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\title % (optional, use only with long paper titles)
{Projective geometries arising from Elekes-Szab\'o problems}

\author%[Author, Another] % (optional, use only with lots of authors)
{Martin Bays\\
Joint work with Emmanuel Breuillard}
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%   affiliation.

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\date{13-11-2018 \\
Oxford}
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\begin{document}
\maketitle
%Frame:
  %\titlepage
  %  \tableofcontents
  %  % You might wish to add the option [pausesections]
  %.

% Plan:
%   Strongly minimal sets and acl geometry
%   Modularity
%   Example: generic torus in CCM as pure group
%   Strongly minimal pseudofinite \Rightarrow  locally modular
%   Pseudofinite subsets of fields and \bdl
%     (use monster model presentation, without spelling it out?)
%   Coherence
%   cgp as then yielding minimality
%   magically, also exactly what's needed for SzT to yield modular geometry
%   (noting crucial that it applies to things outside the coherent set)
%   Counterexample without cgp
%   Statement
%   Converse

\begin{frame}
  \frametitle{Elekes-Szab\'o}
  \begin{itemize}\item Suppose $f \in \C[X,Y,Z]$ is an irreducible polynomial in which each of 
  $X,Y,Z$ appears.
  \item Set $V := \{ (x,y,z) \in \C^3 : f(x,y,z) = 0 \}$.
  \item Consider intersections with finite ``grids'' $A \times B \times C$ with 
  $|A|,|B|,|C| \leq  N \in \N$.
  \item We have \[ |V \cap (A\times B\times C)| \leq  O(N^2) .\]
  \item Say $V$ ``admits no powersaving'' if for no $\epsilon>0$ do we have
  \[ |V \cap (A\times B\times C)| \leq  O(N^{2-\epsilon}) .\]
  \item Example: if $f(x,y,z) = z-x-y$ then arithmetic progressions $A=B=C:= 
  [-m,m]$ witness that $V$ admits no powersaving.
  \end{itemize}
  \begin{theorem}[Elekes-Szab\'o 2012]
    $V$ admits no powersaving iff $V$ is in co-ordinatewise algebraic 
    correspondence with the graph of addition on a 1-dimensional algebraic 
    group.
  \end{theorem}
\end{frame}

% Define on board co-ordinatewise correspondence, via generics, preparing to 
% modify to allow permutations and arbitrary co-ordinate varieties

\begin{frame}
  \frametitle{Pseudofinite dimension}

  Hrushovski ``On Pseudo-Finite Dimensions'' (2013)

  \begin{itemize}\item
    $\U \subseteq  \P(\omega)$ non-principal ultrafilter.
  \item
    $K := \C^\U$.
  \item
    $X \subseteq  K^n$ is \defnstyle{internal} if $X = \prod_{s\rightarrow \U}X_s$ for some 
    $X_s \subseteq  \C^n$,
    and \defnstyle{pseudofinite} if each $X_s$ is finite.

  \item
    For $X$ internal, set $|X| := \lim_{s\rightarrow \U} |X_s| \in \R^\U \cup \{ \infty 
    \}$.
  \item
    $|X| \in \R^\U$ if $X$ is pseudofinite, $|X| := \infty$ else.

  \item Fix $\xi \in \R^\U$ with $\xi > \R$.

  \begin{definition}[Coarse pseudofinite dimension $\bdl$]
    For $X$ internal,
    \[ \writeme{\bdl(X)} = \bdl_{\xi}(X) := \st \left({\log_\xi(|X|)}\right) \in 
    \R_{\geq 0} \cup \{-\infty,\infty\} .\]

  \end{definition}
  %** $\bdl(X) < c \in \R_{>0}$ iff for some $\epsilon>0$ and $A \in \U$,
  %$|X_s| \leq  O((\xi_{0,s})^{c-\epsilon})$ for $s \in A$.
  %** $\bdl(X\times Y)=\bdl(X)+\bdl(Y)$.
  %** $\bdl(X \cup Y) = \max(\bdl(X),\bdl(Y))$.

