\documentclass[8pt]{beamer}
\usepackage[ansinew]{inputenc}
\usepackage{amssymb, color,hyperref}
\usepackage{graphicx}
\usepackage{pgf,tikz}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\usepackage{beamerthemesplit}
\usepackage[french]{babel}
\hypersetup{pdfpagemode=FullScreen,colorlinks=true}

\newtheorem{thm}{Th\'eor\`eme}
\newtheorem{lem}{Lemme}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollaire}
\newtheorem{defi}[lem]{D\'efinition}
\newtheorem{rem}[lem]{Remarque}
\newtheorem{nota}[lem]{Notation}
\newtheorem{ques}[lem]{Question}
\newtheorem{prob}[lem]{Probl\`eme}
\newtheorem{sol}[lem]{Solution}
\newtheorem{conc}[lem]{Conclusion}
\newtheorem{conj}[lem]{Conjecture}
\newtheorem{ex}[lem]{Exemple}
\newtheorem{exs}[lem]{Exemples}
\newtheorem{res}[lem]{R\'esultats}
\newtheorem{fac}[lem]{Fait}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\n#1{{|\hskip-1pt | #1 |\hskip-1pt |}}



\title{Persistance en théorie géométrique des groupes}
\author{Pierre Pansu, Universit\'e Paris-Saclay}
\date{30 avril 2020}


\mode<presentation>{\usetheme{Warsaw}}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\section{Homologie persistante}

\subsection{Théorie de Morse}

\begin{frame}

\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.

\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
\end{tikzpicture}

\vglue1.5cm
\end{frame}

\begin{frame}

\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.

\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{xdxdff}{rgb}{0.49019607843137253,0.49019607843137253,1.}
\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
\draw [color=qqqqff,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--9.53893574834596--7.204725445708604E-4*\x)/-1.215816550433607});
\begin{scriptsize}
\draw [fill=xdxdff] (1.215816550433607,-7.846423629771711) circle (2.5pt);
%\draw [fill=xdxdff] (0.7872618515703377,7.8518388393963505) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.4717738310621363,-3.5909521246411895) grid (10.1542698798651,3.526025988724017);
\clip(-0.4717738310621363,-3.5909521246411895) rectangle (10.1542698798651,3.526025988724017);
%\draw [line width=2.4pt,color=qqqqff] (1.0356416721159134,-3.220276181236752)-- (1.0603534016762093,3.1800617748798747);
\draw [fill=qqqqff] (1.0356416721159134,-3.220276181236752) circle (2.5pt);
\end{tikzpicture}

\vglue1.5cm
\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.

\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
\draw [color=ffqqqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(-3.5297423048062013-0.04048064139028318*\x)/1.6176430886669881});
\begin{scriptsize}
\draw [fill=ffqqqq] (0.38888706107772114,-2.191759558895437) circle (2.5pt);
%\draw [fill=ffqqqq] (1.602119377577962,2.210595983608359) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.4717738310621363,-3.5909521246411895) grid (10.129558150304808,3.526025988724017);
\clip(-0.4717738310621363,-3.5909521246411895) rectangle (10.129558150304808,3.526025988724017);
\draw [line width=2.4pt,color=qqqqff] (1.,-3.195564451676456)-- (1.,-0.7738149547674621);
%\draw [line width=2.4pt,color=ffqqqq] (2.,-0.7738149547674621)-- (2.,0.98072);
\draw [fill=ffqqqq] (2.,-0.7738149547674621) circle (2.5pt);
\end{tikzpicture}

\vglue1.5cm
\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.


