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\def\ind #1#2#3{#1 \mathbin{\mathop{\downarrow}_{#2}} #3}
\def\LS{\rm LS}

\def\Sscr{{\cal S }}


%\def\gaS{\rm ga-S}
%\def\gaS{{\cal S}}
%\def\gaS{\rm ga-S}
\def\gaS{{\mathbb{S}}}
\def\aut{\rm aut}
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\newtheorem{sexample}[stheorem]{Example}
\newtheorem{sexercise}[stheorem]{Exercise}
\newtheorem{sapp}[stheorem]{Applications}










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\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\indd #1#2#3{#1 \mathbin{\mathop{\down}^d_{#2}} #3}
\def\bar#1{\overline{#1}}
%\newcommand{\qed}{\hfill\mbox{$\Box$}}
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\def\td{d_f}
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\def\alphabar{{\boldmath\alpha}}
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\def\Pscr{{\cal P }}
\def\Rscr{{\cal R }}
\def\Cscr{{\cal C }}
\def\Escr{{\cal E }}
\def\Ascr{{\cal A }}
 \def\mod{{\rm mod}}
 \def\Aind{A_{\rm ind}}
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\def\bP{\mbox{\boldmath $P$}}
\def\bbK{\mbox{\boldmath ${\overline { K}}$}_{-1}}
\def\bboK{\mbox{\boldmath ${\overline { K}}$}_{0}}
\def\subi{\leq_{i}}
%\def\th{\rm Th }
\def\Th{\mathop{\rm Th}}
\def\th{\mathop{\rm Th}}
\def\Ex{\mathop{\rm Ex}}
\def\bD{\mathop{\bf D}}
\def\stp{\mathop{\rm stp}}
\def\span{\mathop{\rm span}}
%\newcommand{\qed}{\hfill\mbox{$\Box$}}
\def\hqt{\rm ht}
\def\at{\rm at}
\def\icl{\rm icl}
%\def\acl{\rm acl}
\def\diag{\rm Diag}
\def\rest{\mid}
\def\Nbar{\overline{N}}
\def\Mbar{\overline{M}}
\def\Kbar{\overline{K}}
 \def\dom{\mathop{\rm dom}}
\def\join{\vee}
\def\meet{\wedge}
\def\kappabar{\overline{\kappa}}

\def\grp{{\rm grp}}
\def\span{{\rm span}}
\def\fld{{\rm fld}}
\def\dcl{{\rm dcl}}
\def\rcl{{\rm rcl}}
\def\acl{{\rm acl}}
\def\cl{{\rm cl}}
\def\pr{{\rm pr}}
\def\ex{{\rm ex}}
\def\st{{\rm st}}
\def\gp{{\rm gp}}
\def\qf{{\rm qf}}
\def\ccl{{\rm ccl}}
\def\acf{{\rm acf}}
\def\str{{\rm str}}
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\def\tcl{{\rm acf}}
\def\rg{\mathop{\rm rg}}
\def\bij{\mathop{\rm bij}}

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\def\LS{\rm LS}
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\newcommand{\CC}{\mathbb{\cal C}}

\def\subm{\prec_{\bK}}

\def\grpf #1 #2{{\rm grp}_{#2}(#1)}
\def\spanf #1 #2{{\rm span}_{#2}(#1)}
\def\fldf #1 #2{{\rm fld}_{#2}(#1)}
\def\dclf #1 #2{{\rm dcl}_{#2}(#1)}
\def\rclf #1 #2{{\rm rcl}_{#2}(#1)}
\def\aclf #1 #2{{\rm acl}_{#2}(#1)}
\def\acff #1 #2{{\rm acf}_{#2}(#1)}
\def\strf #1 #2{{\rm str}_{#2}(#1)}
%\def\tclf #1 #2{{\rm tcl}_{#2}(#1)}
\def\tclf #1 #2{{\rm acf}_{#2}(#1)}

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%\input{jbmacaec}
\usetheme{Berkeley}
\title{First Order and Infinitary Model Theory:\\Obtaining Amalgamation}
\author{John T. Baldwin} \institute[University
of Illinois at Chicago]

\begin{document}
\date{\today}

\frame{\titlepage}



\AtBeginSection[]






\begin{frame}
\frametitle{Ecclesiates 1:1-2, 9}



 1   The words of the Preacher, the son
of David, king in Jerusalem.

2   Vanity of vanities, saith the Preacher, vanity of vanities;
all is vanity.

%3   What profit hath a man of all his labor which he taketh under
%the sun?

%4   One generation passeth away, and another generation cometh:
%but the earth abideth for ever.

%\end{frame}

%\begin{frame}

%5   The sun also ariseth, and the sun goeth down, and hasteth to
%his place where he arose.

%6 The wind goeth toward the south, and turneth about unto the
%north; it whirleth about continually, and the wind returneth again
%according to his circuits.

%7   All the rivers run into the sea; yet the sea is not full: unto
%the place from whence the rivers come, thither they return again.


%8   All things are full of labor; man cannot utter it: the eye is
%not satisfied with seeing, nor the ear filled with hearing.

\smallskip

9 The thing that hath been, it is that which shall be; and that
which is done is that which shall be done: and there is no new
thing under the sun.

%10 Is there any thing whereof it may be said, See, this is new? it
%hath been already of old time, which was before us.

%11 There is no remembrance of former things; neither shall there
%be any remembrance of things that are to come with those that
%shall come after.

\end{frame}

\begin{frame}
 \frametitle{Dialectical Model Theory}

{\bf Thesis} Model theory is a technique for solving problems in
mainstream mathematics, usually involving structures of small
cardinality.

\medskip

{\bf Antithesis}  Model theory is a branch of mathematics in its
own right.  In particular, it explores the `higher infinite'.


\medskip

{\bf Synthesis}  The insights developed in exploring the
Antithesis repeatedly enrich the Thesis.  This leads to new `pure'
explorations.

\end{frame}

\begin{frame}
 %\frametitle{Two Directions after Morley}
 \frametitle{The 70's}


%\begin{enumerate}
%\item  Classify the countable models of $\aleph_1$-categorical
%theory.  (Baldwin-Lachlan, Zilber, geometric stability theory)


%\pause \item

Stability theory developed
\begin{enumerate}
\item abstractly with the stability classification \item
concretely by finding the stability class of important
mathematical theories and using the techniques of the abstract
theory.


\end{enumerate}
%\end{enumerate}

The absoluteness of fundamental notions such as
$\aleph_1$-categoricity and stability liberated first order model
theory from set theory.

\end{frame}


\begin{frame}\frametitle{In the Background}

At the same time and largely unnoticed, Shelah developed the
fundamentals of stability theory for infinitary logic.

\medskip

It was not until Zilber's exploration of complex exponentiation in
the 1990's that the significance of this work for mainstream
mathematics was realized.

\end{frame}


\begin{frame}\frametitle{Submitted by 1978}

\begin{enumerate}
\item

Sh:3
    Shelah, Finite diagrams stable in power (1970)
 \item Sh:10
    Shelah, Stability, the f.c.p., and superstability; model theoretic properties of
    formulas in first order theory --  (1971)
\item Sh:48
    Shelah, Categoricity in $\aleph _{1}$ of sentences in $L_{\omega _{1},\omega}(Q)$ --
   (1975)

\item    Sh:54
    Shelah, The lazy model-theoretician's guide to stability --
     (1975)
\item Sh:72
    Shelah, Models with second-order properties. I. Boolean algebras with no definable
    automorphisms
\item    Sh:87a, 87b
    Shelah, Classification theory for nonelementary classes, I.
    The number of uncountable models of $\psi \in L_{\omega _{1},\omega }$.
    Part A, Part B  (1983)

\end{enumerate}


\end{frame}

\begin{frame}
 %\frametitle{Two Directions after Morley}
 \frametitle{The Higher Infinite}

 Large enough models so that pathologies have disappeared and the essential uniformity of classes of models are apparent.

\pause
\medskip

Today the higher infinite means:

greater than $\aleph_\omega$ or even,
\medskip

Greater than $\aleph_1$!
\end{frame}

%\begin{frame}%



\section{The Amalgamation Property}

\begin{frame}
\frametitle{Generalized Amalgamation}

The crucial notion is \myalert{generalized amalgamation}.

\medskip

In this talk, we discuss developments of generalized amalgamation
and connections between the first order and infinitary versions.

\end{frame}
%\section{The Amalgamation Property}

\begin{frame}
\frametitle{ AMALGAMATION PROPERTY }
 The class $\bK$ satisfies the {\em
amalgamation property}  if for any
situation with $A,M , N \in \bK$: \\

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\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\end{picture}
\end{center}
 \vskip .3in

there exists an $N_1\in \bK$ such that \\
\vskip .3in  \unitlength=1.00mm \linethickness{0.4pt}
\begin{center}
\begin{picture}(21.20,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(23.00,11.50){\makebox(0,0)[cc]{$N_1$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
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\put(5.50,9.00){\vector(3,-2){5.00}}
\put(15.50,6.00){\vector(3,2){5.00}}
\end{picture}\
\end{center}
\end{frame}


\begin{frame}
\frametitle{ SET AMALGAMATION PROPERTY }
 The class $\bK$ satisfies the {\em
 set amalgamation property}  if for any
situation with $M , N \in \bK$ and $A \subset M, A \subset N$: \\

\unitlength=1.00mm \linethickness{0.4pt} \begin{center}
\begin{picture}(15.00,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\end{picture}
\end{center}
 \vskip .3in

there exists an $N_1\in \bK$ such that \\
\vskip .3in  \unitlength=1.00mm \linethickness{0.4pt}
\begin{center}
\begin{picture}(21.20,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(23.00,11.50){\makebox(0,0)[cc]{$N_1$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(15.50,16.00){\vector(3,-2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\put(15.50,6.00){\vector(3,2){5.00}}
\end{picture}\
\end{center}
\end{frame}




\begin{frame}
 \frametitle{Is there a difference?}

For a complete first order theory, Morley taught us:
\smallskip

There is no difference. \pause

\smallskip

Tweak the language and we obtain set amalgamation.

\smallskip

(Tweak: put predicates for every definable set in the language)

\end{frame}





\begin{frame}
 \frametitle{There is a difference!}

Zilber's examples of quasiminimal excellent classes have
amalgamation over models but the interesting examples do
\myalert{not} have set amalgamation.

\end{frame}

\begin{frame}
 \frametitle{A simple example}

There is a first order theory $T$ with a prime model $M$ such
that: \begin{enumerate} \item $M$ has no proper elementary
submodel \item but $M$ contains an infinite set of indiscernibles.
\end{enumerate}

\pause

\begin{block}{Inhomogeneity}

 Construct an $L_{\omega_1, \omega}$-sentence $\psi$ whose
models are partitioned into two sets; on one side is an atomic
model of $T$, on the other is an infinite set.