  \item Note that internality is closed under cardinality quantifiers:
    if $R \subseteq  K^n \times K^m$ is internal and $\alpha \in \R^\U$,
    then $\{ \y \in K^n : \exists_{\geq \alpha} \x. R(\x,\y) \}$ is internal.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{$\Lint$ monster}
  \begin{itemize}\item $\Lint$: predicate for each internal $X \subseteq  K^n$.
  %** Note $\{a\}$ is internal for $a \in K$.
  \item $\Km \succ  K$ monster model in $\Lint$.
  \item For $\phi \in \Lint$, set $\bdl(\phi) := \bdl(\phi(K))$.
  \item $\bdl$ has a unique extension to $(\Lint)_{\Km}$ such that
    \begin{align*}
      \tp(\b) &\mapsto  \bdl(\phi(\x,\b)) \\
      S_{\y}(\emptyset ) &\rightarrow  \{-\infty\} \cup \R \cup \{\infty\}
    \end{align*}
    is well-defined and continuous for each $\phi(\x,\y) \in \Lint$.
  \item Explicitly, $\bdl(\phi(\x,\a)) := \sup \{ q \in \Q : \Km \vDash  
  \exists_{\geq \xi^q} \x.\; \phi(\x,\a) \}$.

  \pause

  \item For $\Phi$ a partial type, $\bdl(\Phi) := \inf \{ \bdl(\phi) : \Phi \vDash  
  \phi \}$.
  \item $\bdl(a/C) := \bdl(\tp(a/C))$.

  \begin{fact}
    For $C \subseteq  \Km$ small and $a,b \in \Km^{<\omega}$,
    \begin{itemize}\item({\bf Invariance}): $a\equiv _C b \Longrightarrow  \bdl(a/C)=\bdl(b/C)$.
    \item({\bf Additivity}): $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
    \item({\bf Existence}): A partial type $\Phi$ over $C$ has a realisation
    $a \in \Phi(\Km)$ with $\bdl(a/C)=\bdl(\Phi)$.
    \end{itemize}
  \end{fact}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{$\acl^0$}

  For $V$ a variety over $\Km$, eliminate the imaginary to consider elements 
  of $V(\Km)$ as tuples from $\Km$, i.e.\ $V(\Km) \subseteq  \Kmeq$.
  
  \begin{definition}
    Superscript 0 means: reduct to $\operatorname{ACF}_\C$.

    ($\C \leq  \C^{\U} = K \prec  \Km$)


    \begin{itemize}\item $\writeme{ d^0(A/B) } := \trd(\C(A,B)/\C(B))$ (for $A,B \subseteq  \Kmeq$)
    \item $\writeme{ \acl^0(B) } := \{ a \in \Kmeq : d^0(a/B) = 0 \} =$ 
    field-theoretic algebraic closure of $\C(B)$ in $\Kmeq$.
    \end{itemize}
  \end{definition}
  \begin{remark}
    $a \in \acl^0(B) \Longrightarrow  \bdl(a/B) = 0$.
  \end{remark}
\end{frame}

\begin{frame}
  \frametitle{Coherence (1-dimensional version)}
  % Stability theory teaches us that given a dimension function, we should 
  % look for pregeometries. Can't hope to say much about \bdl in general, but 
  % here's a case where we can.

  \begin{definition}
    $P \subseteq  \Km$ is \defnstyle{coherent} if
    for any tuple $\a \in P^{<\omega}$,
    \[ \bdl(\a) = d^0(\a) .\]
  \end{definition}

  In other words, $\bdl$ is equal on $P^{<\omega}$ to the dimension function 
  of the pregeometry $(P;\acl^0)$.

  \pause
  \begin{lemma}
    Say an irreducible complex variety $V \subseteq  \mathbb{A}^n$ ``admits no powersaving'' 
    if for no $\epsilon>0$ do we have a bound
    \[ |V(\C) \cap \prod_i A_i| \leq  O(N^{\dim(V)-\epsilon}) ,\]
    for $A_i \subseteq  \C$ with $|A_i| = N$.

    Then $V$ admits no powersaving iff
    there exists a generic point $\a \in V(\Km)$ with $\{a_1,\ldots ,a_n\}$ 
    coherent.
  \end{lemma}
\end{frame}

\begin{frame}
  \frametitle{Coarse general position}
  %More generally:

  \begin{definition}
    $a \in \Kmeq$ is in \defnstyle{coarse general position} (or is 
    \writedefnstyle{cgp}) if for any $B \subseteq  \Km$,
    \[ d^0(a/B) < d^0(a) \Longrightarrow  \bdl(a/B) = 0 .\]
  \end{definition}

  Any $a \in \Km$ is cgp.

  \begin{definition}
    Let $W$ be an irreducible variety over $\C$.

