\def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}}

\def\hexnbr#1{\ifnum#1<10 \number#1\else
 \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else
 \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi}
\font\tenmathx=mathx10
\font\eightmathx=mathx8
\font\sevenmathx=mathxm7
\font\fivemathx=mathxm5
\newfam\mathxfam
  \textfont\mathxfam=\tenmathx
  \scriptfont\mathxfam=\sevenmathx
  \scriptscriptfont\mathxfam=\fivemathx
\def\mathx{\fam\mathxfam\tenmathx}
\def\mathxtype{\hexnbr\mathxfam}

\def\overacute{\mathaccent"0\mathxtype79}
\def\overobtuse{\mathaccent"0\mathxtype7D}

% This file is a solution template for:
% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.
% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice. 
\mode<presentation>
{
% \setbeamertemplate{background canvas}[vertical shading][bottom=red!10,
% top=blue!10]
  \usetheme{Warsaw}
  \usefonttheme[onlysmall]{structurebold}
}
% or whatever

\usepackage{amsmath,amssymb}
\usepackage[latin1]{inputenc}
\usepackage{colortbl}
\usepackage[english]{babel}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.

\title[\RGBColor{1 1 1}{\ 
\kern-190pt Jean-Pierre Demailly (Grenoble), 07/06/2018\kern60pt
Structures géométriques et physique de l'Univers}]
% (optional, use only with long paper titles)
{Structures géométriques et\\ physique de l'Univers}

%% \subtitle{Presentation Subtitle} % (optional)

\author[] % (optional, use only with lots of authors)
{Jean-Pierre Demailly}

\institute[]{Institut Fourier, Universit\'e Grenoble Alpes, France}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[]% (optional)
{7 juin 2018\\
Présentation au Collège Louis Aragon\\
Villefontaine}

%%\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
% out. 

% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

\definecolor{ColClaim}{rgb}{0,0,0.8}
\def\claim#1{{\color{ColClaim}#1}}

\definecolor{Alert}{rgb}{0.8,0,0}
\def\alert#1{{\color{Alert}#1}}

\def\srelbar{\vrule width0.6ex height0.65ex depth-0.55ex}
\def\merto{\mathrel{\srelbar\kern1.3pt\srelbar\kern1.3pt\srelbar
    \kern1.3pt\srelbar\kern-0.78ex\raise0.3ex\hbox{${\scriptscriptstyle>}$}}}

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\tors}{\operatorname{torsion}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\reg}{\operatorname{reg}}
\renewcommand{\div}{\operatorname{div}}

\newcommand{\bB}{{\mathbb B}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bD}{{\mathbb D}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\cA}{{\mathcal A}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cX}{{\mathcal X}}

\newcommand{\dbar}{\overline\partial}
\newcommand{\ddbar}{\partial\overline\partial}
\newcommand{\ovl}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\lra}{\longrightarrow}
\newcommand{\bul}{{\scriptscriptstyle\bullet}}

% mathematical operators
\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
\newcommand{\codim}{\mathop{\rm codim}\nolimits}
\newcommand{\diam}{\mathop{\rm diam}\nolimits}
\newcommand{\Id}{\mathop{\rm Id}\nolimits}
\newcommand{\Sing}{\mathop{\rm Sing}\nolimits}
\newcommand{\Supp}{\mathop{\rm Supp}\nolimits}
\newcommand{\Vol}{\mathop{\rm Vol}\nolimits}
\newcommand{\rank}{\mathop{\rm rank}\nolimits}
\newcommand{\pr}{\mathop{\rm pr}\nolimits}

\newcommand{\NS}{\mathop{\rm NS}\nolimits}
\newcommand{\GG}{{\mathop{\rm GG}\nolimits}}
\newcommand{\NE}{\mathop{\rm NE}\nolimits}
\newcommand{\ME}{\mathop{\rm ME}\nolimits}
\newcommand{\SME}{\mathop{\rm SME}\nolimits}
\newcommand{\alg}{{\rm alg}}
\newcommand{\nef}{{\rm nef}}
\newcommand{\num}{\nu}
\newcommand{\ssm}{\mathop{\Bbb r}}
\newcommand{\smallvee}{{\scriptscriptstyle\vee}}