\smallskip

$\psi$ is categorical in all infinite cardinalities but no model
is $\aleph_1$-homogeneous because there is a countably  infinite
maximal indiscernible set.
\end{block}

\end{frame}
\begin{frame}
 \frametitle{J. Knight's construction}


Three unary predicates, $W, F,I$  partition the set.
\smallskip
Let $W$ and $I$ be countably infinite sets and fix an isomorphism
$f_0$ between them.

\smallskip

$F$ is the collection of all bijections between $W$ and $I$ that
differ from $f_0$ on only finitely many points.

\smallskip

 Add also a
successor function on $W$ so that $(W,S)$ is isomorphic to
$\omega$ under successor and the evaluation predicate $E(n,f,i)$
which holds if and only if $n \in W, f\in F, i\in I$ and $f(n)
=i$.

\smallskip

The resulting structure $M$ is atomic and minimal. Since every
permutation of $I$ with finite support extends to an automorphism
of $M$, $I$ is a set of indiscernibles.
\end{frame}


\begin{frame}
 \frametitle{An orthogonal notion}

 \begin{block}{Definition (R. Knight)}

 A sentence $\sigma$ of $L_{\omega_1,\omega}$  is \myalert{amalgamative}
 in a fragment $
\Delta$ if for any complete $\Delta$-types, $p(\xbar,y),
q(\xbar,z), r(\xbar)$ such that $r$ is the common restriction of
$p$ and $q$, there is an $s(\xbar,y,z)$ that restricts to both $p$
and $q$.

\smallskip

For any sentence $\sigma$, let $\Delta^\alpha$ ($\alpha <
\omega_1$) be the Morley/Scott expansions of $\Delta$.

\end{block}


\begin{block}{Theorem (R. Knight)}

If $\sigma$ has uncountably many models then for a cub $C$, for
any $\alpha \in C$, $\sigma$ is amalgamative in $\Delta^\alpha$.


\end{block}

\end{frame}





\begin{frame}
\frametitle{Quasiminimal Excellence}

A class $(\bK,\cl)$ is {\em quasiminimal excellent} if $\cl$ is a
combinatorial geometry which satisfies on each $M \in \bK$:

\begin{enumerate}

\item there is a unique type of a basis,

\item  a technical homogeneity condition:\\
$\aleph_0$-homogeneity over $\emptyset$ and over models.


\item and the `excellence condition' which follows.

\end{enumerate}

Conditions 1 and 2 are \myalert{sufficient} for
$\aleph_1$-categoricity.
\end{frame}

\begin{frame}
\frametitle{Necessary Notation}

 In the
following definition it is essential that $\subset$ be understood
as \myalert{ proper} subset.

\begin{definition}\begin{enumerate}
\item For any $Y$, $\cl^-(Y)= \bigcup_ {X\subset Y}\cl(X)$.
\item  We call  $C$  (the union of) {\em an $n$-dimensional $\cl$-independent system} if
$C = \cl^-(Z)$ and $Z$ is an independent set of cardinality $n$.
\end{enumerate} \index{independent system}

\end{definition}

\end{frame}

\begin{frame}
\frametitle{Essence of Excellence}


\begin{block}{There is a primary model over any finite independent
system.}
\end{block}



 Let
$C \subseteq H \in \bK$ and let $X$ be a finite subset of $H$.  We
say $\tp_{\qf}(X/C)$ is {\em defined over} \index{defined over}
the finite $C_0$ contained in $C$ if it is determined by its
restriction to $C_0$.

\begin{block}
{Quasiminimal Excellence}

Let $G \subseteq H,H' \in \bK$ with $G$ empty or in $\bK$. Suppose
$Z \subset H-G$ is an $n$-dimensional independent system,
$C=\cl^-(Z)$,  and $X$ is a finite subset of $\cl(Z)$. Then there
is a  finite $C_0$ contained in $C$ such that $\tp_{\qf}(X/C)$ is
defined over $C_0$.

\end{block}









\end{frame}

\begin{frame}
\frametitle{ QM EXCELLENCE IMPLIES CATEGORICITY }

QM Excellence implies by a direct limit argument:


\begin{lemma}\label{1.1} An isomorphism between independent $X $  and $Y$ extends to an isomorphism of
 $\cl(X)$ and $\cl(Y)$. \end{lemma}

 This gives categoricity in all uncountable powers if the closure of finite sets is countable.



\end{frame}







\begin{frame}
 \frametitle{Methodology}

Amalgamation for independent families of models is a key to
 Zilber's algebraic proofs of quasiminimal excellence.

 \medskip

 But Shelah had already seen a bigger picture.

\end{frame}

\begin{frame}
 \frametitle{General Setting}

An amalgamation \myalert{problem} is a functor $\mathscrnew{a}$
taking $\Pscr(n)^-$ into a family $\Ascr$ of subsets of models in
a class.  Some notion of independence is assumed.

\pause \medskip An amalgamation \myalert{solution} is an extension
${\overline{\mathscrnew{a}}}$ of $\mathscrnew{a}$ to map $\Pscr(n)$ into $\Ascr$F
such that:

$$\{{\overline{\mathscrnew{a}}}(u):u \subset n\}$$ is an independent system
contained in $\overline{\mathscrnew{a}}(n)$.

\medskip

The formulation and examples are derived from Hrushovski and
Kolesnikov.
\end{frame}



\begin{frame}
 \frametitle{3 properties}

\begin{block}{$n$-existence}

Every $\mathscrnew{a}$ from $\Pscr(n)^-$ has a solution.
\end{block}




\begin{block}{Weak $n$-Uniqueness}

Any two solutions for any $\mathscrnew{a}$ are
\myalert{compatible} over $\{{\overline{\mathscrnew{a}}}(u):u
\subset n\}$.
\end{block}



\begin{block}{$n$-Uniqueness}

Any two solutions for any $\mathscrnew{a}$ are
\myalert{isomorphic} over $\{{\overline{\mathscrnew{a}}}(u):u
\subset n\}$.
\end{block}

\end{frame}



\begin{frame}
 \frametitle{Choices of $\Ascr$}

\begin{enumerate}
\item Zilber: {\bf QM-excellence}: closed subsets of a combinatorial geometry
\item Shelah: {\bf Excellence 1}: \myalert{full} atomic models of a first order theory;
\item Shelah: {\bf Excellence 2}: atomic models of a first order theory;
\item Shelah: {\bf NOTOP}: models of a superstable theory with NDOP;
    \item Kim-Pillay:  Independence property;
\item Hrushovski: {\bf groupoid paper}: algebraically closed sets;
\item Kolesnikov, Kim, Tsuboi: {\bf $n$-simple}: boundedly closed sets
\end{enumerate}

\end{frame}
\begin{frame}
 \frametitle{Increasing $n$}

 $n$-existence and $n$-uniqueness imply $n+1$ existence.

 \medskip

 The excellent case shows a close connection between uniqueness
 and goodness.

 \end{frame}
\begin{frame}
 \frametitle{Consequences of $n$-amalgamation}

 \begin{description}

\item [Shelah]: existence and classification of models
 \item [Kolesnikov:] Hierarchy of $n$-simplicity
\item [Hrushovski:] Elimination of Imaginaries, \ldots
\item  [Kim,Depiro, Millar:] Group configurations
\end{description}
\end{frame}

\section{Excellence}


\begin{frame}
\frametitle{$L_{\omega_1,\omega}$ and `first order' model theory}
\begin{block}

The models of a complete sentence in  $L_{\omega_1,\omega}$ can be
represented as:


 $\bK$ is the class of atomic models (realize only
principal types) of a first order theory.
\end{block}


 We study $S_{at}(A)$ where $A \subset M \in \bK$ and

 $p\in S_{at}(A)$ means $Aa$ is atomic if $a$ realizes $p$.
\end{frame}

\begin{frame}
\frametitle{Two Examples}

 \begin{block}{}

$T_1$ is the theory of an infinite set under equality. $M \models
T$.  $p$ asserts $x \neq m$ for every $m\in M$.  Then $p\in
S_{at}(A)$.

\medskip

(Marcus/J. Knight): There is a model $M$ which is atomic, minimal
and contains an infinite indiscernible set.

\smallskip

Every $p\in S_{at}(M)$ is realized in $M$.
\end{block}
\end{frame}


\begin{frame}
\frametitle{$L_{\omega_1,\omega}$: The General Case}

\begin{block}{Quasiminimality is the rank one case}

Any geometry has a notion of independent $n$-system.
\end{block}

\begin{block}{In the more general setting}

Splitting gives an analogous notion of independent $n$-system.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Goodness}
\begin{block}{Definition}
A set $A$ is {\em good} if the isolated types are dense in
$S_{at}(A)$.
\end{block}
\smallskip
For countable $A$, this is the same as $|S(A)| = \aleph_0$.
\smallskip

%\myalert{Eventually}, one shows that

Are there prime models over
good sets?

\pause


YES, In $\aleph_0$ and $\aleph_1$

but not generally above $\aleph_1$ (J. Knight, Kueker, Laskowski-Shelah).

\medskip

Yes, if excellent.



\end{frame}


\begin{frame}
\frametitle{Excellence}

\begin{block}{Definition}\label{excdef}\index{good} \index{excellent}
\begin{enumerate}
\item $\bK$ is {\em $(\lambda,n)$-good} if for
any  independent $n$-system $\Sscr$ (of models of size $\lambda$),
the union of the nodes is good. \vskip .2in That is, there is a
prime model over any countable independent $n$-system. \vskip .2in

\item $\bK$ is {\em excellent} if it is $(\aleph_0,n)$-good  for every $n<
\omega$.
\end{enumerate}
\end{block}

\end{frame}

%\end{document}

\begin{frame}
\frametitle{Excellence implies large models}

Shelah proved:

\begin{block}{Theorem}
Let $\lambda$ be infinite  and $n<\omega$. Suppose $\bK$ has
$(<\lambda,\leq n+1)$-existence and is $(\aleph_0,n)$-good. Then
$\bK$ has $(\lambda,n)$-existence.
\end{block}

\pause \medskip This yields:

\begin{block}{Theorem (ZFC)}
If an atomic class $\bK$ is excellent and has an uncountable model
then  \begin{enumerate}
\item it has models of arbitrarily large cardinality;
\item if it is categorical in one uncountable power it is categorical in all uncountable powers.
    \end{enumerate}
\end{block}


\end{frame}
\begin{frame}
\frametitle{Existence only }

What can one conclude from existence only?


\end{frame}
\begin{frame}
\frametitle{`Universal' Classes -propagation}

Let $\bK$ be defined by universal sentences and the omission of
types.

\begin{block}{Theorem {B-Kolesnikov-Shelah}}

For any $s< \omega$, if $\bK_k$ has the $(\lambda,\leq s+1)$
disjoint amalgamation property,
 then it has the
$(\lambda^+,\leq s)$-disjoint amalgamation property.
\end{block}

Thus $(\aleph_0,<\aleph_0)$ disjoint amalgamation implies $\bK$
has arbitrarily large models and disjoint amalgamation (of pairs).

