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\title{A history of the Zilber--Pink conjecture}
\author{Jiewei Xiong}
\institute{University of Reading}
\date{12th October 2026}

\begin{document}

\frame{
\titlepage
}

\frame{
\centering
\begin{minipage}{.35\textwidth} \includegraphics[width=\textwidth]{BZc.jpg} \footnotesize Boris \textcolor{Silver}{Zilber} (1949--) \end{minipage} \begin{minipage}{.35\textwidth} \includegraphics[width=\textwidth]{RPc.jpg} \footnotesize Richard \textcolor{Pink}{Pink} (1959--) \end{minipage}
}

\frame{
\frametitle{1922: starting point}
\begin{minipage}{.2\textwidth}
\includegraphics[width=\textwidth]{LJM.jpg} \tiny Louis Joel Mordell (1888--1972) \\ \includegraphics[width=\textwidth]{GFc.jpg} Gerd Faltings (1954--)
\end{minipage} \ \begin{minipage}{.78\textwidth}
\begin{itemize}
\item An elliptic curve is a smooth projective algebraic curve of genus \(1\).
\item Mordell (1922) proved the Mordell--Weil theorem for elliptic curves: the group of rational points \(E(\Q)\) of an elliptic curve \(E/\Q\) is a finitely generated abelian group.
\item Recall that a height function on the points is the key ingredient of the proof. 
\item In the same paper Mordell conjectured that curves\(/\Q\) of genus \(>1\) have only finitely many rational points, now a theorem due to Faltings (1983), later reproved by Bombieri (1990) using a relatively elementary theory of heights after Vojta.
\end{itemize}
\end{minipage}
}

\frame{
\frametitle{1938: Chabauty's conjecture}
\begin{minipage}{.75\textwidth}
In his paper \textit{Sur les \'equations diophantiennes li\'ees aux unit\'es d'un corps de nombre alg\'ebriques fini} (1938), Chabauty considered the following problem.

Let \(K\) be a number field of degree \(n\) and \(\omega_i\) its integral basis. He wanted to solve for \(x_i\in\Z\) the equation
\[ \tag{$\dagger$}
\operatorname{Norm}(x_1\omega_1+\cdots+x_n\omega_n)=\pm 1,
\]
i.e. coefficients of the units of \(\cO_K\). The question is:
\end{minipage} \ \begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{CC.jpg} \tiny ``On peut conjecturer que ces conditions suffisantes sont aussi n\'ecessaires...'' \end{minipage}
\begin{quote}
After imposing more simultaneous equations \(F_i(x_1,\ldots,x_n)=0\), when are there infinitely many solutions?
\end{quote}
 In other words, denoting \(V=V(F_1,F_2\ldots,F_m)\subset\A^n\) and by \(\Gamma\) the solutions to \((\dagger)\), when is \(V\cap\Gamma\) infinite? Motivated by Thue's theorem, Chabauty conjectured the answer to be \alert{if and only if \(V\), except maybe a finite number of points, is a union of sublattices of \(\Gamma\) and their cosets}.
}

\frame{
\frametitle{1960: Lang's formulation of Chabauty's conjecture}
\begin{minipage}{.75\textwidth}
Let \(G/\C\) be a torus (some \(\G_m^n\)) or an abelian variety, \(V\) a subvariety of \(G\), and \(\Gamma\) a finitely generated subgroup of \(G(\C)\). If \(V(\C)\cap\Gamma\) has infinitely many points, then \(V\) contains a finite number of translations of subgroup varieties of \(G\), containing all but finitely many points of \(V(\C)\cap\Gamma\).
\end{minipage} \ \begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{SLc.jpg} \tiny Serge Lang (1927--2005) \end{minipage}
\hspace{1ex}


Chabauty himself proved a related result (1941): if \(C\) is a curve of genus \(g\ge 2\) over \(\Q\) and \(\rk J(C)(\Q)<g\), then \(C(\Q_p)\cap\overline{J(\Q)}\) is finite, hence \(C(\Q)\) is too.

