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\title{Modular forms in a problem of spectral theory}
\author{Fernando Chamizo}
\institute{Universit\'e du Littoral C\^ote d'Opale}
\date{May 30, 2022}


\begin{document}


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\begin{miblock}
{\footnotesize\color{gray!50!black}\textbf{Short abstract.} The Weyl law is a remarkable formula in spectral theory that relates the asymptotic behavior of the frequencies of an oscillating manifold and its volume. A proof of a sharp version of the Weyl law for the Lie group $\textrm{SO}(N)$ is presented using some properties of the classical modular forms and basic facts about some lattice point problems. No prerequisites are assumed about spectral theory or Lie groups.}
\end{miblock}

\vfill

{\scriptsize\textbf{Reference work.}
Chamizo, F.; Granados, J. A short proof of a sharp Weyl law for the special orthogonal group. \emph{J. Spectr. Theory} 10 (2020), no. 1, 311--322.}


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\section{What is Weyl's law?}
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\frametitle{What is Weyl's law?}

\begin{miblock}
\textbf{Physical formulation.}
When a membrane of area $A$ oscillates, the frequencies
$\nu_1,\nu_2,\dots$ of the stationary waves satisfy
\[
 \frac 1N
 \#\big\{ n\,:\, \nu_n< N\big\}
 \sim \pi A
 \qquad\text{(for $N$ large)}.
\]
\end{miblock}

\begin{center}
\begin{tabular}{cccc}
	\includegraphics[height=32.12pt]{./images/mode22_1.png}
	&
	\includegraphics[height=32.12pt]{./images/mode22_2.png}
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	&
	\includegraphics[height=32.12pt]{./images/mode22_4.png}
\end{tabular}
\end{center}

\

\pause
{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
{\footnotesize\textbf{Old problem (M. Kac):} So, we can ``hear'' the area of a drum. A refinement allows to hear the length of its boundary too. Can we hear its whole  shape? Solved in the negative in 1992.}
\end{block}}




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\frametitle{What is Weyl's law?}

\begin{miblock}
\textbf{Mathematical formulation.}
$\Omega\subset\R^2$ smooth (except for corners), $\lambda_n$ eigenvalues for $-\Delta u=\lambda u$ in $\Omega$, $u|_{\partial\Omega}=0$. Then
\[
 \mathcal{N}(\lambda):=
 \#\big\{ n\,:\, \lambda_n< \lambda\big\}
 \sim
 \frac{|\Omega|}{4\pi}\lambda.
\]
It generalizes to dimension $d$ and manifolds as
\[
 \mathcal{N}(\lambda)
 \sim
 C_d\lambda^{d/2}
 \qquad\text{with}\quad
 C_d=
 \frac{2\,\text{Vol}(M)}{d(4\pi)^{d/2}\Gamma(d/2)}.
\]
\end{miblock}


{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=light_blue!15!white}
\begin{block}{}
{\small It also extends to other elliptic operators.}
\end{block}}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!85!mybackg}
\begin{block}{}
{\footnotesize\textbf{Big problem:} Prove good bounds for the error term.}
\end{block}}

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\section{Why is it true?}
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\frametitle{Why is it true?}

{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
{\footnotesize Recall linear algebra: If $A$ is a (positive) self-adjoint on a subspace $V$
\[
 \lambda_n= \sup_{\vec{x}\in V-\{\vec{0}\}}
 \frac{\langle A\vec{x},\vec{x}\rangle}{\|\vec{x}\|^2}
\]
$V_{-}\subset V\subset V_{+}$ $\Rightarrow$ $\lambda_n\downarrow$ in $V_{-}$ and  $\lambda_n\uparrow$ in $V_{+}$.}
\end{block}}