\def\ovl{\overline}
\def\build#1^#2_#3{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}}
\def\bibitem[#1]#2#3{\medskip{\bf[#1]} #3}

\begin{document}

% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%%\AtBeginSubsection[]
%%{
%% \begin{frame}<beamer>
%%    \frametitle{Outline}
%%    \tableofcontents[currentsection,currentsubsection]
%%  \end{frame}
%%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}

\begin{frame}
  \pgfdeclareimage[height=1cm]{acad-logo}{acad-logo}
  \pgfuseimage{acad-logo}
  \pgfdeclareimage[height=1cm]{uga-logo}{uga-logo}
  \pgfuseimage{uga-logo}
  \titlepage
\end{frame}

%%\begin{frame}
%%  \frametitle{Outline}
%%  \tableofcontents
%% You might wish to add the option [pausesections]
%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%% \section*{Basic concepts}
%%\def\pause{}

\begin{frame}
 \frametitle{Géométrie de l'espace}
 Selon Einstein et sa théorie de la relativité générale (1907--1915), 
 \alert{l'espace est courbé en raison de la distribution de matière,} 
 qui induit un champ gravitationnel
 \vskip8pt
 \hbox{$\strut$\kern-8mm
 \pgfdeclareimage[height=5.4cm]{espace-temps-courbe}{espace-temps-courbe}
 \pgfuseimage{espace-temps-courbe}}
\end{frame}

\begin{frame}
 \frametitle{Découvertes des ondes gravitationnelles (2015)}
 \pgfdeclareimage[height=5cm]{ondes-grav}{ondes-grav}
 \pgfuseimage{ondes-grav} \strut
 \vskip4pt
 Le 14 septembre 2015, on a détecté l'arrivée sur Terre d'ondes
 gravitationnelles, provoquée par la fusion de deux trous noirs
 de 36 et 29 masses solaires, situés à 1,3 milliards d'années-lumière.
 Dans le choc, 3 masses solaires ont été converties en énergie~!
\end{frame}

\begin{frame}
 \frametitle{Comment calcule-t-on la courbure ?}
 Une courbe et son \alert{cercle osculateur} de rayon $r$
 \vskip8pt
 \hbox{\strut\kern1.5cm
 \pgfdeclareimage[height=6cm]{cercle-osculateur}{cercle-osculateur}
 \pgfuseimage{cercle-osculateur}}
 \vskip4pt
 Si le rayon \alert{$r=\infty$}, la courbure $K$ est \alert{nulle}.
\end{frame}

\begin{frame}
 \frametitle{Coefficients de courbure d'une surface}
\strut\kern-5mm Les deux courbures d'une surface dans un espace de dimension~3
 \vskip5pt
 \hbox{\strut\kern1.5cm
 \pgfdeclareimage[height=5.5cm]{surface-courbures}{surface-courbures}
 \pgfuseimage{surface-courbures}}
 $$K_1={1\over r_1}>0,\kern3cm K_2={1\over r_2}<0$$
\end{frame}

\begin{frame}
 \frametitle{La courbure moyenne}
 Courbure moyenne : \alert{$\displaystyle M={1\over 2}(K_1+K_2)$}\\
 Une bulle de savon ``libre'' est de courbure moyenne nulle en tout point~:
 \alert{$K_1=-K_2$,~~~ $M=0$}
 \vskip5pt
 \hbox{%
 \pgfdeclareimage[height=5cm]{bulle-catenoide}{bulle-catenoide}
 \pgfuseimage{bulle-catenoide}\kern1.2cm
 \vbox{Catenoide:\\ \alert{$x= a~\cosh u~\cos\theta$\\
  $y= a~\cosh u~\sin\theta$\\ $z= a u$}}}
\end{frame}

