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\title % (optional, use only with long paper titles)
{Projective geometries arising from Elekes-Szabó 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{31-05-2018 \\\
Lyon}
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%{Friday}

!newop ACF
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!newop Aut
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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 => 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

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

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

Frame:
  \frametitle{Pseudofinite dimension}

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

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

  **
    For $X$ internal, set $|X| := \prod_{s-->\U} |X_s|$.
  **
    $|X| (- \R^\U$ if $X$ is pseudofinite, $|X| := \infty$ else.

  ** Fix $\xi (- \R^\U$ with $\xi > \R$.

  Definition[Coarse pseudofinite dimension $\bdl$]:
    For $X$ internal,

    \[ \bdl(X) = \bdl_{\xi}(X) := \st ({ \frac{\log(|X|)}{\log(\xi)} }) (- 
    \R_{>=0} \cup \{-\infty,\infty\} .\]

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

  ** Note that internality is closed under cardinality quantifiers:
    if $R (= K^n \times K^m$ is internal and $\alpha (- \R^\U$,
    then $\{ \y (- K^n : \exists_{>=\alpha} \x. R(\x,\y) \}$ is internal.
  *e)*
  .

Frame:
  \frametitle{$\Lint$ monster}
  ***) $\Lint$: predicate for each internal $X (= K^n$.
  ** $\Km ~> K$ monster model in $\Lint$.
  ** For $\phi (- \Lint$, set $\bdl(\phi) := \bdl(\phi(K))$.
  ** $\bdl$ has a unique extension to $(\Lint)_{\Km}$ such that
    *\[a*
      \tp(\b) &|-> \bdl(\phi(\x,\b)) \\\
      S_{\y}({}) &--> \{-\infty\} \cup \R \cup \{\infty\}
    *a\]*
    is well-defined and continuous for each $\phi(\x,\y) (- \Lint$.
  ** Explicitly, $\bdl(\phi(\x,\a)) := \sup \{ q (- \Q : \Km |= 
  \exists_{>=\xi^q} \x.\; \phi(\x,\a) \}$.
  ** For $\Phi$ a partial type, $\bdl(\Phi) := \inf \{ \bdl(\phi) : \Phi |= 
  \phi \}$.
  ** $\bdl(a/C) := \bdl(\tp(a/C))$.

  Fact:
    For $C (= \Km$ small and $a,b (- \Km^{<\omega}$,
    **(i) $a==_C b ==> \bdl(a/C)=\bdl(b/C)$.
    **(ii) $\bdl(ab/C)=\bdl(a/bC)+\bdl(b/C)$.
    **(iii) A partial type $\Phi$ over $C$ has a realisation
    $a (- \Phi(\Km)$ with $\bdl(a/C)=\bdl(\Phi)$.
    *ee*
    .
  *e)*
  .

Frame:
  \frametitle{$\acl^0$}

  We have $\C <= \C^{\U} <= \Km$.
  
  Definition:
    Superscript 0 means: reduct to $\"ACF_\C$.

    Work in $\Kmeq := \{ \"ACF-\textrm{imaginaries} \}$

    {\small (or, essentially equivalently, $\Kmeq := \Km^{<\omega}$)}.

    ***) $d^0(B) := \trd(B/\C)$
    ** $a (- \acl^0(B)$ iff $d^0(a/B) = \trd(a/\C(B)) = 0$.
    ** $\Cb^0(a/B) := \Cb^{ACF}(a/\C(B))$
    *e)*
    .
  Remark:
    $a (- \acl^0(B) ==> \bdl(a/B) = 0$.
    .
  .

Frame:
  \frametitle{Coherence}
  % 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.

  Definition:
    $P (= \Km$ is \defnstyle{coherent} if
    for any tuple $\a (- P^{<\omega}$,
    \[ \bdl(\a) = d^0(\a) .\]
    .

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

Frame:
  \frametitle{Coherence}
  %More generally:

  Definition:
    $a (- \Kmeq$ is in \defnstyle{coarse general position} (or is 
    \defnstyle{cgp}) if for any $B (= \Km$,
    \[ d^0(a/B) < d^0(a) ==> \bdl(a/B) = 0 .\]
    .

  Any $a (- \Km$ is cgp.