\pause
\begin{miblock}
Assume $\Omega = \bigcup Q_j$, $Q_j= \text{disjoint squares}$ {\tiny (ask Archimedes)}
\end{miblock}

\begin{miblock}
\[
 \uncover<3->{\begin{array}{c}
 V_{-}=
 \big\{
 u|_{\partial Q_j}=0
 \big\}
 \\[5pt]
 \begin{tabular}{c}
	\includegraphics[height=46.501pt]{./images/tiles_d.jpg}
 \end{tabular}
 \end{array}}
 \quad
 \begin{array}{c}
 V=
 \big\{
 u|_{\partial \Omega}=0
 \big\}
 \\[5pt]
 \begin{tabular}{c}
	\includegraphics[height=46.501pt]{./images/domsq.png}
 \end{tabular}
 \end{array}
 \quad
 \uncover<3->{\begin{array}{c}
 V_{+}=
 \big\{
 \frac{\partial u}{\partial n}
 \big|_{\partial Q_j}=0
 \big\}
 \\[5pt]
 \begin{tabular}{c}
	\includegraphics[height=46.501pt]{./images/tiles_n.jpg}
 \end{tabular}
 \end{array}}
\]
\uncover<3->{In $V_{\pm}$ the problem decouples. It is enough to study it on each tile.}
\end{miblock}

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{\setbeamercolor{block title}{bg=red}
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The whole problem reduces to prove Weyl's law for the unit square.
\end{block}}

{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
\vspace{-10pt}
\[
 \begin{array}{c|c|c|}
  & u=0\text{ on $\partial Q$} &\partial u/\partial n=0\text{ on $\partial Q$}
  \\
  \hline
  \rule{0pt}{12pt}\text{eigenfunctions} & \sin(\pi m x)\sin(\pi n y) & \cos(\pi m x)\cos(\pi n y)
  \\
  \hline
  \rule{0pt}{12pt}\text{eigenvalues} & \pi^2(m^2+n^2)\text{\color{blue}\quad\footnotesize $m,n> 0$} & \pi^2(m^2+n^2)\text{\color{blue}\quad\footnotesize $m,n\ge 0$}
  \\
  \hline
 \end{array}
\]
\end{block}}

\vspace{-15pt}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
\[
 \mathcal{N}_Q(\lambda)\leftrightarrow \#\Big\{(m,n)\in(\mathbb{Z}^+)^2\text{ in the circle }x^2+y^2\le \frac{\lambda}{\pi^2}\Big\}
\]
\end{block}}

\begin{miblock}
\[
 \begin{tabular}{c}
  \includegraphics[height=80.331pt]{./images/circsq.png}
 \end{tabular}
 \qquad
 \mathcal{N}_Q(\lambda)\sim \frac{\lambda}{4\pi}.
\]
\end{miblock}

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\section{Our result in words}
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\frametitle{Our result in words}


\vspace{-10pt}

\begin{miblock}
We prove Weyl's law with error term for the manifold corresponding to the (Lie group of) rotations in $\R^N$.
\end{miblock}

\begin{center}
 \begin{tabular}{c}
 \includegraphics[height=42.4241pt]{./images/sph1.png}
 \end{tabular}
 \
 \begin{tabular}{c}
 \includegraphics[height=42.4241pt]{./images/sph2.png}
 \end{tabular}
 \
 \begin{tabular}{c}
 \includegraphics[height=42.4241pt]{./images/sph3.png}
 \end{tabular}
 \
 \begin{tabular}{c}
 \includegraphics[height=42.4241pt]{./images/sph4.png}
 \end{tabular}
 \
 \begin{tabular}{c}
 \includegraphics[height=42.4241pt]{./images/sph5.png}
 \end{tabular}
\end{center}


\pause

\vspace{-10pt}

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Two assets:
\begin{enumerate}
 \item
 Except for $N=4,5,6,7$, the error term is sharp {\scriptsize(read the fine print below)}.
 \item
 Knowing the ingredients, the proof is very short.
\end{enumerate}
\end{block}}

\pause

\vspace{-5pt}

{\setbeamercolor{block title}{bg=red}
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{\footnotesize To say the whole truth, for $N=8,9$  the sharpness is in the exponent, a logarithmic factor could be improved. For $N>9$ no improvement can be made.}
\end{block}}

