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\begin{document}

\begin{frame}[fragile]
\begin{center}
\textbf{Periods and the conjectures of Grothendieck \\and of Kontsevich-Zagier} \\
\bigskip
\bigskip
Pedro A. Castillejo Blasco\\
\bigskip
10.06.2020
\bigskip
\bigskip
\bigskip

\begin{figure}[h]
	\centering
	\includegraphics[height = 100 pt, keepaspectratio = true]{Bridget_Bishop.png}
\end{figure}
\textit{{\scriptsize In the memory of Bridget Bishop (1632 â€“ 10 June 1692), \\ first woman executed for witchcraft during the Salem witch trials 328 years ago}}\\

\end{center}
\end{frame}

\begin{frame}[fragile]
\begin{figure}[h]
	\centering
	\includegraphics[width = \textwidth, keepaspectratio = true]{the-last-judgement.png}
\end{figure}
\end{frame}

\begin{frame}[fragile]

Pope Clement VII commissioned Michelangelo \\
\begin{wrapfigure}{R}{0.27\textwidth}
\includegraphics[width=0.27\textwidth]{pope-clement-vii.png}
\caption{\begin{scriptsize}Clement VII.\end{scriptsize}}
\end{wrapfigure}
to decorate the altar wall of Sistine Chapel\\
 \pause
 \bigskip
Less known: in 1533, Clement VII was told \\
about the Copernican system... \pause \\
And he was actually very happy about it!
\end{frame}

\begin{frame}[fragile]
\textbf{Heliocentric model of Copernicus}\\
\begin{wrapfigure}{R}{0.3\textwidth}
\centering
\includegraphics[width=0.3\textwidth]{Copernicus-1.png}
\caption{\begin{scriptsize}Heliocentric model\end{scriptsize}}
\end{wrapfigure}
\textcolor{white}{.}
\begin{itemize}
\item Planetary orbit is a circle \pause
\item The Sun is at the center of the orbit \pause
\item The speed of the planet in the orbit\\
is constant
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\begin{figure}[h]
	\centering
	\includegraphics[width = \textwidth, keepaspectratio = true]{Copernicus-2.png}
\end{figure}
\end{frame}

\begin{frame}[fragile]
\begin{figure}[h]
	\centering
	\includegraphics[width = \textwidth, keepaspectratio = true]{spanish-inquisition.jpg}
	\caption{\begin{scriptsize}Inquisition asking for Galileo in 1615. Meanwhile, in the Holy Roman Empire...\end{scriptsize}}

\end{figure}

\end{frame}

\begin{frame}[fragile]
Copernicus was actually wrong!! \\ \pause
Kepler (1619) described better the orbits of the solar system...
\begin{itemize}
\item Planetary orbit is \sout{a circle} an ellipse \pause
\item The Sun is at \sout{the center} a focal point of the orbit \pause
\item The speed of the planet in the orbit \sout{is constant} varies, but the \textit{area speed} is constant \pause
\item The orbital \textcolor{blue}{period} of a planet is \textcolor{blue}{algebraically} related to the semi-axis of its orbit
\end{itemize}
\footnotesize{(For the experts: the square of the period is directly proportional to the cube of the semi-axis)}
\end{frame}

\begin{frame}[fragile]
\textbf{Sir Isaac Newton (1642 â€“ 1727): mathematician, scientist... and alchemist!}\\
\begin{wrapfigure}{R}{0.3\textwidth}
\centering
\includegraphics[width=0.3\textwidth]{newton.png}
\caption{\begin{scriptsize}Fortunately he was not born in Salem...\end{scriptsize}}
\end{wrapfigure}
Derived Kepler's laws from his law of \\ universal gravitation. \\ \pause
(Principia, Problem XXIII) ``Find the place of a body moving in a given elliptic trajectory at any assigned time'' \\ \pause
\textbf{Theorem} (Newton, 1687):
The position of a planet in its orbit can't be universally found by means of equations of any number of finite terms and dimensions.\\ \pause
\bigskip

Reformulation in modern language:\\
\textbf{Theorem} (Newton, 1687; reformulation \\ by Arnold, 1987, Newton's proof essentially\\ works):\\
There exists no algebraically integrable convex non-singular algebraic curve.
\end{frame}