    A $\Km$-definable set $X \subseteq  W(\Km)$ with $\bdl(X) \in \R_{>0}$ is 
    \defnstyle{cgp} if
    for any $W' \subsetneq  W$ proper subvariety over $\Km$,
      \[ \bdl(X \cap W') = 0 .\]
  \end{definition}

  If $X$ is cgp, then any $a \in X$ is cgp.

  %``No internal structure''.
\end{frame}

\begin{frame}
  \frametitle{Coherence}

  \begin{definition}
    $P \subseteq  \Kmeq$ is \writedefnstyle{coherent} if
    \begin{itemize}\item every $a \in P$ is cgp, and
    \item for any tuple $\a \in P^{<\omega}$,
    $ d^0(\a)=\bdl(\a)$.
    \end{itemize}
  \end{definition}

  Then $(P;\acl^0)$ is a pregeometry,
  and if $d^0(a)$ is constant for $a \in P$,
  then $\bdl$ is proportional on $P^{<\omega}$ to its dimension function.

  \begin{example}
    Let $G$ be a complex semiabelian variety, e.g.\ $G=(\C^\times)^n$.

    Let $\gamma \in G(\C)$ generic.

    Let $X := \prod_{s \rightarrow  \U} \{-s\cdot\gamma,\ldots ,s\cdot\gamma\}$, and set 
    $\xi$ such that $\bdl(X) = \dim(G)$.
    Then $X$ is cgp, since $|X \cap W'| < \aleph_0$ by uniform Mordell-Lang.

    Now $\bdl(X^3 \cap \Gamma_+) = 2\bdl(X)$.

    Let $(a,b,c) \in X^3 \cap \Gamma_+$ with $\bdl(abc) = 2\bdl(X)$.

    Then $\{a,b,c\}$ is coherent.
  \end{example}
\end{frame}

\begin{frame}
  \frametitle{Szemer\'edi-Trotter bounds}
  Suppose $a,e \in \Kmeq$ satisfy (working over some $C \subseteq  \Km$):

  If $e' \equiv  e$ and $e' \equiv ^0_a e$ but $e' \neq  e$, then $\bdl(a/ee') = 0$.

  \begin{lemma}[Trivial bound]
    Assume such a $e'$ exists (i.e.\ $e \notin \dcl(a))$).

    Then $\bdl(a/e) \leq  \frac12\bdl(a)$.
    %{\em Proof on board.}
  \end{lemma}
  \begin{proof}
    Take $e' \equiv _a e$ with $e' \neq  e$ and $\bdl(e'/ab) = \bdl(e'/a)$.

    Then
    $0 = \bdl(a/ee') = \bdl(aee') - \bdl(ee') = 2\bdl(ab) - \bdl(a) - 
    \bdl(ee') \geq  2\bdl(a/e) - \bdl(a)$ .
  \end{proof}
  \pause

  %Lemma[Elekes-Szab\'o, Fox-Pach-Sheffer-Suk-Zahl]:
  \begin{lemma}[Szemer\'edi-Trotter bound due to Elekes-Szab\'o]
    %With $\epsilon_0 := \frac1{4n_2-1} > 0$ and
    %$y^+:=\max\{0,y\}$,
    %\begin{align*} \bdl(X) \leq  \max
    %&( [\frac{1}{2}\bdl(X_1) + \bdl(X_2)] - \epsilon_0 [\bdl(X_2) - 
    %\frac{1}{2}\bdl(X_1)]^{+} \\
    %&,\bdl(X_1), \bdl(X_2) ) . \end{align*}

    %In particular,
    %If $\bdl(X_2) > \frac12\bdl(X_1) > 0$, then
    %$\bdl(X) < \frac12\bdl(X_1) + \bdl(X_2)$.

    Assume $\bdl(e) > \frac12\bdl(a) > 0$.

    Then $\bdl(a/e) < \frac12\bdl(a)$.
  \end{lemma}

  Hrushovski: Szemer\'edi-Trotter bounds correspond to modularity.
\end{frame}


\begin{frame}
  \frametitle{Modularity}
  \begin{recall}
    \begin{itemize}\item A \defnstyle{geometry} is a pregeometry with $\cl(\emptyset )=\emptyset $ and 
    $\cl(\{x\})=\{x\}$.