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\section{Lie groups and eigenvalues}
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\frametitle{Lie groups and eigenvalues}

\begin{miblock}
Essentially a Lie group is a manifold that is a group too i.e., we have a (differentiable) way of multiplying points.
\end{miblock}

\vspace{-10pt}

{\setbeamercolor{block title}{bg=red}
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\begin{center}
\begin{tabular}{c}
\multicolumn{1}{l}{$S^1$}
\\[-10.001pt]
\includegraphics[height=52.3152pt]{./images/S1.png}
\\
{\scriptsize$p\mapsto z=e^{i\theta}$}
\end{tabular}
\qquad
\begin{tabular}{c}
\multicolumn{1}{l}{\hspace{-10pt}$S^2$}
\\[-10.001pt]
\includegraphics[height=52.3152pt]{./images/S2.png}
\\
{\scriptsize (hairy ball theorem)}
\end{tabular}
\quad\
\begin{tabular}{c}
\multicolumn{1}{l}{\hspace{10pt}$S^3$}
\\[-10.001pt]
\includegraphics[height=52.3152pt]{./images/S3.png}
\\
{\scriptsize$p\mapsto\text{quaternion } \|\mathbf{q}\|=1$}
\end{tabular}
\end{center}
\end{block}}



{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
{\footnotesize The typical examples of compact Lie groups are the \emph{classical groups}: $\text{SO}(N)$ and allied groups.
\[
 \text{SO}(N)
 \ \leftrightarrow\
 \text{orthogonal $N\times N$ matrices}
 \ \leftrightarrow\
 \text{rotations in $\R^N$.}
\]
In fact, Lie groups are locally matrix groups.}
\end{block}}

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\begin{frame}
\frametitle{Lie groups and eigenvalues}

\begin{miblock}
\textbf{Good property of the Lie groups.}
Analytic and geometric problems are translated into linear algebra problems.
\[
 \text{Lie group}
 \ \longleftrightarrow\
 \text{Lie algebra}.
\]
\end{miblock}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize \textbf{Example:} Lie group $SO(N)$ $\to$ Lie algebra $\mathfrak{so}(N)=\,$ antisymmetric matrices.

Straight line in $\mathfrak{so}(N)$ $\to$ geodesic $\gamma(t)=e^{tA}$ in $SO(N)$.}
\end{block}}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize \textbf{Much more involved example:} Eigenvalues of $-\Delta$ for $SO(N)$ $\to$ properties related to the \emph{root system}, eigenvalues of
$X\mapsto AX-XA$ in $\mathfrak{so}(N)$.

\

\underbar{Output}: Explicit (but quite complicate) formulas for the eigenvalues of $-\Delta$ and their multiplicities for  the classical groups.}
\end{block}}



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\section{Modular forms come on the scene}
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\frametitle{Modular forms come on the scene}

\begin{miblock}
 \textbf{Recall:} Modular form $f:\mathbb{H}\to\C$,
 $f(z)=\sum_{k=0}^\infty a_k e^{2\pi i kz}$ such that
 \[
  f\big(\gamma(z)\big)
  =
  C_\gamma
  (cz+d)^r f(z)
  \qquad\text{for}\quad
  \gamma(z)=\frac{az+b}{cz+d}
 \]
 with $\gamma\in \Gamma$, where $\Gamma=\text{SL}_2(\Z)$ or $[\text{SL}_2(\Z):\Gamma]<\infty$.
\end{miblock}

\vspace{-8pt}

\pause

{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
\begin{tabular}{p{0.75\textwidth}}
{\footnotesize \underbar{Names}: $r$ is called the \emph{weight} and $C_\gamma$ the \emph{multiplier system}.}
\\[8pt]
{\footnotesize \underbar{Geometric interpretation for $r=2$ and $C_\gamma=1$}: $f(z)\, dz$ is a differential form on the Riemann surface $\Gamma\backslash\mathbb{H}$.}
\\[8pt]
{\footnotesize \underbar{Cusp forms}: They are the modular forms holomorphic at $\infty$.}
\end{tabular}
\
\begin{tabular}{c}
  \includegraphics[height=66.712pt]{./images/fdom.png}
 \end{tabular}
\end{block}}


