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\begin{document}

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\section*{geometric stability theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{basic_concepts}{Basic concepts}\dotfill \pageref*{basic_concepts} \linebreak
\noindent\hyperlink{minimal_and_strongly_minimal_sets}{Minimal and strongly minimal sets}\dotfill \pageref*{minimal_and_strongly_minimal_sets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Geometric stability theory} is the principal part of the branch of [[model theory]] called \textbf{[[geometric model theory]]}. It was introduced in works of [[Boris Zilber]], Gregory Cherlin, [[Ehud Hrushovski]], Anand Pillay, and others.

Geometric stability theory has largely to do with the model-theoretic classification of [[structure in model theory|structures]] in terms of dimension-like quantities that can be axiomatized in terms of notions of [[combinatorial geometry]] such as [[matroid]]. Key guiding examples include [[vector spaces]] and [[algebraically closed fields]], and many of the guiding concepts have an [[algebraic geometry|algebro-geometric]] flavor (e.g., Morley rank as a generalization of Krull dimension).

Such a structural geometric approach can make geometric stability theory an attractive key of entry into modern model theory for those mathematicians who are not already logicians.

\hypertarget{basic_concepts}{}\subsection*{{Basic concepts}}\label{basic_concepts}

Perhaps the key axiomatic notions of geometric stability theory (which at the outset don't seem particularly wedded to logic) are those of \emph{pregeometry} and \emph{geometry}.

\begin{defn}
\label{}\hypertarget{}{}
Let $X$ be a set. A \textbf{pregeometry} on $X$ is a [[closure operator]] (i.e., a [[monad]] $cl \colon P X \to P X$ on the [[power set]]), satisfying the following two conditions:

\begin{itemize}%
\item The monad $cl$ is [[finitary functor|finitary]], i.e., $A \in X$ and $a \in cl(A)$, then there is a finite $A_0 \subseteq A$ such that $a \in cl(A_0)$.


\item (Exchange condition) If $A \in P X$, $a,b \in X$, and $a \in cl(A\cup\{b\})$, then $a \in cl(A)$ or $b \in cl(A \cup \{\a\})$. (Cf. [[matroid]])



\end{itemize}
A \textbf{geometry} is a pregeometry such that $cl(\emptyset) = \emptyset$ and $cl(\{x\}) = \{x\}$ for all $x \in X$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item Let $X$ be a [[vector space]], and let $cl$ be the monad on $P X$ whose algebras are vector subspaces of $X$. Clearly $cl$ is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus $cl$ is a pregeometry.


\item Similarly, let $X$ be a [[projective space]] $\mathbb{P}V$, and let $cl$ be the monad on $P X$ whose algebras are projective subspaces. Then $cl$ is a geometry (the closure of a point is a point). Any pregeometry $cl$ gives rise to a geometry in a similar way, in the sense that a pregeometry $cl$ induces a geometry on the image of the function $X \to P X$, $x \mapsto cl(\{x\})$, as explained in Remark \ref{project}.


\item Let $X$ be an algebraically closed field; let $cl$ be the monad on $P X$ whose algebras are algebraically closed subfields. Then $cl$ is a pregeometry. That the exchange condition is satisfied is a classical result credited to Steinitz\footnote{Although oddly enough, as explained at the \href{http://www-history.mcs.st-andrews.ac.uk/Biographies/Steinitz.html}{MacTutor biography page}, what is called the Steinitz exchange condition was set out by Ernst Steinitz in a 1913 publication on convergent series and apparently not (?), as might be supposed, in his \emph{Algebraische Theorie der K\"o{}rper} (Crelle's Journal, 1910). His 1913 lemma was for \emph{vector spaces}.} .



\end{itemize}
\end{example}
\begin{defn}
\label{}\hypertarget{}{}
Given a pregeometry $(X, cl)$, a subset $A \in P X$ is \textbf{independent} if for all $a \in A$, $a \notin cl(A - \{a\})$. An independent set $A$ said to be a \textbf{basis} for $Y \in P X$ if $Y \subseteq cl(A)$. All bases of $Y$ have the same cardinality, called the \textbf{dimension} of $Y$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Given a pregeometry $(X, cl)$ and a subset $Y \subseteq X$, there is a restriction pregeometry $cl^Y: P(Y) \to P(Y)$ defined by the formula

\begin{displaymath}
cl^Y(S) = cl(S) \cap Y.
\end{displaymath}
It is immediate that $A \subseteq Y$ is independent in $(X, cl)$ iff it is independent in $(Y, cl^Y)$, and that it is a basis for $Y$ in $(X, cl)$ iff it is a basis for $Y$ in $(Y, cl^Y)$. By this observation, to prove that any $Y \in P(X)$ has a well-defined dimension, we may assume without loss of generality that $Y = X$. That a pregeometry $X$ has well-defined dimension was proven \href{nlab/show/matroid#welldefined}{here}.