On the other hand, the full Chabauty conjecture implies Mordell...
}

\frame{
\frametitle{Chabauty implies Mordell}
\begin{enumerate}
\item Take a curve \(C\) of genus \(g\ge 2\), fix an embedding \(i:C\to J(C)\), and take \(G=J(C)\), \(V=i(C)\), \(\Gamma=J(C)(\Q)\). Suppose per contra that \(C(\Q)\subset i(C)(\C)\cap J(C)(\Q)\) has infinitely many points. 
\item Then by Chabauty, \(i(C)\) contains a finite number of translations of subgroup varieties of \(J(C)\), but \(\dim i(C)=1\) so \(i(C)\) itself must be one; 
\item however, a \(1\)-dimensional subgroup variety of an abelian variety is an elliptic curve, which has genus \(1<g\), contradiction.
\end{enumerate}
}

\frame{
\frametitle{1965: Manin--Mumford}
In the 60's, Manin was looking at Picard--Fuchs equations and the problem of the intersection of a curve with the torsion of its jacobian naturally arises. Mumford told this to Lang and the latter formulated a conjecture again. 

\begin{thm}[Manin--Mumford conjecture]
Let \(G/\C\) be an abelian variety, \(V\) a closed irreducible subvariety of \(G\), and \(G_\tor\) the torsion points of \(G(\C)\). If \(V\cap G_\tor\) is Zariski dense in \(V\), then \(V\) is a translation of an abelian subvariety by a point in \(G_\tor\).
\end{thm}

Eventually proved by Raynaud in 1983.

 In the same paper (1965), Lang reproduced a proof of Ihara--Serre--Tate of an elementary special case of Chabauty (if an irreducible plane curve \(V\) has infinitely many points whose coordinates are pairs of roots of unity, then \(V\) is defined by \(x^ry^s=\zeta\) for \(r,s\in\Z\) and \(\zeta\) a root of unity).
}

\frame{
\frametitle{Around the same time: Lang conjectures again}
The Chabauty case Lang proved correspond to \(G=\G_m^2\) and
\[
\Gamma=\left\{(z_1,z_2)\in\G_m^2(\C)=(\C^\times)^2:\exists n,m\in\Z_{>0}:z_1^n=z_2^m=1\right\}.
\]
In 1974, Liardet proved the same thing, replacing \(\Gamma\) by a more general
\[
\Gamma=\left\{(z_1,z_2)\in\C^2:\exists n,m\in\Z_{>0}:z_1^n,z_2^m\in\Gamma_0\right\}
\]
where \(\Gamma_0\) is any finitely generated multiplicative subgroup of \(\C\).  Inspired by this, Lang proposed another conjecture.

\begin{thm}[Mordell--Lang conjecture, 1991]
Let \(G/\C\) be a semiabelian variety, \(V\) the same, \(\Gamma_0\) a finitely generated subgroup of \(G(\C)\), and \(\Gamma\coloneq\{g\in G(\C):\exists n\in\Z_{>0}:ng\in\Gamma_0\}\) (the \(\Gamma\) above written additively), the ``torsion hull'' of \(\Gamma_0\). If \(V\cap\Gamma\) is Zariski dense in \(V\), then \(V\) is a translation of a semiabelian subvariety by a point in \(\Gamma\).
\end{thm}
}

\frame{
\frametitle{Mordell--Lang conjecture}
\begin{itemize}
\item Manin--Mumford is recovered by taking \(G\) to be an abelian variety and \(\Gamma_0\) the trivial group. 
\item Liardet's theorem is recovered by taking \(G\) to be a torus and \(V\) a curve. Chabauty is recovered by considering the irreducible components of the Zariski closure of \(V(\C)\cap\Gamma_0\). 
\item The statement for higher dimensions, that is, the full Mordell--Lang for tori, is a result due to Laurent (1984). 
\item If we again take \(\Gamma_0\) to be the trivial group, but \(G\) any commutative group variety, the statement still holds true: this is a theorem of Hindry (1988). 
\item All of the above, together with work of Faltings and Vojta, led to the proof by McQuillan of the full Mordell--Lang in 1995.
\end{itemize}
}