{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
{\small
  If $f$ is modular, $|\Im z|^{r/2}|f(z)|$ is invariant under $\Gamma$  and  it is uniformly bounded if $f$ is a cusp form.}
\end{block}}


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\frametitle{Modular forms come on the scene}

{\setbeamercolor{block title}{bg=red}
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\begin{block}{}
{\small\textbf{Example.} $\theta(z)=\sum_{k=-\infty}^\infty e^{2\pi ik^2z}$ is a modular form of weight $1/2$,
\[
 \theta
 \Big(
 \frac{az+b}{cz+d}
 \Big)
 =
  C_\gamma
 (cz+d)^{1/2}\theta(z)
  \qquad\text{for}\quad
  4\mid c.
\]
$C_\gamma\leftrightarrow$\,Legendre symbol, (Poisson summation formula).

Raising to $d$ $\to$
$ \theta^d(z)
 =
 \sum_{\vec{k}\in\Z^d} e^{2\pi i(k_1^2+\dots +k_d^2)z}$
modular of weight $d/2$.}
\end{block}}

\pause
\begin{miblock}
 \textbf{Generalization.} For $P=P(x_1,\dots,x_d)$ homogeneous harmonic polynomial
 ($\sum \partial_j^2P=0$)
 \[
 \Theta(z)
 =
 \sum_{\vec{k}\in\Z^d}
 P(\vec{k})
 e^{2\pi i(k_1^2+\dots +k_d^2)z}
 \]
 is modular of weight $\frac12 d+\deg P$ and cusp form if $P\ne \text{constant}$.
\end{miblock}


\end{frame}



\section{Summing up}
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\begin{frame}
\frametitle{Summing up}

\vspace{-12pt}

\begin{miblock}
\begin{enumerate}\setlength{\itemsep}{9pt}
 \item
 Weyl's law is the asymptotic formula for the counting function of the eigenvalues and it involves the volume.
 \item
 Our result is Weyl's law for $\text{SO}(N)$ with optimal error term.
 \item
 Lie groups are manifolds with a group structure. For $\text{SO}(N)$ and the rest of the classical compact Lie groups it is possible to obtain an explicit formula for the eigenvalues.
 \item
 Modular forms are holomorphic functions on the upper half plane given by Fourier series and satisfying a certain symmetry condition under a group of linear fractional transformations.
 \item
 If we raise the Jacobi theta function to a power and we put a homogeneous harmonic polynomial as coefficient, we get a modular form.
\end{enumerate}
\end{miblock}



\end{frame}



\section{Our result in formulas}
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\begin{frame}
\frametitle{Our result in formulas}

\begin{miblock}
\textbf{Theorem.}
For the manifold $M=\text{SO}(N)$
\[
 \mathcal{N}(\lambda)
 =
 C_d\lambda^{d/2}
 +
 \begin{cases}
  O\big(\lambda^{d/2-1}\big) &\text{if } n>4
  \uncover<2->{\color{dark_green}\quad\text{\footnotesize (sharp)}}
  \\
  O\big(\lambda^{d/2-1}\log\lambda\big) &\text{if } n=4
  \uncover<2->{\color{dark_green}\quad\text{\footnotesize (sharp up to log)}}
 \end{cases}
\]
where $C_d=\text{constant in Weyl's law}$, $d=\text{dimension}$, $n=\text{\emph{rank} as Lie group}$:
\[
 d =\frac{N(N-1)}{2}
 \qquad\text{and}\qquad
 n =
 \begin{cases}
  N/2&\text{if } 2\mid N
  \\
  (N-1)/2&\text{if } 2\nmid N
 \end{cases}
\]
\end{miblock}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize \textbf{Remark.} The case $n=1$ is trivial and $n=2,3$ are related to some old conjectures in number theory.}
\end{block}}