\definecolor{qqyqqq}{rgb}{0.,0.5019607843137255,0.}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
\draw [color=qqzzqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--1.2500754380912373--0.04986166149311333*\x)/-1.8858105162620982});
\begin{scriptsize}
\draw [fill=qqzzqq] (1.8858105162620982,-0.7127466265048878) circle (2.5pt);
%\draw [fill=qqyqqq] (0.11123190406771578,0.7110603280540482) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.4717738310621362,-3.5909521246411895) grid (10.1542698798651,3.526025988724017);
\clip(-0.4717738310621362,-3.5909521246411895) rectangle (10.1542698798651,3.526025988724017);
\draw [line width=2.4pt,color=qqqqff] (1.,-3.195564451676456)-- (1.,-0.329);
\draw [line width=2.4pt,color=ffqqqq] (2.,-0.77381)-- (2.,-0.329);
%\draw [line width=2.4pt,color=qqzzqq] (3.,-0.329)-- (3.,0.41235);
\draw [fill=qqzzqq] (3.,-0.329) circle (2.5pt);

\end{tikzpicture}

\vglue1.5cm
\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.


\definecolor{qqyqqq}{rgb}{0.,0.5019607843137255,0.}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
%\draw [color=qqzzqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--1.2500754380912373--0.04986166149311333*\x)/-1.8858105162620982});
\draw [color=qqzzqq] (-3.,0.7110603280540482)-- (3.,0.7110603280540482);
\begin{scriptsize}
%\draw [fill=qqzzqq] (1.8858105162620982,-0.7127466265048878) circle (2.5pt);
\draw [fill=qqyqqq] (0.11123190406771578,0.7110603280540482) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.4717738310621362,-3.5909521246411895) grid (10.1542698798651,3.526025988724017);
\clip(-0.4717738310621362,-3.5909521246411895) rectangle (10.1542698798651,3.526025988724017);
\draw [line width=2.4pt,color=qqqqff] (1.,-3.195564451676456)-- (1.,0.41235);
\draw [line width=2.4pt,color=ffqqqq] (2.,-0.77381)-- (2.,0.41235);
\draw [line width=2.4pt,color=qqzzqq] (3.,-0.329)-- (3.,0.41235);

\end{tikzpicture}

\vglue1.5cm
\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.


\definecolor{qqyqqq}{rgb}{0.,0.5019607843137255,0.}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{qqwuqq}{rgb}{0.,0.39215686274509803,0.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=2.0cm,y=0.2cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.13624334457163684,-8.811460225088029) grid (2.1272497832273203,8.572166996407981);
\draw[->,color=black] (-0.13624334457163684,0.) -- (2.1272497832273203,0.);
\foreach \x in {,0.5,1.,1.5,2.}
\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->,color=black] (0.,-8.811460225088029) -- (0.,8.572166996407981);
\foreach \y in {-5.,5.}
\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
\clip(-0.13624334457163684,-8.811460225088029) rectangle (2.1272497832273203,8.572166996407981);
\draw[line width=1.2pt,color=qqwuqq,smooth,samples=100,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{2.0*sin((3.1415926535*(\x))*180/pi)-4.0*sin((2.0*3.1415926535*(\x))*180/pi)+3.0*sin((3.0*3.1415926535*(\x))*180/pi)});
%\draw [color=qqzzqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--1.2500754380912373--0.04986166149311333*\x)/-1.8858105162620982});
\draw [color=ffqqqq] (-3.,2.210595983608359)-- (3.,2.210595983608359);
\begin{scriptsize}
%\draw [fill=qqzzqq] (1.8858105162620982,-0.7127466265048878) circle (2.5pt);
%\draw [fill=qqyqqq] (0.11123190406771578,0.7110603280540482) circle (2.5pt);
\draw [fill=ffqqqq] (1.602119377577962,2.210595983608359) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{cqcqcq}{rgb}{0.7529411764705882,0.7529411764705882,0.7529411764705882}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
\draw [color=cqcqcq,, xstep=1.0cm,ystep=1.0cm] (-0.4717738310621362,-3.5909521246411895) grid (10.1542698798651,3.526025988724017);
\clip(-0.4717738310621362,-3.5909521246411895) rectangle (10.1542698798651,3.526025988724017);
\draw [line width=2.4pt,color=qqqqff] (1.,-3.195564451676456)-- (1.,1.);
\draw [line width=2.4pt,color=ffqqqq] (2.,-0.77381)-- (2.,1.);
\draw [line width=2.4pt,color=qqzzqq] (3.,-0.329)-- (3.,0.41235);
%\draw [fill=qqzzqq] (3.,-0.329) circle (2.5pt);
\end{tikzpicture}

\vglue1.5cm

\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.


