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%\input{jbmacaec}
\usetheme{Berkeley}
%\title{Categoricity\\Asian Logic Conference\\Singapore\\June %22, 2009}
\title{Can categorical classes in $L_{\omega_1,\omega}$ be bounded in size?\\
UIC Seminar: 2025}
\date{\today}






\author{John T. Baldwin\\
University of Illinois at Chicago}


%\institute[University of Illinois at Chicago]

\begin{document}



\frame{\titlepage}



\date{\today}
\author{John T. Baldwin}
\institute[University of Illinois at Chicago]




\begin{frame}{}
\tableofcontents
\end{frame}
%\AtBeginSection[]
    
\section{Two Alternatives}

\begin{frame}
\frametitle{Formal and Formalism-Free Mathematics}

\begin{block}{ Formal}
\begin{enumerate}
\item Explicit vocabulary,  syntax and notion of truth

e.g. $L_{\omega,\omega}, admissible fragments, L_{\lambda,\omega}(Q),
L_{2nd}$ etc. `elementary' submodel.

\item Natural axiomatizations of much of mathematics: axiom of Archimedes
    around structures up to the continuum.

\item Lindenbaum algebra and Stone space: Shelah for anything beyond first
    order \cite{Sh48}
\end{enumerate}
\end{block}
\begin{block}{Formalism-Free \cite{Kenbook}}
\begin{enumerate}
    \item standard mathematics including

     AEC: fixed vocabulary; abstract properties of `elementary' submodel

        \item `Galois' types
\end{enumerate}



\end{block}
\end{frame}

\begin{frame}
\frametitle{ ABSTRACT ELEMENTARY CLASSES defined}


\begin{definition}\label{aec} 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 following conditions.


 \begin{itemize}
\item  { \bf A1}. If $M\subm N$ then $M\subseteq N$.

\item { \bf A2}. $\subm $ is a partial order on $\bK$.

\item { \bf A3}. If $\langle A_i:i< \delta\rangle$ is
$\subm$-increasing chain:
\begin{enumerate}
\item $\bigcup_{i<\delta}A_i \in \bK$;
\item for each $j<\delta$, $A_j \subm \bigcup_{i<\delta}A_i$
\item if each $A_i \subm  M \in \bK$ then \\ $\bigcup_{i<\delta}A_i
\subm M$.
\end{enumerate}
\end{itemize}
\end{definition}

\end{frame}
\begin{frame}
\begin{itemize}
\item { \bf A4}. If $A,B,C \in \bK$, $A \subm C$, $B \subm C$ and $A\subseteq
 B$ then $A \subm B$.

\item { \bf A5}.  There is a L\"owenheim-Skolem number
$\LS(\bK)$ such that if $A \subseteq B \in K$ there is a $A' \in
\bK$ with $A \subseteq A' \subm B$ and\\ $|A'|< \LS(\bK)+|A|$.
\end{itemize}


\end{frame}
\section{Historical Background}
Historical Background
%\section{Do $\aleph_1$-categorical classes have `big' models?}
%Do $\aleph_1$-categorical theories have `big' models

\begin{frame}
\frametitle{When do $\aleph_1$-categorical theories (AEC) have a bounded
size of models?}


%\myalert{How big is big?}

In the mid-70's Shelah answered my question as to whether a sentence of
$L_{\omega_1,\omega}(Q)$ could be {\em categorical in the philosophers
sense}, have only one model.  In different papers he proved  in different
ways that $\aleph_1$-categorical such sentence has a model in $\aleph_2$.




\bigskip

Two questions: Under what conditions does
%
%WRONG FIRST QUESTION = omega stability or different cardinality   -- have
%%arbitrarily large?
% \begin{enumerate}  \item
 a sentence of $L_{\omega_1,\omega}$  (with LN $\aleph_0$)  that is $\aleph_1$-categorical
 have   models in $\aleph_2$, $2^{\aleph_0}$, or even larger?


 More generally, {\em Grossberg's  question} Must an aec categorical in
 $\lambda$ with $I(\bK,\lambda^+) < 2^{\lambda^+}$ have a model in $\lambda^{++}$?

\smallskip 
 We already know the second is independent of ZFC.  As follows.
 
% But \cite{MazYang} have
% prove the result for $\lambda < 2^{\aleph_0}$ assuming ap in $\lambda$,
% $\lambda$-stable and $(<\lambda^+,\lambda)$-local.

% We focus first on what else is needed to get $\omega$-stability from
% $\aleph_1$-categoricity.

\end{frame}

\begin{frame}\frametitle{One Completely General Result}

WGCH($\lambda$): $2^\lambda< 2^{\lambda^+}$

Let $\bK$ be an abstract elementary class (AEC).

\smallskip

\begin{theorem}\label{amalholds}[WGCH $(\lambda)$ ] Suppose $\lambda \geq \LS(\bK)$ and $\bK$ is
$\lambda$-categorical.  If amalgamation fails in $\lambda$ there are
$2^{\lambda^+}$ models in $\bK$ of cardinality $\kappa = \lambda^+$.
\end{theorem}



Uses [$\hat \Theta_{\lambda^+}(S)$]  (weak diamond) for many $S$.

\smallskip

$\lambda$-categoricity plays a fundamental role.

\smallskip

No really specific  model theoretic hypothesis but a \myalert{set-theoretic} one?

Definitely not provable in ZFC for AEC (even for $L_{\omega_1,\omega}(Q_1)$ maybe for $L_{\omega_1,\omega}$).

\end{frame}






\begin{frame}
\frametitle{THE counterexample: $\Phi$}


 $\bK$ is the  models  in a vocabulary with two unary
relations $P$, $Q$ and two binary relations $E$, $R$ which satisfy:

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 $P\times Q$ so that each element of $Q$ codes a
    subset of $P$.

\item $R$ induces the independence property on $P \cup Q$.

%\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}

This class is axiomatized by a  sentence $\Phi$ in
$L_{\omega_1,\omega}(Q_1)$.

\end{frame}

\begin{frame}
\frametitle{Properties of models of $\Phi$}
\begin{block}{amalgamation, $\omega$-stablilty, and arbitrarily large models FAIL}
%\begin{enumerate}
%\item
%Yes for AEC, even for $L_{\omega_1,\omega}(Q)$.
%
%There is a sentence in $L_{\omega_1,\omega}(Q)$ that

 Under MA  $\Phi$ is $\aleph_1$-categorical but is not $\omega$-stable,
 fails amalgamation in $\aleph_0$, and  has no models beyond the continuum.

%\pause
%\smallskip
%\item $L_{\omega_1,\omega}$: open - equivalent to absoluteness by results
  %  below.
%\end{enumerate}

\medskip

Shelah suggested a variant, axiomatized in $L_{\omega_1,\omega}$ with the
same properties in $\aleph_0$.  Laskowski showed that sentence had at  least
$2^{\aleph_0}$ models in $\aleph_1$.

\medskip

The AEC attached to $\Phi$ (\cite[6.3]{Sh88r}) is the $\bK_3$.  

$M,N\in \bK$, $M\leq_{\bK_3}N$ if $M\subseteq N$ and $[a]^M = [b]^N$.

\cite{Sh88,Sh87b,Sh88r},\cite[\S 17]{Baldwincatmon}


\end{block}

\end{frame}




\section{The $L_{\omega_1,\omega}$ case}
The $L_{\omega_1,\omega}$ case\\



\subsection{Syntax and Semantics}
Syntax and Semantics

\begin{frame}
\frametitle{The class of models}

$\bK_T$ is the class of atomic models of the countable first order
theory $T$.

\begin{block}{Definition}
The atomic class $\bK_T$ is \myalert{extendible} if there is a pair
$M\preceq N$ of countable, atomic models, with $N\neq M$.

\end{block}

Equivalently, $\bK_T$ is extendible if and only if there is an
uncountable, atomic model of $T$.

\medskip

 We assume throughout that $\bK_T$ is extendible.  We work in
the monster model of $T$, which is usually not atomic.

