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%Galois-types:

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%the Stone space: write \gaS  instead of   S

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\begin{document}
%\input jbmac
%\hbox{}\vskip -3in
\title{TAMENESS }

\vskip .1in
\author{John T. Baldwin
\\ Department of Mathematics, Statistics and  Computer
Science\\University of Illinois at Chicago\\
\hskip -1in www.math.uic.edu/~jbaldwin}


%def\LARGE{\Large}

\begin{landscape}
%\LARGE
%\Large
%\textwidth 9in \textheight 6in



\begin{slide}
\hbox{} \vskip -3in \maketitle
\end{slide}


\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf TOPICS}

1. Some history

2.  background on tameness

3.  A new sufficient condition and application

4.  Non-tameness

\end{slide}

\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf HISTORY AND CONTEXT}

In 1970 logic was assimilating striking new methods:


recursion theory:  priority method

set theory:  forcing, diamonds, descriptive set theory

model theory:  Morley's theorem

proof theory: ?




\end{slide}

\begin{slide}
\hbox{} \vskip -2.5in \centerline{ \bf FORCING FESTIVAL 1973?}



 East
Coast Model Theory (Yale): small models; focus on first order
theories of `real' structures; definability in individual models

tools: compactness, quantifier elimination in natural languages

West Coast Model Theory (Berkeley, Madison, and Cornell):  Models
of arbitrary cardinality; arbitrary theories: first order and
infinitary

tools: compactness, Erdos-Rado, Ehrenfeucht-Mostowski; quantifier
elimination by fiat; combinatorial and sometimes axiomatic set
theory; Morley rank; indiscernibles

\end{slide}
\begin{slide}
\hbox{}
 %\vskip -2in
  \centerline{ \bf SHELAH'S REVOLUTION}

\begin{enumerate}
\item Clsssification of theories (not models 1974):
\begin{enumerate}
\item frees first order model theory from axiomatic set theory
\item systematizes model theoretic investigations by providing a
general tool
\end{enumerate}
\item monster model -- universal domain
\item dependence relation (forking)
\item analyze types locally
\begin{enumerate}
\item geometry of types
\item nonorthogonality, $p$-regular types, internality,
\end{enumerate}
\item decompose all models into  models of size $|L|$
\item uncountable languages
\end{enumerate}


\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf Marxist Synthesis }


Pillay (2000): `There is only one model theory'.
\end{slide}


\begin{slide}
\hbox{} \vskip -3in \centerline{LET MANY FLOWERS GROW    }

Models of Arithmetic,

finite model theory,

universal algebra,

model theory of alternative logics,

abstract model theory,

admissible model theory

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf IDEOLOGY   }

Can the study of large ($> 2^{\aleph_0}$) structures seriously
impact the understanding of structures discovered in the 19th
century?

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf IDEOLOGY   }

Can the study of large ($> 2^{\aleph_0}$) structures seriously
impact the understanding of structures discovered in the 19th
century?

METHODOLOGICALLY: The answer is decisively yes.  The tools of
classifying theories (stability, superstability) orthogonality,
$p$-regular decompositions, etc., invented by Shelah to study the
uncountable spectrum of {\em arbitrary} first order theories (in
possibly uncountable languages), were converted through geometric
stability theory as fundamental tools used in the modern model
theoretic investigations of algebraic structures.


\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ }

Properly axiomatized categoricity implies quantifier elimination.


Can the connection be stronger?  In a very rough sense, Morley's
theorem says no.  At least for categoricity, in first order logic
the story is over at $\aleph_1$.

Is the answer different for infinitary logic?

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{STRONGER IDEOLOGY } For Shelah and
probably Tarski, this is not an interesting question.  Their goal
is to understand mathematics not just the mathematics that has
already been long investigated.
\end{slide}

\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf ABSTRACT ELEMENTARY CLASSES}

AEC's are `theories' not logics.

The structures are `ensembliste'; Tarski Structures.

{\bf Methodological Problem}  Understand the exact connection
between AEC and the cats-positive bounded-continuous logic
configuration.