\end{frame}

%\end{document}
\begin{frame}
\frametitle{`Universal' Classes -Limitations}

\begin{block}{Theorem {B-Kolesnikov-Shelah}}
There is a `universal' $\bK_k$ such that:
\begin{enumerate}
\item
$\bK_k$ has no models of cardinality greater than
$\beth_{k}$.
\item
$\bK_k$ has the disjoint amalgamation property  on models of
cardinality less than or equal to $\aleph_{({k}-3)}$.
\item
$\bK_k$ has models of cardinality $\aleph_{({k}-1)}$.
%\item
%there is a cardinal $\mu$ strictly less than  $\beth_{k}(\kappa)$
%where disjoint amalgamation fails.
\end{enumerate}
\end{block}


\begin{block}{Theorem {B-Kolesnikov-Shelah}}

For every countable ordinal $\alpha$, there is a `universal' class
$\bK_\alpha$ that has disjoint amalgamation over pairs of models
of size $\leq \aleph_{\alpha}$. But $\bK$ has no model of power
$\beth_{(2^{\aleph_0})^+}$.

\end{block}

And consistently, $\aleph_{\alpha}$ can be replaced by
$\beth_{\alpha}$.


\end{frame}

\begin{frame}\frametitle{Abstract Elementary Classes}



Consider a class of $L$-structures and a
notion of `strong submodel' $\subm$.

\smallskip

$(\subm$, $(\bK,\subm )$
is
an {\em abstract elementary class} if both $\bK$ and the binary
relation $\subm$ are closed under isomorphism and satisfy a
collection of conditions generalizing those of J\'onnson for
constructing homogeneous universal models.



\smallskip In particular, the
class must be closed under $\subm$-increasing chains.

\smallskip

A further crucial
requirement is the existence of a L\"owenheim number for the
class.
\end{frame}

%\end{document}

\begin{frame}

\frametitle{  GALOIS TYPES:  Algebraic Form}

Suppose $\bK$ has the amalgamation property.  Then there is a
monster model $\Mscr$.

\medskip

\begin{definition}  Let $M \in \bK$, $M \subm \MM$ and $a\in \MM$.
The Galois type of $a$ over $M$ is the orbit of $a$ under the
automorphisms of $\MM$ which fix $M$.
\end{definition}
\vskip .2in


We say a Galois type $p$ over $M$  is realized in $N$ with $M
\subm N \subm \MM$ if $p \cap N \neq \emptyset$.
\end{frame}




\begin{frame}\frametitle{Tameness-Algebraic form}

Grossberg and VanDieren focused on the idea of studying `tame'
abstract elementary classes:


  Suppose $\bK$ has the amalgamation property.\smallskip

$\bK$ is $(\chi,\mu)$-tame if for any model $M$ of cardinality
$\mu$ and any $a,b \in \Mscr$:

\pause \medskip

If for every $N\subm M$ with $|N| \leq \chi$ there exists $\alpha
\in \aut_N(\Mscr)$ with $\alpha(a) =b$,

\smallskip

then there exists $\alpha \in \aut_M(\Mscr)$ with $\alpha(a) =b$.
\end{frame}



\begin{frame}\frametitle{Consequences of Tameness}

Suppose $\bK$ has arbitrarily large models and amalgamation.

\begin{theorem}[Grossberg-Vandieren]  If $\lambda > \LS(\bK)$, $\bK$ is
$\lambda^+$-categorical and $(\lambda,<\infty)$-tame then $\bK$ is
categorical in all $\theta \geq \lambda^+$.
\end{theorem}

\medskip

\begin{theorem}[Lessmann]  If $\bK$ with $\LS(\bK) = \aleph_0$ is
$\aleph_1$-categorical and $(\aleph_0,\infty)$-tame then $\bK$ is
categorical in all uncountable cardinals
\end{theorem}

\end{frame}

\begin{frame}
\frametitle{Tameness is essential!}


{\bf Theorem.} [Hart-Shelah / Baldwin-Kolesnikov] For each $3 \leq
k<\omega$ there is an $L_{\omega_1,\omega}$ sentence $\phi_k$ such
that:



\begin{enumerate}
\item $\phi_k$ has the disjoint amalgamation property;
\item Syntactic types determine Galois types over models of cardinality
at most $\aleph_{k-3}$;
\item But there are syntactic types over models of size
$\aleph_{k-3}$ that split into $2^{\aleph_{k-3}}$-Galois types.

\medskip \pause

\item $\phi_k$ is categorical in $\mu$ if  $\mu\leq \aleph_{k-2}$;
\item $\phi_k$ is not $\aleph_{k-2}$-Galois stable;
\item But for $m\leq k-3$, $\phi_k$ is  $\aleph_{m}$-Galois stable;
\item $\phi_k$ is not categorical in any $\mu$ with
$\mu>\aleph_{k-2}$.

\end{enumerate}

\end{frame}


\begin{frame}
\frametitle{Two Problems }
\begin{enumerate}
\item Is there a finitely axiomatizable strongly minimal set?
    \item Is there an $\aleph_0$-homogeneous (over models) quasiminimal class which is \myalert{not} excellent?
        \end{enumerate}



\end{frame}






\begin{frame}
\frametitle{Enough categoricity implies Excellence }

VWGCH: $2^{\aleph_n} < 2^{\aleph_{n+1}}$ for $n< \omega$.

\pause
\medskip


\begin{block}{ VWGCH: Shelah 1983}

  An atomic class $\bK$ that has at least one uncountable model and
is categorical in $\aleph_n$ for each $n< \omega$ is
excellent.
\end{block}


%It is open whether VWGCH suffices to delete `very' from `very %few'.

\begin{block}{Show by induction:}

 Very few models in $\aleph_n$ implies
$(\aleph_0,n-2)$-goodness.
\end{block}
\end{frame}


\section{Covers}

\begin{frame}
\frametitle{ Covers of  Algebraic Groups}

{\bf Definition}  A cover %$\Ascr^*$
of a commutative algebraic group $\mathbb{A}(\CC)$ is a  short
exact sequence


\begin{eqnarray}0 \rightarrow Z^N \rightarrow V\ {\stackrel{\exp}{\rightarrow}}
\mathbb{A}(\CC)\rightarrow 1.
\end{eqnarray}




where $V$ is a $\QQ$ vector space and $\mathbb{A}$ is an algebraic
group, defined over $k_0$ with the full structure imposed by
$(\CC,+,\cdot)$.

\smallskip

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

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

\end{frame}



\begin{frame}
 \frametitle{Covers}

  There is an $L_{\omega_1,\omega}$-sentence
$\Sigma$ such that there is a $1$-$1$ correspondence between
models of $\Sigma$ and short exact sequences described above

\pause \medskip

\begin{theorem}\label{coverqm}
If $\mathbb(A)(\CC) = (\CC,\cdot)$, $\Sigma$ is quasiminimal
excellent with the countable closure condition and categorical in
all uncountable powers.\end{theorem}

\pause \medskip


But, the uncountable model $H$ contains a maximal infinite
indiscernible set. Thus, $H$ is not $\aleph_1$-homogeneous.


\end{frame}





\begin{frame}
\frametitle{ Axiomatizing Covers}

Let $\mathbb{A}$ be a commutative algebraic group over an
algebraically closed field $F$.

Let $T_A$ be the first order theory asserting:

\begin{enumerate}
\item $(V,+,f_q)_{q\in \QQ}$ is a $\QQ$-vector space.
\item The complete first order theory of
$\mathbb{A}(F)$ in a language with a symbol for each $F$-definable
variety.
\item $\exp$ is a group homomorphism from $(V,+)$ to $(\mathbb{A}(F),\cdot)$.
\end{enumerate}


$T_A + \Lambda = \ZZ^N$ asserts the kernel of $\ex$ is standard.

\medskip
In general, VWGCH implies categoricity is equivalent
to `arithmetic' properties of $\Ascr$.
\end{frame}







\begin{frame}
\frametitle{Notation}

\begin{definition} A {\em multiplicatively closed divisible
subgroup } associated with $a\in\CC^*$, $a^{\QQ}$, is a {\bf
choice} of a multiplicative subgroup isomorphic to $\QQ$
containing $a$  .
\end{definition}

\medskip

$\mathbb{A}_{n}(F) =\{a\in \mathbb{A}(F): a^n =1\}$




\medskip


{\bf Fact.} For any acf $F$, $\mathbb{A}_{n}(F) \approx (Z/nZ)^N$.


\medskip

For any $a_1,\ldots a_n$, let $k_a = k_0(\mathbb{A}_{\rm tors},
a_1^{\QQ}, \ldots a_n^{\QQ})$.

\end{frame}



\begin{frame}
\frametitle{$\aleph_1$-categorical iff}


\begin{block}{Algebraic Conditions for $\omega$-stability and homogeneity}

\begin{enumerate}

\item there is a $d$ such that for any $m \in \NN$,
${\rm Gal} ({\tilde {k_0}}:k_0)$ has at most $d$ orbits on
$$\{\langle a_1, \dots a_N\rangle : a_1, \dots a_N {\rm \
generate\ } \mathbb{A}_{m}\}.$$

\item Let $F_0 \subset F(\mathbb{A})$ be a countable acf, $b_1,
\ldots b_n$ be multiplicatively independent over
$\mathbb{A}(F_0)$.  There is an $\ell \in \NN$ such that:
$$
 {\rm Gal}(F_0(b_1^{\frac{1}{m\ell}}, \ldots
 , b_k^{{1}\over{m\ell}}):F_0(b_1^{{1}\over{\ell}}\ldots
 ,b_k^{{1}\over{\ell}})) \approx (\ZZ/m\
 \ZZ)^{Nk}.$$


\end{enumerate}


\end{block}

\end{frame}





\begin{frame}
\frametitle{Algebraic Formulations of Model theoretic properties}

%Excellence is equivalent in this context to algebraic properties
%of the groups $\Ascr(k)$ for various countable $k$.  Here are some
%sample properties.



Let $\Sscr = \{F_s: s\subset n\}$ be an independent $n$-system of
\myalert{algebraically closed} fields contained in  a suitable
monster $\Mscr$.

Denote  the \myalert{subfield} of $\Mscr$     generated by
$(\bigcup_{s\subset n} F_s)$ as $k$.

\begin{block}{Canonical completions}

$$\Ascr(k) = A^n \oplus \prod_{s\subset n} \Ascr(F_s)$$

where $A^n$ is a free Abelian group.
\end{block}
\end{frame}


\begin{frame}
\frametitle{Excellence}

The following are equivalent under VWGCH

\begin{enumerate}
\item

The cover of $\mathbb{A}$ is categorical in all uncountable
$\kappa$.

\item The cover of $\mathbb{A}$ is categorical in all $\aleph_n$
for $n< \omega$.

\item The cover of $\mathbb{A}$ is quasiminimal excellent.

\item  $\mathbb{A}$ satisfies  algebraic conditions for $\omega$-stability and homogeneity
and has
canonical completions.