\end{remark}
\begin{remark}
\label{project}\hypertarget{project}{}
There is a standard way of getting a geometry from a pregeometry $(X, cl)$. First, if $cl(\emptyset) \neq \emptyset$, then replace $X$ by $X' \coloneqq X - cl(\emptyset)$, equipped with the restriction pregeometry of the previous remark. Then define an equivalence relation on $X'$ by $x \sim y$ if $cl(\{x\}) = cl(\{y\})$, and define a pregeometry on the quotient set $X'/\sim$ by

\begin{displaymath}
\widehat{cl}(A) \coloneqq \{[b]: b \in cl(A)\}
\end{displaymath}
where $[b]$ denotes the equivalence class of $b$. This abstracts the process of taking a projectivization of a vector space.

\end{remark}
\hypertarget{minimal_and_strongly_minimal_sets}{}\subsubsection*{{Minimal and strongly minimal sets}}\label{minimal_and_strongly_minimal_sets}

We'll suppose for now that $\mathbf{L}$ is a countable [[signature (in logic)|signature]], so that the [[first-order language|language]] it generates consists of countably many formulas. Let $\mathbf{M}$ be an $\mathbf{L}$-structure, with underlying set $M$.

\begin{defn}
\label{}\hypertarget{}{}
For $A \subseteq M$, an element $b \in M$ is \textbf{algebraic} over $A$ if there is a formula $\phi(y, w_1, \ldots, w_n)$ and elements $a_1, \ldots, a_n \in A$ such that $\mathbf{M} \models \phi(b, a_1, \ldots, a_n)$ and

\begin{displaymath}
\{c \in M: \mathbf{M} \models \phi(c, a_1, \ldots, a_n)\}
\end{displaymath}
is finite. The \textbf{algebraic closure} of $A$, denoted $acl(A)$, is the set of elements of $M$ that are algebraic over $A$.

\end{defn}
\begin{prop}
\label{finclos}\hypertarget{finclos}{}
The algebraic closure $A \mapsto acl(A)$ defines a finitary closure operator on $M$.

\end{prop}
\begin{proof}
That $acl$ is monotone (preserves order) is obvious. Also the fact that $acl$ is finitary is easy: given that $b \in acl(A)$ satisfies $\mathbf{M} \models \phi(b, a_1, \ldots, a_n)$ for $a_1, \ldots, a_n \in A$, then similarly $b \in acl(A_0)$ for $A_0 = \{a_1, \ldots, a_n\}$.

Taking $\phi(y, w)$ to be the equality predicate $y = w$, we see $A \subseteq acl(A)$.

For idempotence of $acl$, suppose $b_1, \ldots, b_k \in acl(A)$ and $\mathbf{M} \models \psi(c, b_1, \ldots, b_k)$ where $\{y \in M: \mathbf{M} \models \psi(y, b_1, \ldots, b_k)\}$ has exactly $m$ elements. Write down a formula $F_{m, \psi}(x_1, \ldots, x_k)$ that says $\{y \in M: \mathbf{M} \models \psi(y, x_1, \ldots, x_k)\}$ has at most $m$ elements (by adding some extra inequalities, we can make this ``exactly $m$ elements''):

\begin{displaymath}
F_{m, \psi}(x_1, \ldots, x_k) \coloneqq \exists_{y_1, \ldots, y_m} \forall_y \psi(y, x_1, \ldots, x_k) \Leftrightarrow (y = y_1) \vee \ldots \vee (y = y_m).
\end{displaymath}
Now for $i = 1, \ldots, k$, let $\phi_i$ be a formula witnessing $b_i \in acl(A)$, i.e., $\mathbf{M} \models \phi_i(b_i, a_{i, 1}, \ldots, a_{i, n_i})$ where $a_{i, k} \in A$ and $\{x \in M: \mathbf{M} \models \phi_i(x, a_{i, 1}, \ldots, a_{i, n_i})\}$ has finitely many elements. Then

\begin{displaymath}
\exists_{x_1, \ldots, x_k} F_{m, \psi}(x_1, \ldots, x_k) \wedge \psi(y, x_1, \ldots, x_k) \wedge \bigwedge_{i=1}^k \phi_i(x_i, a_{i, 1}, \ldots, a_{i, n_i})
\end{displaymath}
is a formula with parameters in $A$ that witnesses $c \in acl(A)$. This proves the idempotence of $acl$.