\begin{frame}
 \frametitle{Métrique riemannienne / Tenseur de courbure}
  Bernhard Riemann (1826--1866) / espace à $n$ dimensions
 \vskip5pt
 \hbox{
 \pgfdeclareimage[height=4.5cm]{riemann}{riemann}
 \pgfuseimage{riemann}\kern1cm
 \pgfdeclareimage[height=4.5cm]{levi-civita}{levi-civita}
 \pgfuseimage{levi-civita}}
 \vskip-15pt
\alert{ $$
 ds^2=\sum g_{\alpha\beta}(x)\,dx^{\alpha}dx^{\beta}\kern2cm
 R^{\delta}_{\alpha\beta\gamma},\quad 1\le\alpha,\beta,\gamma,\delta\le n.
 $$}
 \vskip-10pt
\hbox{\strut \raise15pt\hbox{Métrique riemannienne}\kern 1cm 
\vbox{Tenseur de courbure de Riemann\\ 
(calcul précisé par Levi-Civita)}\kern-5mm}
\end{frame}

\begin{frame}
 \frametitle{Tenseur de Ricci / équation d'Einstein}
 Le tenseur de Ricci est une sorte de ``courbure moyenne'' :
\alert{$$R_{\alpha\beta}=\sum_\gamma  R^{\gamma}_{\alpha\beta\gamma}$$}
\hbox{\strut\kern2cm
\pgfdeclareimage[height=4cm]{albert-einstein}{albert-einstein}
 \pgfuseimage{albert-einstein}}
 \vskip2pt
 \'Equation d'Einstein (1879--1955) de la relativité générale
\alert{$$\displaystyle R_{\alpha\beta}-\Big(\Lambda+{1\over 2}R\Big)
g_{\alpha\beta}= {8\pi G\over c^4}T_{\alpha\beta}.$$}
\end{frame}

\begin{frame}
 \frametitle{\'Equation d'Einstein en mathématiques}
Equation d'Einstein ``simplifiée'' (univers vide !!)
\vskip-10pt
\alert{$$\displaystyle R_{\alpha\beta}=\lambda g_{\alpha\beta},\qquad
\lambda=\hbox{constante}.$$}Vérifiée
(avec $\lambda=0$, $R_{\alpha\beta}\equiv 0$) par la\\
\alert{variété de Calabi-Yau} 6-dimensionnelle définie dans
$\bC\bP^4$ par
\vskip7pt
\hbox{\strut\kern3cm
\pgfdeclareimage[height=3.5cm]{calabi-yau}{calabi-yau}
 \pgfuseimage{calabi-yau}}
\vskip-17pt
$$z_0^5+z_1^5+z_2^5+z_3^5+z_4^5-5a\,z_0z_1z_2z_3z_4=0,\quad
z_0,z_1,z_2,z_3,z_4\in\bC,$$
\vskip-10pt
\hbox{(avec 1 paramètre $a$ de déformation).
\alert{Yau: on a bien Ricci${}\equiv 0$.}\kern-5pt}
\end{frame}

\begin{frame}
 \frametitle{\kern-4pt 
Variétés de Calabi-Yau, sièges des champs de forces?\kern-4pt}
\pgfdeclareimage[height=4.5cm]{calabi-yau2}{calabi-yau2}
 \pgfuseimage{calabi-yau2}
\vskip4pt
Notre univers aurait 6 dimensions supplémentaires ultra-miscroscopiques
(${}\simeq 10^{-35}\,$m) qui seraient le siège des
champs de force (théorie des cordes)... \alert{sous forme d'une variété
de Calabi-Yau de dimension complexe 3}.\\
Celle-ci étant de dimension réelle 6, 
ceci amène à un univers de $4+6=\alert{10}$ \alert{dimensions}
au total.
\end{frame}

\end{document}