\end{enumerate}



\end{frame}



\begin{frame}
 \frametitle{Refining the Model theoretic analysis}

%Zilber shows with extremely basic algebraic arguments:

Consider $T_A + \Lambda = Z^N$:


\begin{enumerate}[1)]
\item {all $\mathbb{A}$}
\begin{enumerate}[a)]
\item has $2$-amalgamation;
\item has arbitrarily large models;
\end{enumerate}
\item Identify algebraic properties corresponding exactly to small, $\omega-stable$, $\aleph_0$-homogeneity over models, excellence.

\item Do these properties differentiate algebraic groups in a natural way?

\end{enumerate}

\end{frame}

\begin{frame}
 \frametitle{Will more model theory help?}

\begin{block}{Application of Theorem of Grossberg-Kolesnikov}

If an $\omega$-stable atomic class has $(\aleph_0,n)$-weak
uniqueness for all $n$, then it is $(\aleph_0,\infty)$-tame.

\end{block}

\end{frame}

\begin{frame}
 \frametitle{Still more model theory}

Recall
\begin{theorem}[Lessmann]  If $\bK$ with $\LS(\bK) = \aleph_0$ is
$\aleph_1$-categorical and $(\aleph_0,\infty)$-tame then $\bK$ is
categorical in all uncountable cardinals
\end{theorem}

So if we establish weak-uniqueness in $(\aleph_0,n)$ for all $n$
for an algebraic group $\Ascr$, we can conclude the `arithmetic'
consequences on $\Ascr(k)$ for all $k$

(modulo Shelah's theorem and VWGCH)





\end{frame}

\begin{frame}
\frametitle{Is  WCH  necessary?}
\begin{block}

Does  MA + $\neg$ CH imply there is a sentence of
$L_{\omega_1,\omega}$ that is $\aleph_1$ categorical but
\begin{itemize}
\item[a)] is not $\omega$-stable
\item[b)] does not satisfy amalgamation even for countable models.
\end{itemize}
\end{block}

There is such an example in $L_{\omega_1,\omega}(Q)$ but Laskowski
showed the example proposed for $L_{\omega_1,\omega}$ by Shelah
(and me) fails.
\medskip

Is $\aleph_1$-categoricity of a sentence of
$L_{\omega_1,\omega}$ absolute?



\end{frame}



\begin{frame}
 \frametitle{Concluding the reading}


10 Is there any thing whereof it may be said, See, this is new? it
hath been already of old time, which was before us.

11 There is no remembrance of former things; neither shall there
be any remembrance of things that are to come with those that
shall come after.

\end{frame}

\end{document}
\begin{frame}
\frametitle{Choosing Roots}

\begin{definition} A {\em multiplicatively closed divisible
subgroup } associated with $a\in\CC^*$, is a {\bf choice} of a
multiplicative subgroup isomorphic to $\QQ$ containing $a$  .
\end{definition}
\begin{definition} $b_1^{{1}\over{m}} \in b_1^{\QQ},
\ldots b_\ell^{{1}\over{m}} \in b_\ell^{\QQ} \subset \CC^*$, {\em
determine the isomorphism type} of $b_1^{\QQ}, \ldots b_\ell^{\QQ}
\subset \CC^*$ over $F$ if given subgroups of the form $c_1^{\QQ},
\ldots c_\ell^{\QQ} \subset \CC^*$ and $\phi_m$ such that
$$\phi_m:F(b_1^{{1}\over{m}} \ldots b_\ell^{{1}\over{m}}) \rightarrow F(c_1^{{1}\over{m}}
 \ldots c_\ell^{{1}\over{m}})$$
 is a field isomorphism it extends to
$$\phi_{\infty}:F(b_1^{\QQ}, \ldots b_\ell^{\QQ}) \rightarrow F(c_1^{\QQ}, \ldots c_\ell^{\QQ}).$$

\end{definition}

\end{frame}



\begin{frame}
\frametitle{A finiteness condition}
\begin{block}{Thumbtack Lemma}

 For any $b_1, \ldots b_\ell  \subset \CC^*$, there
exists an $m$ such that  $b_1^{{1}\over{m}} \in b_1^{\QQ}, \ldots
b_\ell^{{1}\over{m}} \in b_\ell^{\QQ} \subset \CC^*$, {\em
determine the isomorphism type} of $b_1^{\QQ}, \ldots b_\ell^{\QQ}
\subset \CC^*$ over $F$.
\end{block}


\begin{block}{Zilber}

The Thumbtack Lemma is equivalent to $T_A + \Lambda = \ZZ^N$ is
quasiminimal excellent.
\end{block}
\end{frame}


\begin{frame}
 \frametitle{Program}


 We want to show that if a semi-Abelian
  variety satisfies the `arithmetical conditions' given in
  statement ii) of Theorem 1 and statements iii) and v) of
  Theorem 3 of \cite{Zilbersa} hold, then one can deduce statement
  ii) of Theorem 3 of \cite{Zilbersa}.

  The argument has the following steps:

  \begin{enumerate}
  \item Statement ii) of Theorem 1 and statements iii) and v) of
  Theorem 3 of \cite{Zilbersa} imply $T_A + \Lambda = Z^N$ is
  $\aleph_1$-categorical is proved by Zilber in
  \cite{Zilbercovers}
  \item By the arguments here $T_A + \Lambda = Z^N$ has
  arbitrarily large models and is $(\aleph_0,\infty)$-tame.
  \item By Lessmann \cite{Lessmannupcat1}, $T_A + \Lambda = Z^N$
  is categorical in all cardinalities.
  \item Note $T_A + \Lambda = Z^N$ is axiomatized in
  $L_{\omega_1,\omega}$.
  \item By Shelah \cite{Sh87a,Sh87b}, $T_A + \Lambda = Z^N$ is
  excellent.
  \item Now apply Theorem 3 of \cite{Zilbersa}.
  \end{enumerate}

  This suggests further work on when the hypotheses of step 1)
  hold and on really understanding step 4.
\end{frame}







\begin{frame}
 \frametitle{The goal of Frames}

 \begin{block}{Shelah wants:}

If there is a sufficiently strong
 `stability theory' on $\bK_{\kappa}$

 then the generated AEC is `controlled' in a strong way by
 $\bK_{\kappa}$.

 \end{block}

 He proposes to deduce the existence of such classes from categoricity.

 Can we find them `in nature'?


\end{frame}



\begin{frame}
 \frametitle{Three approaches to eventual categoricity}

\begin{enumerate}
\item Work from the bottom up.  Existence of models must be
earned. 87a, 87b, Zilber (WGCH) Beginning at $\omega$ is crucial.
\item  Begin above the Hanf number for existence.
Ehrehfeucht-Mostowski models are a powerful tool.
\begin{enumerate}
\item $(ap)^+$ -394,
\item (with tameness Grossberg-VanDieren and Lessmann)
\end{enumerate}
 \item the
frame approach: combine the two.
\end{enumerate}


\end{frame}


%\begin{frame}
%\frametitle{Ehrenfeucht-Mostowski models}

%Assume categoricity:
%\begin{enumerate}
%\item needed to deduce stability (pace Keisler)
%\item needed to deduce superstability ($\kappa(T)$-form)
%\end{enumerate}

%\end{frame}


%\section{The Lower Infinite}




\begin{frame}
 \frametitle{The stability spectrum}

 For a countable $T$ there are six functions $f_i(\kappa)$  which
 give all the possible stability spectra.

 \medskip

 This might appear `combinatorial' but \myalert{stable} (i.e.
 cofinally stable) implies the existence of a well-behaved
 dependence relation.

 \end{frame}

\begin{frame}
 \frametitle{The saturation  spectrum}

The same functions compute the cardinals in which a first order
theory has a saturated model.


\bigskip
This can be viewed a property of the group of automorphisms of a
monster model.

\end{frame}

\begin{frame}
 \frametitle{Some Problems}

 \begin{enumerate}
 \item For eventually categorical classes, what happens below.

 \item What exactly is the Hanf number for categoricity?
\item Is there one, without assuming ap?
 \item Are there Hanf numbers for amalgamation, tameness,
 compactness \ldots ??
 \item Use the technology of finitary classes to investigate e.g.
 Vaught's conjecture.
% \item Stability spectrum of homogeneous classes is known
 %\myalert{above} the Hanf number. What about below?
 \end{enumerate}

 \pause

What model theoretic of infinitary logic properties are absolute?

 \end{frame}









\section{Connections with Set Theory}

\section{The Weak Continuum Hypothesis}
%\section{Background}% on $L_{\omega_1,\omega}$}
%\subsection{Scott's Isomorphism Theorem}% $\aleph_0$-categoricity}
%\subsection{Morley's  Theorem}% $\leq \aleph_1$ Countable Models}




\begin{frame}
\frametitle{The Weak Generalized Continuum Hypothesis}
\begin{block}{Setting}

ZFC is the base theory throughout.
\end{block}

\begin{block}{Axiom  WGCH:  Weak GCH}
For every cardinal $\lambda$, $2^\lambda < 2^{\lambda^+}$.
\end{block}

\begin{block}{Axiom  VWGCH:  Very Weak GCH}

For every cardinal $\lambda$ \myalert{with $\lambda <
\aleph_\omega$}, $2^\lambda < 2^{\lambda^+}$.
\end{block}

\end{frame}











%\subsection{Weak Diamond}
%\begin{frame}
%\frametitle{Definition: Devlin-Shelah Weak Diamond}
%\begin{block}{$\Phi_\lambda$ is the
%proposition:}

% For any function $F: 2^{<\lambda} \rightarrow 2$
%there exists $g\in 2^\lambda$ such that for every $f\in 2^\lambda$
%the set $$\{\delta < \lambda: F(f\restriction \delta) =
%g(\delta)\}$$ is stationary.
%\end{block}

%\begin{block}


 %For every $X\subset \lambda$ and $\alpha< \lambda$,

% Weak-$\diamond$ predicts whether $X \cap \alpha$ is in
% one side or another of a partition of $\Pscr(\alpha)$.
% \end{block}


% \end{frame}



\begin{frame}
\frametitle{Crucial Fact}

 Weak diamond is the operative form of WGCH.

\medskip

\begin{block}{$2^\lambda < 2^{\lambda^+}$ if and only if Weak-$\diamond$ on
$\lambda^+$  }


\end{block}

%\Phi_{\lambda^+}$:

\end{frame}

\section{Model Theoretic Background}

\begin{frame}
\frametitle{Model Theoretic Context}

\begin{block}

 For the next section, $\bK$ is the class of models of a
sentence $\psi$ in $L_{\omega_1,\omega}$.

\medskip

We write $M \subm N$ where $\subm$ is elementary submodel in the
smallest fragment $L^*$ containing $\psi$.


\medskip

We will sketch how to study this situation as the class of
\myalert{atomic} models of a first order theory.

%$\psi$ is \myalert{complete} if for every sentence $\phi$ of
%$L_{\omega_1,\omega}$

% $\psi \models \phi$ or $\psi \models \not
%\phi$.


\end{block}




\end{frame}

\begin{frame}
\frametitle{More Background}





A model $M$ is {\em small} if it realizes only countably many
$L_{\omega_1,\omega}$-types over the empty set. \vskip .2in

$M$ is small if and only $M$ is Karp-equivalent to a countable
model.