{\bf A complete sentence of  has such a representation by 
Chang's trick: Expanding the language by introducing predicates for countable conjunctions  (theory $T^*$)
and making them correct by omitting types.

The $L_{\omega_1,\omega}$-class is the reducts of atomic models of $T^*$.}





\end{frame}


\begin{frame}
\frametitle{$\omega$-stability in Atomic Classes}
%
%\begin{block}
%
%The models of a \myalert{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 (in an expanded
%language).
%\end{block}
%
%$(\bK,\subm)$ is the class of atomic models of a first order
%theory under elementary submodel.
%
%\smallskip

\begin{block}{Definitions}

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



$\bK$ is $\omega$-stable if for every countable
\textcolor{clifton}{model} $M$, $S_{at}(M)$ is countable.
\end{block}

\medskip
But, there may be $A \subseteq M$, $p \in S_{at}(A)$ that has no extension to
$S_{at}(M)$.


Note also $\phi$ may be $\kappa$-stable in this sense while the associated
AEC is not $\kappa$-stable (for Galois types) \cite{BKHS}.

\end{frame}

%\begin{frame}
%\frametitle{Atomic Models of First order theories}
%
%Is the property:
%
%\smallskip
%
%`the class of atomic models of a complete first order theory $T$ is $\aleph_1$-categorical'
%
%
%\smallskip
%an absolute property of $T$?
%\end{frame}



\begin{frame}\frametitle{First order absoluteness}

\begin{block}{Theorem (Morley-Baldwin-Lachlan)}
A first order theory $T$ in a countable language is $\aleph_1$ categorical iff

\begin{enumerate}
\item $T$ has no 2-cardinal models and
\item $T$ is $\omega$-stable.
\end{enumerate}
\end{block}


1) is arithmetic and 2) is $\Pi^1_1$.

\begin{block}{Fact}
A first order theory $T$ in a countable language whose class of atomic models
satisfies 1) and 2) is $\aleph_1$-categorical.

\medskip
I emphasize Morley because it is his direction:

 `$\aleph_1$-categorical implies $\omega$-stable' that is problematic for $L_{\omega_1,\omega}$.
\end{block}


%Morley minimal extension
\end{frame}


%\section{The $L_{\omega_1,\omega}$ case}
%The $L_{\omega_1,\omega}$ case




\begin{frame}
\frametitle{Getting $\omega$-stability: I}

\begin{block}{Theorem: Keisler/Shelah } $\bK = \mod(\psi)$, $\psi\in L_{\omega_1,
\omega}$
\begin{enumerate} \item  {\em (Keisler) ZFC } If some uncountable model in $\bK$ realizes uncountably many
types  (in a countable fragment) over  $\emptyset$ then $\bK$ has $2^{\aleph_1}$ models in $\aleph_1$.
\item {\em (Shelah) ($2^{\aleph_0}< 2^{\aleph_1}$}) If $\bK$ has
    $<2^{\aleph_1}$ models
of cardinality $\aleph_1$, then $\bK$ is $\omega$-stable.
\end{enumerate}
\end{block}


\begin{block}{Two uses of WCH}



\begin{enumerate}[A]
\item WCH implies AP in $\aleph_0$. Thus, if $\bK$ is not $\omega$-stable there
is a countable model $M$ and an uncountable $N\in \bK$ which realizes uncountably many types over $M$.

\pause
\item  By Keisler, $\th_M(M)$ has $2^{\aleph_1}$ models.
From
WCH we conclude $\th(M)$ has $2^{\aleph_1}$ models.



\end{enumerate}
\end{block}
\end{frame}











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

Morley's original {\em first order} proof using Hanf number for omitting types, EM-models, and the Skolem hull gives:

\begin{block}{Theorem}
If a complete first order theory has arbitrarily large models and is $\aleph_1$-categorical then it is $\omega$-stable.
\end{block}


More generally,
\begin{block}{Theorem}
An $\aleph_1$-categorical atomic class $\bK$ that has arbitrarily large models
 and amalgamation in $\aleph_0$  is $\omega$-stable.
\end{block}

Tradeoff: $\beth_{\omega_1}$ for weak CH 



\end{frame}





%{\em Locally} the models of  $\aleph_1$-categorical
%$L_{\omega_1,\omega}$-sentences} Understanding  {\em Locally} the  models of
%$\aleph_1$-categorical $L_{\omega_1,\omega}$-sentences

%\subsection{Classes of Atomic Models and Geometry}
%{Classes of Atomic Models}









\begin{frame}
\frametitle{A new notion of closure}



\begin{block}{Definition}\label{pcldef}   An atomic tuple
 $\cbar$ is  in the pseudo-algebraic closure of the finite, atomic
set $B$ ($\cbar  \in \pcl(B)$) if

for every atomic model $M$ such that $B\subseteq M$, {and}
$M\cbar$ is atomic, $\cbar\subseteq M$.

\medskip

 When this occurs, and $\bbar$
is any enumeration of $B$ and $p(\xbar,\ybar)$ is the complete type
of $\cbar\bbar$, we say that {\em $p(\xbar,\bbar)$ is
pseudo-algebraic.}

\end{block}



\end{frame}







\begin{frame}\frametitle{Example I}

 Our notion, $\pcl$ of \emph{algebraic} differs from
the classical first-order notion of algebraic as the following
examples show:

\begin{block}{Example}\label{ex1}
	Suppose that an atomic model M consists of two sorts.  The
$U$-part is countable, but non-extendible (e.g., $U$ infinite, and
has a successor function $S$ on it, in which every element has a
unique predecessor).
  On the other sort, $V$ is an infinite set with no structure
(hence arbitrarily large atomic models).  Then,   an element
$x_0\in U$ is not algebraic over $\emptyset$ in the normal sense but
is in $\pcl(\emptyset)$.


\end{block}
\end{frame}



\begin{frame}\frametitle{Example II}

\begin{block}{Example}
Let $L={A,B,\pi,S}$ and $T$ say that $A$ and $B$ partition
	the universe with $B$ infinite,
	$\pi:A\rightarrow B$ is a total surjective
	function and $S$ is a successor function on $A$ such that
	every $\pi$-fiber is the union of $S$-components. 	
	$K_T$ is the class of $M\models T$ such that every
	$\pi$-fiber contains exactly one $S$-component.
	Now choose elements $a,b\in M$ for such an $M$ such that
	$a\in A$ and $b\in B$ and $\pi(a) =b$. Clearly, $a$ is not
algebraic
	over $b$ in the classical sense, but $a\in \pcl(b)$.



\end{block}
\end{frame}





\begin{frame}
\frametitle{Definability of pseudo-algebraic closure}


Strong $\omega$-homogeneity of the monster model of $T$ yields:


\begin{block}{Fact}\label{pclinv}
If $p(\xbar,\ybar)$ is the complete type of $\cbar\bbar$, then
$$\cbar\in\pcl(\bbar)\quad\hbox{if and only
if}\quad\cbar'\in\pcl(\bbar')$$
for any $\cbar'\bbar'$ realizing $p(\xbar,\ybar)$.

In particular,
the truth of  $c\in \pcl(\bbar)$ does not depend on an ambient atomic
model. \end{block}


Further, since a model which atomic over the empty set is also atomic
over any finite subset, moving $M$ to $N$ we have:

\begin{block}{Fact}\label{weakonemore} If $\cbar\not\in\pcl(B)$, witnessed by $M$ then for
every countable, atomic $N\supset B$,  there is a realization
$\cbar'$ of $p(\xbar,B)$ such that $\cbar'\not\subseteq N$.