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%\draw [color=qqzzqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--1.2500754380912373--0.04986166149311333*\x)/-1.8858105162620982});
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\draw [line width=2.4pt,color=qqqqff] (1.,-3.195564451676456)-- (1.,3.);
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\draw [fill=qqqqff] (6.,3.) circle (2.5pt);
\draw (3,-2) node[anchor=north west] {$H_0$};
\draw (6,-2) node[anchor=north west] {$H_1$};

\end{tikzpicture}

\vglue1.5cm

\end{frame}

\begin{frame}
\begin{prob}
$f:M\to\R$ fonction de Morse. $H_k(f<s)$ est sensible à une petite perturbation de $f$. Corriger cela ? $im(H_k(f<s)\to H_k(f<t))$ est moins sensible.
\end{prob}
Un \emph{code-barre} est construit : une barre naît à l'instant $t$ avec un élément de $H_k(f<t)$ qui n'appartient pas à $im(H_k(f<s)\to H_k(f<t))$ pour un $s<t$, et survit tant que l'image de cet élément dans $H_k(f<u)$ ne s'annule pas.


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\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
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%\draw [color=qqzzqq,domain=-0.13624334457163684:2.1272497832273203] plot(\x,{(--1.2500754380912373--0.04986166149311333*\x)/-1.8858105162620982});
\draw (-3.,8.2) -- (3.,8.2);
\begin{scriptsize}
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%\draw [fill=qqyqqq] (0.11123190406771578,0.7110603280540482) circle (2.5pt);
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\end{scriptsize}
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\draw [line width=2.4pt,color=qqqqff] (6.,3.)-- (6.,3.5);
\draw [line width=2.4pt,color=ffqqqq] (2.,-0.77381)-- (2.,1.);
\draw [line width=2.4pt,color=qqzzqq] (3.,-0.329)-- (3.,0.41235);
%\draw [fill=qqzzqq] (3.,-0.329) circle (2.5pt);
\draw (3,-2) node[anchor=north west] {$H_0$};
\draw (6,-2) node[anchor=north west] {$H_1$};

\end{tikzpicture}\pause

\begin{thm}
Si $\|f-g\|_\infty$ est petit, les deux codes-barres sont voisins.
\end{thm}

\end{frame}

\subsection{Complexes de Rips et Vietoris}


\begin{frame}

\begin{prob}
On échantillonne une surface. On construit le complexe de Rips ($k$-faces = $k+1$-uplets de diamètre $<r$) ou de Vietoris (nerf du recouvrement par les $r$-boules) $X_r$. $H_k(X_r)$ est sensible aux fluctuations de l'échantillonnage. Corriger cela ? $im(H_k(X_r)\to H_k(X_s))$ est moins sensible. 
\end{prob}\pause

\begin{center}
\includegraphics[width=2in]{Vietoris.jpg}

Echantillonnage d'un triangle $T$. $\forall r$, $H_\cdot(X_r)\not=H_\cdot(T)$. \pause

$im(H_\cdot(X_r)\to H_\cdot(X_s))$ est une meilleure approximation de $H_\cdot(T)$.

En sus, il contient une information métrique.
\end{center}
D'où l'intérêt des codes-barres, un par groupe d'homologie.
\hfill {\tiny Crédit image : Daniel Perez}

\end{frame}

\subsection{Application pratique}

\begin{frame}

\begin{prob}
D'après Chazal-da Silva-Glisse-Oudot 2016 : 60 images à classifier. 
\end{prob}

\begin{center}
\includegraphics[width=2.5in]{CdSGO.jpg}
\end{center}\pause

Leur solution : échantillonner, calculer Rips/Vietoris, calculer les codes-barres. A gauche : les distances entre codes-barres. A droite, une projection de cet espace métrique dans $\R^2$.
\end{frame}

\section{Fonctions de Morse sur les groupes}

\subsection{Distance à l'origine}

\begin{frame}

Dans les graphes de Cayley des présentations de groupes, la distance $g\mapsto |g|$ à $e$ atteint parfois des maxima locaux.