\frame{
\frametitle{Things get modular}
\begin{minipage}{.55\textwidth}
Consider the family of elliptic curves
\[
E_j:y^2=4x^3-\frac{27j}{j-1}x-\frac{27j}{j-1}
\]
parameterised by \(j\in\Qb\setminus\{0,1\}\). We want to know for which \(j\) does \(E_j\) have CM. Recall that this means \(\End(E_j)\) is an order \(\cO\) in an imaginary quadratic field \(\Q(\sqrt{-d})\). 

There are infinitely many such \(j\). But if one bounds \([\Q(j):\Q]\) the answer is finite. Indeed, \([\Q(j):\Q]\) is the class number of \(\cO\) (Kronecker 1880s), and there are only finitely many discriminants with given class number (Heilbronn 1934).
\end{minipage}\quad \begin{minipage}{.41\textwidth} \includegraphics[height=.6\textwidth]{CFG.jpg} \includegraphics[height=.6\textwidth]{LK.jpg} \\ \tiny \textbf{Gauss} (1798): ``... series determinantium, quibus eadem classificatio data [...] respondet, semper abrumpi videtur [...] Demonstrationes autem \textit{rigorosae} harum obseruationum perdifficiles esse videntur.'' \\ \textbf{Kronecker} (1882): ``es war ein ganz neues, \"uberraschendes und interessantes Phänomen, als mir bei der Besch\"aftigung mit der complexen Multiplication der elliptischen Functionen (im Winter 1856) Gattungen algebraischer Zahlen vor die Augen traten, welche in der angegebenen Weise mit den Gattungen von Quadratwurzeln negativer ganzer Zahlen associirt sind. Eine solche der Gattung \(\sqrt{-n}\) associirte Gattung \(\Gamma\) liefert [...] die s\"ammtlichen, nach der \textit{Kummer}schen Bezeichnung, idealen Divisoren der Gattung \(\sqrt{-n}\); ihre Ordnung ist gleich der Classenanzahl f\"ur die Gattung \(\sqrt{-n}\), und es haben \"uberhaupt alle tieferen, auf die Composition und Classeneintheilung bez\"uglichen Eigenschaften der Gattung \(\sqrt{-n}\) in den elementaren Eigenschaften der associirten Gattung \(\Gamma\), so zu sagen, ihr Abbild.'' \end{minipage}
}

\frame{
\frametitle{1989: Andr\'e and his G-functions}
Using G-functions, Andr\'e was able to produce a higher dim analogue:
\begin{minipage}{.75\textwidth}
\begin{thm}[1989]
Let \(f:A\to C\) be a family of abelian varieties over a curve\(/K\), with some nice properties. In particular \(A_{\overline\eta}\) is a simple abelian variety of odd dim \(g>1\). Then there are only finitely many \(s\) with \([K(s):K]\le d\) bounded such that there is no ring embedding \(\End A_s\hookrightarrow M_g(\Q)\). In fact, the height \(h(s)\) is bounded effectively by a power of \(d+1\).
\end{thm}
\end{minipage} \ \begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{YAc.jpg} \tiny Yves Andr\'e (1959--)  \end{minipage}

\begin{itemize}
\item Bombieri's Hasse principle (1981): A set of algebraic numbers where there is a nontrivial relation between some G-functions has bounded height.
\item Choosing basis \(\omega_i\) for \(H_\dR^1(A/C)\) and \(\gamma_j\) for \(R^1f_\ast\C\), we get a matrix of periods \(\Omega_{ij}=\int_{\gamma_j}\omega_i\). In coordinate of \(C\), their Taylor expansions are G-functions.
\item Construct nontrivial relations.
\end{itemize}
}