\end{slide}


\begin{slide}
\hbox{} \vskip -3in

\centerline{\bf MONSTER MODEL} \vskip .3in {\bf Definition. } The
model
 $M$ is {\em $\kappa$-$({\bK}, \subm)$-homogeneous}
 if
 $A\leq M, A\leq B\in \bK$ and $|B| \leq \kappa$ implies there exists
 $B'\leq M $
such that $ B\cong_{A} B'$.

\vskip .3in

Standard arguments show:

\begin{theorem} If a class  $(\boK, \subm)$  has the amalgamation property
and the joint embedding property, for every $\kappa$, there is a
structure $M_\kappa$ which is
 $\kappa$-$({\bK},\subm)$-homogeneous.
\end{theorem}

For sufficiently large $\kappa$, this is the {\bf Monster model}.

\end{slide}



\begin{slide}
\hbox{} \vskip -3in \centerline{GALOIS TYPES}

Define:

$$(M,a,N) \cong (M,a',N')$$

if there exists $N''$ and strong embeddings $f,f'$ taking $N, N'$
into $N''$ which agree on $M$ and with

$$f(a) =f'(a').$$
\end{slide}
\begin{slide}
\hbox{} \vskip -3in `The Galois type of $a$ over $M$ in $N$' is
the same as `the Galois type of $a'$ over $M$ in $N'$'

if $(M,a,N)$ and $(M,a',N')$ are in the same class of the
equivalence relation generated by $\cong$.
\end{slide}

\begin{slide}
\hbox{} \vskip -3in\centerline{AMALGAMATION PROPERTY} If
$(\bK,\subm)$ has the amalgamation property:

\begin{enumerate}
\item $\cong$ is an equivalence relation.
\item $(\bK,\subm)$ has a monster model.
\end{enumerate}

$\gaS(M)$ denotes the set of Galois types over $M$.




\end{slide}


\begin{slide}
\hbox{} \vskip -3in\centerline{GALOIS TYPES IN THE MONSTER}

If $(\bK,\subm)$ has a monster model $ \Mscr$, $$\gatp(a/M) =
\gatp(a'/M)$$
 if $a$ and $a'$ are conjugate under an automorphism
of $\Mscr$ fixing $M$.
\end{slide}


\begin{slide}
\hbox{} \vskip -3in\centerline{\bf SYNTACTIC TYPES}

Fix a logic $\Lscr$.  (E.g. $L_{\omega,\omega},
L_{\omega_1,\omega}, L_{\omega_1,\omega}(Q)$.



 Then

$$\tp(a/M,N) = \tp(a'/M,N')$$

if $a$ satisfies the same $\Lscr$-formulas with parameters from
$M$ in $N$ as

$a'$ satisfies in $N'$.

\end{slide}


\begin{slide}
\hbox{} \vskip -3in\centerline{ \bf KEY PROPERTIES}

Compactness: If $\langle p_i:i<\delta\rangle$ is an increasing
sequence of syntactic types $\bigcup_{i<\delta} p_i$ is a
syntactic type.

Locality:  If $M = \bigcup_{i<\delta}M_i$, $p\neq q \in S(M)$ then
there exists an $i$ such that $p\restriction M_i \neq
q\restriction M_i $.

Tameness: If $p,q \in S(M)$ and $p\restriction N = p\restriction
N$ for every small submodel $N$, then $p=q$.
\end{slide}


\begin{slide}
\hbox{} \vskip -3in\centerline{ \bf TAMENESS PARAMETERS}

If for every model $M$ of cardinality $\mu$,

if $p\restriction N = q\restriction N$ for every  submodel $N$
with $|N| \leq \kappa$ implies $p=q$;

we say $(\bK,\subm)$ is $(\mu,\kappa)$-tame.

No assumption of amalgamation.

\end{slide}



\begin{slide}
\hbox{} \vskip -3in\centerline{\bf HANF NUMBERS}


\begin{notation}\label{Hnumbers}
\begin{enumerate}
\item  Let $H(\lambda,\kappa)$ be the least cardinal $\mu$ such
that if a first order theory $T$ with  $|T|=\lambda$ has models of
every cardinal less than $\mu$ which omit each of a set $\Gamma$
of types, with $|\Gamma| = \kappa$, then there are arbitrarily
large models of $T$ which omit $\Gamma$.
\item Write  $H(\kappa)$ for $H(\kappa,\kappa)$.