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\section{Weyl's law as a lattice point problem}
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\begin{frame}
\frametitle{Weyl's law as a lattice point problem}

\begin{miblock}
 Formula for the eigenvalues and their multiplicities ($N$ even):
 \[
    \lambda_{\vec{b}}=
    \sum_{j=1}^n x_j^2
    -
    K_n
    \qquad\text{and}\qquad
    m
    =
    \prod_{i=0}^{n-1}
    \prod_{j=i+1}^{n-1}
    \frac{(x_{n-j}^2-x_{n-i}^2)^2}{(j^2-i^2)^2}
 \]
 where $n=\frac 12 N$, $K_n=\frac 16n(n-1)(2n-1)$, $\vec{b}\in\Z^n$,
 \[
    b_1\ge b_2\ge\dots\ge |b_n|
    \qquad\text{and}\qquad
    x_j=b_j+n-j.
 \]
\end{miblock}

{\setbeamercolor{block title}{bg=red}
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{\footnotesize The formula is slightly more involved for $N$ odd.}
\end{block}}


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\begin{frame}
\frametitle{Weyl's law as a lattice point problem}

Symmetry in $b_j$ + elementary argument $\Rightarrow$

\begin{miblock}
 \textbf{Lemma} {\small(relation to lattice point theory)}
 For $N$ even ($n=N/2$)
 \[
  \mathcal{N}(\lambda)
  =
    \frac{1}{2^{n-1}n!}
    \sum_{\vec{x}\in \Z^n}
    m(\vec{x})
    \chi_R(\vec{x})
 \]
 with $\chi_R=$\,charac. function of the ball of radius $R=\sqrt{\lambda+K_n}$.
 \medskip

 And something similar when $N$ is odd.
\end{miblock}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize Regularization+Poisson summation (as in other LPP) $\to$ not so good idea}

{\footnotesize $\widehat{\chi}_R$ has large ``wavefronts'' when $n$  grows (Morris {\&} Taheri 2017)}
\end{block}}


\end{frame}





\section{The proof}
%%%%%%%%%
% SLIDE %
%%%%%%%%%
\begin{frame}
\frametitle{The proof}


\begin{miblock}
\textbf{Target:}
Estimate
$\displaystyle{S=\sum_{\vec{x}\in \Z^n}
m(\vec{x})
\chi_R(\vec{x})}$,
\qquad$m=\text{hom. poly.}$
\end{miblock}

\vspace{-10pt}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize\textbf{Fact from analysis:} 
$Q= \text{hom. poly.}$, $\deg Q=g$ {\footnotesize(even)}
\[
 Q =
 \|\vec{x}\|^{2g}P_0
 +
 \|\vec{x}\|^{2g-2}P_2
 +
 \dots
 +
 \|\vec{x}\|^{2}P_{g-2}
 +
 P_{g}.
\]

\vspace{-5pt}
with $P_j=\text{har. poly.}$, $\deg P_j=j$.}
\end{block}}


\begin{miblock}
\[
	S=
	C
	\sum_{\substack{\vec{x}\in \Z^n\\ \|\vec{x}\|\le R}}
	\|\vec{x}\|^{2g}
	+
	O\Big(
	\sum_{\substack{\vec{x}\in \Z^n\\ \|\vec{x}\|\le R}}
	\|\vec{x}\|^{2g-2\ell}
	P_{2\ell}(\vec{x})
	\Big)
	=C\mathcal{M}+O(\mathcal{E}).
\]
\end{miblock}

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\textbf{Main:} 
$\mathcal{M}=\sum_{k\le R^2}k^{g}r_n(k)$, \hfill $r_n= \#\text{\small repr. as sum of $n$ squares}$}

For $n>4$ {\footnotesize ($n= 4$)}, $r_n(k)$ very regular $\to$ sharp asymptotic formula.
\end{block}}