%\begin{frame}[fragile]
%\begin{columns}
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%        \visible&lt;1-&gt;{
%            Ellipse: $C: x^2/a^2 + y^2/b^2 = 1$.
%            We compute the area $A:= \int_C dx dy$
%        }
%        \visible&lt;2-&gt;{
%            \[
%                f: [-a, a] \rightarrow \mathbb{R}: \; x \mapsto b \sqrt{1 -
%                x^2/a^2}
%            \]
%        }
%            \[\begin{array}{ll}
%            \visible&lt;3-&gt;{A &amp; =  2 \displaystyle\int_{-a}^a b \sqrt{1 - x^2/a^2} \; dx\\}
%            \visible&lt;4-&gt;{  &amp; = \frac{b}{a} \displaystyle\int_{-a}^a 2 \sqrt{a^2 - x^2} \; dx \\}
%            \visible&lt;5-&gt;{  &amp; =  \pi ab}
%            \end{array}\]
%    \end{column}
%    \begin{column}{0.3\textwidth}
%        \begin{overlayarea}{\textwidth}{.8\textheight}
%            \begin{figure}
%                \centering
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%\end{frame}

\begin{frame}[fragile]
\begin{itemize}
\item \textcolor{blue}{Rational function}: $f = p/q$, with $p,q \in \bbQ[x_1, \ldots, x_n]$. Write $f \in \bbQ(x_1, \ldots, x_n)$ \pause
\item \textcolor{blue}{$\bbQ$-semi-algebraic set}: finite union or intersection of sets of the form
\[ S = \{ x \in \bbR^n | \; r(x) \geq 0, \; r \in \bbQ[x_1, \ldots, x_n]\} \]
\end{itemize}

\begin{definition}[period as a complex number]
\visible&lt;3-&gt;{A \textit{period} is a complex number $z \in \textcolor{blue}{P_C} \subset \bbC$ whose real and imaginary parts can be written as absolutely convergent integrals of the form
\[ \int_S f \; dx_1 \ldots dx_n,\]
where $f\in \bbQ(x_1, \ldots, x_n)$  and $S$ is a $\bbQ$-semi-algebraic set.
}
\visible&lt;3-&gt;{In other words,
\[ z = \int_{S_1} f_1 + \left( \int_{S_2} f_2 \right) i\] 
}
\end{definition}

\end{frame}

\begin{frame}[fragile]
\begin{example}[some periods]
\begin{itemize}
\item&lt;1-&gt; Algebraic numbers are periods (exercise)
\item&lt;2-&gt; $\pi = 3.1415926 \ldots$ is a period:
\[ \pi = \int_{ \{ x^2 + y^2 \leq 1\}} dx dy \pause = \int_{-\infty}^\infty \frac{dx}{x^2+1}\]
\item&lt;3-&gt; Logarithms of rational numbers $q &gt; 1$ are periods:
\[ \log(q) = \int_1^q \frac{dx}{x}\]
\end{itemize}

\end{example}

\visible&lt;4-&gt;{
\begin{block}{Still open questions}
\begin{itemize}
\item Is $1/\pi$ a period?
\item Is $e = 2.71828 \ldots$ a period?
\end{itemize}
\end{block}
}
\end{frame}

\begin{frame}[fragile]
\textbf{Fact}: set of periods $P_C$ is a ring\\ \pause

Linear relations between (representations of) periods (a.k.a. rules of integration):
\begin{itemize}
\item \textcolor{blue}{Bilinearity}: $\int_S (f\textcolor{olive}{+}g) = \int_S f \textcolor{olive}{+} \int_S g$;\\ $\int_{S \textcolor{olive}{\sqcup} T} f = \int_S f \textcolor{olive}{+} \int _T f$ \pause
\item \textcolor{blue}{Change of variable}: for $\textcolor{olive}{\phi}$ algebraic, $\int_S f = \int_{\textcolor{olive}{\phi}^{-1}(S)} (f \circ \textcolor{olive}{\phi}) |Jac(\textcolor{olive}{\phi})|$ \pause
\item \textcolor{blue}{Stokes}: $\int_S \textcolor{olive}{d}\omega = \int_{\textcolor{olive}{\partial} S} \omega$ \pause
\end{itemize}
\begin{block}{Kontsevich-Zagier period conjecture}
The linear relations between representations of periods come all from the above rules.
\end{block}