\item For a similarity type $\tau$, $H(\tau)$ means
$H({|\tau|})$.
\item $H_1 = H(\tau)$; $H_2 = H(H(\tau))$.
\end{enumerate}
\end{notation}


\end{slide}
\begin{slide}
\hbox{} \vskip -3in\centerline{\bf PRESENTATION THEOREM}



\begin{theorem}[The Presentation
Theorem]\label{pres}\index{Presentation Theorem}
 If $K$ is  an AEC with Lowenheim number $\LS(\bK)$
(in a  vocabulary $\tau$ with $|\tau| \leq \LS(\bK)$), there is a
vocabulary $\tau'$ with cardinality $|\LS(\bK)|$, a first order
$\tau'$-theory $T'$ and a set $\Gamma$ of $2^{\LS(\bK)}$ types
such that:

$$\bK =\{M'\restriction \tau: M' \models T' {\ \rm and \ } M'
{\ \rm omits \ } \Gamma\}.$$

Moreover, if $M'$ is a $\tau'$-substructure of $N'$ where $M', N'$
satisfy $T'$ and omit $\Gamma$ then $M'\restriction \tau \subm
N'\restriction \tau$.
\end{theorem}



\end{slide}
\begin{slide}
\hbox{} \vskip -3in\centerline{\bf }




%\begin{lemma}\label{ott}\index{omitting types theorem}
% $H(\kappa,2^{\kappa}) \leq \beth_{(2^\kappa)^+}$.
%\end{lemma}

By Shelah's presentation theorem:



\begin{corollary}\label{hnumaec} If $\bK$ is an AEC with
L\"{o}wenheim-Skolem number $\kappa$ and $\bK$ has a model of
cardinality at least $H(\kappa,2^{\kappa})$ (which is at most ${
\beth}_{(2^\kappa)^+}$), then $\bK$ has arbitrarily large models.
\end{corollary}




When $\LS(\bK) = |\tau(\bK)|= \kappa$, $H(\kappa)=H_1$ is the Hanf
number for all AEC with the same L\"owenheim number.

Same Hanf number for omitting Galois types.

%$H_2$ denotes

\end{slide}
\begin{slide}

\hbox{}\vskip -3in\centerline{} \centerline{ \bf TWO THEMES}


    What is the eventual behavior of categoricity?  So assume
    there exist arbitrarily large models?

    Does categoricity in small cardinals (below $\beth_{\omega_1}$
    or better $\aleph_{\omega}$ imply {\bf existence} and categoricity
    eventually?

    Yes, for excellence. Excellence yields upward
    L\"owenheim-Skolem.






\end{slide}
\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf CONSEQUENCES I}


{\bf Theorem} (Shelah)  If $(\bK,\subm)$  has amalgamation and is
categorical in $\lambda^+
> H_2$ and $\bK$ is
 then $(\bK,\subm)$ is categorical on
the interval $[H_2,\lambda^+)$.


This uses the derivation of tameness from categoricity described
above.


\end{slide}

\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf CONSEQUENCES II}


Grossberg and Vandieren began the investigation of stability
spectrum for tame AEC with:

\begin{theorem}
$\bK$ is stable in $\mu >H_1$ and $(\infty,\chi)$-tame for some
$\chi<H_1$, then $\bK$ is stable in $\lambda$ if $\lambda^\mu =
\lambda$.
\end{theorem}



\end{slide}

\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf }



Baldwin, Kueker, VanDieren proved the following:


\begin{theorem}\label{up stab}
Let $\bK$ be an AEC with L\"{o}wenheim-Skolem number $\leq\kappa$.
Assume that $\bK$ satisfies the amalgamation property and is
$\kappa$-weakly tame and Galois-stable in $\kappa$.  Then, $\bK$
is Galois-stable in $\kappa^{+n}$ for all $n<\omega$.
\end{theorem}

With one further hypothesis we get a very strong conclusion in the
countable case.

\begin{corollary}\label{genbasecase} Let $\bK$ be an AEC that
satisfies the amalgamation property with L\"{o}wenheim-Skolem
number $\aleph_0$ that is $\omega$-local and $\aleph_0$-tame. If
$\bK$ is $\aleph_0$-Galois-stable then $\bK$ is Galois-stable in
all cardinalities.
\end{corollary}

\end{slide}



\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf CONSEQUENCES III}

{\bf Theorem} (Grossberg-VanDieren)  If $(\bK,\subm)$ is
categorical in $\lambda^+ > \LS(\bK)^+$ and $\bK$ is
$(\infty,\LS(\bK)$-tame, then $(\bK,\subm)$ is categorical on the
interval $[\lambda^+, \infty)$.