    \item The geometry of a pregeometry $(P;\cl)$ is
    $(\{ \cl(x) : x \in P \};\cl)$.
    \end{itemize}
  \end{recall}

  \begin{definition}
    \begin{itemize}\item A geometry $(P,\cl)$ is \defnstyle{modular} if for $a,b \in P$ and $C 
    \subseteq  P$,
    \\ if $a \in \cl(bC) \setminus \cl(C)$
    then there exists $c \in \cl(C)$ such that $a \in \cl(bc)$.

    \item Say $a,b \in P$ are \defnstyle{non-orthogonal} if $a \in \cl(bC) \setminus 
    \cl(C)$ for some $C \subseteq  P$.
    %A geometry $(P,\cl)$ is \defnstyle{locally modular} if $(P,\cl_a)$ is 
    %modular for any $a \in P$, where $\cl_a(C) = \cl(Ca)$.
    \end{itemize}
  \end{definition}

  %If $V$ is a vector space over a division ring $F$,
  %then its projective geometry $\P_F(V)$ is modular.

  %$\omega$-categorical, and more generally pseudofinite, strongly minimal 
  %sets have locally modular geometries.

  \begin{fact}[Veblen-Young co-ordinatisation theorem]
    The modular geometries of dimension $\geq 4$ in which every two points are 
    non-orthogonal are precisely the projective geometries $\P_F(V)$ of vector 
    spaces of dimension $\geq 4$ over division rings.
    %(by the coordinatisation theorem of projective geometry)
  \end{fact}
\end{frame}

\begin{frame}
  \frametitle{Linearity}
  \begin{lemma}
    Suppose $P \subseteq  \Kmeq$ is coherent, $a_1,a_2,b_1,\ldots ,b_n \in P$, and:
    \begin{itemize}\item $d^0(a_1) = k = d^0(a_2)$;
    \item $a_1 \in \acl^0(a_2\b) \setminus \acl^0(\b)$ and $a_1 \notin \acl^0(a_2)$.
    \end{itemize}
    Let $e := \Cb^{\ACF}(\a/\C(\b))$.
    Then $d^0(e) = k$.

    Moreover, $e$ is cgp, so $\{e\}$ is coherent.
  \end{lemma}
  \pause
  \begin{proof}
<2->
    \begin{overprint}
      \onslide<2>
      By cgp and canonicity, if $e' \equiv  e$ and $e' \equiv ^0_{\a} e$ but $e' \neq  e$,
      then $\bdl(\a/ee') = 0$.

      But $\bdl(\a/e) \geq  \bdl(\a/\b) = d^0(\a/\b)= \frac12d^0(\a) = 
      \frac12\bdl(\a)$.

      So by the Szemer\'edi-Trotter bounds, we have $\bdl(e) \leq  
      \frac12\bdl(\a)$.

      Now $e \in \acl^0(\b)$ and $\b$ is coherent, and it follows that $d^0(e) 
      \leq  \bdl(e)$.

      So $d^0(e) \leq  \bdl(e) \leq  \frac12\bdl(\a) = k$.

      \ldots 

      \onslide<3>
      \ldots 

      It remains to show that $e$ is cgp.

      Suppose $B \subseteq  \Kmeq$ and $e \nind ^0 B$;
      we show $\bdl(e/B) = 0$.

      Let $\a' = a_1'a_2'$ such that $\a' \equiv _e \a$ and $\a' \dind_e B$.
      So $e \in \acl^0(\a')$.
      Then $\a' \nind ^0 B$.
      So since $\a'$ is coherent, $\bdl(\a'/B) \leq  \bdl(a_i') = k$.

      Meanwhile, $\bdl(\a'/e) = \bdl(\a') - \bdl(e) = d^0(\a') - d^0(e) = k$.
      So $\bdl(e/B) = \bdl(\a'/B) - \bdl(\a'/eB) = \bdl(\a'/B) - \bdl(\a'/e) 
      \leq  k - k = 0$.
    \end{overprint}
  \end{proof}
\end{frame}

\begin{frame}
  \frametitle{Modularity of coherence}

  \begin{definition}
    For $P \subseteq  \Kmeq$,
    $\ccl(P) := \{ x \in \acl^0(P) : \{x\} \;\textrm{is coherent}\}$.
  \end{definition}

  \begin{lemma}
    If $P$ is coherent, so is $\ccl(P)$.
  \end{lemma}

  \begin{proposition}
    Suppose $P = \ccl(P)$ is coherent.