\end{frame}

\begin{frame}[fragile]
Ayoub constructs periods in a different way:\\
Fix $\sigma: K \hookrightarrow \bbC$
\begin{itemize}
\item $\textcolor{blue}{\calO_{K-alg}(\overline{D}^{\textcolor{olive}{n}})} :=$ $K$-vec. sp. of formal power series on $(z_1, \ldots, z_{\textcolor{olive}{n}})$, convergent on a polydisk $D$ of radius $r &gt;1$, algebraic over $K(z_1, \ldots, z_{\textcolor{olive}{n}})$ \pause
\item $\textcolor{blue}{\calO_{K-alg}(\overline{D}^{\textcolor{olive}{\infty}})} := \bigcup_{\textcolor{olive}{n}} \calO_{K-alg}(\overline{D}^{\textcolor{olive}{n}})$ \pause
\end{itemize}
We integrate elements of $\calO_{K-alg}(\overline{D}^{\infty})$  on the unit hypercube:
\begin{footnotesize}
\[ \textcolor{blue}{\int_\square}:  \calO_{K-alg}(\overline{D}^{\infty}) \rightarrow \bbC,\]
\end{footnotesize}
\noindent which is a linear map, and for $K= \bbQ$ we recover the previous periods!  \pause
If  $f \in \calO_{K-alg}(\overline{D}^n)$,
\begin{footnotesize}
\[ \textcolor{blue}{\int_\square} f := \int_{[0,1]^n} f \; dz_1 \cdots dz_n \in P_C.\] \pause
\end{footnotesize}
\begin{example}
The element $g:= 2z - 1 \in {\calO_{\bbQ-alg}(\overline{D}^{1})}$ maps to zero: $\int_{[0,1]}(2z-1)dz = (z^2-z)|_1 - (z^2-z)|_0 = 0$.\\ \pause
Note that $g = \frac{df}{dz} - f(1) + f(0)$ for $f(z) = z^2$.
\end{example}
\end{frame}

\begin{frame}[fragile]

In general, for any $f\in \calO_{K-alg}(\overline{D}^{\infty})$, the element
\[\frac{\partial f}{\partial z_i} - f|_{z_i = 1} + f|_{z_i = 0} \]
maps to zero. \pause
\begin{block}{Kontsevich-Zagier period conjecture (Ayoub's formulation)}
The kernel of 
\[\int_\square: \calO_{\bbQ-alg}(\overline{D}^{\infty}) \rightarrow \bbC	 \]
is spanned by the elements of the form
\[\frac{\partial f}{\partial z_i} - f|_{z_i = 1} + f|_{z_i = 0}. \]


\end{block}

\end{frame}





\begin{frame}[fragile]{Grothendieck's point of view}


$\textcolor{blue}{K} \hookrightarrow \bbC$, $\textcolor{blue}{X}/K$ smooth variety, $X^{\textcolor{blue}{\an}}$ its analytification.\\
Grothendieck's \textcolor{blue}{period isomorphism}:
\visible&lt;1-&gt;{
\[ H^*_{dR}(X) \otimes_K \bbC \rightarrow H^*_{sing}(X^\an, \bbQ) \otimes_\bbQ \bbC \]}
\visible&lt;2&gt;{
Bhatt-Morrow-Scholze (2018) obtained information of the torsion part (for $X$ proper and smooth, but this is unrelated with today's topic)
}
\end{frame}

\begin{frame}[fragile]{Grothendieck's point of view}
$\textcolor{blue}{K} \hookrightarrow \bbC$, $\textcolor{blue}{X}/K$ smooth variety, $X^{\textcolor{blue}{\an}}$ its analytification.\\
Grothendieck's \textcolor{blue}{period isomorphism}:
\visible&lt;1-&gt;{
\[ H^*_{dR}(X) \otimes_K \bbC \rightarrow H^*_{sing}(X^\an, \bbQ) \otimes_\bbQ \bbC \]
}
\visible&lt;2-&gt;{
We can describe it via the \textcolor{blue}{period pairing}:\\
\[ \int: H^*_{dR}(X) \otimes H_{sing, *}(X^\an, \bbQ) \longrightarrow \bbC \]
}
\visible&lt;2-&gt;{
where
\[[\omega]\otimes [\gamma] \mapsto \int_\gamma \omega \]
}
\visible&lt;3-&gt;{
\noindent \textbf{Definition}:  The \textit{period field} of a smooth variety $X$ is the subfield $Per(X)$ of
$\bbC$ generated by the image of the period pairing:
\[\textcolor{blue}{Per(X)} := K\left(\im\left(\int\right) \right)\]
}
\end{frame}