\frame{
\frametitle{Andr\'e--Oort conjecture}
Shimura varieties are a natural inventory for base varieties that parameterise abelian varieties. For example,
\begin{itemize}
\item \(Y(1)(\C)=\SL_2(\Z)\backslash\bH\) is a moduli space of elliptic curves, and
\item \(\cA_g(\C)=\Sp_{2g}(\Z)\backslash\cH_g\) is a moduli space of principally polarised abelian varieties of dimension \(g\).
\end{itemize}

The general notion for a point on a Shimura variety \(S\) whose fibre has CM is a \alert{special point}. In Shimura theory, they are equivalent to \alert{special subvarieties} of dim \(0\), and every special subvariety contains a dense subset of special points. 

But the theorem above tells us that if a subvariety \(V\) has infinitely special points, the arithmetic of its points gets arbitrarily complex, and should have a geometric explanation. More precisely, the converse of the density of special points should be true.
\begin{conj}[Andr\'e--Oort, 1st ver, 1989]
Let \(S\) be a Shimura variety and \(C\) a curve in \(S\). If \(C\) contains infinitely many special points, then \(C\) is special.
\end{conj}
}

\frame{
\frametitle{What about Oort?}
\begin{minipage}{.75\textwidth}
Coleman conjectured in 1987 that there are only finitely many complete smooth curves \(C/\C\) of fixed genus \(g\ge 4\) such that \(J(C)\) has CM.
\vspace{1ex}

Oort found that the following conjecture can reduce Coleman to modular statement, now called the Coleman--Oort conjecture.
\begin{thm}[Andr\'e--Oort, 2nd ver, 1994]
If \(V\subset\cA_g\) contains a set of special points Zariski dense in \(V\), then \(V\) is special.
\end{thm}
\end{minipage} \ \begin{minipage}{.22\textwidth} \includegraphics[width=\textwidth]{RFCc.jpg} \tiny Robert Frederick Coleman (1954--2014) \end{minipage}

\hspace{1ex}

So the natural generalisation of both is
\begin{conj}[Andr\'e--Oort, final ver]
Let \(S\) be a Shimura variety and \(V\) an irreducible subvariety of \(S\). If \(V\) contains a set of special points Zariski dense in \(V\), then \(V\) is special.
\end{conj}
}

\frame{
\frametitle{Partial results of Andr\'e--Oort}
\begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{SJE.jpg} \tiny Bas Edixhoven (1962--2022) \\ \includegraphics[width=\textwidth]{DMc.jpg} \tiny David William Masser (1948--) \end{minipage} \ \begin{minipage}{.75\textwidth}
\begin{itemize}
\item Edixhoven did \(S=Y(1)^2\) assuming GRH for imaginary quadratic fields (1996), then Andr\'e unconditionally (1998) using Masser's work on transcendence (1975). (Edixhoven's method owes to Hindry.)
\item His method was extended, eventually leading to a proof of the full Andr\'e--Oort by Klingler--Ullmo--Yafaev (2014) under GRH for CM fields. 
\item Earlier, Pila introduced Pila--Zannier and pushed Andr\'e's result to all \(Y(1)^n\) (2011).
\item Pila then did \(\cA_2\) with Tsimerman (2013) and the latter pushed to all \(\cA_g\) (2018).
\item There is now apparently a full unconditional proof by Pila--Shankar--Tsimerman (announced 2021, latest draft 2024).
\end{itemize}
\end{minipage}
}

\frame{
\frametitle{Turn of the millennium: unlikely intersections}
\begin{minipage}{.22\textwidth} \includegraphics[width=\textwidth]{PT.jpg} \tiny Paula Tretkoff (1957--) \\ \includegraphics[width=\textwidth]{UZ.jpg} Umberto Zannier (1957--) \end{minipage} \ \begin{minipage}{.76\textwidth}
Unlikely intersections, where \textit{unlikely} means dimensionally unexpected, were first investigated (1999) and named (2008) by Bombieri--Masser--Zannier.