\vskip .2in

 $\phi$ is complete for $L_{\omega_1,\omega}$ if for every
sentence $\psi$ of $L_{\omega_1,\omega}$, either $\phi \rightarrow
\psi$ or $\phi \rightarrow \neg\psi$.

\vskip .2in

Note that a sentence is complete if and only if it is a Scott
sentence; so every model of a complete sentence is small.


\end{frame}

\begin{frame}
\frametitle{Passing to Atomic}

A model $M$ is \myalert{atomic} if every finite sequence from $M$
realizes a principal type over $\emptyset$.

\begin{block}{Theorem}

Let  $\psi$ be a  \myalert{ complete}  sentence in
$L_{\omega_1,\omega}$ in a countable vocabulary $\tau$.  Then
there is a countable vocabulary $\tau'$ extending $\tau$ and a
complete first order $\tau'$-theory $T$ such that reduct is a
$1$-$1$ map from the {\em atomic} models of $T$ onto the models of
$\psi$.
\end{block}



\end{frame}






\begin{frame}
\frametitle{ AMALGAMATION PROPERTY }
 The class $\bK$ satisfies the {\em
amalgamation property}  if for any
situation with $A,M , N \in \bK$: \\

\unitlength=1.00mm \linethickness{0.4pt} \begin{center}
\begin{picture}(15.00,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\end{picture}
\end{center}
 \vskip .3in

there exists an $N_1$ such that \\
\vskip .3in  \unitlength=1.00mm \linethickness{0.4pt}
\begin{center}
\begin{picture}(21.20,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(23.00,11.50){\makebox(0,0)[cc]{$N_1$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(15.50,16.00){\vector(3,-2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\put(15.50,6.00){\vector(3,2){5.00}}
\end{picture}\
\end{center}
\end{frame}




\begin{frame}
\frametitle{Failure of amalgamation yields many models}

\begin{block}{Theorem (WGCH: Shelah)}

If $\bK$ is $\lambda$-categorical and amalgamation fails in
$\lambda$ there are $2^{\lambda^+}$ models in $\bK$ of cardinality
$ \lambda^+$.
\end{block}

\end{frame}

\begin{frame}
\frametitle{Upward L\"owenheim Skolem}

\begin{block}{Definition}

$\aleph_\alpha$ is characterized by $\phi_\alpha$ if  there is a
model of $\phi_\alpha$ with cardinality $\aleph_\alpha$ but no
larger model.
\end{block}

\begin{block}{Known}

Morley:  If $\phi$ has a model of cardinality at least
$\beth_{\omega_1}$, $\phi$ has arbitrarily large models.

\medskip

Hjorth:  If $\alpha$ is countable $\aleph_\alpha$ is
characterizable.

\end{block}

\begin{block}{Conjecture}

Shelah:  If $\kappa$ is characterized by $\phi$, $\phi$ has
$2^{\lambda}$ models in some $\lambda \leq \kappa$.

\end{block}



 \end{frame}

\begin{frame}
\frametitle{Few models and smallness}


\begin{block}{Theorem (Keisler)}

If $\bK$ has less than $2^{\aleph_1}$ models of cardinality
$\aleph_1$  then every model of $\bK$ realizes only countably many
types over the empty set in the countable fragment $L^*$.
\end{block}
\end{frame}



\begin{frame}
\frametitle{Few models in $\aleph_1$  implies completeness}






\begin{block}{Theorem ( Shelah)}

If the $L_{\omega_1,\omega}$-$\tau$-sentence
%$L_{\omega_1,\omega}$-sentence
$\psi$ has a model of cardinality $\aleph_1$ which is $L^*$-small
for every countable $\tau$-fragment $L^*$ of
%$L_{\omega_1,\omega}(Q)$,
$L_{\omega_1,\omega}$, then $\psi$ has a small model of
cardinality $\aleph_1$.
\end{block}

\end{frame}



\begin{frame}
\frametitle{$\kappa$-Categoricity implies completeness ????}



 Thus, any $\aleph_1$-categorical sentence of $L_{\omega_1,\omega}$ can be
replaced (for categoricity purposes) by considering the atomic
models of a first order theory.  ($EC(T,Atomic)$-class)  But this
result uses properties of $\aleph_1$ heavily.

\begin{block}{Question}

If the %$L_{\omega_1,\omega}$-$\tau$-sentence
$L_{\omega_1,\omega}$-sentence $\psi$ is $\kappa$-categorical,
must the model in cardinality $\kappa$ be small?
%
% has a model of cardinality
%$\kappa$ which is $L^*$-small for every countable $\tau$-fragment
%$L^*$ of
%$L_{\omega_1,\omega}(Q)$,
%$L_{\omega_1,\omega}$, must $\psi$ have a $\tau$-small model of
%cardinality $\kappa$?
\end{block}

\end{frame}


\begin{frame}
\frametitle{Categoricity Transfer in $L_{\omega_1,\omega}$}

An atomic class $\bK$ is \myalert{excellent} if it is
$\omega$-stable and satisfies certain amalgamation properties for
finite systems of models.

\begin{block}{ZFC: Shelah 1983}

 If $\bK$ is  an \myalert{excellent}
$EC(T,Atomic)$-class then if it is categorical in one uncountable
cardinal, it is categorical in all uncountable cardinals.
\end{block}

\begin{block}{VWGCH: Shelah 1983}

If an  $EC(T,Atomic)$-class $\bK$ is  categorical  in $\aleph_n$
for all  $n< \omega$, then it is excellent.
\end{block}
\end{frame}




\begin{frame}
\frametitle{Excellence gained: more precisley}


\begin{block}{VWGCH: Shelah 1983}

 An atomic class $\bK$ that has at
least one uncountable model and with  $I(\bK,\aleph_n)\leq
2^{\aleph_{n-1}}$ for each $n< \omega$ is excellent.

\end{block}
\end{frame}
\begin{frame}
\frametitle{Context}
\begin{block}

$\bK$ is the class of atomic models (realize only principal types)
of a first order theory.

 We study $S_{at}(A)$ where $A \subset M \in \bK$ and

 $p\in S_{at}(A)$ means $Aa$ is atomic if $a$ realizes $p$.



\end{block}

\end{frame}



\begin{frame}
\frametitle{reprise: Few models and smallness}


\begin{block}{Theorem (Keisler)}

If $\bK$ has less than $2^{\aleph_1}$ models of cardinality
$\aleph_1$  then every model of $\bK$ realizes only countably many
types over the empty set in the countable fragment $L^*$.
\end{block}
\end{frame}







\begin{frame}
\frametitle{$\omega$-stability I}
\begin{block}{Definition}

The atomic class $\bK$ is $\lambda$-stable
 \index{$\lambda$-stable} if for every $M\in \bK$ of
cardinality $\lambda$, $|S_{\at}(M)| = \lambda$.

\end{block}

\begin{block}{Corollary (Shelah) CH}

If $\bK$ is $\aleph_1$-categorical and $2^{\aleph_0} <
2^{\aleph_1}$ then $\bK$ is $\omega$-stable.
\end{block}




\end{frame}


\begin{frame}
\frametitle{$\omega$-stability II}



\begin{block}{Consequences}

\begin{enumerate}
\item This gets $\omega$-stability without assuming arbitrarily
large models.
\item We only demand few types over \myalert{models}, not arbitrary sets; this is  crucial.
\item But, apparently uses CH twice! (for amalgamation and type
counting)
\end{enumerate}
\end{block}

\end{frame}


\begin{frame}
\frametitle{$\omega$-stability III}



\begin{block}{Getting $\omega$-stability}

\begin{enumerate}
\item Assume arbitrarily large models; use Ehrenfeucht-Mostowski
models
\item Keisler-Shelah using CH.
\item Diverse classes (Shelah)

\end{enumerate}
\end{block}

\end{frame}

\section{Is  WCH is necessary?}

\begin{frame}
\frametitle{Fundamental question}

Let $\phi$ be a sentence of $L_{\omega_1,\omega}$

Are the following properties absolute for cardinal-preserving
forcing? :
\smallskip

\begin{block}
{Almost surely true}

$\phi$ is $\omega$-stable

$\phi$ is excellent.
\end{block}

e.g. excellence is $\pi^1_2$.  `For very configuration of n
independent countable models there is a primary extension.'

\pause

\begin{block}{Real question}

$\phi$ is $\aleph_1$-categorical

\end{block}

\smallskip



\end{frame}

\begin{frame}
\frametitle{Is  WCH is necessary?}
\begin{block}

Does  MA + $\neg$ CH imply there is a sentence of
$L_{\omega_1,\omega}$ that is $\aleph_1$ categorical but
\begin{itemize}
\item[a] is not $\omega$-stable
\item[b] does not satisfy amalgamation even for countable models.
\end{itemize}
\end{block}

There is such an example in $L_{\omega_1,\omega}(Q)$ but Laskowski
showed the example proposed for $L_{\omega_1,\omega}$ by Shelah
(and me) fails.



\end{frame}


\begin{frame}
\frametitle{Towards Counterexamples}


For any model $M\in \bK$,
\begin{enumerate}
\item $P$ and $Q$ partition $M$.
\item $E$ is an equivalence relation on $Q$.
\item $P$  and each equivalence class of $E$ is
denumerably infinite.
\item $R$ is a relation on $P\times Q$ that is extensional on $P$.
That is, thinking of $R$ as the `element' relation, each member of
$Q$ denotes a subset of $P$.
\item  For every set $X$ of $n$ elements $X$ from $P$ and every subset
$X_0$ of $X$ and each equivalence class in $Q$, there is an
element of that equivalence class that is $R$-related to every
element of $X_0$ and not to any element of $X-X_0$.
\item Similarly, for every set of $n$ elements $Y$ from $Q$ and every subset
$Y_0$ of $Y$, there is an element of $P$ that is $R$-related to
every element of $Y_0$ and not to any element of $Y-Y_0$.
\end{enumerate}

\end{frame}


\begin{frame}
\frametitle{An AEC counterexample} Fix the class $\bK$ as above
and for $M,N \in \bK$, define $M \subm N$

if  $P^M=P^N$ and for each $m \in Q^M$, $\{n\in N: mEn\}=\{n\in M:
mEn\}$ (equivalence classes don't expand).

This class does  \myalert{not} have finite character.
% (Trlifaj's talk).

\end{frame}




 \begin{frame}
 \frametitle{$L_{\omega_1,\omega}$ vrs $L_{\omega_1,\omega}(Q)$}

 \myalert{Without} $Q$ -transfer is proved from VWGCH.

 \myalert{With} $Q$, $\aleph_1$-categoricity is not absolute.

 The most convincing example (Zilber) is \myalert{with} $Q$.


\pause

Does \myalert{finitary} capture the difference?