\begin{frame}[fragile]{Grothendieck's point of view}
Example:  $X = \textcolor{blue}{\bbG_m} = \Spec(\bbQ[T, T^{-1}])$, $X^\an = \textcolor{blue}{\bbC^*}$. \\
\[ H^*_{dR}(X) \otimes_\bbQ \bbC \rightarrow H^*_{sing}(X^\an, \bbQ) \otimes_\bbQ \bbC \] \pause
\begin{itemize}
\item $H_{sing,1}(\bbC^*, \bbQ) = \bbQ$ is generated by the class of the loop
\[ \textcolor{blue}{\gamma}: \; [0,1] \rightarrow \bbC^*: \; \textcolor{olive}{t} \mapsto e^{2\pi i\textcolor{olive}{t}}\] \pause
\item $H^1_{dR}(\bbG_m) = \bbQ$ is generated by $\textcolor{blue}{\omega} = dT/T$ \pause
\end{itemize}
Then, period pairing maps
\[ [\omega]\otimes [\gamma] \mapsto \int_\gamma \omega = \int_0^1 e^{-2 \pi i t}d(e^{2 \pi i t}) = 2 \pi i \int_0^1 dt = 2 \pi i,\]
so we see that $\bbQ(2 \pi i) \subset Per(\bbG_m)$. \pause Some extra work yields an equality. 
\end{frame}

\begin{frame}[fragile]{Grothendieck's point of view}
Note that in the previous example,
\[ tr.deg(Per(\bbG_m)/\bbQ) = 1\] \pause
\begin{block}{Grothendieck's period conjecture}
If $K/\bbQ$ is an algebraic field extension, $X/K$,
\[ tr.deg(Per(X)/\bbQ) = \dim G_\mot(X)\]
\end{block} \pause
What is the motivic Galois group of $X$???

Two approaches: via Nori motives, via Voevodsky motives
\end{frame}

\begin{frame}[fragile]{Motivic Galois group of a variety $X$?}
\begin{block}{Recall}

\textbf{Absolute Galois group} of a field $K \hookrightarrow \bbC$:
\[ \Gal(K^\sep/ K) := \lim_{\substack{\longleftarrow \\ L/ {K} \mbox{ \footnotesize{finite}}}} \textcolor{olive}{\Aut}_K(L) \]
\pause
Grothendieck's \textbf{Ã©tale fundamental group} of a scheme $X$:\\ \pause
Geometric point $\bar{x}: \Spec(\Omega) \rightarrow X$, then we have a \textit{fiber functor}: \pause
\[ \Fib_{\bar{x}}: (Y \rightarrow X) \text{ fin. Ã©tale } \mapsto \Hom_X(\Spec(\Omega), Y).\]
$\pi_1^{\etale}(X, \bar{x}) := \textcolor{olive}{\Aut}(\Fib_{\bar{x}})$ \\
\end{block}
\pause
\textcolor{blue}{Tannakian category} $(\calC, \omega)$ over $\bbQ$, where
\begin{itemize}
\item $\calC$ ($\bbQ$-linear) abelian tensor category (+ some assumptions)
\item $\omega: \calC \rightarrow \fVec_{\bbQ}$ exact faithful tensor functor, called fiber functor
\end{itemize}

\end{frame}


\begin{frame}[fragile]{Motivic Galois group of a variety $X$?}
Tannakian category $(\calC, \omega)$ over $\bbQ$ induces an affine group scheme
\[ G := \textcolor{olive}{\Aut}^\otimes(\omega) \]
called the \textcolor{blue}{Tannakian fundamental group} of $(\calC, \omega)$. \pause
\begin{example} For any affine group scheme $G/\bbQ$, \\
$\Rep_G$: fin. dim. representations of $G$ over $\bbQ$.
\[ (\Rep_G, \omega)\]
$\omega$ is the forgetful functor $\Rep_G \rightarrow \fVec_{\bbQ}$.
\end{example}\pause
For any Tannakian category $(\calC, \omega)$, we have an equivalence of categories:
\[ (\calC, \omega) \longleftrightarrow \Rep_G \]