{\bf Theorem} (Lessmann)  If $(\bK,\subm)$ is categorical in
$\lambda^+
> \LS(\bK)=\aleph_0$ and $\bK$ is $(\infty,\LS(\bK)$-tame, then
$(\bK,\subm)$ is categorical on the interval $[\lambda^+,
\infty)$.

\end{slide}






\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf SUFFICIENT
CONDITIONS I}

If $(\bK,\subm)$ is quasiminimal excellent in the sense of Zilber
then Galois types = syntactic types.

(Corollary to proof of categoricity transfer.)

\end{slide}



\begin{slide}\hbox{}\vskip -3in

\centerline{ \bf SUFFICIENT CONDITIONS II}
\begin{theorem}\label{gettame}  If $\bK$ is $\lambda$-categorical for $\lambda >
H_1$, then for any $\mu < \lambda$, $\bK$ is $(\mu, H_1)$-weakly
tame.
\end{theorem}

(Weakly means the large model is assumed to be saturated.)

\end{slide}



\begin{slide}\hbox{}\vskip -3in

\centerline{ \bf SUFFICIENT CONDITIONS III}

The Hrushovski construction (even with infinitary input) gives
rise to an abstract elementary class;

$(\bK, \subm, d)$

$d$ is dimension function which imposes a combinatorial geometry
and a notion of strong submodel which gives an abstract elementary
class.

\end{slide}



\begin{slide}\hbox{}\vskip -3in

\centerline{ \bf GEOMETRY}

\begin{assumption}\label{dprop}
\begin{enumerate}
\item $\delta$ and so $d$ map into the integers.
\item $d_N$ satisfies $d_N(X) \leq |X|$.
\item If $X\subseteq Y$, $d_N(X) \leq d_N(Y)$.
\end{enumerate}
\end{assumption}









\begin{definition}  Let $a \in \cl_N(X)$ if $d_N(a/X) =0$.
\end{definition}












\end{slide}



\begin{slide}\hbox{}\vskip -3in



Much more restrictive hypotheses than  general AEC or
Grossberg-Kolesnikov (who earlier deduced tameness from similar
hypotheses assuming the existence of an abstract independence
relation).

 For each $\abar$ in $N$, there is a finite $\mbar_{\abar}$
with $$d(\abar/M) = d(\abar/\mbar_{\abar}).$$


\begin{lemma}\label{kappat} Let $M \subm N_1, N_2$ with $|N_j-M| =\kappa$.  There are
$N_1', N_2', M'$ with $M' \subset N_j$ and $|N_j-M| = |M'|
=\kappa$ such that each $N'_j$ is independent from $M$ over $M'$.


\end{lemma}
\end{slide}
\begin{slide}
\hbox{}\vskip -3in\centerline{} \centerline{ \bf }
\begin{theorem}\label{bvz}(B,Villaveces,Zambrano)
If $(\bK,\subm,d)$  in a countable language satisfies
\begin{enumerate}
\item free amalgamation
\item free extension over independent pairs
\item weak $3$-amalgamation.
\end{enumerate}
then $(\bK,\subm,d)$ is $(\infty,\aleph_0)$-tame.
\end{theorem}

Note hypotheses are about arbitrarily large models.




\end{slide}

\begin{slide}
\hbox{} \vskip -3in \centerline{\bf UNIVERSAL COVERS}


When is the exact sequence:
\begin{eqnarray}0 \rightarrow Z \rightarrow V \rightarrow A \rightarrow 0.
\end{eqnarray}

categorical where $V$ is a $\QQ$ vector space and $A$ is a
semi-abelian variety.