    \end{block}

\end{frame}





%\begin{frame}
%\frametitle{Stronger Version}
%
%\begin{block}{Lemma} \label{onemore} Let $N$ be an atomic model containing $\bbar\abar$.
%If $\bbar $ is not    pseudoalgebraic over $\abar$ then
%$\tp(\bbar/\abar)$ is realized in $N-\pcl(\abar\bbar)$.
%\end{block}
%
%Proof.  Let $M_1$ be a countable submodel of $N$ containing
%$\abar\bbar$ and $M_0$ an elementary submodel of $M_1$ containing
%$\abar$ but not $\bbar$. Note $M_0 \approx M_1$. Let $M_2$ be the
%image of $M_1$ under an automorphism $f$ of the monster taking $M_0$
%to $M_1$. Then $f(\bbar)$ is not in $\pcl(\abar\bbar)$ (It's in
%$M_2-M_1$.). Since $M_2$ is atomic over $\abar\bbar$, there is an
%embedding $g$ of $M_2$ into $N$ realizing $\tp(\bbar/\abar)$ by
%$g(f(\bbar))$ not in $\pcl(\abar\bbar)$ since $\pcl$ is invariant
%under automorphism.
%
%
%\end{frame}
%
%
%
%
%
%
%
%%\begin{frame}
%%\frametitle{pcl = qcl: sometimes}
%%
%% Zilber \cite{Zilbercatex, Baldwincatmon} introduced the abstract notion of a
%%quasiminimal (excellent) class and proved such classes are categorical in all
%%uncountable powers.  In general, these classes are axiomatized in $L_{\omega_1,\omega}(Q)$ (\cite{Kirbyqm}) and the quasiminimal closure is distinct from
%%our notion of $\pcl$.
%%
%%
%% However, in some cases, most notably \cite{ZilberBays},
%%the study of covers of certain algebraic groups  e.g. \cite{ZilberBays, Baysthesis},  the countability of the quasiminal closure is expressible in
%%$L_{\omega_1,\omega}$ and then $\pcl = \qcl$ and quasiminimal = pseudominimal.
%%
%%\begin{block}{Lemma}
%%$a \in \pcl(\bbar)$ if and only $\tp(a/\bbar)$ is realized only
%%countably many times in any model of $T$.
%%\end{block}
%%
%%{\bf check this. is pseudoclosure a model -- no so doesn't satisfy Zilber}
%%
%%Iterating the last result, a type not realized in the pseudoclosure is
%%realized arbitrarily often.
%%
%%But if $p(x,\bbar)$ is pseudoalgebraic, all realizations of $p$ must
%%be in any countable model $M$ containing $\bbar$.
%%%
%%%
%%%
%%%  However, in some cases, most notably \cite{ZilberBays},
%%%the study of covers of certain algebraic groups  e.g. \cite{ZilberBays, Baysthesis},  the countability of the quasiminal closure is expressible in
%%%$L_{\omega_1,\omega}$ and then $\pcl = \qcl$.
%%
%%
%%
%%
%%\end{frame}
%
%
%
%
%
%
%
%\begin{frame}
%\frametitle{Countable closure property}
%
%
%\begin{block}{Lemma}\label{getcnt}  For
%any  finite $\abar$, any $N\in \bK_T$, $\pcl_N(\abar) = N\cap
%\pcl(a)$ satisfies $|\pcl_N(a)| =\aleph_0$.
%
%\end{block}
%
%Proof. By the last Lemma, if $\bbar$ is algebraic over $\abar$ then
%for any $N\in \bK_T$ (i.e. $N$ is atomic), $\tp(\bbar/\abar)$ is
%realized only countably many times in $N$.  Whether $\bbar \in
%\pcl_N(\abar)$ depends only on $\tp(\abar\bbar)$.
%
%\medskip
%
%This type must be atomic and there are only countably many atomic
%types of finite sequences. So $\pcl_N(\abar)$ is countable.
%
%\medskip
%
%%quasiclosure not quasiminimal
%
%
%\end{frame}

%\section{Interlude: The model theory again}



\begin{frame}

\frametitle{Pseudo-minimal sets}

\begin{block}{Definition}
\begin{enumerate}
\item  A {\em possibly incomplete}  type $q$ over $\bbar$ is {\em
    pseudominimal} if  for any finite,  $\bbar^*\supseteq \bbar$,
    $\abar\models q$, and  $\cbar$ such that $\bbar^*\cbar \abar$
    is atomic, if  $\cbar \subset \pcl(\bbar^*\abar)$, and $\cbar
    \not\in\pcl(\bbar^*)$,  %$c\not\in\pcl(\bbar^*)$,
    then $\abar \in\pcl(\bbar^*\cbar)$.
    \item $M$ is pseudominimal if $x=x$ is pseudominimal in $M$.
 \end{enumerate}
\end{block}

I.e, $\pcl$ satisfies exchange (and more); we have a geometry.

\end{frame}


\begin{frame}
\frametitle{`Density'}



\begin{block}{Definition}

$\bK_T$ satisfies {\em `density' of
 pseudominimal
 types}
 if for every atomic $\ebar$ and atomic type $p(\ebar,\xbar)$ there
 is a $\bbar$ with $\ebar\bbar$ atomic and $q(\ebar,\bbar,\xbar)$
 extending $p$ such that $q$ is pseudominimal.
 \end{block}

 So density fails if there is a single type $p(\ebar,\xbar)$ over which
 exchange fails.

\end{frame}



%\end{document}
%\begin{frame}
%\frametitle{Failing `density'}
%
%\begin{block}{Lemma}
%	$\bK_T$ \myalert{fails `density' of pseudominimal types} if,
%after naming a finite tuple $\ebar$, there is a complete 1-type
%$\tilde p(x)$ over $\ebar$ such that
%
%for any finite, atomic $\bbar$ containing $\ebar$ and complete
%$q(\ebar,\bbar,\xbar)$
% extending $\tilde p$
% there are  a finite atomic
%$\bbar^*\supset \bbar$, $\abar\models q$, and  $\cbar$ such that
%
%$\bbar^*\cbar \abar$ is atomic,  $\cbar \subset \pcl(\bbar^*\abar)$,
%$\cbar\not\in\pcl(\bbar^*)$, and $\abar\not\in\pcl(\bbar^*)$, but
%$\abar\not\in\pcl(\bbar^*\cbar)$.
%\end{block}
%
%I.e. $\pcl$ fails exchange locally.
%
%
%
%\end{frame}
\begin{frame} \frametitle{Method: `Consistency implies Truth':I}

\cite{BL}

Let $\phi$ be a $\tau$-sentence in $L_{\omega_1,\omega}(Q)$ such that
it is consistent that $\phi$ has a model.



Let $A$ be the countable $\omega$-model of set theory, containing
$\phi$, that thinks $\phi$ has an uncountable model.




\medskip
Construct $B$, an uncountable model of set theory, which is an
elementary extension of $A$,
such that $B$ is correct about uncountability. Then the model of
$\phi$ in $B$ is actually an uncountable model of $\phi$.