  Definition:
    $P (= \Kmeq$ is \defnstyle{coherent} if
    ***) every $a (- P$ is cgp, and
    ** for any tuple $\a (- P^{<\omega}$,
    \[ d^0(\a)=\bdl(\a) .\]
    *e)*
    .

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

Frame:
  \frametitle{Example}

  Definition:
    Let $W$ be an irreducible variety over $\C$.

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

  If $X$ is cgp, then any $a (- X$ is cgp.

  %``No internal structure''.

  Example:
    Let $G$ be a complex semiabelian variety, e.g. $G=(\C^\times)^n$.

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

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

    Also $\bdl(X^3 \cap \Gamma_+) = 2\bdl(X)$.
    So if $(a,b,c) (- X^3 \cap \Gamma_+$ with $\bdl(abc) = 2\bdl(X)$,
    then $\{a,b,c\}$ is coherent.
    .
  .

Frame:
  \frametitle{Szemerédi-Trotter bounds}

  Suppose $X_1 (= \Km^{n_1}$ and $X_2 (= \Km^{n_2}$ are $//\\$-definable,
  and $V (= \Km^{n_1+n_2}$ is $\Km$-Zariski closed.

  Let $X := (X_1 \times X_2) \cap V$.

  Suppose thazetat for $b,b' \in X_2$ with $b \neq b'$, we have $\bdl(X(b) 
  \cap X(b')) = 0$.

  Remark:
    We have the trivial bound $\bdl(X) <= \frac12\bdl(X_1) + \bdl(X_2)$.
    {\em Proof on board.}
    .

  %Lemma[Elekes-Szabó, Fox-Pach-Sheffer-Suk-Zahl]:
  Lemma[Elekes-Szabó]:
    %With $\epsilon_0 := \frac1{4n_2-1} > 0$ and
    %$y^+:=\max\{0,y\}$,
    %*\[a* \bdl(X) <= \max
    %&( [\frac{1}{2}\bdl(X_1) + \bdl(X_2)] - \epsilon_0 [\bdl(X_2) - 
    %\frac{1}{2}\bdl(X_1)]^{+} \\\
    %&,\bdl(X_1), \bdl(X_2) ) . *a\]*

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

  Hrushovski: such bounds correspond to modularity.
  .

Frame:
  \frametitle{Linearity}
  Lemma:
    Suppose $P (= \Kmeq$ is coherent, $a_1,a_2,b_1,...,b_n (- P$, and:
    ***) $d^0(a_1) = k = d^0(a_2)$
    ** $a_1 \|/^0 a_2$
    ** $a_1 \+/^0_{\b} a_2$.
    *e)*
    Let $e := \Cb^0(\a/\b)$.
    Then $d^0(e) = k$.

    %Moreover, $\{e\}$ is coherent.
    .
  \pause
  Proof:
    $X_1 := \tp(\a)$, $X_2 := \tp(e)$, $V := \loc^0(\a e)$.

    By cgp and canonicity, $\bdl(X(e_1)\cap X(e_2)) = 0$ for $e_1 /= e_2 (- 
    X_2$.

    Meanwhile, $\bdl(X) - \bdl(X_2) >= \bdl(\a/e) >= \bdl(\a/\b) = 
    d^0(\a/\b)= \frac12d^0(\a) = \frac12\bdl(X_1)$.

    So by Szemerédi-Trotter bounds, must have $\bdl(X_2) <= \frac12\bdl(X_1)$.

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

    So $d^0(e) <= \bdl(e) = \bdl(X_2) <= \frac12\bdl(X_1) = k$.
    .
  .

Frame:
  \frametitle{Modularity}
  Recall:
    ***) A \defnstyle{geometry} is a pregeometry with $\cl({})={}$ and 
    $\cl(\{x\})=\{x\}$.

    ** The geometry of a pregeometry $(P;\cl)$ is
    $(\{ \cl(x) : x (- P \};\cl)$.
    *e)*
    .

  Definition:
    ***) A geometry $(P,\cl)$ is \defnstyle{modular} if for $a,b (- P$ and $C 
    (= P$,
    \\\ if $a (- \cl(bC) \\ \cl(C)$
    then there exists $c (- \cl(C)$ such that $a (- \cl(bc)$.