\smallskip

\textbf{Exemple}. $G=\mathbb{Z}$ avec le système générateur $\{2,3\}$. 

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\draw [shift={(3.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(5.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(7.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(9.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw [shift={(11.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
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\draw [shift={(6.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(8.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
\draw [shift={(10.,5.)}] plot[domain=4.068887871591405:5.355890089177974,variable=\t]({1.*5.*cos(\t r)+0.*5.*sin(\t r)},{0.*5.*cos(\t r)+1.*5.*sin(\t r)});
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\draw [shift={(15.,-0.5)}] plot[domain=0.6435011087932844:2.498091544796509,variable=\t]({1.*2.5*cos(\t r)+0.*2.5*sin(\t r)},{0.*2.5*cos(\t r)+1.*2.5*sin(\t r)});
\draw (-2,1.4) node[anchor=north west] {-2};
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\draw (11.98,1.4) node[anchor=north west] {5};
\draw (13.98,1.4) node[anchor=north west] {6};
\begin{scriptsize}
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\draw [fill=ffqqqq] (15.,1.) circle (2.5pt);
\draw [fill=qqffqq] (17.,1.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

Autre vue, montrant la distance à $0$:

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\draw (11.,-4.)-- (13.,-3.);
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\draw (12.,-4.)-- (13.,-3.);
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\draw (14.,-3.)-- (15.,-2.);
\draw (15.,-3.)-- (16.,-2.);
\begin{scriptsize}
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\draw [fill=qqqqff] (10.,-5.) circle (2.5pt);
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\draw [fill=qqqqff] (11.,-4.) circle (2.5pt);
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\draw [fill=qqqqff] (16.,-2.) circle (2.5pt);
\draw [fill=qqqqff] (17.,-2.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

\end{frame}

\begin{frame}

\textbf{Définition}. La \emph{profondeur} d'un élément $g$ de $G$ relativement à un système générateur $S$ est la distance de $g$ au complémentaire de la boule $B(e,|g|)$. 

\medskip

\textbf{Exemple}. L'allumeur de réverbère \emph{(lamplighter group)} a pour présentation $\langle a,t\,|\,a^2,\,[a^{t^n},a],\,\forall n\in\mathbb{Z}\rangle$. 

\smallskip
Pour cette présentation, la profondeur n'est pas bornée.

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\begin{center}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.5cm,y=0.5cm]
%\draw [color=cqcqcq,, xstep=0.25cm,ystep=0.25cm] (1.8211111111111111,3.722222222222222) grid (12.47,6.62);
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\draw (8.994444444444449,6.1133333333333315) node[anchor=north west] {$|g|=14$};
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%\begin{scriptsize}
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\end{center}

L'élément $g_n$ satisfait $|g_n|=6n+1$. $\forall h\in\partial B(g_n,n)$, $|h|\leq |g_n|-n$.\pause

Le code-barre pour $H_0(f<t)$ :

\begin{center}
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\draw [line width=3.6pt] (0.,2.)-- (3.36,1.98);
\draw [line width=3.6pt] (1.,3.)-- (4.6,3.);
\end{tikzpicture}
\end{center}

\end{frame}


\begin{frame}

\textbf{Exemple (T. Riley and A. Warshall 2006)}. Il existe un groupe de type fini avec des profondeurs non bornées pour un système générateur, et des profondeurs bornées pour un autre.\pause

\medskip

\textbf{Définition}. La \emph{profondeur relative} d'un élément $g$ est $prof(g)/|g|$.

\begin{ques}
Existe-t-il des groupes tels que les profondeurs relatives soient non bornées ? 
\end{ques}

C'est un invariant de quasiisométrie.

\vskip1cm

{\tiny Correction après exposé : cette question est stupide, car $prof(g)/|g|\leq 3$ (merci Y. Cornulier). Reste à trouver une variante qui est invariant de quasiisométrie .}

\end{frame}

\subsection{Longueur des lacets}

\begin{frame}

Sur l'espace des lacets du graphe de Cayley, la longueur peut être vue comme une fonction de Morse. La profondeur des minima locaux est mesurée par l'\emph{altitude homotopique}, i.e. la plus petite fonction $ah$ telle que tout lacet $w$ est homotopable à $0$ via des lacets de longueur $\leq ah(|w|)$.