\end{frame}

 %    \begin{frame}
%\frametitle{Building many atomic models of $T$ by ultralimits of models of set theory}
%
%The first technical tool  is the iterated generic
%elementary embedding induced by the nonstationary ideal on
%$\omega_{1}$, which we will denote by $NS{\omega_1}$.
%
%\begin{block}{The ultrafilter}
%Forcing with the Boolean algebra
%$(\Pscr(\omega_1)^{M}$ over a ZFC model $M$ gives rise to an $M$-normal
%ultrafilter $U$ on $\omega_{1}^{M}$ (i.e., every regressive function
%on $\omega_{1}^{M}$ in $M$ is constant on a set in $U$).
%\end{block}
%
%\end{frame}
%
%
%\begin{frame}
%\frametitle{The Ultrapower}
%
%
%
%Given such
%$M$ and $U$, we can form the generic ultrapower $Ult(M,U)$, which
%consists of all functions \myalert{in} $M$ with domain $\omega_{1}^{M}$,
%
%\medskip
%
% where
%for any two such functions $f$, $g$, and any relation $R$ in
%$\{=,\in\}$, $f R g$ in $Ult(M,U)$ if and only if $\{ \alpha <
%\omega_{1}^{M} \mid f(\alpha) R g(\alpha)\} \in U$.
%
%\medskip
%\begin{block}{Nota Bene}
%If $M$ is countable, $Ult(M,U)$ is countable.
%\end{block}
%
%
%By convention, we
%identify the well-founded part of the ultrapower Ult$(M, U)$ with its
%Mostowski collapse.
%
%
%
%\end{frame}
%
%\begin{frame}
%\frametitle{Iterations}
%\begin{block}{Definition}\label{itdef} Let $M$ be a model of
%a sufficient finite fragment of ZFC and let $\gamma$ be an ordinal less than or equal to
%$\omega_{1}$.
%
%An \myalert{iteration} $J_X$  of $M$ of length $\gamma$ consists of models
%$$M_{\alpha}:\hskip .1in (\alpha \leq \gamma), $$ sets $$G_{\alpha}:\hskip .1in (\alpha <
%\gamma),$$ and a commuting family of elementary embeddings
%$$j_{\alpha\beta} \colon M_{\alpha} \to M_{\beta}: \hskip .1in (\alpha \leq \beta
%\leq \gamma)$$
%
%
%such that for $J_X$ we choose the ultrafilter $U_\alpha$
%such that $P^{M_\alpha} \in U_\alpha$ iff $\alpha \in X$.
%%\begin{itemize}
%%\item $M_{0} = M$,
%%\item each $G_{\alpha}$ is an $M_{\alpha}$-generic filter for
%%    $(\pons)^{M_{\alpha}}$,
%%\item each $j_{\alpha\alpha}$ is the identity mapping,
%%\item each $j_{\alpha(\alpha + 1)}$ is the ultrapower embedding
%%    induced by $G_{\alpha}$,
%%\item for each limit ordinal $\beta \leq \gamma$, $M_{\beta}$ is
%%    the direct limit of the system $$\{ M_{\alpha}, j_{\alpha
%%    \delta} : \alpha \leq \delta < \beta \},$$ and for each
%%    $\alpha < \beta$, $j_{\alpha\beta}$ is the induced embedding.
%%\end{itemize}
%\end{block}
%
%
%\end{frame}
\begin{frame}
\frametitle{Main Theorem}

\begin{block}{Goal Theorem \cite{BLSmanymod}}\label{nonstrthm}  If $\bK_T$ fails
`density of pseudominimal types'
    then $\bK_T$ has $2^{\aleph_1}$ models of cardinality $\aleph_1$.
    \end{block}

    We prove this in two steps
    \begin{enumerate}

    \item
   Force to construct a model $(M,E)$ of set theory in which a model of $T$
codes model theoretic and combinatorial information sufficient to guarantee
the non-isomorphism of its image in the different ultralimits.
 \item Apply Skolem ultralimits  of the models of set theory from 1)
 to construct $2^{\aleph_1}$ atomic models of $T$ with cardinality $\aleph_1$ in $V$.
        \end{enumerate}

        \end{frame}




%\begin{frame}
%\frametitle{Proof sketch:}
%
%Fix a countable transitive model $M$ of ZFC  and choose
%$S\subseteq\omega_1^M\setminus\{0\}$ such that $$(M,\in)\models `\hbox{$S$ is
%stationary/costationary'}$$
%
%Construct $(I,<,P,E)\in {\bf  I}^*$ is an $\aleph_1$ dense linear order with a unary predicate $P$ and an equivalence relation $E$ such that $I/E$ is dense and both $P/E$ and $\neg P/E$ are dense.
%
%\begin{block}{Theorem}  \label{forcing}  Suppose $\delta(x)$ is a complete, non-pseudo algebraic formula with no pseudo-minimal extension.
%%For every $(I,<,P,E)\in {\bf I}^*$
%There is a c.c.c.\ forcing $\QQ_I$ such that in $M[G]$, there is a full, atomic $N_I\models T$ and
%%striated by $(I,<)$ such that in $M[G]$
%%we identify structure $N^*$ in a vocabulary $\tau^*$ such that:
%\medskip
%
%Ultralimits of $M[G]$ by iterations $J_X$ and $J_Y$
%give rise to non-isomorphic atomic models $N_X$ and $N_Y$ of $T$ if
%$X-Y$ is stationary.
%\end{block}
%%such that applying the ultralimit construction we obtain for each $X \subseteq \omega_1$, an elementary extension $M^X$, which is a model of $ZFC$ with cardinal $\aleph_1$.
%
%%\begin{enumerate}
%%\item  For every suitable initial segment $J\subseteq I$, $N_J\preceq N_I$;
%%\item  If  $t\in I$ and $P(t)$ holds,
%%then $a_{t,0}$ catches and has unbounded reach in $N_{<t}$;
%%\item  If $J\subseteq I$ is seamless, then for every $b\in N_I\setminus N_{J}$,
%%if $b$ catches $N_J$, then $b$ has bounded effect in $N_J$.
%%\end{enumerate}
%
% \end{frame}




%\begin{frame}
%\frametitle{Idea of proof}
%
%        Step 1) generalizes the well-known result that there are $2^{\aleph_1}$
%$\aleph_1$-like linear orders of cardinality $\aleph_1$.
%
%  The key idea of that
%argument (e.g. \cite{Markerbook})  is to code a stationary set of cuts which have a least upper
%bound. In the current proof, the coding is not so sharp. Instead, we
%force an atomic model of $T$ that codes a stationary set by
%infinitary formulas defined using $\pcl$.
%
%\bigskip
%
%        Step 2) was laid out in a model theory context in \cite{BL} and expounded along with the application in \cite{BLSmanymod}.
%
%\end{frame}


%\begin{frame}
%\frametitle{Extension}
%
%\begin{block}{Definition} $\bK$ is $\pcl$-small if $S_{at}(\pcl(\abar))$ is countable for every finite sequence $\abar$.
%\end{block}
%
%\cite{LasShstrongfail}  show:
%
%\begin{block}{Theorem}%{\cite{LasShstrongfail}
%If $\bK$ has fewer than $2^{\aleph_1}$ models in $\aleph_1$, then
%$\bK$ is $\pcl$-small.
%\end{block}
%
%
%\end{frame}

%\begin{frame}
%\frametitle{Goal of the Forcing}
%
% We will  construct many models
%in two steps. In the first, we work in the model $\Nscr_1$ of
%$ZFC \ldots + MA$ and show how to construct for a pair of disjoint
%stationary sets $S,T$ a model $M^{S,T}$ such that $M^{S,T} \models
%\theta(S,T)$.
%
%\medskip
% $\theta$ will satisfy that if $S_1,S_2$ are each stationary
%subsets of $\aleph_1$ and  $S_1 -S_2$ is stationary and both
%$M^{S_1,T_1}$ and $M^{S_2, T_2}$ satisfy $\theta(S_i,T_i)$ then
%$M^{S_1,T_1}  \not \approx M^{S_2, T_2}$.
%
%\end{frame}
%
%\begin{frame}
%\frametitle{Properties of $\theta(S,T)$} Given $\psi\in
%L_{\omega_1,\omega}(\tau^)$,  the formula $\theta(S,T) \in
%L_{\omega_1,\omega}(Q)(\tau^*)$ holds of model $M^*\in \tau^*$ if
%
%\begin{enumerate}
%
%\item $M^*\restriction \tau \models \psi$
%
%\item $M^*$ admits a filtration as described below.
%
%\item implies a first order $\tau^*$-formula $\theta_1(P_1,P_2)$
%    which expresses:
%
%     \begin{enumerate}[a] \item
%
%        If $\alpha \in P_1$ then there is an $a \in
%                M-M_{J_\alpha}$ which catches $M_{J_\alpha}$
%                but does not strongly catch $M_{J_\alpha}$.
%
%        \item  If $\alpha \in C - (P_1 \cup P_2)$ every $a
%            \in M-M_{J_\alpha}$ which catches $M_{J_\alpha}$
%            strongly catches
%             $M_{J_\alpha}$.
%\end{enumerate}
%\end{enumerate}
%\end{frame}
%\section{Do $\aleph_1$-categorical theories have `big' models?}
%Do $\aleph_1$-categorical theories have `big' models
%
%\begin{frame}
%\frametitle{Must $\aleph_1$-categorical theories have a bounded number of models?}
%
%\myalert{How big is big?}
%
%In the mid-70's Shelah answered my question as to whether a sentence of $L_{\omega_1,\omega}(Q)$ could be {\em categorical in the philosophers sense}, have only one model.  In different papers he proved  in different ways such a theory has a model in $\aleph_2$.
%
%\bigskip
%
%A natural question is whether a sentence of $L_{\omega_1,\omega}$
% that is $\aleph_1$-categorical has a model in $2^{\aleph_0}$.
%\end{frame}


\subsection{Getting Models in the continuum} Getting Models in the continuum
\begin{frame}
\frametitle{Getting models in $2^{\aleph_0}$: Method}
In the novel {\em White Light} \cite{Rucker}, Rudy Rucker proposes a metaphor for
the continuum hypothesis.  One can reach $\aleph_1$ by a laborious climb up the side of Mt.\ ON, pausing at $\epsilon_0$.


