example, in the case of
(computably presentable) infinite dimensional vector
spaces, it turns out that the inherent difficulty of constructing
bases for subspaces coincides exactly with the relation of $wtt$
reducibility, rather than Turing reducibility.  A similar situation
arises in combinatorial group theory, where so-called
\emph{quasi-reducibility}, or $Q$-reducibility, turns out to be a more natural
means of comparing enumerable word problems than ordinary $T$-reducibility.
In the case of groups without enumerable word problems, the notion of
\emph{Ziegler reducibility} (see \cite{zig}) is the natural
one.  Both of these reducibilities arise as abstractions of the
process of effectively using sets of equations true
in one structure to check the truth of equations in another.

We study the upper semilattice of enumerable $Q$-degrees, giving several
results comparing this structure with the
more familiar structure of the enumerable Turing degrees.
We also use coding methods as in \cite{nss}
to show that the elementary theory of
$\code{\mathcal{R}_Q,\qr}$ is undecidable.
The proofs of these results have additional technical interest,
particularly because of the more subtle methods that have to
replace standard permitting techniques for ensuring a constructed
set will be computable from some set given in advance.
Our constructions illustrate the ideas needed
to modify standard finite injury and
infinite injury
constructions in order to deal with the enumerable $Q$-degrees, as
well as shedding some light on the (easier) proofs of the analogous Turing
results, by way of contrast.




%    Bibliography
%

\begin{thebibliography}{99}

\bibitem{nss}
A. Nies, R. Shore, T. Slaman, {\em Standard models of arithmetic
and definability in the enumerable degrees},
Bulletin of Symbolic Logic, Dec. 1996.


\bibitem{zig}
M. Ziegler, {\em Algebraisch abgeschlossen gruppen} in {\em Word
Problems II, The Oxford Book}, ed. S.I. Adian, W.W. Boone, and G.
Higman, North Holland, 1980, pp.449-576.


\end{thebibliography}

\end{document}



Abstract for Dag Normann


Dag Normann

The Superjump as a type constructor

Dag Normann
University of Oslo
Norway


Intuitionistic type theory represents the investigation of complex
set-theoretical constructions using constructive means.
A general philosophical question is how complex constructions and closure
principles we can grasp from an intuitionistic point of view. A special
case was discussed at a workshop in Uppsala in 1996:
Is it possible to grasp the Mahlo-property intuitionistically?

In this talk we will discuss the kind of type constructors needed in order
to obtain a structure of the same classical complexity as the first
recursively Mahlo ordinal. Harrington showed in 1972 that this ordinal
corresponds to the complexity of computations in the Superjump, a type 3
functional described by Gandy ten years earlier.
We will use typed structures of effective domains with totality to simulate
computations in the Superjump. Each application of the Superjump will
correspond to the application of a universe operator in type theory. The
technical result will be the match of the complexity of the constructed
typed structure and the computation theory of the superjump.

The construction will indicate which universes we must axiomatise to exist
in type theory in order to grasp the concept of Mahloness
intuitionistically from a type theory perspective.



Abstract for Deirdre Haskell
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% abstract for talk ASL - Leeds 1997
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\magnification=1200
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\centerline{One-dimensional $p$-adic subanalytic sets}
\smallskip
\centerline{Deirdre Haskell}
\centerline{College of the Holy Cross}
\centerline{Worcester, MA, USA}
\medskip

In their 1998 paper ``$p$-adic and real subanalytic sets'', J.~Denef
and L~van den Dries use model-theoretic techniques to develop
a notion of subanalytic sets over the $p$-adics, analogous to
the notion from algebraic geometry in the reals. They prove quantifier 
elimination for the
$p$-adics in a language containing restricted analytic functions
and a partial division function. Essentially the same proof
can be used for the reals. In particular, subanalytic subsets of the
$p$-adic line are shown to be semialgebraic. In our recent work,
L.~van den Dries, D.~Macpherson and I have extended this last
result to arbitrary models. This shows that the theory of
the $p$-adics with restricted analytic functions is P-minimal, 
analogous to the o-minimality of the reals with restricted
analytic functions.  In my talk, I will review some of the 
background material and discuss some of the methods used to
prove our theorem. 

\end






Title and abstract from Katrin Tent (Wuerzburg)


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\title{Fields and Algebraic Groups}
\author{L.Kramer, K. Tent}
\date{}
\maketitle


We show how the field of definition, $k$, of 
an isotropic absolutely almost simple $k$-group $G$
may be defined in the group $G(k)$ of $k$-rational points
in pure group theoretic language.
Our construction is inspired by the famous paper of Borel-Tits
\cite{BT3}.
However, since we are not dealing with homomophisms, we need
much less of the heavy machinery used by Borel-Tits.
The construction is as follows: inside the center of the unipotent 
radical of a minimal parabolic $k$-subgroup we pick a suitable
1-dimensional $k$-vector space. This defines the additive
group structure of $k$; the action of the center of a Levi $k$-subgroup
$L$ induces the multiplicative structure. Here we make use of
a result of Azad-Barry-Seitz \cite{ABS} concerning the action of
$L$.

This answers a question by
Borovik-Nesin \cite{BN} p. 367 (under the additional assumption 
that $G$ is $k$-isotropic). Easy examples also show that if $G$
is not absolutely simple, then $G(k)$ might be a simple group of 
finite Morley rank without $k$ being algebraically closed since $k$ 
might be the 'wrong' field of
definition. In most such cases one can replace $G$ by an
absolutely simple group $H$ with $H(l)\cong G(k)$ for some suitable 
field extension $l$ of $k$ - thus $l$ has to be algebraically closed.

\begin{thebibliography}{99}

\bibitem{ABS}
H. Azad, M. Barry, G.Seitz,
On the structure of parabolic subgroups,
Comm. Algebra 18, 551--562 (1990)

\bibitem{BT3}
A. Borel, J. Tits,
Homomorphismes ``abtraits'' de groupes alg\'ebriques simples,
Ann. Math. 97, 499--571 (1973)

\bibitem{BN}
A. Borovik, A. Nesin,
{\em Groups of finite Morley rank,}
Oxford Science Publ. 1994

\end{thebibliography}

\end{document}