\end{frame}




 \begin{frame}
 \frametitle{Finitary Classes}

 \begin{enumerate}
 \item an independence theory over sets
 \item an analysis for \myalert{arbitrary} sentences in
 $L_{\omega_,\omega}$ - Vaught's conjecture.
 \item warning: `uniqueness of monster model'

 \item simplicity?
 \item the Hart-Shelah example; other examples of Hytinnen.
\item [Trlifaj] Every $\Nperp$ of pure-injective modules is finitary.
Conjecturally all $\Nperp$'s are finitary
 \end{enumerate}



 \end{frame}



\section{Connections with Algebra and analysis}


\begin{frame}
 \frametitle{More restrictive classes}

\begin{block}{Properties}

Tameness, finitary, amalgamation, Galois-compactness

\end{block}

\begin{block}{NEED EXAMPLES}
In particular, examples connected to core mathematics:
\begin{enumerate}
\item Banach Spaces \item complex exponentiation and semi-abelian
varieties \item Modules and Abelian groups \item locally finite
groups
\end{enumerate}
\end{block}


\end{frame}
\begin{frame}
\frametitle{ ZILBER'S PROGRAM FOR  $(\CC, +, \cdot,\exp)$} \vskip
.1in


Goal: Realize $(\CC, +, \cdot,\exp)$ as a model of an
$L_{\omega_1,\omega}(Q)$-sentence discovered by the Hrushovski
construction.

\vskip .1in
\begin{block}{Done}

 {\bf A.}
Expand $(\CC, +, \cdot)$ by a unary function which behaves like
exponentiation using a Hrushovski-like dimension function. Prove
some $L_{\omega_1,\omega}(Q)$-sentence $\Sigma$ is categorical and
has quantifier elimination. \end{block} \vskip .1in

\begin{block}{Very open}


 {\bf B.} Prove
$(\CC, +, \cdot,\exp)$ is a model of the sentence
 $\Sigma$ found in Objective A.
 \end{block}


\end{frame}

\begin{frame}
\frametitle{Questions}

\begin{enumerate}
\item B is a problem in analysis and number theory \item If a
sentence $\psi$ of $L_{\omega_1,\omega}(Q)$ is categorical up to
$\aleph_\omega$, must it be forever categorical?
\end{enumerate}

\end{frame}


\begin{frame}
\frametitle{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{frame}


\begin{frame}
\frametitle{Contexts}

Trivial case. $A$ is $\aleph_1$-categorical in the Abelian group
language-- e.g. $(Q,\cdot)$

Interesting case.  Work with `field language' on $A$.

Really interesting case: pseudo-exponentiation and Hrushovski
construction.

\end{frame}



\begin{frame}
\frametitle{ Semi-abelian varieties}


In the Ravello volume Zilber obtains
   `arithmetical conditions' on algebraic variety that are
   equivalent to `excellence'




    Thus he is able to conclude - using Shelah's machinery that
    under WGCH,

    Categoricity up to $\aleph_\omega$ implies `arithmetical'
    properties of certain algebraic varieties.

    This argument could be shortened if there were a direct
    algebraic argument for tameness.


\end{frame}



\section{Classification Theory}

\begin{frame}
 \frametitle{Tameness}

 For countable theories $(\aleph_0,\infty)$-tameness is incredibly
 powerful.  It allows the categoricity transfer from a single
 small cardinal (Grossberg, VanDieren, Lessmann).

 \begin{enumerate}
 \item Find other sufficient conditions for tameness.

 \item What is the relation between tameness and excellence
 \end{enumerate}


 \end{frame}


\end{document}


\begin{frame}
 \frametitle{Superstability}

 \begin{enumerate}
 \item Shelah
 \item Grossberg-VanDieren-Villaveces
 \item JTB
 \item Hytinnen-Kes{\"a}l{\"a}
\end{enumerate}
 \end{frame}




\begin{frame}
 \frametitle{More Topics}

\begin{enumerate}
 \item Compute the possible stability spectra of AEC.
  (Even assuming tameness).




 \item Presentation theorems: Can these be viewed as Lindstrom theorems?
 \begin{enumerate}
 \item Shelah
 \item Kirby
 \item Kueker
 \end{enumerate}
\end{enumerate}

\end{frame}

\end{document}



\begin{frame}
 \frametitle{$L_{\omega_1,\omega}(Q)$ as an AEC}

 \begin{block}{Fails union of chains under natural elementary
 submodel}
 \end{block}

 \begin{block}{$\leq^*$}
 Require that `small' definable sets do not grow.

 Gives an AEC with LN $\aleph_1$.

 \end{block}

\begin{block}{Weak models}
 The class of `weak models' with $\leq^*$
 gives an AEC with LN $\aleph_0$.

 But we added models; the argument showing the existence of many
 models don't work (they might be weak models).
 \end{block}
\end{frame}

\begin{frame}
 \frametitle{Approaches to $L_{\omega_1,\omega}(Q)$}

 \begin{enumerate}
 \item ad hoc (there is a model in $\aleph_2$.)
 \item $Q$-aec (Coppola)
 \item frames
 \item deal with sentences that are AEC
\begin{itemize}
\item [a)] for semantic reasons
(Zilber/Kirby)
  \item [b)]for syntactic reasons (Caicedo)
 \end{itemize}
\end{enumerate}

\end{frame}
\section{Amalgamation}

\begin{frame}
\frametitle{ AMALGAMATION PROPERTY }
 The class $\bK$ satisfies the $\lambda$-{\em
amalgamation property}  if for any
situation with $A,M , N \in \bK_\lambda$: \\

\unitlength=1.00mm \linethickness{0.4pt} \begin{center}
\begin{picture}(15.00,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\end{picture}
\end{center}
 \vskip .3in

there exists an $N_1$ such that \\
\vskip .3in  \unitlength=1.00mm \linethickness{0.4pt}
\begin{center}
\begin{picture}(21.20,11.00)(2.00,9.90)
\put(3.00,11.50){\makebox(0,0)[cc]{$A$}}
\put(13.00,4.50){\makebox(0,0)[cc]{$M$}}
\put(23.00,11.50){\makebox(0,0)[cc]{$N_1$}}
\put(13.00,17.50){\makebox(0,0)[cc]{$N$}}
\put(5.50,13.00){\vector(3,2){5.00}}
\put(15.50,16.00){\vector(3,-2){5.00}}
\put(5.50,9.00){\vector(3,-2){5.00}}
\put(15.50,6.00){\vector(3,2){5.00}}
\end{picture}\
\end{center}
\end{frame}

\begin{frame}
\frametitle{ AMALGAMATION PROPERTY -- variants}

$\bK$ has the amalgamation property if  each $K_\lambda$-does.

I.e. no cardinality restrictions.

\medskip

Note that amalgamating over subsets rather than submodels is
strictly stronger.
\medskip

Keisler proved Morley's theorem for classes with arbitrarily large
models and set-amalgamation in 1970.

\end{frame}

\begin{frame}
\frametitle{Finite Diagrams and Homogeneous Model Theory}

\begin{block}{Finite Diagrams}

If $T$ is a first order theory $\bK$ is the class of models of $T$
that omit a certain set $\Gamma$ of finite-types over the empty
set then $\bK$ under first order elementary submodel is called a
\myalert{finite diagram}.

\end{block}


\begin{block}{Atomic Classes}

If $\Gamma$ is all non-principal types, $\bK$ is called an atomic
class.

\end{block}

\begin{block}{Homogenous Model Theory}

is the study of finite diagrams that admit amalgamation over
\myalert{SETS}.
\end{block}

\end{frame}



\begin{frame}
\frametitle{Connections to $L_{\omega_1,\omega} $}

Every sentence of $L_{\omega_1,\omega}$ can be regarded (class of
models is isomorphic) as a finite diagram.

\medskip

\begin{block}{Definition}

 A   sentence $\psi$ in $L_{\omega_1,\omega}$ is called {\em
complete} if for every sentence $\phi$ in $L_{\omega_1,\omega}$,
either $\psi \models \phi$ or $\psi \models \neg\phi$.
\end{block}

Every \myalert{complete}-sentence can be regarded  as an atomic
class.

\end{frame}






\begin{frame}
\frametitle{Categoricity and completeness}

Let $\bK$ be the class of models of an
$L_{\omega_1,\omega}$-sentence $\phi$.

\begin{block}{Theorem}
If $\bK$ has a unique model $M$ with cardinality $\kappa$, and

\begin{enumerate}
\item $\bK$ has arbitrarily large models
\item or $\kappa =\aleph_1$
\end{enumerate}

then some complete $\psi$ implies $\phi$ and $M \models \phi$.

\end{block}

*) But what if $\kappa = \aleph_2$?

\medskip

The proof of this gives a much shorter argument of a result of
Makkai on Vaught's conjecture.

\end{frame}






%\begin{frame}

%\frametitle{Partial answer to *}

%Kueker shows that if $\bK$ is

%finitary (a.p., jep, arb large models and finite character)


%then $\kappa$ can be arbitrary




%\end{frame}






\begin{frame}
\frametitle{Non-homogeneous examples I}

There is a finite diagram (Example 18.14) with:
\begin{enumerate}
\item
 joint embedding,
 \item ap in $K_\lambda$ for
uncountable $\lambda$ \item categorical in all uncountable
$\lambda$ \item but amalgamation of countable models fails.
\end{enumerate}

\medskip


Under WGCH there can be no such atomic class.


\end{frame}


\begin{frame}
\frametitle{Non-homogeneous examples II}

There are $L_{\omega_1,\omega}$ sentences that are categorical in
all powers, and do not satisfy set amalgamation.

\begin{enumerate}
\item Marcus
\item Knight
\item Zilber: covers of algebraic groups
\item Zilber: psuedoexponentiation ($L_{\omega_1,\omega}(Q)$)
\end{enumerate}



\end{frame}

\section{Galois Types}

\begin{frame}
\frametitle{  Model Homogeneity}

 \begin{definition} $M$ is $\mu$-model homogenous if for every $N \subm M$
and every $N' \in \bK$  with $|N'| < \mu$ and $N \subm N'$ there
is a $\bK$-embedding of $N'$ into $M$ over $N$.
\end{definition}
\vskip .2in To emphasize, this differs from the homogenous context
because the $N$ must be \myalert{ in} $\bK$.  It is easy to show:
\vskip .2in
\end{frame}

\begin{frame}

\frametitle{  Monster Model}
\begin{lemma}\label{mhuni} (jep) If $M_1$ and $M_2$ are $\mu$-model homogenous of
cardinality $\mu> \LS(\bK)$ then \\ $M_1 \approx M_2$.
\end{lemma}

\vskip .4in
\begin{theorem}\label{monsexist}  If $\bK$ has the amalgamation property and
$\mu*^{<\mu*}= \mu*$ and $\mu*\geq 2^{ \LS(\bK)}$ then there is a
model $\Mscr$ of cardinality $\mu*$ which is $\mu*$-model
homogeneous.

\end{theorem}

\end{frame}



\begin{frame}

\frametitle{  GALOIS TYPES:  Algebraic Form}

Suppose $\bK$ has the amalgamation property.

\begin{definition}  Let $M \in \bK$, $M \subm \MM$ and $a\in \MM$.
The Galois type of $a$ over $M$ is the orbit of $a$ under the
automorphisms of $\MM$ which fix $M$.
\end{definition}
\vskip .2in


We say a Galois type $p$ over $M$  is realized in $N$ with $M
\subm N \subm \MM$ if $p \cap N \neq \emptyset$.