    Then the geometry of $(P;\acl^0)$ is modular.
  \end{proposition}
\end{frame}

\begin{frame}
  \frametitle{Projective subgeometries in $\operatorname{ACF}$}
  Define $\P(\Km) := \{ \acl^0(x) : x \in \Km \}$.
  \begin{example}
    Suppose $G$ is a 1-dimensional complex algebraic group and $F \leq  \Q \otimes_\Z
    \End(G)$ is a subfield.

    $G(\Km) / G(\C)$ is naturally an $F$-vector space.

    Let $B \subseteq  G(\Km)$ be a set of independent generics, and set
    $V := \left<{B / G(\C)}\right>_F$.

    Define $\eta : \P_F(V) \rightarrow  \P(\Km)$ by
    $\eta(\left<{x/G(\C)}\right>_F) := \acl^0(x)$.

    Then $\eta$ embeds $\P_F(V)$ as a subgeometry of $\P(\Km)$.
  \end{example}
  \begin{theorem}[Evans-Hrushovski 1991]
    Any projective subgeometry of $\P(\Km)$ of dimension at least 3 arises in 
    this way.
  \end{theorem}
\end{frame}

\begin{frame}
  \frametitle{Projective geometries fully embedded in $\operatorname{ACF}^{\eq}$}
  Define $\P(\Kmeq) :=\{ \acl^0(x) : x \in \Kmeq \}$.
  \begin{definition}
    $\eta : \P_F(V) \rightarrow  \P(\Kmeq)$ is a \defnstyle{$k$-dimensional full 
    embedding}
    if for all $\b \in \P_F(V)^{<\omega}$, we have
    $d^0(\eta(\b)) = k\cdot \dim_{\P_F(V)}( \b )$ .
  \end{definition}

  \begin{example}
    If $G$ is a complex abelian algebraic group,
    $F \leq  \Q \otimes_\Z \End(G)$ a division subring,
    $B \subseteq  G(\Km)$ independent generics,
    and $V := \left<{B / G(\C)}\right>_F$.

    Then $\eta(\left<{x/G(\C)}\right>_F) := \acl^0(x)$ is a $\dim(G)$-dimensional full 
    embedding.
  \end{example}

  \begin{theorem}[``Higher Evans-Hrushovski'']
    Suppose $V$ is a vector space of dimension at least 3 over a division ring 
    $F$,
    and $\eta : \P_F(V) \rightarrow  \P(\Kmeq)$ is a $k$-dimensional full embedding.

    Then there are $G$ and embeddings $F \leq  \Q \otimes_\Z \End(G)$ and $V \leq  
    G(\Km) / G(\C)$ such that $\eta$ is as in the example.

    %Furthermore, $G$ is unique up to isogeny.
  \end{theorem}
\end{frame}


\begin{frame}
  \frametitle{Projective geometries fully embedded in $\operatorname{ACF}^{\eq}$}

  \begin{proof}[Proof idea]
    Abelian group configuration yields $G$.
    \newcommand{\phant}{\text{\hphantom{$[0:0:0]$}}}
    \[ \resizebox{1\textwidth}{!}{
      \xymatrix@=1em{
      & & &&&& [0:1:0] \ar@{-}'[ddll][dddlll] \ar@{-}'[dlll][ddllllll]   \\
          & & & [1:1:0] \ar@{-}'[d][dd] & & & \\
          [1:0:0] & \phant & \phant & [2:1:1] & [1:1:1] \ar@{-}'[l][llll] & 
          \phant & \\
          &  & & [1:0:1] &&   & \\
          && & & & & [0:0:1] \ar@{-}'[uull][uuulll] \ar@{-}'[ulll][uullllll] } 
          }
          \]