The set
\[
\{\tau\in\Qb:\tau\text{ and }1-\tau\text{ are multiplicatively dependent}\}
\]
should be sparse when restricting degrees. Tretkoff (n\'ee Cohen) and Zannier (1998) showed an explicit height bound \(\log 2\). BMZ's general theorem is
\begin{thm}[1999a]
Let \(C/\Qb\) be an irreducible curve in \(\G_m^n\) for \(n\ge 2\), not contained in any translate of a proper algebraic subgroup of \(\G_m^n\). Then
\[
C(\Qb)\cap\bigcup_{H\subsetneq\G_m^n\text{ algebraic subgroup}}H(\Qb)
\]
has bounded height (effective).
\end{thm}
\end{minipage}
}

\frame{
\frametitle{But if one bounds the dimension...}
\begin{minipage}{.75\textwidth}
BMZ noticed the following result (original context is reducibility of lacunary polynomials).
\begin{thm}[Schinzel 1989]
Let \(V\subset\A^n\) be a curve or a point. When \(n\ge 3\), there is a constant \(c(V)\) such that if \(\exists\zeta\in k^\times:(\zeta^{k_1},\ldots,\zeta^{k_n})\in V\) for some \(k_i\in\Z\), then either \(\zeta^q=1\) or \(\exists M\in M_{2\times n}(\Z)\) with \(\rk M=2\) and \(v\in\Z^2\) such that \(h(M)\le c(V)\) and \((k_i)=vM\).
\end{thm}
\end{minipage} \ \begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{AS.jpg} \tiny Andrzej Bobola Maria Schinzel (1937--2021) \end{minipage}
Inspired by this, BMZ wrote and proved a ``striking'' result.
\begin{thm}[1999b]
Let \(C\) be a curve in \(\G_m^n\) as in (a). Then
\[
C(\Qb)\cap\bigcup_{\substack{H\subsetneq\G_m^n\text{ algebraic subgroup} \\ \dim H\le n-2}}H(\C)
\]
is finite.
\end{thm}
}

\frame{
\frametitle{Around the same time: the conception of Zilber--Pink, and things get modular again}
\begin{itemize}
\item Already in 1989, Andr\'e was thinking about intersection of a subvariety of a Shimura variety with special subvarieties, but the absence of dimensional concerns led to a faulty conjecture.
\item The year before, Pink finished his PhD at Bonn. His thesis develops the general theory of \textit{mixed Shimura varieties}. 
\item In 2001 at the Franco--Taiwan Arithmetic Festival, in an attempt to unify Andr\'e--Oort and Manin--Mumford, Andr\'e first formulated the mixed Andr\'e--Oort conjecture, which is however not enough to recover Mordell--Lang. 
\item In 2003, Pink proposed a tentative conjecture that tries to combine Mordell--Lang and Andr\'e--Oort but ended up only implying a special case of latter (Hecke orbits, Edixhoven--Yafaev).
\item Finally in 2005, having learned about BMZ's work, Pink formulated Zilber--Pink for mixed Shimura varieties.
\end{itemize}
}

\frame{
\frametitle{2005: Zilber--Pink conjecture for mixed Shimura varieties}
\begin{conj}
Let \(S\) be a mixed Shimura variety and \(V\) an irreducible subvariety of \(S\). If
\[
V\cap\bigcup_{\substack{Z\subset S\text{ special} \\ \dim Z+\dim V<\dim S}}Z
\]
is Zariski dense in \(V\), then \(V\) is contained in a proper special subvariety of \(S\).
\end{conj}
 
A mixed Shimura variety is roughly the universal family lying above a base Shimura variety; for example,
\begin{itemize}
\item the Legendre surface \(L:y^2=x(x-1)(x-\lambda)\) over \(Y(2)\);
\item the universal abelian \(\fA_g\to\cA_g\).
\end{itemize}
}

\frame{
\frametitle{Zilber--Pink implies Manin--Mumford}
\begin{tikzpicture}[scale=.9]
\draw (-5,-1) node {base SV \(\cA_g\)} .. controls (-2,0) and (2,0) .. (5,-1);