%\pause \medskip
% Equivalently,

%Suppose $M \subm N_1, N_2$ and $\abar\in N_1-M$,
%$\bbar\in N_2-M$.  If there exists an $N_3$ and
%\textcolor{clifton}{strong embeddings} $f_1, f_2$ mapping $N_1,
%N_2$ into $N_3$ over $M$ with $f_1(\abar) =f_2(\bbar)$ then
%$\abar$ and $\bbar$ realize the same Galois type over $M$.


\end{frame}
\begin{frame}

\frametitle{ Galois vrs Syntactic Types}


Syntactic types have certain natural locality properties.

\begin{itemize}
\item[]\textcolor{clifton}{locality} Any increasing chain of types has at
most one upper bound;
\item[]\textcolor{clifton}{tameness} two distinct types differ on a finite set;
\item[]\textcolor{clifton}{compactness} an increasing
chain of types has a realization.

\end{itemize}

The translations of these conditions to Galois types do not hold
in general.


\end{frame}








\section{Tameness}



\begin{frame}\frametitle{Tameness}

Grossberg and VanDieren focused on the idea of studying `tame'
abstract elementary classes:

\vskip .2in

\begin{definition}\label{tamedef}\index{tame} \index{weakly tame}
  We say $\bK$ is $(\chi,\mu)$-{\em tame} if for any
 $N \in \bK$ with $|N|=\mu $ if $p,q, \in \gaS(N)$
and for every $N_0 \leq N$ with $|N_0| \leq \chi$, $p\restriction
N_0 =q\restriction N_0$ then $q =p$.
\end{definition}

\end{frame}


\begin{frame}\frametitle{Tameness-Algebraic form}
  Suppose $\bK$ has the amalgamation property.\smallskip

$\bK$ is $(\chi,\mu)$-tame if for any model $M$ of cardinality
$\mu$ and any $a,b \in \Mscr$:

\pause \medskip

If for every $N\subm M$ with $|N| \leq \chi$ there exists $\alpha
\in \aut_N(\Mscr)$ with $\alpha(a) =b$,

\smallskip

then there exists $\alpha \in \aut_M(\Mscr)$ with $\alpha(a) =b$.
\end{frame}



\begin{frame}\frametitle{Consequences of Tameness}

Suppose $\bK$ has arbitrarily large models and amalgamation.

\begin{theorem}[Grossberg-Vandieren]  If $\lambda > \LS(\bK)$, $\bK$ is
$\lambda^+$-categorical and $(\lambda,<\infty)$-tame then $\bK$ is
categorical in all $\theta \geq \lambda^+$.
\end{theorem}

\medskip

\begin{theorem}[Lessmann]  If $\bK$ with $\LS(\bK) = \aleph_0$ is
$\aleph_1$-categorical and $(\aleph_0,\infty)$-tame then $\bK$ is
categorical in all uncountable cardinals
\end{theorem}

\end{frame}







\begin{frame}\frametitle{Two Examples that are not tame}

\begin{enumerate}



\item Coding EXT

A class that is not $(\aleph_{0}, \aleph_{1})$-tame.

A class that is not $(\aleph_{0}, \aleph_{1})$-tame but is
$(2^{\aleph_{0}}, \infty)$-tame.  (Baldwin-Shelah)

 \item  `Hiding the zero'

For each $k<\omega$ a class which is $(\aleph_0,
\aleph_{k-3})$-tame but not $(\aleph_{k-3}, \aleph_{k-2})$-tame.
Baldwin-Kolesnikov (building on Hart-Shelah)




\end{enumerate}
\end{frame}


\begin{frame}\frametitle{Tameness gained}


Baldwin-Shelah (Goodrick)

\begin{theorem}\label{fullex}  There is an AEC with the amalgamation property in a countable language

 with L\"owenheim-Skolem number
$\aleph_0$ which is not $(\aleph_0,\aleph_1)$-tame but is
$(2^{\aleph_0},\infty)$-tame.
\end{theorem}
\end{frame}

\begin{frame}
\frametitle{Syntactic not Galois}


{\bf Theorem.} [Hart-Shelah / Baldwin-Kolesnikov] For each $3 \leq
k<\omega$ there is an $L_{\omega_1,\omega}$ sentence $\phi_k$ such
that:



\begin{enumerate}
\item $\phi_k$ has the disjoint amalgamation property;
\item Syntactic types determine Galois types over models of cardinality
at most $\aleph_{k-3}$;
\item But there are syntactic types over models of size
$\aleph_{k-3}$ that split into $2^{\aleph_{k-3}}$-Galois types.

\medskip \pause

\item $\phi_k$ is categorical in $\mu$ if  $\mu\leq \aleph_{k-2}$;
\item $\phi_k$ is not $\aleph_{k-2}$-Galois stable;
\item But for $m\leq k-3$, $\phi_k$ is  $\aleph_{m}$-Galois stable;
\item $\phi_k$ is not categorical in any $\mu$ with
$\mu>\aleph_{k-2}$.

\end{enumerate}
\end{frame}




\begin{frame}\frametitle{A Fundamental Construction }

\begin{block}{(Baldwin, Lachlan, Marker)}

 Let $G$ be an expansion of a  group and
let $\pi$ map $X$ onto~$G$.
\medskip

 Add to the language a binary function
$t:G \times X \rightarrow X$ for the  fixed-point free action of
$G$ on $\pi^{-1}(g)$ for each $g \in G$.
\medskip

That is, we represent $\pi^{-1}(g)$ as $\{ga: g \in G\}$ for some
$a$ with $\pi(a) =g$. This action of $G$ is strictly
$1$-transitive. This guarantees that each fiber has the same
cardinality as $G$.
\medskip

$\pi$ guarantees the number of fibers is the same as $|G|$.
\medskip

Since there is no interaction among the fibers, categoricity in
all uncountable powers follows if $G$ is categorical.
\end{block}

\end{frame}







\begin{frame}\frametitle{Consequence}

This gives us groups which are categorical but not almost strongly
minimal or even almost quasi-excellent.

\medskip

If  $G$ is the collection of maps from $I$ into $Z_2$ with finite
support, the structure $(I,G,Z_2)$ is quasiminimal excellent and
not first order.


\end{frame}


\begin{frame}\frametitle{Specific Construction }

\begin{block}{(Hart-Shelah, Baldwin-Kolesnikov)}

vocabulary $L'$:  unary predicates $I,K,G,G^*,H,H^*$;

a binary function $e_G$ taking $G \times K$ to $H$;

a function $\pi_G$ mapping $G^*$ to $K$,

a function $\pi_H$ mapping $H^*$ to $K$,

 a 4-ary relation $t_G$
on $K\times G \times G^*\times G^*$,

a $4$-ary relation $t_H$ on $K\times H \times H^*\times H^*$.
\medskip

 vocabulary $L$: Add a $(k+1)$-ary relation $Q$ on
$(G^*)^k \times H^*$.
\end{block}

\end{frame}


\begin{frame}\frametitle{Key points}

$K = [I]^k$;
\medskip

$G$ is finite support functions from $K$ to $G$.

\medskip


$t_G$ and $t_H$ are the actions on $G^*$ and $H^*$ as before.

\end{frame}


\begin{frame}\frametitle{The Crux: Diagonalize}


$Q$ is a $(k+1)$-ary relation on $(G^*)^k \times H^*$.

\medskip

For all $\gamma_1,\dots,\gamma_k\in G$ and all $\delta\in H$ we
have
\begin{multline*}
Q((u_1,x_1),\dots, (u_k,x_k),(u_{k+1},x_{k+1}))
\\
\Leftrightarrow Q((u_1,x_1+\gamma_1),\dots,
(u_k,x_k+\gamma_k),(u_{k+1},x_{k+1}+\delta))
\end{multline*}
if and only if $\gamma_1(u_{k+1})+\dots
+\gamma_k(u_{k+1})+\delta=0\  \mod \ 2$.




\end{frame}


\begin{frame}\frametitle{The Rub}

$\phi_k$ is the Scott sentence of this structure.

\medskip

\begin{block}{We would like to say:}

It admits k-ary but not $(k+1)$-ary amalgamation of countable
structures.

\medskip

Hence, it is not excellent.
\end{block}

\begin{block}{Can only say:}

It admits k-ary but not $(k+1)$-ary amalgamation of
\myalert{solutions} over  countable structures.

\medskip

Hence, it is  not stable, so not categorical, so not excellent.
\end{block}





\end{frame}




\section{$\Nperp$}

 \begin{frame}\frametitle{Whitehead Groups}

\begin{definition} We say $A$ is a {\em Whitehead group} if
$Ext(A,\ZZ) =0$.  That is, every short exact sequence
$$0\rightarrow \ZZ \rightarrow H\rightarrow A \rightarrow 0,$$
splits or in still another formulation, $H$ is the direct sum of
$A$ and $\ZZ$.
\end{definition}

\smallskip

Side question: Under V=L, Whitehead groups are free; hence
PC$\Gamma$. What about in ZFC?
\end{frame}







\begin{frame}
\frametitle{$\Nperp$}

\begin{definition}  \begin{enumerate}
\item $\Nperp = \{A: Ext^i(A,N) =0: i<\omega\}$

\item For $A \subseteq B$ both in  $\Nperp$, $A \subN B$ if $B/A \in
\Nperp$.
\end{enumerate}
\end{definition}

Generalizes the class of Whitehead groups: $Ext(G,\ZZ) = 0$.


\end{frame}

\begin{frame}
\frametitle{Synergy}

Baldwin, Eklof, Trlifaj :

\begin{theorem}\label{isaec}
 \begin{enumerate}
\item  For any module  $N$, %over an hereditary ring $R$,
if the class $(\Nperp, \subN)$ is an abstract elementary class
then  $N$ is a cotorsion module.

\item %(V=L)
For any $R$-module $N$, over a ring $R$,
 if  $N$ is a pure-injective
%cotorsion
 module then the class $(\Nperp, \subN)$
is an abstract elementary class.
\item  For an abelian group $N$, (module over a  Dedekind domain $R$),
the class $(\Nperp, \subN)$ is an abstract elementary class if and
only if $N$ is a cotorsion module.


%\item In any case$(\Nperp,
%\subN)$ has the amalgamation property.
\end{enumerate}
\end{theorem}

\end{frame}

\begin{frame}
\frametitle{An Interesting detail}

We do not know exactly the rings for which the hypothesis of $N$
cotorsion is sufficient  for {\bf A.3}(3). It is sufficient when
$R$ is a Dedekind domain:

\begin{block}{Lemma}

 Let $R$ be a Dedekind domain and $N$ a module. Then the
following are equivalent:
\begin{enumerate}
\item $N$ is cotorsion;
\item $\Nperp = {}^\perp PE(N)$ where $PE(N)$ denotes the pure--injective envelope of $N$;
\item $\Nperp$ is closed under direct limits;
\item {\bf A3}(3) holds for $(\Nperp, \subN)$.
\end{enumerate}
\end{block}

3) concludes closure under limits of \myalert{all} homomorphisms.