Can be viewed as an expansion of $V$ and there is a combinatorial
geometry given by:

$$\cl(X) = \ln(\acl(\exp(X)))$$

\end{slide}
Moreover by direct basic algebraic argument one can show the three
hypotheses.  Just using linear disjointness.  This is free
amalgamation so there are arbitrarily large models.

By Lessmann $\aleph_1$ categoricity implies categoricity in all
powers.

Zilber had proved: categoricity in all uncountable powers

iff `arithmetic' properties on semiabelian variety V'.

We improve this to:

categoricity in $\aleph_1$

iff `arithmetic' properties on semiabelian variety V



\begin{slide}
\hbox{}\vskip -3in \centerline{\bf GOAL }

    The goal of this line of research is to identify those
    properties of the Hrushovski construction which are
    `automatic' and those which must be verified in individual
    cases.


    Next task:  Check the status of the hypotheses of
    Theorem~\ref{bvz} for pseudoexponentiation.

\end{slide}






\begin{slide}
\hbox{}\vskip -3in \centerline{\bf NONTAMENESS}

We want to find a monster model $\Mscr$ and a submodel $N$ with
elements $a,b$  such that for every small $N_0 \subm N$, there is
an automorphism of $\Mscr$ fixing $N_0$ and taking $a$ to $b$ but
there is no such automorphism which fixes $N$.

We find examples by translating problems from Abelian group
theory.




\end{slide}
\begin{slide}
\hbox{}\vskip -3in \centerline{\bf Hart-Shelah} There is an AEC
$(\bK,\subm)$ which is
\begin{enumerate}
\item $\aleph_1$ and $\aleph_0$ categorical.
\item Ap fails  in $\aleph_0$.

\item Many models in all cardinals $\geq \lambda$ for some $\lambda< 2^{\aleph_1}$.
\end{enumerate}


The argument that $(\bK,\subm)$ is not $(\aleph_1,\aleph_0)$-tame
needs some work.  It may be that one has to go to a larger
example. E.g. the Hart-Shelah example which is
$\aleph_3$-categorical.


\end{slide}





\begin{slide}
\hbox{}\vskip -3in \centerline{\bf WHITEHEAD GROUPS}
%\begin{slide}
%\hbox{} \vskip -3in \centerline{ \bf ABSTRACT ELEMENTARY CLASSES}

$A$ is a {\em Whitehead group}  ($W$-group) if $Ext(Z,A) =0$.

That is, every short exact sequence:
\begin{eqnarray}0 \rightarrow Z \rightarrow H \rightarrow A \rightarrow 0.
\end{eqnarray}
splits.

\end{slide}

\begin{slide}
\hbox{}\vskip -3in \centerline{\bf BACKGROUND}

 J.H.C. Whitehead conjectured that every Whitehead
group of cardinality $\aleph_1$ is free. Shelah proved:

1)  (ZFC) There is an $\aleph_1$-free (every countable subgroup is
free) Abelian group of cardinality $\aleph_1$.

2)  The Whitehead Conjecture is independent of ZFC.


\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf THE EXAMPLE}


 Let $\bK$ be the class of structures $M=\langle
G,Z,I,H\rangle$, where each of the listed sets is the solution set
of one of the  unary predicates ($\bG,\bZ,\bI,\bH$).

 $G$ is a
torsion-free Abelian Group.  $Z$ is  a copy of $(Z,+)$. $I$ is an
index set and $H$ is a family of infinite groups.

\end{slide}
\begin{slide}
\hbox{}
%\vskip -3in
\centerline{}

 The vocabulary
also includes function symbols $\bF, \bk$ and $\bpi$, naming
functions $F,k$, and $\pi$.  There is a binary $+$ on $G$ and on
$\ZZ$ and a ternary $+$.

 $F$ maps $H$ onto $I$ and for $s \in
I$, $+(\_,\_,s)$ is a group operation on $H_s =F^{-1}(s)$.
Finally, $\pi$ maps $H$ onto $G$ so that $\pi_s =\pi\restriction
H_s$ is a projection from $H_s$ onto $G$.