\pause
\medskip
Or one can take

\myalert{Cantor's elevator}
 An instantaneous  trip up a shaft at the center of the mountain.

\pause
\pause
\medskip
 For atomic models we take the slightly slower



\myalert{Shelah's elevator} The elevator is a bit slower but has only
countably many floors.  After building   finitely many rooms at each step
%determining a finite amount of information,
we reach
an object of cardinality $2^{\aleph_0}$.


%The underlying construction applies
%for finding atomic models, two-cardinal theorems, a collection of continuum many
%points that are {\em asymptotically similar} (a weak form of indiscernibility), and a coloring with a Borel square of size continuum.

\end{frame}

\begin{frame}
\frametitle{Asymptotic similarity}

\begin{block}{Definition}  \label{asympsim}
Fix an $L$-structure  $M$.
A subset of $M$, indexed by $\{a_\eta\mcolon \eta\in 2^\omega\}$,
    is {\em asymptotically similar} if, for every $k$-ary
    $L$-formula $\theta$, there is an integer $N_\theta$ such that
    for every $\ell\ge N_\theta$, $$M\models
    \theta(a_{\eta_0},\dots,a_{\eta_{k-1}})\leftrightarrow
 \theta(a_{\tau_0},\dots,a_{\tau_{k-1}})$$
whenever $(\eta_0,\dots,\eta_{k-1})$ and $(\tau_0,\dots,\tau_{k-1})$ satisfy:
$\eta_i\myrestriction \ell = \tau_i\myrestriction \ell$ and the $\eta_i\myrestriction \ell$ are distinct. 

\end{block}



\begin{block}{Remark} Asymptotic similarity is a type of indiscernibility; the indiscernibility is only formula by formula.
Let $M=(2^\omega,\{U_i:i<\omega\}$ where the $U_i$ are independent unary predicates.
%Let $M=(2^\omega,U_a)_{a\in 2^{<\omega}}$, where  each $U_a$ is a unary predicate
%interpreted as the cone above $a$, i.e., $U_a(M)=\{\eta\in 2^\omega:a\triangleleft \eta\}$.
The entire universe %$\{\eta:\eta\in 2^\omega\}$ 
is asymptotically similar, although no
two elements have the same 1-type.
\end{block}


\end{frame}


\begin{frame}
\frametitle{Getting models in $2^{\aleph_0}$}


\begin{block}{Theorem \cite{BLlarge}}\label{getcontpcl}
If  a countable first order theory $T$ has an atomic pseudominimal
model $M$ of cardinality $\aleph_1$ then there is an atomic pseudominimal model
$N$ of $T$ which a contains a set of {\em asymptotically similar} elements with cardinality $2^{\aleph_0}$.

Equivalently, if the models of a complete sentence $\Phi$ in
$L_{\omega_1,\omega}$ are pseudominimal and $\Phi$ has an uncountable
model, it has a model in the continuum.
\end{block}

A simple application of the method gives Borel models in the continuum of
any theory with trivial definable closure.
\end{frame}

%\begin{frame}
%\frametitle{State of the art: Eventual behavior}
%
%\cite{ShVas}  (simplified for ease of reading)
%\begin{block}{Theorem}
%Let $\kappa$ be a strongly compact cardinal and let $\psi$ be an $L_{\kappa,\omega}$-sentence. If $\psi$ is categorical in some $\mu \geq \beth_{(2^{\kappa})^+}$, then $\psi$ is categorical in all
%$\mu \geq \beth_{(2^{\kappa})^+}$
%\end{block}
%
%Extending \cite{Sh87a,Sh87b}:
%
%\begin{block}{}
%For any countable AEC, very few models below $\aleph_\omega$ and
%$2^{\aleph_n} < 2^{\aleph_{n+1}}$ for $n<\omega$ imply
%
%\begin{enumerate}
%\item excellence, thus arbitrarily large models,
%\item and so categoricity in any cardinal implies categoricity in all
%uncountable cardinals.
%\end{enumerate}
%
%%
%%
%%Let $\bK$ be an AEC and suppose %let $\lambda \geq LS(\bK)$ be such
%% that $2^{\aleph_n} < 2^{\aleph_{n+1}}$ for $n<\omega$
%%holds. Assume that $\bK$ is categorical in $\aleph_0$ and $\aleph_1$,
%%$\bK_{\aleph_2} \neq \emptyset$, and
%very few: $I(\bK, \aleph_{n}) \leq 2^{\aleph_{n-1}}$ for $n<\omega$.
%%Then K is categorical in all $\mu\geq\lambda$.
%\end{block}
%
%\end{frame}

\subsection{$\bK_{(2^{{\aleph_0})^+} \neq \emptyset}$ implies $\omega$-stability}
$\bK_{(2^{{\aleph_0})^+}}\neq \emptyset$ implies $\omega$-stability

\begin{frame}
\frametitle{Goal Theorem \cite{BLSwhendoes}}
\begin{theorem}  \label{largemodel}  If an atomic class $\At$ is $\aleph_1$-categorical
and has a model of size $(2^{\aleph_0})^+$ then $\At$ is $\omega$-stable.
\end{theorem}


Old and new definitions:

\begin{block}{Definition}
\begin{enumerate}
\item  {\em  A type $p\in S_{at}(M)$ {\em splits over $F\subseteq M$} if
    there are tuples $\bbar,\bbar'\subseteq M$ and a formula
    $\phi(\xbar,\ybar)$ such that $\tp(\bbar/F)=\tp(\bbar'/F)$, but
    $\phi(\xbar,\bbar)\wedge\neg\phi(\xbar,\bbar')\in p$.

\item We call $p\in S_{at}(M)$ {\em constrained} if $p$ does not split over
    some finite $F\subseteq M$ and {\em unconstrained} if $p$ splits over
    every finite subset of $M$.

\item $C_M:=\{p\in S_{at}(M): p \ \hbox{is constrained}\}$, for  an atomic
    model $M$, . We say $\At$ has {\bf only constrained types} if
    $S_{at}(N)=C_N$ for every atomic model $N$.}

    \end{enumerate}
\end{block}

\end{frame}

\begin{frame}
\frametitle{Basic Properties}
\begin{block}{Lemma}  \label{dropunconstrained}
%Suppose $At$ contains an uncountable
%model.

\begin{enumerate}
\item If $M$ is a countable atomic model and $p\in S_{at}(M)$ then $p$ is
    realized in an atomic extension of $M$.

\item  For any atomic models $M\preceq N$ and   finite $A\subseteq M$,
    then for any  $q\in S_{at}(N)$ that does not split over $A$, the
    restriction $q\myrestriction{M}$ does not split over $A$; and any $p\in
    S_{at}(M)$
 that does not split over $A$ has a unique non-splitting extension $q\in
 S_{at}(N)$.