    ** Say $a,b (- P$ are \defnstyle{non-orthogonal} if $a (- \cl(bC)$ for 
    some $C (= P$.
    %A geometry $(P,\cl)$ is \defnstyle{locally modular} if $(P,\cl_a)$ is 
    %modular for any $a (- P$, where $\cl_a(C) = \cl(Ca)$.
    *e)*
    .

  %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.

  Fact[Veblen-Young co-ordinatisation theorem]:
    The modular geometries of dimension $>=4$ in which every two points are 
    non-orthogonal are precisely the projective geometries $\P_F(V)$ of vector 
    spaces of dimension $>=4$ over division rings.
    %(by the coordinatisation theorem of projective geometry)
    .
  .

Frame:
  \frametitle{Canonical base is cgp}
  Lemma:
    Suppose $P$ is coherent, $a_1,a_2,b_1,...,b_n (- P$, and:
    ***) $d^0(a_1) = k = d^0(a_2)$
    ** $a_1 \|/^0 a_2$
    ** $a_1 \+/^0_{\b} a_2$.
    *e)*
    Let $e := \Cb^0(\a/\b)$.
    Then $d^0(e) = k$.

    %If $P$ is coherent, $a_1,a_2,b_1,...,b_n (- P$,
    %$d^0(a_1) = k = d^0(a_2)$
    %and $a_1 \|/^0 a_2$ but $a_1 \+/^0_{\b} a_2$.
    %Let $e := \Cb^0(\a/\b)$.
    %Then $d^0(e) = k$.

    {\bf Moreover, $\{e\}$ is coherent.}
    .
  \pause
  Proof:
    We already saw $\bdl(e) = d^0(e)$; it remains to show that $e$ is cgp.

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

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

    Meanwhile, $\bdl(\a'/e) = \bdl(\a') - \bdl(e) = d^0(\a') - d^0(e) = k$.
    So $\bdl(e/B) = \bdl(\a'/B) - \bdl(\a'/eB) = \bdl(\a'/B) - \bdl(\a'/e) <= 
    k - k = 0$.
    .
  .

Frame:
  \frametitle{Modularity of coherence}

  Definition:
    For $P (= \Kmeq$,
    \[ \ccl(P) := \{ x (- \acl^0(P) : \{x\} \;\textrm{is coherent}\} .\]
    .

  Lemma:
    If $P$ is coherent, so is $\ccl(P)$.
    .

  Proposition:
    Suppose $P = \ccl(P)$ is coherent.

    Then the geometry of $(P;\acl^0)$ is modular.
    .
  .

Frame:
  \frametitle{Projective subgeometries in $\"ACF$}
  Define $\P(\Km) := \{ \acl^0(x) : x (- \Km \}$.
  Example:
    Suppose $G$ is a 1-dimensional complex algebraic group and $F <= \Q (x)_\Z
    \End(G)$ is a division subring.

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

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

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

    Then $\eta$ embeds $\P_F(V)$ as a subgeometry of $\P(\Km)$.
    .
  Theorem[Evans-Hrushovski 1991]:
    Any projective subgeometry of $\P(\Km)$ of dimension at least 3 arises in 
    this way.
    .
  .

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

  Example:
    If $G$ is a complex abelian algebraic group,
    $F <= \Q (x)_\Z \End(G)$ a division ring,
    $A (= G(\Km)$ independent generics,
    and $V := <{ A / G(\C) }>_F$.

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

  Theorem[``Evans-Hrushovski for $\Kmeq$'']:
    Suppose $V$ is a vector space of dimension at lesat 3 over a division ring 
    $F$,
    and $\eta : \P_F(V) --> \P(\Kmeq)$ is a $k$-dimensional full embedding.

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

    %Furthermore, $G$ is unique up to isogeny.
    .
  .


Frame:
  \frametitle{Projective geometries fully embedded in $\"ACF^{\eq}$}

  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 $F (--> \Q (x)_\Z \End(G)$ and $V (--> G(\Km) / G(\C)$.
    .
  .

Frame:
  \frametitle{Elekes-Szabó consequences}
  Definition:
    Say a finite subset $X$ of a variety $W$ is \defnstyle{$\tau$-cgp} if for 
    any proper subvariety $W' (!= W$ of complexity $<=\tau$, we have
    $ |X \cap W'| < |X|^{\frac1\tau}$ .
    .