The kernel of each $\pi_s$ is isomorphic to $Z$ via a map
$\bk(\_,s)$ where $k: \ZZ\times I \mapsto H$.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{}

 $M_0 \subm M_1$ if

  $M_0$ is a substructure of
$M$,

but $\bZ^{M_0} = \bZ^{M}$

and $\bG^{M_0}$ is a pure subgroup of $\bG^{M_1}$.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf FACTS}






\begin{definition} We say the AEC  $(\bK,\subm)$ {\em admits closures} if for every $X \subseteq M \in
\bK$, there is a minimal closure of $X$ in $M$.  That is, $
M\restriction \bigcap\{N: X \subseteq N \subm M\}= \cl_M(X) \subm
M$.

\end{definition}

 The class $(\bK,\subm)$  is an abstract elementary class that admits
 closures and has the amalgamation property.


 \end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf NOT LOCAL   }


\begin{lemma}\label{notloc}  $(\bK,\subm)$ is not $(\aleph_1,\aleph_1)$-local.
That is, there is an $M^0 \in \bK$ of cardinality $\aleph_1$ and a
continuous increasing chain of models $M^0_i$ for $i< \aleph_1$
 and two distinct types
$p,q \in \gaS(M^0)$ with $p\restriction M^0_i = q \restriction
M_i$ for each $i$.
\end{lemma}

Let $G$ be an Abelian group of cardinality $\aleph_1$ which is
$\aleph_1$-free but not a Whitehead group.  There is an $H$ such
that,

$$0\rightarrow \ZZ \rightarrow H \rightarrow G \rightarrow 0$$
is exact but does not split.
\end{slide}


\begin{slide}
\hbox{} \vskip -3in \centerline{\bf WHY?}

Let $M_0 = \langle G, \ZZ,a, G \oplus Z\rangle$

$M_1 = \langle G, \ZZ,\{a,t_1\}, \{G \oplus Z,H\}\rangle$

$M_2 = \langle G, \ZZ,\{a,t_2\}, \{G \oplus Z,G \oplus Z\}\rangle$

Let $p =\tp(t_1/M^0,M^1)$ and $q = \tp(t_2/M^0,M^2)$.

Since the exact sequence for $\bH^{M^2}$ splits and that for
$\bH^{M^1}$ does not, $p \neq q$.
\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf NOT  $\aleph_1$-LOCAL }

But for any countable $M'_0\subm M_0$, $p\restriction M'_0 =
q\restriction M'_0$, as

\begin{eqnarray}0 \rightarrow Z \rightarrow H'_i\rightarrow G' \rightarrow 0.
\end{eqnarray}
splits.

$G' = \bG(M'_0)$, $H' = \pi^{-1}(t_i, G')$.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf NOT  $\aleph_0$-TAME }

It is easy to see that if $(\bK,\subm)$ is
$(\aleph_1,\aleph_0)$-tame then it is $(\aleph_1,\aleph_1)$-local,
so

$(\bK,\subm)$ is not $(\aleph_1,\aleph_0)$-tame.

So in fact, $(\bK,\subm)$ is not $(\chi,\aleph_0)$-tame for any
$\chi$.\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf NOT $\kappa$-TAME}

With some use of diamonds, for each successor cardinal $\kappa$,
there is a $\kappa$-free but not free group of cardinality
$\kappa$ which is not Whitehead.  This shows that, consistently,

For arbitrarily large $\kappa$,

$(\bK,\subm)$ is not $(\kappa^+,\kappa)$-tame for any $\kappa$.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf BECOMING TAME }

Grossberg and Van Dieren asked for $(\bK,\subm)$, and $\mu_1 <
\mu_2$ so that $(\bK,\subm)$ is not $(\geq \mu_1, \mu_1)$-tame but
is $(\geq \mu_2, \mu_2)$-tame.



This requires finding a way to bound the cardinality of $G$
without imposing too much structure on $G$.

Even more interesting.  Is it possible to find categorical
examples which only become tame near the Hanf number.
\end{slide}

\begin{slide}
\hbox{} \vskip -3in \centerline{LOCALITY AND COMPACTNESS}

The existence of a sup for an increasing chain of Galois types is
also problematic. Every $\omega$-chain has a sup.

Locality up to $\kappa$ implies compactness at $\kappa^+$.