%\item  For any atomic models $M\preceq N$, if $p\in S_{at}(M)$ does not split over $A$, then there is a unique
%(constrained) extension $q\in S_{at}(N)$ of $p$ that also does not split over $A$.
%%constrained $p\in C_M$ has a constrained extension $q\in C_N$.

\item   If some atomic $N$ has an unconstrained $p\in S_{at}(N)$, then for
    every countable $A\subseteq N$, there is a countable $M\preceq N$ with
    $A\subseteq M$ for which the restriction $p\myrestriction{M}$ is
    unconstrained.

\item  $\At$ has only constrained types if and only if $S_{at}(M)=C_M$ for
    every/some countable atomic model $M$.
\end{enumerate}
\end{block}


Straightforward proofs using atomicity.
\end{frame}


\begin{frame}
\frametitle{Limit types}
\begin{block}{Definition} For $|N| = \aleph_1$,
a type  $p\in S_{at}(N)$ is a {\rm limit type} if the restriction
$p\myrestriction {M}$
    is realized in $N$ for every countable $M \preceq N$.
\end{block}

Trivially, for every $N$, every  type in $S_{at}(N)$ realized in $N$ is a
limit type. Since we allow $M = N$ in the definition of a limit type, if $M$
is countable, then the only limit types in $S_{at}(M)$ are those realized in
$M$.

%\begin{block}{Definition \cite{MazYang}} An AEC is {\rm $(<\aleph_1, \aleph_0)$-local}
% if for every
%increasing chain $\langle M_i: i< \aleph_1\rangle$ of countable structures in
%$\bK$, if $M \bigcup_{<\aleph_1} M_i$, for any $p,q \in S(M)$ if
%$p\restriction M_i = q \restriction M_i $ for all $i$ then $p =q$.
%\end{block}

%Also, if $(M_\alpha:\alpha\in\omega_1)$ is a filtration of $N$, then a type
%$p\in S_{at}(N)$ is a limit type if and only if $N$ realizes
%$p\myrestriction{M_\alpha}$ for cofinally many $\alpha$.

\end{frame}






%\begin{frame}
%\frametitle{2/3 of the proof}
%
%$M$ is the unique countable model
%
%\begin{enumerate}
%\item $\aleph_1$-categoricity implies $C_M$ is countable. \cite[2.2.5]{BLSwhendoes}.
%\item
%
%
%\end{frame}
\begin{frame}
\frametitle{ `Consistency implies Truth': II}




 Note that there are no additional
 assumptions on $\At$, other than the existence of an uncountable, atomic
 model.

\begin{block}{KEY Theorem:}    If $\At$ admits an uncountable, atomic model,
 then there is some  $N\in\At$ with $|N| =\aleph_1$ for which every limit type
  in $S_{at}(N)$ is constrained.
\end{block}

So if $\aleph_1$ -categorical: limit = constrained on the model in $\aleph_1$.

\end{frame}

\begin{frame}
\frametitle{Linking a largish model with $\omega$-stability}
\begin{block}{Theorem}  \label{largemodel}  If an atomic class $\At$ is $\aleph_1$-categorical
and has a model of size $(2^{\aleph_0})^+$ with a relatively
$\aleph_1$-saturated submodel of cardinality continuum, then
$S_{At}(M)$ has only constrained types

{\bf Pf.}  Choose a $c$ realizing an unconstrained type and use relative $\aleph_1$-saturation to build
an unconstrained limit type. 
This   contra the KEY.


Similarly argue that if there is a unconstrained type over a countable model then
there is a model in $\aleph_1$ with an unconstrained limit type. Apply KEY again
\cite[Theorem 2.4.4]{BLSwhendoes}
\end{block}

\end{frame}

%\sidebar{ May 11 Need this proof but not here}

\begin{frame}\frametitle{Relatively saturated models exist!}

\begin{proof}  Let $M^{**}$ be an atomic model of size $(2^{\aleph_0})^+$.
We construct a  \emph{relatively} $\aleph_1$-saturated elementary
substructure $M^*\preceq M^{**}$ of size $2^{\aleph_0}$ as the union of a
continuous chain $(N_\alpha:\alpha\in\omega_1)$ of elementary substructures
of $M^{**}$, each of size $2^{\aleph_0}$, where, for each $\alpha<\omega_1$
and each of the $2^{\aleph_0}$ countable $M\preceq N_{\alpha}$,
$N_{\alpha+1}$ realizes each of the at most $2^{\aleph_0}$ $p\in S(M)$ that
is realized in $M^{**}$. 
%$\omega$-stability is immediate from
%$(1)\Rightarrow(4)$ in Lemma~\ref{omegastbchar}.
\end{proof}


Note there is no reason to think any of these models are even $\aleph_1$-saturated,
until we conclude $\omega$-stability.
\cite[Theorem 2.4.5]{BLSwhendoes}
\end{frame}

%\begin{frame}
%\frametitle{What is the right analogy?}
%\begin{enumerate}
%\item The weak diamond WGCH theory
% \item Zilber quasiminimality
% \pause
%
%\item uncountable first order theories: `superstable' but not necessarily $\omega$-stable.
%
%    \item continuous
%    \pause
%    \item The MA  `$\aleph_1$-categoricity doesn't yield structure' approach.
%    \end{enumerate}
%
%    \end{frame}





%\section{The AEC Side}
%The AEC Side
%\begin{frame}\frametitle{Larger models from stability and locality}
%
%\begin{block}{Definition \cite{MazYang}} \begin{enumerate}
%\item  Let $|M| =\lambda$. $p \in S(M)$ is $\lambda$-extendible if for every
%    $M'\in \bK_{\lambda}$ with $M\leq M'$, there is a $q\in S^{na}(M')$ (not
%    realized in $M'$) extending $p$.
%    \item $q$ is $\lambda$-unique (quasiminimal) if it is
%    $\lambda$-extendible and has a unique  $\lambda$-extendible extension `to any $M' \geq
%    M$.
%
%\end{enumerate}
%\end{block}
%
%\begin{block}{Theorem \cite{MazYang}} Suppose $\lambda< 2^{\aleph_0}$. Let $\bK$
%be an AEC with $\lambda \geq LS(\bK)$. If $\bK_{\lambda}$ has amalgamation,
%no maximal model, and is stable (for $\lambda$-algebraic types) in $\lambda$.
%Then:
%%\begin{enumerate}
%%\item
%
%i) Each $\lambda$-extendible type in any $M\in \bK_\lambda$ extends to a
%    $\lambda$-unique (quasiminimal) type.
%
%
%ii) Moreover, if $\bK$ is $(< \lambda^+, \lambda)$-local, then $\bK$ has
%    a model of cardinality $\lambda^{++}$.
%
%\end{block}
%\end{frame}
%
%
%\begin{frame}\frametitle{Distinction from BLS}
%Take $\lambda = \aleph_0$
%
%Orthogonal to earlier. \begin{enumerate} \item EARLIER:
%$\aleph_1$-categoricity assumed, $L_{\omega_1,\omega}$
%
%From model in $(2^{\aleph_0})^+$ and $\aleph_1$-categoricity get
%$\omega$-stable.
%
%\item HERE: $\omega$-stability and amalgamation assumed; AEC, locality
%
%From locality get a
%    model  in $\aleph_2$.
%\end{enumerate}
%
%%      But the common theme: From conditions on small models find larger ones.
%
%\end{frame}
%
%
%\begin{frame}\frametitle{Common elements I}
%
%\begin{block}{Definition \cite{MazYang}} $\bK_\lambda$ is  $(\lambda^+,\lambda)$-local if for every
%increasing chain with $\bigcup_{\alpha<\lambda^+} M_\alpha = M$ with each
%$|M_\alpha| = \lambda$, if $p\neq q \in S(M)$, for some $\alpha$,
%$p\myrestriction \alpha \neq p\myrestriction \alpha$.
%\end{block}