\end{frame}






%%%%%%%%%
% SLIDE %
%%%%%%%%%
\begin{frame}
\frametitle{The proof}



{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\textbf{Error:} 
$\mathcal{E}=\sum_{k\le R^2}k^{g-\ell}a_k$, \hfill $a_k= \text{\small $k$-th Fourier coeff. of $\Theta$}$}
\[
 \begin{array}{l}
  \Theta(z)
  =
  \sum_{\vec{k}\in\Z^d}
  P(\vec{k})
  e^{2\pi i(k_1^2+\dots +k_d^2)z}
  =\sum_{k=1}^\infty a_k e^{2\pi i k z}
  \\[7.23pt]
  \Theta\text{ cusp form }
  \Rightarrow |\Im z|^{r/2}|\Theta(z)|\text{ unif. bounded}
  \\[7.23pt]
  |\Theta(x+iy)|\le C y^{-r/2}\text{ as }y\to 0^+
  \qquad\text{\color{dark_blue}\scriptsize ($r=d/2+\deg P$)}
 \end{array}
 \quad
 \begin{tabular}{c}
  \includegraphics[height=62.712pt]{./images/fdom.png}
 \end{tabular}
\]
\end{block}}


{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
\[
 \sum_{k=1}^K
 a_k
 =
 \int_0^1
 \Theta(x+iy)
 \sum_{k=1}^K
 e^{-2\pi ik(x+iy)}
 dx
 \qquad
 \text{under control}
\]
\end{block}}


{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
Choosing $y=1/K$, the error term $\mathcal{E}$ is proved to be negligible. \qed
\end{block}}

\end{frame}


\section{The low-dimensional cases}
%%%%%%%%%
% SLIDE %
%%%%%%%%%
\begin{frame}
\frametitle{The low-dimensional cases}

\begin{miblock}
\[
 \begin{array}{|c|c|c|c|c|}
  {\color{blue}n}&
  {\color{blue}\text{group}}&
  {\color{blue}\text{dim.}}&
  {\color{blue}\text{problem}}&
  {\color{blue}\text{bound for $\mathcal{E}$}}
  \\
  \hline
  \rule{0pt}{13.017pt}1& 
  \text{SO}(2),\ \text{SO}(3) &
  1,\ 3&
  \text{trivial}&
  \lambda^{(d-1)/2} \text{ \color{dark_green!70!black}\footnotesize (sharp)}
  \\
  \hline
  \rule{0pt}{13.017pt}2& 
  \text{SO}(4),\ \text{SO}(5) &
  6,\ 10&
  \text{circle problem}&
  \lambda^{(d/2-1)+27/82}
  \\
  \hline
  \rule{0pt}{13.017pt}3& 
  \text{SO}(6),\ \text{SO}(7) &
  15,\ 21&
  \text{sphere problem}&
  \lambda^{(d/2-1)+1/4}\log \lambda
  \\
  \hline
 \end{array}
\]
\end{miblock}

\

{\setbeamercolor{block title}{bg=red}
\setbeamercolor{block body}{fg=black,bg=white!55!mybackg}
\begin{block}{}
{\footnotesize\textbf{Trivial case $N=2$:} $SO(2)\cong S^1$  (an angle $\alpha$ determines a plane rotation).}

{\footnotesize Eigenfunctions $\to $ $e^{ik\alpha}$,\quad, $\Delta=-\frac{d^2}{d\alpha^2}$,\quad eigenvalues $\to $  $k^2$
\[
 \mathcal{N}(\lambda)
 =
 \#\{k\in\Z\,:\, k^2\le \lambda\}
 =
 2\lambda^{1/2}+O(1).
\]}
\end{block}}


\end{frame}





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\begin{frame}
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\begin{tikzpicture}
\invisible{\draw[step=1cm,help lines] (-.35,0) grid (10.35,-8);}
\node[gray,font=\tiny] at (3.4,-7) {All the images are made by myself or public domain or CC from the Wikipedia};
\node at (5,-1.8) {These slides will be available in};
\draw[dark_blue] (-.35,-2.5) -- (10.35,-2.5);
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\node at (5,-4.5) {\bf Thank you for your attention!};
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\end{document}