\end{proof}
As before, this notion of algebraic closure $acl: P(M) \to P(M)$ can be restricted to a closure operator $acl^X: P(X) \to P(X)$ for a subset $X \subseteq M$, via the definition $acl^X(A) \coloneqq X \cap acl(A)$ for $A \subseteq X$. One often writes just $acl$ instead of $acl^X$, provided that $X$ is understood.

\begin{defn}
\label{}\hypertarget{}{}
For an $\mathbf{L}$-structure $\mathbf{M}$, let $D \subseteq M^n$ be definable. $D$ is \textbf{minimal} if the only definable subsets of $D$ are finite or cofinite in $D$. Slightly abusing language, if $\phi(x_1, \ldots, x_n, a_1, \ldots, a_k)$ is a formula with parameters that defines $D$, we also say $\phi$ is minimal. We say $D$ (or $\phi$) is \textbf{strongly minimal} if it is minimal in any elementary extension of $\mathbf{M}$.

A theory $\mathbf{T}$ is \emph{strongly minimal} if for any model $\mathbf{M}$ of $\mathbf{T}$, the underlying set $M$ (definable by the formula $x = x$) is strongly minimal.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item Let $K$ be an algebraically closed field. (Elimination of quantifiers, Chevalley's theorem, etc.) Conclusion: ACF is a strongly minimal theory.


\item Divisible torsionfree abelian groups.


\item Non-example of dense linear orders.



\end{enumerate}
\end{example}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Baldwin, Lachlan)} The algebraic closure operator on a minimal set $X$ is a pregeometry.

\end{theorem}
\begin{proof}
Let $A \subseteq X$ and suppose $c \in acl(A \cup \{b\}) - acl(A)$ for $b, c \in X$; we want to show $b \in acl(A \cup \{c\})$. Thus, suppose $\phi(c, b)$ is a formula with parameters from $A$ such that $\mathbf{M} \models \phi(c, b)$ and $card(\{x \in X: \mathbf{M} \models \phi(x, b)\}) = n$, a finite number. As in the proof of Proposition \ref{finclos}, let $\psi(w)$ be a formula that says $card(\{x \in X: \mathbf{M} \models \phi(x, w)\}) = n$. This is a formula with parameters from $A$ and $\psi(b)$ is satisfied in $\mathbf{M}$.

If $\{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\}$ is finite, then since $b$ belongs to this set, $\phi(c, y) \wedge \psi(y)$ would witness $b \in acl(A \cup \{c\})$ and we would be done. So suppose otherwise. Then this set is cofinite in $X$, so that

\begin{displaymath}
card(X - \{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\}) = m
\end{displaymath}
for some finite $m$, and we can again write down a formula $\chi(x)$ with parameters from $A$ that says

\begin{displaymath}
card(X - \{y \in X: \mathbf{M} \models \phi(x, y) \wedge \psi(y)\}) = m.
\end{displaymath}
If $\chi(x)$ defines a finite subset of $X$, then since $\chi(c)$ is satisfied, we would reach the conclusion that $c \in acl(A)$, a contradiction. Hence $\chi(x)$ defines a subset cofinite in $X$. We may therefore choose $n+1$ elements $a_1, \ldots, a_{n+1} \in X$ such that $\chi(a_i)$ is satisfied. By our supposition,

\begin{displaymath}
B_i \coloneqq \{u \in X: \mathbf{M} \models \phi(a_i, u) \wedge \psi(u)\}
\end{displaymath}
is cofinite for $i = 1, \ldots, n+1$. Hence $\bigcap_{i=1}^{n+1} B_i$ is inhabited, say by an element $b'$. We have at least $n+1$ elements $x \in X$ such that $\phi(x, b')$ is satisfied, namely $x = a_1, \ldots, a_{n+1}$. But now this contradicts the fact $\psi(b')$.

\end{proof}
=--

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[stability in model theory]]


\item [[Zariski geometry]]


\item [[matroid]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Anand Pillay, \emph{Geometric stability theory}, Oxford Logic Guides \textbf{32}


\item slides from conference ``\href{http://www.maths.ox.ac.uk/events/borisfest}{Geometric model theory}'', Oxford 2010: directory \href{http://people.maths.ox.ac.uk/bays/misc/borisfest/notes}{html}


\item Misha Gavrilovich, \emph{Model theory of universal covering space of complex algebraic varieties}, thesis, \href{http://people.maths.ox.ac.uk/bays/misha-thesis.pdf}{pdf}


\item [[Boris Zilber]], \emph{Elements of Geometric Stability Theory}, 2003 (\href{http://people.maths.ox.ac.uk/zilber/est.pdf}{pdf})



\end{itemize}


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