    Version due to Faure of the fundamental theorem of projective geometry 
    (semilinearity of projective morphisms)
    yields embeddings $F \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \Q \otimes_\Z \End(G)$ and $V \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} G(\Km) / G(\C)$.
  \end{proof}
\end{frame}

\begin{frame}
  \frametitle{Elekes-Szab\'o consequences}
  \begin{definition}
    Say a \underline{finite} subset $X$ of a variety $W$ is 
    \defnstyle{$\tau$-cgp} if for any proper subvariety $W' \subsetneq  W$ of 
    complexity $\leq \tau$, we have
    $ |X \cap W'| < |X|^{\frac1\tau}$ .
  \end{definition}

  \begin{definition}
    If $V \subseteq  \prod_i W_i$ are irreducible complex algebraic varieties,
    with $\dim(W_i) = m$ and $\dim(V) = d m$,
    say $V$ \defnstyle{admits a powersaving} if for some $\tau$ and 
    $\epsilon>0$
    there is a bound
    \[ \left|{\prod_i X_i \cap V}\right| \leq  O(N^{d-\epsilon}) \]
    for $\tau$-cgp $X_i \subseteq  W_i$ with $|X_i| \leq  N$.
  \end{definition}
  \begin{lemma}
    $V$ admits no powersaving iff exists coherent generic $\a \in V(\Km)$.
  \end{lemma}
\end{frame}

\begin{frame}
  \frametitle{Elekes-Szab\'o consequences}

  \begin{definition}
    $H \leq  G^n$ is a \defnstyle{special subgroup} if $G$ is a commutative 
    algebraic group and $H = \ker(A)^o$ for some $A \in \operatorname{Mat}(F \cap \End(G))$ 
    for some division subalgebra $F \leq  \Q \otimes_{\Z} \End(G)$.
  \end{definition}
  \begin{theorem}
    $V \subseteq  \prod_i W_i$ admits no powersaving iff it is in co-ordinatewise 
    algebraic correspondence with a product of special subgroups.
  \end{theorem}
  \begin{proof}
    $\Longrightarrow $: Take coherent generic; apply modularity of coherence and ``higher 
    Evans-Hrushovski''.

    $\Leftarrow $: see below.
  \end{proof}
\end{frame}

\begin{frame}
  \frametitle{Sharpness}
  \begin{fact}[Amitsur-Kaplansky]
    Any division subring $F \subseteq  \Mat_n(\C)$ has finite dimension over its centre.
  \end{fact}
  \begin{corollary}
    Any finitely generated subring of a division subring $F \subseteq   
    \End^0(G)$ is contained in a finitely generated subring $\O \subseteq  F$ which is 
    \defnstyle{constrainedly filtered}: there are finite $\O_n \subseteq  \O$ such that
    \begin{enumerate}[(CF0)]
    \item $\O_n \subseteq  \O_{n+1}$; $\bigcup_{n \in \N} \O_n = \O$
    \item[(CF1)] $\exists k.\; \forall n.\; \O_n+\O_n \subseteq  \O_{n+k}$;
    \item[(CF2)] $\forall a \in \O.\; \exists k.\; \forall n.\; a\O_n \subseteq  
    \O_{n+k}$;
    \item[(CF3)] $\forall \epsilon>0.\; \frac{|\O_{n+1}|}{|\O_n|} \leq  
    O(|\O_n|^\epsilon)$.
    \end{enumerate}
  \end{corollary}

  (e.g.\ $\Z = \bigcup_n [-2^n,2^n]$ is constrainedly filtered.)

  Let ``$X_k := \prod_{s \rightarrow  \U}(\sum_{i=1}^s \O_{s-k}\gamma_i)$'' with 
  $\gamma_i \in G$ generic independent.
  
  Then $X := \bigcap_k X_k$ is an $\O$-submodule and $\bdl(X)=\bdl(X_0)$ and 
  $X$ is cgp.
  
  So any special subgroup defined using $\O$ admits no powersaving.
\end{frame}

\begin{frame}
  \frametitle{Application: Generalised sum-product phenomenon}

  \begin{corollary}
    Let $(G_1,+_1)$ and $(G_2,+_2)$ be one-dimensional non-isogenous connected 
    complex algebraic groups,
    and for $i=1,2$ let $f_i : G_i(\C) \rightarrow  \C$ be a rational map.
    Then there are $\epsilon,c>0$ such that if $A \subset \C$ is a finite set 
    lying in the range of each $f_i$, then setting $A_i=f_i^{-1}(A) \subseteq 
    G_i(\C)$ we have
    \[ \max( |A_1 +_1 A_1|, |A_2 +_2 A_2| ) \geq  c|A|^{1+\epsilon} .\]
  \end{corollary}
  \begin{proof}
    Else, considering $\{ (x,y,f_1^{-1}(x)+_1f_1^{-1}(y), 
    f_2^{-1}(x)+_2f_2^{-1}(y)) : x,y \in \C \}$, get group $(G;+)$ such that 
    $\Gamma_{+_i}$ is in co-ordinatewise correspondence with $\Gamma_+$, 
    $i=1,2$.
    