There are NO known sufficient conditions for locality.

Variants of the Whitehead construction should give (consistently)
a class that is not $(\aleph_2,\aleph_2)$-compact.


\end{slide}



\begin{slide}
\hbox{} \vskip -3in \centerline{\bf  TAMENESS AND AMALGAMATION }

How closely are tameness and amalgamation intertwined?


\end{slide}



\begin{slide}
\hbox{} \vskip -3in \centerline{\bf  TWO PROPERTIES}


$\bK$ is {\em model complete} if $N \subset M$ and $N \in \bK$,
implies $N \subm M$.


\begin{definition} We say the AEC  $(\bK,\subm)$ {\em admits closures} if for every $X \subseteq M \in
\bK$, there is a minimal closure of $X$ in $M$.  That is, $
M\restriction \bigcap\{N: X \subseteq N \subm M\}= \cl_M(X) \subm
M$.

\end{definition}

\end{slide}



\begin{slide}
\hbox{} \vskip -2.5in \centerline{\bf  Non-tame with A.P.}




 \begin{theorem}      For any AEC $(\bK,\subm)$ which admits closures, we can
 assign $(\bK ',\subm ')$ which has the amalgamation property and
  and is
 no more tame than $\bK$.
 \end{theorem}


IDEA:  Expand the language by adding a new binary relation.  Form
a complete graph on each $\bK$-structure. $\bK'$ is the  smallest
AEC containing these expansions such that each finite complete
graph is extended to a minimal model that expands to a member of
$\bK$ (which is also a complete graph).

$N' \subm M$ if $\cl'_{N'}(A) =\cl'_{M'}(A)$ for every finite $A$.

Get ap by no edges amalgamation of the underlying graphs.

\end{slide}



\begin{slide}
\hbox{} \vskip -3in \centerline{\bf  }


\begin{definition}\label{newclass}
Let $\bK$ be an AEC with a relational  vocabulary $\tau$. The
vocabulary $\tau'$ of $\bK'$ is obtained by adding one additional
binary relation $R$. We say the domain of a $\tau'$-structure $A$
is an $R$-set if $R$ defines a complete graph on $A$.
\begin{enumerate}
\item The class $\bK'$ is
 those $\tau'$-structures $M$ which assign a $\tau'$-structure $M_A =
 \cl'_M(A)$ to each finite subset $A$ of $M$ such that:
 \begin{enumerate} \item  If the finite subset $A$ of $M$ is an $R$-set there is
 a $\tau'$-structure $M_A$ such that  $A \subseteq M_A
 \subseteq M$ with $M_A \leq \LS(\bK)$, $M_A$ is an $R$-set, and
 $M_A\restriction \tau \in \bK$.
 \item If $N\subset M$ satisfies the conditions on $M_A$ in
 requirement 1), then $M_A\restriction \tau \subm N\restriction
 \tau$.
 %\item We denote $N_A$ by $\cl_M(A)$.
 \end{enumerate}
 \item  If $M_1 \subseteq M_2$ are each in $\bK'$, then
 $M_1 \subm ' M_2$ if for each finite $R$-set $A$ in $M_1$,
$\cl'_{M_1}(A)=\cl'_{M_2}(A)$.
 \end{enumerate}
\end{definition}
\end{slide}

\begin{slide}
\hbox{} \vskip -3in \centerline{Connections}

Motivated by issues in databases Baldwin-Benedikt introduced the
notion of `small' to study embedded finite model theory.

Casanovas and Ziegler extended and clarified this work; providing
a framework which included Poizat's theory of `belles paires'.

Baldwin and Baizhanov refined this further:

\begin{theorem}  If $(M,A)$ is uniformly weakly benign and
the $\#$-induced theory on $A$ is stable then $(M,A)$ is stable.
\end{theorem}

The beautiful work of Van den Dries and Ghuryikan could have
applied these results.

\end{slide}




\begin{slide}
\hbox{} \vskip -3in \centerline{\bf  REFERENCES }

My monograph at http://www2.math.uic.edu/~jbaldwin/model.html

contains much of this.  The actual papers discussed here may not
appear on the web for a couple of months.