\draw (-4,-.7) -- (-5.5,3) node[above] {mixed SV \(\fA_g\)};
\draw (-2,-.35) -- (-3,3);
\draw[red] (-1,-.3) node {CM} -- (-2,3);
\draw (0,-.25) -- (0,3.5) node[right] {\(A\supsetneq V\)};
\draw (2,-.35) -- (3,3);
\draw[red] (3,-.5) node {CM} -- (5,3); 
\draw (4,-.7) -- (5.5,3);

\fill[blue] (-4.6652,.941145) circle(2pt) node[above] {torsion};
\fill[blue] (-2.47,1.2247) circle(2pt) node[below] {torsion};
\fill[purple] (-1.5,1.35) circle(2pt) node[above] {torsion};
\fill[blue] (0,1.54375) circle(2pt) node[below] {torsion};
\fill[blue] (2.668,1.888) circle(2pt) node[above] {torsion};
\fill[purple] (4.5,2.125) circle(2pt) node[below] {torsion};
\fill[blue] (5.18,2.212953) circle(2pt) node[above] {torsion};

\draw[red] (-5,.8979) node[below] {torsion section \(T\)} -- (6,2.31875);
\end{tikzpicture}

\begin{itemize}
\item \(\dim \cA_g=\frac{g(g+1)}{2}=\dim T\)
\item \(\dim \fA_g=\dim\cA_g+g=\frac{g(g+3)}{2}\)
\item \(\dim V<\dim A=g\), so \(\dim V+\dim T<\dim\fA_g\), an unlikely intersection
\end{itemize}
}

\frame{
\frametitle{Zilber--Pink implies Andr\'e--Oort}
\begin{enumerate}
\item Let \(S\) be a MSV and \(V\subsetneq S\) irreducible. Denote by \(\cS\) the set of special point in \(S\).
\item In particular, \(\dim V<\dim S\), so
\[
V\cap\cS=V\cap\bigcup_{\substack{Z\subset S\text{ special} \\ \dim Z=0}}Z\subset V\cap\bigcup_{\substack{Z\subset S\text{ special} \\ \dim Z+\dim V<\dim S}}Z
\]
is Zariski dense in \(V\) by assumption.
\item Hence by Zilber--Pink, write \(V\subset T\) where \(T\subsetneq S\) is special. 
\item But then, considering \(T\) as a Shimura variety and denoting by \(\cS_T\) the special points of \(T\), the intersection \(V\cap\cS_T=V\cap\cS\) is again Zariski dense in \(V\).
\item This iterative shrinking of the ambient Shimura variety must halt at when \(V=T\) is special, since at each step the dimension drops strictly.
\end{enumerate}
}

\frame{
\frametitle{Partial results of Zilber--Pink}
\begin{itemize}
\item The family \(Y(1)^n\): \(n=1\) is trivial, \(n=2\) is Andr\'e--Oort, \(n=3\) and curves \(C\) is the first interesting scenario.
\begin{itemize}
\item Habegger--Pila (2012): \(C/\Qb\) is \textit{asymmetric}, that is, the restriction of coordinates \(x_1,\ldots,x_n\) to \(C\) have distinct degrees, up to at most one exception (ignoring constant ones) (Pila--Zannier) 
\item Daw--Orr (2025): \(\overline C^{\text{Zar}}\) in the compactification \((\bP^1)^n\) of \(Y(1)^n\) passes through the point \((\infty,\infty,\ldots,\infty)\), or \(C\) is a line (G-functions)
\item Daw--Orr--Papas (2026): \(n=3\), \(\overline C^{\text{Zar}}\backslash C\) contains a modular point
\end{itemize}
\item The family \(\cA_g\): again, \(n=2\) and curves is the first case not covered by Andr\'e-Oort.
\begin{itemize}
\item Daw--Orr (2021, '22, '23): \(\overline C^{\text{Zar}}\) intersects \(\cA_0\) in the compactification \(\cA_2\sqcup\cA_1\sqcup\cA_0\) of \(\cA_2\) 
\end{itemize}
\item Geometric Zilber--Pink: there is a finite set \(\Sigma\) of special subvarieties of \(S\) such that \(\langle W\rangle_{\text{ws}}\) for any weakly optimal \(W\subset V\) is a fibre of some \(Y\in\Sigma\)
\begin{itemize}
\item Habegger--Pila (2016) for \(Y(1)^n\)
\item Daw--Ren (2018) for all Shimura varieties
\item Gao (2020) for mixed Shimura varieties of Kuga type
\end{itemize}
\end{itemize}
}