    But then (by Ziegler) $G_i$ is isogenous to $G$.
  \end{proof}

  Similarly in higher dimension, with a cgp assumption.
\end{frame}

\begin{frame}
  \frametitle{Application: Intersections of varieties with approximate 
  subgroups}
  \begin{theorem}
    $\Gamma \leq G(\Km)$ a $\emptyset $-$\bigwedge$-definable subgroup of a 
    1-dimensional algebraic group $G$, with $\bdl(\Gamma) = \dim(G)$.

    Then any coherent tuple $\gammatup \in \Gamma^n$ is generic in a coset of 
    an algebraic subgroup of $G^n$.
  \end{theorem}
  Similarly in higher dimension, with a cgp assumption.

  \begin{corollary}
    Let $G$ be a commutative complex algebraic group.
    Suppose $V$ is a subvariety of $G^n$ which is not a coset of a subgroup.
    Then there are $N, \epsilon, \eta>0$ depending only on $G$ and the 
    complexity of $V$ such that if $A \subseteq  G$ is a finite subset such 
    that $A-A$ is $\tau$-$cgp$ and $|A+A| \leq  |A|^{1+\epsilon}$ and $|A| 
    \geq  N$, then
    $|A^n \cap V| < |A|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
  \end{corollary}
\end{frame}

\begin{frame}
  \frametitle{Diophantine connection}
%  Corollary:
%    Let $G$ be an algebraic group over $\C$.
%    Let $\Gamma \leq  G(\Km)$ be a $\emptyset $-definable cgp subgroup,
%    and $\bdl(\Gamma)=\dim G$.
%
%    Suppose $\xitup \in \Gamma^n$ is coherent.
%    Then $\locus^{G^n}(\xitup/\C)$ is a coset of an algebraic subgroup.
%    .

  \begin{example}
    $G=E$ complex elliptic curve.

    $E[\infty] := \bigcup_m E[m]$ torsion subgroup.

    Suppose $V \subseteq  E^n$ is an irreducible closed complex subvariety such that 
    $V(\C) \cap E[\infty]$ is Zariski dense in $V$. Let $d := \dim(V)$.

    By Manin-Mumford, $V$ is a coset of an algebraic subgroup.
    Hence for any $\epsilon>0$, for arbitrarily large $r \in \N$,
    \[ |V(\C) \cap E[r!]^n| \geq  \left|{E[r!]}\right|^{d-\epsilon} .\]

    Suppose conversely that we only know this consequence of Manin-Mumford on 
    the asymptotics of the number of torsion points in $V$. Then $V$ has a 
    coherent generic non-standard torsion point, and so by above theorem $V$ 
    is a coset.

    Similarly for Mordell-Lang.
  \end{example}
\end{frame}

\begin{frame}
  \frametitle{Relaxing general position}

  \begin{remark}
    $V :=$ graph of $(a_1,b_1)*(a_2,b_2) = (a_1+a_2+b_1^2b_2^2,b_1+b_2)$,
    $X_i := \{-N^4,\ldots ,N^4\}\times\{-N,\ldots ,N\} \subseteq  \C^2 =: W_i$.

    Then $|X_i^3 \cap V| \geq  \Omega(|X_i|^2)$, but not in coarse general 
    position, and $V$ is not in co-ordinatewise correspondence with the graph 
    of a group operation.
  \end{remark}
\end{frame}

\begin{frame}
  \frametitle{Thanks}
  \pause
  {\em (Bonus slides follow)}
\end{frame}


\begin{frame}
  \frametitle{Elekes-Szab\'o consequences; detailed statement}
  \begin{definition}
    $a \in W(\Km)$ is \defnstyle{dcgp} if $a \in X \subseteq  W(\Km)$ for some 
    $\emptyset $-definable cgp $X$.
  \end{definition}
  \begin{theorem}
    Given $V \subseteq  \prod_i W_i$, TFAE
    \begin{enumerate}[(a)]\item $V$ admits no powersaving.
    \item Exists coherent generic $\a \in V(\Km)$ with $a_i$ dcgp in $W_i$.
    \item Exists coherent generic $\a \in V(\Km)$.
    \item $V$ is in co-ordinatewise algebraic correspondence with a product of 
    special subgroups.
    \end{enumerate}
  \end{theorem}
  \begin{proof}
    $(a) \Leftrightarrow  (b)$: ultraproducts.