%\begin{block}{Definition 3.7. \cite{MazYang}} $p = \gtp(a/M,N)$ is a $\lambda$-unique
%type if (1) $p = gtp(a/M,N)$ has the $\lambda$-extension property. (2) For
%every $M' \in \bK_\lambda$ that $\leq_{\bK}$ extends $M$, $p$ has at most one
%extension $q \in S_{\lambda−ext(M')}$. In this case we say that $p \in
%S_{\lambda−ext(M)}$.
%
%\end{block}
%\begin{frame}\frametitle{Possible dup}
%\begin{block}{Definition \cite{BLSwhendoes}}
%\begin{enumerate}
%\item A type  $p\in S_{at}(N)$ is a {\em limit type} if the restriction
%    $p\mr{M}$ is realized in $N$ for every countable $M\preceq N$
%
%and constrained if $p$ does not split over a finite set.
%
%
%\end{enumerate}
%\end{block}
%
%
%
%\end{frame}
%
%
%\begin{frame}\frametitle{Common elements II}
%
%\begin{block}{Theorem \cite{MazYang}} Suppose $\lambda< 2^{\aleph_0}$ . Let $\bK$
%be an abstract elementary class with $\lambda \geq LS(\bK)$. Assume $\bK$ has
%amalgamation in $\lambda$, no maximal model in $\lambda$, and is stable in
%$\lambda$.
%
%{\color{red} check hypotheses} Then there is a $\lambda$-unique type in
%$S(N)$.
%
%
%\end{block}
%
%
%
%\begin{block}{Theorem}  \label{constrainedexists}  If $\At$ admits an uncountable, atomic model,
% then there is some  $N\in\At$ with $|N| =\aleph_1$ for which every limit type in $S_{at}(N)$ is constrained.
%\end{block}
%
%\end{frame}
\section{Further Context}
Further Context
\subsection{Amost quasiminimal excellent classes}

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





$$
\exp: (\CC,+) {\twoheadrightarrow}   
(\CC,\times). $$


$$j: H  {\twoheadrightarrow} .C $$



$$p: C  {\twoheadrightarrow} 
S(\CC).$$ 

Zilber conjecture that the most complicated (Shimura Varieties) were (almost) quasiminimal exellent and so uncountably categorical.
The many partial results/methods are represented in the next chart.

\end{frame}

\begin{frame}
\frametitle{Approaches to categoricity of covers}
\begin{center}

{\footnotesize
%\begin{flushleft}
\begin{tabular}{|c| c   |c   |l |c|}
\hline
 & topic & paper & method/context & section \\
\hline
    1 & Complex exponentiation & \cite{Zilbercatex} & quasiminimality & §\ref{intro}\\
\hline \hline
2 & cov mult group & \cite{Zilbercovers} & quasiminimality & §\ref{intro}\\
3 & & \cite{BaysZil} & quasiminimality & \\

\hline
4 & $j$-function &\cite{Harristhe} & background & §\ref{aroundjfunction} \\
\hline
5 & Modular/Shimura Curves & \cite{DawHarris} & quasiminimality & §\ref{modshicurves} \\
6 &  Modular/Shimura Curves &\cite{DawZil0} &quasiminimality &\\
 \hline \hline
7 & Abelian Varieties & \cite{BGH} &  finite Morley rank groups & §\ref{2sort}\\
8 & Abelian Varieties & \cite{BHP} & fmr \& notop & §\ref{abelianvarieties}\\
\hline
9 &   Shimura {\em varieties} & \cite{Etrev} & notop & §\ref{Shivar}\\
\hline\hline
10 & Smooth varieties & \cite{Zilbomin} & o-quasiminimality & §\ref{smoothvar}\\
% \hline
% transcendence theory & various & $DCF_0$\\
\hline\hline
\end{tabular}
%\end{flushleft}
}


\end{center}
\end{frame}

\begin{frame}
\frametitle{Almost quasiminimal excellence: I}
\begin{definition}
[Quasiminimal structure]\label{qmdefstr}
A structure  $M$ is {\em
quasiminimal} if every first
order ($L_{\omega_1,\omega}$) definable subset of $M$ is
countable or cocountable.  Algebraic closure is generalized by saying
$b \in \acl'(X)$ if there is a first order formula with {\bf
countably many} solutions over $X$ which is satisfied by $b$.
% Zilber showed that
%if $M$ is quasiminimal then  $(M,\acl')$ is a closure system.
\end{definition}

\end{frame}

\begin{frame}\frametitle{Almost quasiminimal excellence: II}
%\sidebar{jb Nov 24 10:50 am:  I think the point is that the
%trial definition is replaced by ccp which is weaker (unless unique generic type is assumed. But ccp is sufficient for categoricity even though the trial definition fails.}

\begin{definition}
[Quasiminimal excellent geometry]\label{qmdef}
 Let $\bK$ be a class of $L$-structures such that $M \in \bK$ admits a
 closure relation $\cl_M$  mapping $X \subseteq M$ to $\cl_M(X) \subseteq M$
  that satisfies the following properties.

\smallskip
\begin{enumerate}
\item \textbf{Basic Conditions}
\begin{enumerate}
\item
 Each $\cl_M$ defines a pregeometry on $M$.
\item For each $X\subseteq M$, $\cl_M(X) \in \bK$.

 \item countable closure property (ccp): If $|X| \leq \aleph_0$ then  $|\cl(X)| \leq \aleph_0$.

%\item \textbf{Homogeneity}

%\begin{enumerate}
%\item %\textbf{Homogeneity over $\emptyset$:}
%A class $\bK$ of models has {\bf $\aleph_0$-homogeneity over $\emptyset$}  (Definition~\ref{qmdef}) if the  models of $\bK$
%are pairwise qf-back and forth equivalent (Definition~\ref{bfdef})
%
%%If $f$ is a partial monomorphism from $H\in \bK$ to $H'\in \bK$
%%taking $X \cup \{y\} $ to $X' \cup \{y'\} $ then $y\in \cl_H(X)$
%%iff $y'\in \cl_{H'}(X')$.
%
%\item
%A class $\bK$ of models has \textbf{$\aleph_0$-homogeneity over models}
%if for any $G
% \in \bK$ with $G$ empty or a countable member of
%$\bK$, any $H,H'$ with $G\leq H, G\leq H'$,  $H$ is qf-back and forth equivalent with $H'$ over $G$.
%\end{enumerate}
\end{enumerate}

\end{enumerate}
\end{definition}
\end{frame}


\begin{frame}\frametitle{Almost quasiminimal excellence: II}
\begin{enumerate}
  \setcounter{enumi}{3}
\item \textbf{Homogeneity}

\begin{enumerate}
\item %\textbf{Homogeneity over $\emptyset$:}
A class $\bK$ of models has {\bf $\aleph_0$-homogeneity over $\emptyset$}  (Definition~\ref{qmdef}) if the  models of $\bK$
are pairwise qf-back and forth equivalent %(Definition~\ref{bfdef})

%If $f$ is a partial monomorphism from $H\in \bK$ to $H'\in \bK$
%taking $X \cup \{y\} $ to $X' \cup \{y'\} $ then $y\in \cl_H(X)$
%iff $y'\in \cl_{H'}(X')$.

\item
A class $\bK$ of models has \textbf{$\aleph_0$-homogeneity over models}
if for any $G
 \in \bK$ with $G$ empty or a countable member of
$\bK$, any $H,H'$ with $G\leq H, G\leq H'$,  $H$ is qf-back and forth equivalent with $H'$ over $G$.
\end{enumerate}


\item $\bK$ is an {\em almost quasiminimal excellent geometry} if the universe of any model $H\in \bK$ is in $\cl(X)$ for any maximal $\cl$-independent set $X \subseteq H$.


\item We call a class   which satisfies these conditions an {\em almost quasiminimal excellent geometry } \cite{BHHKK}.