\end{frame}


\begin{frame}[fragile]{Motivic Galois group of a variety $X$?}
%Grothendieck's theory of motives (`cohomology essence' of a variety):\\
\textbf{Conjecture} (Grothendieck): for $\sigma: K \hookrightarrow \bbC$, there \textit{exists} a category of \textit{mixed motives} $MM(K)$ such that:
\begin{itemize}
\item&lt;2-&gt; different cohomology theories should factor through $MM(K)$:
\begin{footnotesize}
\[\begin{tikzcd}[ampersand replacement=\&amp;]
\Var_K \arrow{r} \&amp; MM(K) \&amp; H^*_{sing}(X^\an, \bbQ) \\
X \arrow{r} \&amp; M(X) \arrow{ru} \arrow{r} \arrow{rd} \arrow{rdd} \&amp; H^*_{dR}(X) \\
\&amp; \&amp; H^*_\ell(X) \\
\&amp; \&amp; \vdots
\end{tikzcd}
\]
\end{footnotesize}
\item&lt;3-&gt; Morphisms come from geometry
\item&lt;4-&gt; Tannakian category, fiber functor should give singular cohomology (a.k.a. Betti realization)
\end{itemize}

\end{frame}


\begin{frame}[fragile]{Motivic Galois group of a variety $X$?}

Study of $MM(K)$ is very open, although we have two candidates:\pause
\begin{itemize}
\item Nori motives 
\begin{itemize}
\item Nice Tannakian structure and cohomology realizations, not clear geometric nature of morphisms \pause
\item Nori's absolute motivic Galois group $\textcolor{blue}{G_{N, \mot}^\abs(K)}$
\end{itemize} \pause
\item Voevodsky motives (actually a derived category)
\begin{itemize}
\item Nice realizations and geometric nature of morphisms, not clear how to get a Tannakian category \pause
\item Ayoub (2014) constructed here an absolute motivic Galois group $\textcolor{blue}{G_{A, \mot}^\abs(K)}$ \pause
\end{itemize}
\end{itemize}
Hope: yield same theory.\\ 
\textbf{Evidence} (Choudhury, Gallauer 2017):  if $K/\bbQ$ algebraic,
\[G_{N, \mot}^\abs(K) \cong G_{A, \mot}^\abs(K)\]

\end{frame}



\begin{frame}[fragile]{Motivic Galois group of a variety $X$!}

For a variety $X/K$, consider
\[ \textcolor{blue}{\langle M(X) \rangle} \subset MM(K) \] 
(smallest full Tannakian subcategory of $MM(K)$ containing $M(X)$ and stable under taking subobjects and quotiens)\\ \pause
$\textcolor{blue}{G_\mot(X)}:=$ Tannakian fundamental group of $\langle M(X) \rangle$
\[ G_{mot}(X) := \Aut^\otimes H_{sing}|_{\langle M(X) \rangle}\]

\end{frame}



\begin{frame}[fragile]
\begin{center}
Kontsevich-Zagier conjecture \\($\int_\square:\calO_{\bbQ-alg}(\overline{D}^{\infty}) \rightarrow \bbC $ has the expected kernel)
\visible&lt;1-&gt;{ \[\Downarrow} \visible&lt;2&gt;{ \Uparrow\]}
Grothendieck period conjecture for $X/\bbQ$
\[ tr.deg(Per(X)/\bbQ) = \dim G_\mot(X)\]
\visible&lt;2&gt;{+ $\calO_{\bbQ-alg}(\overline{D}^{\infty})$ modulo the expected kernel of $\int_\square$ is an integral domain}