  Definition:
    If $V (= \prod_i W_i$ 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
    \[ |{ \prod_i X_i \cap V }| <= O(N^{d-\epsilon}) \]
    for $\tau$-cgp $X_i (= W_i$ with $|X_i| <= N$.
    .
  Lemma:
    $V$ admits no powersaving iff exists coherent generic $\a (- V(\Km)$.
    .
  .

Frame:
  \frametitle{Elekes-Szabó consequences}

  Definition:
    $H <= G^n$ is a \defnstyle{special subgroup} if $G$ is a commutative 
    algebraic group and $H = \ker(A)^o$ for some $A (- \"Mat(F \cap \End(G))$ 
    for some division subalgebra $F <= \Q (x)_{\Z} \End(G)$.
    .
  Theorem:
    $V (= \prod_i W_i$ admits no powersaving iff it is in co-ordinatewise 
    algebraic correspondence with a product of special subgroups.
    .
  .

Frame:
  \frametitle{Elekes-Szabó consequences; detailed statement}
  Definition:
    $a (- W(\Km)$ is \defnstyle{dcgp} if $a (- X (= W(\Km)$ for some 
    ${}$-definable cgp $X$.
    .
  Theorem:
    Given $V (= \prod_i W_i$, TFAE
    ***(a) $V$ admits no powersaving.
    **(b) Exists coherent generic $\a (- V(\Km)$ with $a_i$ dcgp in $W_i$.
    **(c) Exists coherent generic $\a (- V(\Km)$.
    **(d) $V$ is in co-ordinatewise algebraic correspondence with a product of 
    special subgroups.
    *ee*
    .
  Proof:
    $(a) <==> (b)$: ultraproducts.

    $(b) ==> (c)$: clear.

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

    $(d) ==> (b)$: see below.
    .
  .

Frame:
  \frametitle{Example}
  ***) $G := (\C^\times)^4$.
  ** $\Q (x)_{\Z} \End(G) ~= \Q (x)_{\Z} \Mat_4(\Z) ~= \Mat_4(\Q)$.
  ** $ \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.
  ** $\HZ = \Z[i,j,k] (= \HQ$ acts on $G$ by endomorphisms:
  *\[a* n\cdot(a,b,c,d) &= (a^n,b^n,c^n,d^n); \\\
  i\cdot(a,b,c,d) &= (b^{-1},a,d^{-1},c); \\\
  j\cdot(a,b,c,d) &= (c^{-1},d,a,b^{-1}); \\\
  k\cdot(a,b,c,d) &= (d^{-1},c^{-1},b,a). *a\]*
  ** Then 
  *\[a* V := &\{ (x,y,z_1,z_2,z_3) (- G^5 \\\
  &: z_1=x+y,\; z_2=x+i\cdot y,\; z_3=x+j\cdot y \} *a\]*
  is a special subgroup of $G^5$.
  *e)*
  .
  
Frame:
  \frametitle{Example (continued)}
  ***) $V := \{ (x,y,z_1,z_2,z_3) (- G^5 : z_1=x+y,\; z_2=x+i\cdot y,\; 
  z_3=x+j\cdot y \}$ is a special subgroup of $G^5$.

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

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

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

  Let ``$X_k := \prod_{s --> \U}(\sum_{i=1}^s \O_{s-k}\gamma_i)$'' with 
  $\gamma_i (- G$ 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.
  .

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, 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.
  .

Frame:
  \frametitle{Application: Intersections of varieties with approximate 
  subgroups}
  Theorem:
    $\Gamma \leq G(\Km)$ a ${}$-$\bigwedge$-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 $G^n$.
    .
  Similarly in higher dimension, with a cgp assumption.

  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
    $|A^n \cap V| < |A|^{\frac{\dim(V)}{\dim(G)} - \eta}$.
    .
  .

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

  Example:
    $G=E$ complex elliptic curve.

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

    Suppose $V (= E^n$ 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 $r (- \N$,
    \[ |V(\C) \cap E[r!]^n| >= |{ E[r!] }|^{d-\epsilon} .\]

    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.
    .
  .

Frame:
  \frametitle{Relaxing general position}

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

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

Frame:
  \frametitle{Thanks}
  .

\end{document}


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)
  ***)
    Let $n>1$, and let
    $\Lambda <= \C^n$ be a generic lattice $\Lambda ~= \Z^{2n}$.
    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.

  **
    Manin kernels in DCF.

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

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