\end{enumerate}


\end{frame} 

\begin{frame}\frametitle{Terminology}

%An almost quasiminimal excellent geometry with strong submodel taken as $A\leq M$, if $\acl_M(A) =A$,
%gives an \emph{abstract elementary class} (AEC).
%%\footnote{See~\cite{Bilgi} for the early history of
%%the model theory of AECs.}. 
%But
%the distinct notion of a quasiminimal AEC (defined in terms of $\leq$ rather than any axioms) is due to \cite{Vaseyqmaec}.

The {\bf almost} is essential because the structures are two-sorted.

But in the quasi-minimal covers: Galois type are quantifier-free first order types (in a suitably Morleyized theory).

\end{frame}

\subsection{Arbitrarily large models, bounded categoricity}
Arbitrarily large models, bounded categoricity

\begin{frame}
\frametitle{Galois may properly refine Syntactic}


\begin{block}{\bf Theorem Hart-Shelah/B-Kolesnikov}  For each $2\leq k<\omega$ there is an
$L_{\omega_1,\omega}$-sentence $\phi_k$ such~that:
\begin{enumerate}
\item $\phi_k$ is categorical  in $\mu$ if  $\mu\leq \aleph_{k-2}$;
\item $\phi_k$ is not $\aleph_{k-2}$-Galois stable;
\item $\phi_k$ is not categorical in any $\mu$ with $\mu>\aleph_{k-2}$;
\item $\phi_k$ has the disjoint amalgamation property in every $\kappa$;
\item For $k >2$,
\begin{enumerate}
\item $\phi_k$ is $(\aleph_{0},\aleph_{k-3})$-tame; indeed, syntactic
    first-order types determine Galois types over models of cardinality
    at most $\aleph_{k-3}$;
\item $\phi_k$ is $\aleph_m$-Galois stable for $m\leq k-3$;
\item  $\phi_k$ is not $(\aleph_{k-3},\aleph_{k-2})$-tame.
\end{enumerate}
\end{enumerate}
\end{block}
\cite{Baldwincatmon, BKHS} refining an example of \cite{HartShelah}.
\end{frame}

\begin{frame}
\frametitle{Galois may properly refine Syntactic: II}
\begin{enumerate}
\item $\phi_2$ is $\aleph_0$-categorical but not $\omega$-Galois stable
nor categorical in any power. ($\omega$-synactic stability unclear in paper.)

\item $\phi_3$ is categorial in $\aleph_1, \aleph_2$ and never again:

In $\aleph_0$, syntactic = Galois and $\omega$-stable.
\end{enumerate}

Thus, Morley's result that $\omega$-stable implies $\kappa$-stable for all $\kappa$
is gone (for Galois and likely? syntactic).

\end{frame}

\begin{frame}
\frametitle{Questions/Problems}
Let $\phi$ be a complete sentence of $L_{\omega_1,\omega}$.

\begin{enumerate}
\item  Give a definition of a `complete' that eliminates uninformative counterexamples \cite{BKsoul,BHK2}.
\item  Do the BLS results on $L_{\omega_1,\omega}$ generalize at all? E.g. to analytic classes?
\cite{BL}
\item If $\phi$ characterizes $\kappa> \aleph_0$, must $\phi$ have $2^\kappa$
models in $\kappa$?
\item For $\kappa <\aleph_{\omega_1}$, describe an explicit sentence that characterizes $\kappa$. \cite{BKLas}

\end{enumerate}

\begin{theorem}\cite{BaldwinShelahhanfmaxzfc}  There is a complete sentence  $\phi$  of $L_{\omega_1,\omega}$ such that $\phi$ has maximal
models in a set of cardinals $\lambda$ that is cofinal in the first measurable $\mu$ while $\phi$ has
no maximal models in any $\chi \geq \mu$.
\end{theorem}
\end{frame}



%\begin{frame}\frametitle{Few models necessary}
%%
%%Collectively, these results give a correct proof of Theorem~0.6 of
%%\cite{LaskowskiShelahatom} and improve \cite{Hjorthchar} by giving an
%%explicit sentence characterizing each $\aleph_r$ for $r<\omega$.
%
%\begin{block}{Theorem \cite{BKLas}} \label{sumup} For every $r \ge 1$, there is a class ${\bf At^r}$
%satisfies:
%\begin{enumerate}
%
%\item there is a  model of size $\aleph_r$, but no larger models;
%
%\item every model of size $\aleph_r$ is maximal, and so $2$-amalgamation is
%    trivially true in $\aleph_r$;
%
%
%\item disjoint $2$-amalgamation holds up to $\aleph_{r-2}$;
%%\item The situation for atomic models (or $\hat \bK^{r}$)
%%    $\aleph_{r-1}$ is unclear
%%    \begin{enumerate}
%\item 2-ap fails in $\aleph_{r-1}$.
%
%\item Each of the classes %$\hat {\bK^r}$ and ${\bf At}^r$
%have
%    $2^{\aleph_s}$ models in $\aleph_s$ for $1 \leq s \leq r$.
%%     In addition,
%%    $\hat {\bK^r}$ has $2^{\aleph_0}$ models in $\aleph_0$.
%\end{enumerate}
%
%
%
%
%
%\end{block}
%
%\end{frame}
\begin{frame}[allowframebreaks] {References}
%\bibliography{C:/Users/jbald/texmf/bibtex/bib/local/ssgroups}

%\bibliography{C:\users\jbald\boxsync\base\mytex\ssgroups.bib}
%\bibliographystyle{alpha}

\bibliography{C:/mylatex/ssgroups.bib}
\bibliographystyle{alpha}
%\bibliography{ssgroups.bib}
%\bibliographystyle{alpha}
\end{frame}
\end{document}




***************************



















\begin{frame}
\frametitle{The Theorem} \begin{block}{Main Theorem} If $\bK_T$ fails
`density of pseudo-minimal types' (fails algebraic symmetry)
    then $\bK_T$ has $2^{\aleph_1}$ models of cardinality $\aleph_1$.
\end{block}

\begin{block}{Proof Outline}



\begin{enumerate}
\item Start with a model $\Nscr_0$ of enough set theory and an
    infinitary $\tau$-sentence $\psi$ that fails `density'. \item
    Force a generic extension $\Nscr_1$ of $\Nscr_0$ that
    satisfies Martin' axiom, MA.
     \item Expand the vocabulary $\tau$ to a $\tau^*$ that allows
         the description of filtrations and define an
         $L_{\omega_1,\omega}(Q)$ $\tau^*$-formula
         $\theta(P_1,P_2)$ that relies on the properties of the
         pseudoclosure.


   % catching/strong
%    catching.
\end{enumerate}

\end{block}
 \end{frame}


\begin{frame}
\frametitle{Proof Outline Continued}
\begin{enumerate}\setcounter{enumi}{3}
\item


    In $M=\Nscr_1$, using the fact that $\psi$ `fails density',
    force with an $\aleph_1$-like dense linear order to get a
    generic filter $G$. Conclude that in $\Nscr_1$ for each pair
    of stationary sets $S,T$ there is a model $M^{S,T}[G]$ such
    that if $M^{S,T}[G]\models\theta(P_1,P_2)$ then $P_1\cap P_2
    = \emptyset$, $S\subseteq P_1$ and $T\subseteq P_2$
     \item Expand $\Nscr_1$ to a vocabulary $\tau'$ (including
         $\epsilon$) by interpreting the symbols of $\tau^*$ on
         the model constructed in step 4. Construct an elementary
         extension $\Nscr_2$  of $\Nscr_1$ such that `stationary'
         is absolute between $\Nscr_2$ and $V$.


         \item  In $\Nscr_2$ choose $2^{\aleph_1}$ pairs of
             disjoint stationary sets $(S^\eta,T^\eta)$ such that
             the entire set of $S^\eta$ is pairwise disjoint
             modulo the ideal on non-stationary sets. This
             implies the $M^{S^\eta,T^\eta}[G]$ are pairwise
             non-isomorphic in $\Nscr_2$. Since stationary is
             absolute between $\Nscr_2$ and $V$, in $V$ there are
         $2^{\aleph_1}$ models of $\psi$.


\end{enumerate}


 \end{frame}