\end{center}
\end{frame}


\begin{frame}[fragile]{Geometric version}
Power series of periods:
\[ \sum_{i \geq 0} f_i \cdot \textcolor{olive}{t}^i\]
where $f_i$ are periods in the sense of Ayoub. \pause
\begin{itemize}
\item \textcolor{blue}{Laurent series of periods} $:=\calO(\overline{D}^{n})[[\textcolor{olive}{t}]][t^{-1}]$ \pause
\item $\textcolor{blue}{\calO_{K-alg}^{\textcolor{blue}{\dagger}}(\overline{D}^{\textcolor{olive}{n}})} :=$ Laurent series \textit{algebraic} over $K(z_1, \ldots, z_{\textcolor{olive}{n}})(\textcolor{olive}{t})$ \pause
\item $\textcolor{blue}{\calO_{K-alg}^\dagger(\overline{D}^{\textcolor{olive}{\infty}})} := \bigcup_{\textcolor{olive}{n}&gt;0} \calO_{K-alg}^\dagger(\overline{D}^{\textcolor{olive}{n}})$ \pause
\end{itemize}
We have an integration map $\textcolor{blue}{\int_\square}: \calO_{K-alg}^\dagger(\overline{D}^{{\infty}}) \rightarrow \bbC((t))$:
\[ \sum_{i &gt;&gt; - \infty} f_i \cdot t^i \mapsto \sum_{i &gt;&gt; - \infty} \left(\int_\square f_i \right) \cdot t_i\]
\end{frame}

\begin{frame}[fragile]{Geometric version}
As in the previous case, $\int_\square$ maps elements of the form
\[ \frac{\partial f}{\partial z_i} - f|_{z_i=1} + f|_{z_i=0}\]
to zero (here $f \in \calO_{K-alg}^\dagger(\overline{D}^{{\infty}})$ (call them elements of the \textit{first kind}). \\ \pause Is there anything else in the kernel of
\[\int_\square: \calO_{K-alg}^\dagger(\overline{D}^{{\infty}}) \rightarrow \bbC((t))?\]\pause
Ayoub (Annals '15 + Tohoku - to appear) answers this question:
\end{frame}


\begin{frame}[fragile]
\begin{theorem}[Geometric version of the KZ conjecture]
Assume $K$ contains the number $\pi$ in its algebraic closure. Then, the kernel of 
\[ \int_\square:  \calO_{K-alg}^\dagger(\overline{D}^{{\infty}}) \rightarrow \bbC((t))\]
is generated by
\begin{itemize}
\item Elements of the first kind, \textbf{and}\pause
\item Elements of the form
\[ g \cdot h,\]
where
\begin{itemize}
\item $g \in \calO_{K-alg}(\overline{D}^{{\infty}})$ such that $\int_\square g = 0$ \pause
\item $h \in \calO_{K-alg}^\dagger(\overline{D}^{{\infty}})$ such that $g$ and $h$ don't depend simultaneously on the same variable \\ \pause
(in other words: $\frac{\partial g}{\partial z_i} \neq 0 \Rightarrow \frac{\partial h}{\partial z_i} = 0$)
\end{itemize}
\end{itemize}
\end{theorem}


\end{frame}


\begin{frame}[fragile]
\begin{figure}[h]
	\centering
	\includegraphics[height = \textheight, keepaspectratio = true]{FOTOMONTAJE-PA.png}
\end{figure}
\end{frame}


\end{document}

% Template for commutative diagram
\[
\begin{tikzcd}
â€¢ \ar[r] \ar[d] &amp; â€¢ \ar[d]\\
â€¢ \ar[r] &amp; â€¢
\end{tikzcd}
\]

% Template for functions
\[
\begin{tikzcd}[    ,row sep = 0ex
    ,/tikz/column 1/.append style={anchor=base east}
    ,/tikz/column 2/.append style={anchor=base west}
    ]
â€¢ \arrow{r}  &amp; â€¢ \\
â€¢ \arrow[mapsto]{r} &amp; â€¢
\end{tikzcd}
\]

% Figure

\begin{figure}[h]
	\centering
	\includegraphics[height = 200 pt, keepaspectratio = true]{pic.jpg}
	\caption{This is it.}
	\label{label}
\end{figure}


% Function by pieces
\[   
f(x) := 
     \begin{cases}
       1 &amp;\quad\text{ if } |x|_\infty \leq \sqrt{2},\\
       0 &amp;\quad\text{ else.} \\
     \end{cases}
\]


% Template for sequence of equalities

\[
\begin{array}{ll}
f(x,y) &amp; =   \\
&amp; =
\end{array}
\]

% Template for long sets

\[
  Set = \left\lbrace x \in X \;\middle|\;
  \begin{tabular}{@{}l@{}}
    blahblah blahblah\\
    blahblah blahblah
   \end{tabular}
  \right\rbrace
\]</pre></body></html>