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\textbf{Exemple}. Pour les groupes hyperboliques, $ah$ est linéaire.

\bigskip

\textbf{Exemple}. Pour les groupes de Novikov-Boone (problème des mots insoluble), $ah$ est non récursive.\pause

\begin{ques}
Les codes-barres des fonctions longueurs sur les espaces de lacets des graphes de Cayley, qu'ont ils à dire ?
\end{ques}

\end{frame}

\section{Changement d'échelle}

\subsection{Changer de système générateur}

\begin{frame}

Faire varier $r$ dans le complexe de Rips ou celui de Vietoris, c'est multiplier la distance ambiante par $1/r$ et décider que les boules de rayon $1$ sont contractiles.\pause

\medskip

Dans un groupe, ça revient à remplacer le système générateur $S=B(e,1)$ par $S_r=B(e,r)$.

\medskip

\textbf{Exemple} L'espace métrique $Cay(\mathbb{Z}^2,S_r)$ tend vers $\R^2$ muni de la métrique $\ell^1$ quand $r\to\infty$. 

\vskip-0.5cm

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\end{center}\pause

\vskip-0.5cm

\begin{ques}
Quels invariants du groupe convergent lorsque $r$ tend vers $+\infty$ ?
\end{ques}

\end{frame}

\subsection{Groupes nilpotents}

\begin{frame}

La profondeur ? \pause Probablement oui.

\smallskip

L'altitude homotopique $ah$ ? \pause Probablement oui.

\smallskip

Un cadre pour poser la question : les groupes nilpotents. Pour eux,
$$
\lim_{r\to\infty}Cay(G,S_r)=G_\infty\text{ au sens de Gromov-Hausdorff},
$$
où $G_\infty$ est un groupe de Lie nilpotent autosimilaire ($G_\infty$ admet un groupe à $1$ paramètre d'automorphismes homothétiques pour une métrique géodésique).\pause

\begin{minipage}{4.7cm}
\textbf{Exemple: groupe d'Heisenberg} 

\smallskip

$G=\{\begin{pmatrix}1&x&z\\0&1&y\\0&0&1\end{pmatrix}\,;\,x,y,z\in\mathbb{Z}\}$.

\smallskip

$\delta_t\begin{pmatrix}1&x&z\\0&1&y\\0&0&1\end{pmatrix}=\begin{pmatrix}1&tx&t^2 z\\0&1&ty\\0&0&1\end{pmatrix}$.\\

\smallskip

Dans cet exemple, $G<G_\infty$.
\end{minipage}\hskip.5cm
\begin{minipage}{4.5cm}
\includegraphics[width=2in]{std1.png}
\end{minipage}

\end{frame}

\subsection{Fonction de Dehn à deux variables}

\begin{frame}

\textbf{Définition}. La \emph{fonction de Dehn} de la présentation $G=\langle S\,|\,R\rangle$ est la plus petite fonction $Dehn^G$ tel que tout mot $w$ trivial dans $G$ s'écrit comme un produit d'au plus $Dehn^{S,R}(|w|)$ conjugués d'éléments de $R$.\pause

\smallskip

C'est un invariant de quasiisométrie : $Dehn^{S,R}\sim Dehn^{S',R'}$ mais avec des constantes dépendant des présentations. \pause

\begin{thm}[C. Llosa-Isenrich, G. Pallier et R. Tessera 2019]
Il existe un groupe nilpotent de type fini $G$ tel que $Dehn^G(n)\sim n^3$ et $Dehn^{G_\infty}(n)\sim n^4$.
\end{thm}
Le candidat, un produit central de groupes filiformes, avait été suggéré par Y. Cornulier.\pause

\bigskip

Pour comprendre ce phénomène, il faut étudier la fonction de Dehn à deux variables $(n,r)\mapsto Dehn^{S_r,R_r}(n)$, $S_r=B(e,r)$, $R_r=R\cap B(e,10r)$.

\end{frame}

\end{document}