    $(b) \Longrightarrow  (c)$: clear.

    $(c) \Longrightarrow  (d)$: modularity of coherence + ``higher Evans-Hrushovski''.

    $(d) \Longrightarrow  (b)$: see below.
  \end{proof}
\end{frame}

\begin{frame}
  \frametitle{Example}
  \begin{itemize}\item $G := (\C^\times)^4$.
  \item $\Q \otimes_{\Z} \End(G) \cong  \Q \otimes_{\Z} \Mat_4(\Z) \cong  \Mat_4(\Q)$.
  \item $ \HQ = ( \Q[i,j,k] : i^2=j^2=k^2=-1;\; ij=k;\; jk=i;\; ki=j )$
  embeds in $\Mat_4(\Q)$ via the left multiplication representation.
  \item $\HZ = \Z[i,j,k] \subseteq  \HQ$ acts on $G$ by endomorphisms:
  \begin{align*} n\cdot(a,b,c,d) &= (a^n,b^n,c^n,d^n); \\
  i\cdot(a,b,c,d) &= (b^{-1},a,d^{-1},c); \\
  j\cdot(a,b,c,d) &= (c^{-1},d,a,b^{-1}); \\
  k\cdot(a,b,c,d) &= (d^{-1},c^{-1},b,a). \end{align*}
  \item Then 
  \begin{align*} V := &\{ (x,y,z_1,z_2,z_3) \in G^5 \\
  &: z_1=x+y,\; z_2=x+i\cdot y,\; z_3=x+j\cdot y \} \end{align*}
  is a special subgroup of $G^5$.
  \end{itemize}
\end{frame}
  
\begin{frame}
  \frametitle{Example (continued)}
  \begin{itemize}\item $V := \{ (x,y,z_1,z_2,z_3) \in G^5 : z_1=x+y,\; z_2=x+i\cdot y,\; 
  z_3=x+j\cdot y \}$ is a special subgroup of $G^5$.

  \item ``Approximate $\HZ$-submodules'' witness that $V$ admits no powersaving:
  \item $H_N := \{ n + mi + pj + qk : n,m,j,k \in [-N,N] \} \subseteq  \HZ$
  \item $g \in G$ generic
  \item $X_N := H_N \cdot g = \{ h\cdot g : h \in H_N \} \subseteq  \HZ \cdot g \subseteq  G$.
  \item Then (by uniform Mordell-Lang), for $W \subsetneq  G$ proper closed of complexity 
  $\leq \tau$, $|W \cap \HZ g| \leq  O_\tau(1)$.
  \item So $\forall \tau.\; \forall N>>0.\;$ $X_N$ is $\tau$-cgp in $G$.
  \item But $i\cdot X_N = X_N = j\cdot X_N$, so $|X_N^5 \cap V| \geq  
  \Omega(|X_N|^2)$.
  \end{itemize}
\end{frame}
\end{document}


\begin{frame}
  \frametitle{Examples}

  Examples of naturally arising projective geometries:
  % actually these are all minimal 1-based groups
  % (any non-trivial modular U-rank 1 type is non-orthogonal to such)
  \begin{itemize}\item
    Let $n>1$, and let
    $\Lambda \leq  \C^n$ be a generic lattice $\Lambda \cong  \Z^{2n}$.
    Then the complex torus $T = \C^n/\Lambda$ has no infinite complex analytic 
    subsets.

    Pillay: $T$ is a modular strongly minimal set in the theory CCM of compact 
    complex manifolds.

  \item
    Manin kernels in DCF.

  \item
    In ACFA, $\{ \sigma(x) = \eta(x) \}$ for an appropriate endomorphism 
    $\eta$ of an abelian variety.

  \item
    Structure induced from $ACF$ on roots of unity or on the torsion subgroup 
    of a simple abelian variety (Manin-Mumford).
  \end{itemize}
  Any $\omega$-categorical, and more generally any pseudofinite, strongly 
  minimal set is locally modular.
\end{frame}