\frame{
\frametitle{2016: Klingler's Hodge theoretic generalisation}
In 2016 at the summer school on motives and complex multiplication in Ascona, Klingler proposed by far the most general version of the Zilber--Pink conjecture, for \textit{variations of Hodge structures}.

The prototype is still a family of varieties \(X\to S\). The Hodge filtrations of \(H^n(X_s,\C)\) vary, giving rise to a map \(\phi:S^\an\to\Gamma\backslash D\), which is holomorphic, horizontal, and locally liftable. In general we look at any \(S\) with such a map.

\begin{itemize}
\item A special subvariety of \(S\) looks like \(\phi^{-1}(\Gamma'\backslash D')^\circ\).
\item A special \(Z\) is \textit{atypical} if \(\dim \phi(Z)>\dim\phi(S)+\dim\Gamma'\backslash D'-\dim\Gamma\backslash D\).
\end{itemize}

\begin{conj}[Zilber--Pink for variations of mixed Hodge structures]
The union of atypical special subvarieties (the \textit{atypical locus}) of \(S\) is not Zariski dense in \(S\), or equivalently, the atypical locus is a finite union.
\end{conj}
}

\frame{
\frametitle{Background}
\begin{minipage}{.75\textwidth}
\begin{conj}[Hodge 1950]
Let \(X/\C\) be smooth projective. Then every class in \(H^{2c}(X,\Q)\cap H^{c,c}(X)\) is a rational combination of classes of \(X\) coming from subvarieties of codimension \(c\).
\end{conj}
Only known case is \(c=1\).

In the situation of a family \(X\to S\), one can ask the relationship between
\[
\{s\in S:X_s\text{ gains extra subvarieties}\}
\]
and
\[
\{s\in S:X_s\text{ has more Hodge classes than expected}\}.
\]
Hodge says they are the same. Theory says the latter is the union of special subvarieties (the Hodge locus).
\end{minipage} \ \begin{minipage}{.23\textwidth} \includegraphics[width=\textwidth]{AW.jpg} \tiny \textbf{Weil} (1977): ``One may now ask whether imposing a certain Hodge class upon a generic member of an algebraic family of polarised algebraic varieties amounts to an algebraic condition upon the parameters. [...] It seems that a positive answer to that question would follow [...] from the Hodge conjecture (if true), but even from a weaker version of it [...]'' \end{minipage}
}

\frame{
\hspace*{\fill} \includegraphics[height=.25\textwidth]{EC.jpg} \includegraphics[height=.25\textwidth]{PDc.jpg} \includegraphics[height=.25\textwidth]{AK.jpg} \hspace*{\fill} \\
\hspace*{\fill} \tiny Eduardo Cattani (1946--), Pierre Deligne (1944--), Aroldo Kaplan (1942--2026) \hspace*{\fill}

\normalsize 
\begin{itemize}
\item Cattani--Deligne--Kaplan (1995), landmark: the Hodge locus is a countable union of subvarieties.
\item Brosnan--Pearlstein--Schnell (2010): pushed to admissible mixed.
\item Zilber--Pink says the atypical part is moreover finite, hence algebraic. The typical part is either empty or dense in \(S\), and there's a good criterion for which one happens (Baldi--Klingler--Ullmo 2023).
\end{itemize}
}

\end{document}