\end{slide}

\end{landscape}
\end{document}


\end{document}

We modify the previous example by fixing the group $G$ in advance.
Moreover, we fix a pure filtration of $G$ and of each extension
$H_i$ of $G$.

That is, if
\begin{eqnarray}0 \rightarrow Z \rightarrow H \rightarrow A \rightarrow 0.
\end{eqnarray}
is a short exact sequence, we fix $G_i:i< \aleph_1$, $H_i:i<
\aleph_1$ and $\pi_i:H_i \rightarrow G_i$.


\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{\bf NEW EXAMPLE}


 Let $\bK$ be the class of structures $M=\langle L,
G,Z,I,H\rangle$, where each of the listed sets is the solution set
of one of the  unary predicates ($\bL, \bG,\bZ,\bI,\bH$).

 $G$ is a
torsion-free Abelian Group.  $Z$ is  a copy of $(Z,+)$. $I$ is an
index set and $H$ is a family of infinite groups.

\end{slide}


\begin{slide}
\hbox{} \vskip -3in \centerline{ }

The vocabulary also includes function symbols $\bF, \bk$ and
$\bpi$, naming functions $F,k$, and $\pi$. There is a binary $+$
on $G$ and on $\ZZ$ and a ternary $+$.

 $F$ maps $H$ onto $I$ and for $s \in
I$, $+(\_,\_,s)$ is a group operation on $H_s =F^{-1}(s)$.
Finally, $\pi$ maps $H$ onto $G$ so that $\pi_s =\pi\restriction
H_s$ is a projection from $H_s$ onto $G$.

The kernel of each $\pi_s$ is isomorphic to $Z$ via a map
$\bk(\_,s)$ where $k: \ZZ\times I \mapsto H$.

\end{slide}


\begin{slide}
\hbox{} \vskip -3in \centerline{ } But now, $M=\langle L,
G,Z,I,H\rangle$ is in $\bK$ if and only if

$L$ is an ordinal $\alpha \leq \aleph_1$.

$G= G_\alpha$, For each $s$, $H_s = H_\alpha$, $\pi_s =
\pi_\alpha$.

Thus we have $\aleph_1 + |EXT(Z,G)|$ basic structures; each
corresponding to a short exact sequence.

The other models are disjoint unions of these.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{}

 $M_0 \subm M_1$

if
\begin{enumerate}
 \item $M_0$ is a substructure of
$M$,


 \item $L_0=\alpha$ is an initial segment of $L_1=\beta$.


\item $\bZ^{M_0} = \bZ^{M}$

\item $\bG^{M_0} = G_\alpha$ is a pure subgroup of $\bG^{M_1}=
G_\beta$.

\item For each $s$, $\bH_s^{M_0} = H_\alpha$ is a pure subgroup of $\bH_s^{M_1}=
H_\beta$.
\end{enumerate}

\end{slide}




\begin{slide}
\hbox{} \vskip -3in \centerline{Connections}

Motivated by issues in databases Baldwin-Benedikt introduced the
notion of `small' to study embedded finite model theory.

Casanovas and Ziegler extended and clarified this work; providing
a framework which included Poizat's theory of `belles paires'.

Baldwin and Baizhanov refined this further:

\begin{theorem}  If $(M,A)$ is uniformly weakly benign and
the $\#$-induced theory on $A$ is stable then $(M,A)$ is stable.
\end{theorem}

The beautiful work of Van den Dries and Ghuryikan could have
applied these results.

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ }



\end{slide}


\end{landscape}
\end{document}

\begin{slide}
\hbox{} \vskip -3in \centerline{ \bf ABSTRACT ELEMENTARY CLASSES}
 A class of $L$-structures, $(\bK,\subm )$,
is said to be an {\em abstract elementary class: AEC} if both
$\bK$ and the binary relation $\subm$ are closed under isomorphism
and satisfy the Jonss\'on conditions plus

\end{slide}
\begin{slide}
\hbox{} \vskip -3in \centerline{ } If $A,B,C \in \bK$, $A \subm
C$, $B \subm C$ and $A\subseteq
 B$ then $A \subm B$;


More careful formulation of unions of chains;

Existence of Lowenheim number.

\end{slide}