\end{frame}





\section{Some problems}

\begin{frame}
\frametitle{Some Problems}
\end{frame}

\begin{frame}
\frametitle{What causes tameness?}

\begin{enumerate}
\item syntactic types - galois types
\item ${\rm ga-tp}(a/M) = \tp(\abar/M)$ for a `countable' $\abar$.
($\Nperp$, Abelian groups under pure substructure)
\item excellence (more precisely, the existence of a `nonforking
notion with stationary types and extension (Grossberg-Kolesnikov))
\item ????
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Geometric Model Theory in $L_{\omega_1,\omega}$}

Zilber showed:

\begin{theorem} If in an $\aleph_1$-categorical theory $T$ some
(every) strongly minimal set has trivial geometry then  $T$ is
almost strongly minimal?
\end{theorem}

Is the analog true in $L_{\omega_1,\omega}$ with almost
quasiminimal excellence replacing almost strongly minimal  ?

\end{frame}

\begin{frame}
\frametitle{Problems on the `Lower Infinite'}

lower infinite = below $\beth_{\omega_1}$

\begin{enumerate}
\item Around Vaught's conjecture
\item Bounds on existence
\item Bounds on categoricity
\end{enumerate}
\end{frame}


\subsection{Vaught's conjecture}

\begin{frame}
\frametitle{Around Vaught's conjecture}

\begin{block}{The number of models in $\aleph_1$: $L_{\omega,\omega}$ }


\begin{theorem} If a first order theory is a counterexample to the
Vaught conjecture then it has $2^{\aleph_1}$ models of cardinality
$\aleph_1$.
\end{theorem}
\end{block}

\end{frame}

\begin{frame}
\frametitle{Proof outline} This is easy from two difficult
theorems:

\begin{theorem}[Shelah] If a first order $T$ is  not $\omega$-stable
$T$ has $2^{\aleph_1}$ models of cardinality $\aleph_1$.
\end{theorem}

This argument uses many descriptive set theoretic techniques. See
Shelah's book  or Baldwin's paper.


\begin{theorem}[Shelah]  An $\omega$-stable first order theory
satisfies Vaught's conjecture.
\end{theorem}

\end{frame}
\begin{frame}
\frametitle{Does the previous theorem extend to
$L_{\omega_1,\omega}$?}

%The number of models in $\aleph_1$:
%$L_{\omega_1,\omega}$ }










Keisler showed:


\begin{theorem}\label{keissmall} For any $L_{\omega_1,\omega}$-sentence $\psi$ and
any fragment $L^*$ containing $\psi$, if $\psi$ has fewer than
$2^{\aleph_1}$ models of cardinality $\aleph_1$ then for any $M
\models \psi$ of cardinality $\aleph_1$, $M$ realizes only
countably many $L^*$-types over the empty set
\end{theorem}

%; i.e. $\psi$ is scattered. \vfill\newpage \centerline{ \LARGE\bf
%T$2^{\aleph_0}< 2^{\aleph_1}$ \vskip .3



 Shelah observed that Theorem~\ref{keissmall} implies:

\begin{fact} ($2^{\aleph_0}< 2^{\aleph_1}$) If a \myalert{complete}
sentence $\psi \in L_{\omega_1,\omega}$ is not
$\omega$-stable it has $2^{\aleph_1}$ models of cardinality
$\aleph_1$.
\end{fact}

\end{frame}
\begin{frame}
\frametitle{Develop $\omega$-stability for finite diagrams}

Even assuming ($2^{\aleph_0}< 2^{\aleph_1}$):

\smallskip

Few models in $\aleph_1$ does \myalert{not} imply amalgamation in
$\aleph_0$.

\smallskip

Does few models in $\aleph_1$  imply $\omega$-stability in any
reasonable sense.

\smallskip

Does VC hold for $\omega$-stable sentences in
$L_{\omega_1,\omega}$?  For excellent classes?

\smallskip


This is meaningless for complete sentences. But it fits the
context of Hytinnen-Kesala.

\end{frame}

\subsection{Bounds on existence}
\begin{frame}
\frametitle{Characterizing $\aleph_1$}

\begin{theorem}[Hjorth    ] For every $\alpha< \omega_1$ there is a
sentence in $L_{\omega_1,\omega}$ whose maximal model has
cardinality $\aleph_\alpha$.
\end{theorem}

\end{frame}
\begin{frame}
\frametitle{Proof outline for $\aleph_1$}

vocabulary: $S_n$ binary; $R_k$ is $k+2$-ary.

A set of universal sentences guarantee that every model $M$
satisfies:

\begin{enumerate}
\item The $S_n$ are symmetric and partition $M^{[2]}$.
\item For all $a,b$ for some $n$, $S_n(a,b)$.
\item \begin{eqnarray*}
\bigwedge_m [R_k(a,b,c_1,\ldots c_k) \rightarrow \\(S_m(d,a_0)
\wedge S_m(d,a_1) \rightarrow d \in \{c_1, \ldots c_k\})]
\end{eqnarray*}
\end{enumerate}

Defining $f(a,b) = n$ if $S_n(a,b)$\\ maps  $M^2$ into $\omega$.

In the generic model for each $a,b$ there is finite $C$: $f(c,a)
=f(c,b)$ iff $c\in C$.
\end{frame}
\begin{frame}
\frametitle{Consequences}
\begin{block}{No model in $\aleph_2$}

If $|N| = \aleph_2$ and $M\prec N$, $|M| = \aleph_1$:
\smallskip

For each $a,b\in M$ there is finite $C\subset M$: $f(c,a) =f(c,b)$
iff $c\in C$ (for all $c \in N$!).

\medskip

So if $e \in N-M$, for every $a,b\in M$, $f(e,a) \neq f(e,b)$.
\medskip That is $f(e,-)$ is a $1$-$1$ map from $N$ into $\omega$.
\end{block}
\end{frame}
\begin{frame}
\frametitle{Fra{\"i}ss\'e construction and $L_{\omega_1,\omega}$}
\begin{block}{A model in $\aleph_1$}

The basic Fra{\"i}ss\'e construction yields an atomic model; so
the atomic models of its first order theory are axiomatized in
$L_{\omega_1,\omega}$.

Marker (mainly) and I observed:

\medskip
For any Fra{\"i}ss\'e construction with \myalert{disjoint}
amalgamation:

The generic has a proper atomic elementary extension and so there
is an uncountable atomic model.

\end{block}

\end{frame}
\begin{frame}
\frametitle{Prove or Disprove}

\begin{theorem}[Shelah technique] If an atomic class is
$\omega$-stable and has a model in $\aleph_1$ it has one in
$\aleph_2$.
\end{theorem}

\medskip

Problem: Modify Hjorth's construction by replacing Fra{\"i}ss\'e
with Hrushovski to  get:

\medskip

An $\omega$-stable atomic class characterizing $\aleph_2$.



\end{frame}

\subsection{Bounds on categoricity}

\begin{frame}
\frametitle{Downward Categoricity}

\begin{theorem}[Shelah] If $\bK$ is categorical in $\lambda^+$
and satisfies \myalert{amalgamation and joint embedding} with
$\lambda \geq H_2$ then $\bK$ is categorical on $[H_2,\lambda)$.
\end{theorem}

What about non-successor cardinals for the hypothesis?

\medskip  What is the best lower bound?

Is it $\aleph_1, \aleph_\omega, \beth_{\omega_1}, H_2 =
\beth_{(\beth_{\omega_1})^+}$?
\end{frame}


\begin{frame}
\frametitle{Necessity of hypotheses}

Fact. If $\bK$ has amalgamation and is categorical in $\kappa$
above $H_1$, then $\bK_{\geq\kappa}$  has jep.

\medskip

But for $\alpha< \omega_1$, it is easy to construct sentences
$\phi_\alpha$ of $L_{\omega_1,\omega}$ that are categorical in
$\kappa$ iff $\kappa \geq \beth_\alpha$.

\end{frame}






\end{document}


\begin{frame}

Sh:3
    Shelah, Finite diagrams stable in power -- Annals Math Logic 2 (1970) 69-118
Sh:10
    Shelah, Stability, the f.c.p., and superstability; model theoretic properties of
    formulas in first order theory -- Annals Math Logic 3 (1971) 271-362
Sh:48
    Shelah, Categoricity in $\aleph _{1}$ of sentences in $L_{\omega _{1},\omega}(Q)$ --
    Israel J Math 20 (1975) 127-148

    Sh:54
    Shelah, The lazy model-theoretician's guide to stability --
    Logique et Analyse 18 (1975) 241-308

    Sh:87a
    Shelah, Classification theory for nonelementary classes, I. The number of uncountable models of $\psi \in L_{\omega _{1},\omega }$.
    Part A -- Israel J Math 46 (1983) 212-240

Sh:87b
    Shelah, Classification theory for nonelementary classes, I. The number of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part B -- Israel J Math 46 (1983) 241-273


\begin{frame}
\frametitle{ Categoricity of  Covers}


When is the theory $T_A + \Lambda = \ZZ^N$ of the cover:

\begin{eqnarray}0 \rightarrow Z^N \rightarrow V \rightarrow A \rightarrow 1.
\end{eqnarray}

categorical?

\medskip


Zilber has proved categoricity for $\Ascr = (\CC^{\times},\cdot)$.

In general categoricity, WGCH implies categoricity is equivalent
to `arithmetic' properties of $\Ascr$.




\end{frame}

\begin{frame}\frametitle{Tameness}


\vskip .2in

\begin{definition}\label{tamedef}\index{tame} \index{weakly tame}
  We say $\bK$ is $(\chi,\mu)$-{\em tame} if for any
 $N \in \bK$ with $|N|=\mu $ if $p,q, \in \gaS(N)$
and for every $N_0 \leq N$ with $|N_0| \leq \chi$, $p\restriction
N_0 =q\restriction N_0$ then $q =p$.
\end{definition}

\end{frame}

\begin{frame}
\frametitle{ Categoricity of  Covers}
When is the theory $T_A + \Lambda = \ZZ^N$ of the cover:

\begin{eqnarray}0 \rightarrow Z^N \rightarrow V \rightarrow A \rightarrow 1.
\end{eqnarray}

categorical?

\medskip


Zilber has proved categoricity for $\Ascr = (\CC^{\times},\cdot)$.

In general categoricity, VWGCH implies categoricity is equivalent
to `arithmetic' properties of $\Ascr$.

\end{frame}


\begin{frame}
\frametitle{(Very) Few Models } \begin{block}{Definition}
\begin{enumerate}
\item $\bK$ has {\em few} models in power $\lambda$ if $I(\bK,\lambda)<
2^{\lambda}$.

\smallskip

\item $\bK$ has {\em very few} models in power $\aleph_n$ if
$I(\bK,\aleph_n)\leq {2^{\aleph_{n-1}}}$.
\end{enumerate}
\end{block}

\end{frame}
