\documentclass[8pt]{beamer}
\usepackage[ansinew]{inputenc}
\usepackage{amssymb, color}
\usepackage{graphicx}
\usepackage{beamerthemesplit}
\usepackage{pgf,tikz}
\usepackage{mathrsfs}
\usepackage{multicol}
\usetikzlibrary{arrows}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{pro}[lem]{Proposition}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{dfi}[lem]{Definition}
\newtheorem{rem}[lem]{Remark}
\newtheorem{ntn}[lem]{Notation}
\newtheorem{que}[lem]{Question}
\newtheorem{pbm}[lem]{Problem}
\newtheorem{sol}[lem]{Solution}
\newtheorem{con}[lem]{Conclusion}
\newtheorem{coj}[lem]{Conjecture}
\newtheorem{exa}[lem]{Example}
\newtheorem{exs}[lem]{Examples}
\newtheorem{res}[lem]{Results}
\newtheorem{fac}[lem]{Fact}
\newtheorem{pla}[lem]{Plan of lecture}
\newtheorem{sch}[lem]{Scheme of proof}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\def\C{\mathbb{C}}
\def\E{\mathbb{E}}
\def\P{\mathbb{P}}
\def\F{\mathbb{F}_2}
\def\k{\mathbf{k}}
\def\maprl#1#2{\stackrel{#1,#2}{\rightleftharpoons}}
\def\n#1{{|\hskip-1pt | #1 |\hskip-1pt |}}



\title{Computing homology robustly: The geometry of normed chain complexes}
\author{Pierre Pansu, Universit\'e Paris-Saclay}
\date{October 11th, 2023}


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

\begin{document}

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

\section{Normed chain complexes}

\begin{frame}

\begin{multicols}{2}

A simplicial complex is made of simplices of various dimensions. \emph{Simplicial chains} are linear combinations of simplices. The \emph{boundary} of a simplex is a chain, whence a linear map $\partial$ and its adjoint $d$, which satisfy $\partial\circ\partial=0$ and $d\circ d=0$. 


\begin{center}

\definecolor{zzttqq}{rgb}{0.6,0.2,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4cm,y=0.4cm]
\clip(-2.799791737748933,-5.803887506765698) rectangle (8.054341237019841,2.1887013201094807);
\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (3.317992302575285,0.6099183419612999) -- (4.403405600052162,-1.3964516927686834) -- (5.598266370781756,0.5467288138167195) -- cycle;
\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (0.7195786510397301,-0.6399515157393454) -- (-0.3329433343923935,1.0703967105878536) -- (-1.2878873549933432,-0.6962881800015994) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (6.738688755229687,-1.297777756634422) -- (5.598266370781756,0.5467288138167195) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (5.598266370781756,0.5467288138167195) -- (6.146645138424117,-1.8898213734399907) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (5.598266370781756,0.5467288138167195) -- (6.146645138424117,-1.8898213734399907) -- (6.738688755229687,-1.297777756634422) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (6.738688755229687,-1.297777756634422) -- (6.146645138424117,-1.8898213734399907) -- cycle;
\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (1.0484917714872686,-4.586908961109804) -- (3.6140141109780695,-5.014496017691604) -- (2.701554194561922,-2.578894969425186) -- cycle;
\draw [color=ffqqqq] (3.317992302575285,0.6099183419612999)-- (0.7195786510397301,-0.6399515157393454);
\draw [color=zzttqq] (3.317992302575285,0.6099183419612999)-- (4.403405600052162,-1.3964516927686834);
\draw [color=zzttqq] (4.403405600052162,-1.3964516927686834)-- (5.598266370781756,0.5467288138167195);
\draw [color=zzttqq] (5.598266370781756,0.5467288138167195)-- (3.317992302575285,0.6099183419612999);
\draw [color=zzttqq] (0.7195786510397301,-0.6399515157393454)-- (-0.3329433343923935,1.0703967105878536);
\draw [color=zzttqq] (-0.3329433343923935,1.0703967105878536)-- (-1.2878873549933432,-0.6962881800015994);
\draw [color=zzttqq] (-1.2878873549933432,-0.6962881800015994)-- (0.7195786510397301,-0.6399515157393454);
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (4.403405600052162,-1.3964516927686834);
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (4.403405600052162,-1.3964516927686834);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (4.403405600052162,-1.3964516927686834);
\draw [color=ffqqqq] (-1.2878873549933432,-0.6962881800015994)-- (-0.9907695752874707,-2.9094520468273593);
\draw [color=ffqqqq] (-0.9907695752874707,-2.9094520468273593)-- (1.0484917714872686,-4.586908961109804);
\draw [color=zzttqq] (1.0484917714872686,-4.586908961109804)-- (3.6140141109780695,-5.014496017691604);
\draw [color=zzttqq] (3.6140141109780695,-5.014496017691604)-- (2.701554194561922,-2.578894969425186);
\draw [color=zzttqq] (2.701554194561922,-2.578894969425186)-- (1.0484917714872686,-4.586908961109804);
\draw [color=ffqqqq] (2.701554194561922,-2.578894969425186)-- (4.403405600052162,-1.3964516927686834);
\draw [color=ffqqqq] (6.146645138424117,-1.8898213734399907)-- (3.6140141109780695,-5.014496017691604);
\begin{scriptsize}
\draw [fill=qqqqff] (3.317992302575285,0.6099183419612999) circle (2.5pt);
\draw [fill=qqqqff] (0.7195786510397301,-0.6399515157393454) circle (2.5pt);
\draw [fill=qqqqff] (4.403405600052162,-1.3964516927686834) circle (2.5pt);
\draw [fill=qqqqff] (5.598266370781756,0.5467288138167195) circle (2.5pt);
\draw [fill=qqqqff] (-0.3329433343923935,1.0703967105878536) circle (2.5pt);
\draw [fill=qqqqff] (-1.2878873549933432,-0.6962881800015994) circle (2.5pt);
\draw [fill=qqqqff] (6.738688755229687,-1.297777756634422) circle (2.5pt);
\draw [fill=qqqqff] (6.146645138424117,-1.8898213734399907) circle (2.5pt);
\draw [fill=qqqqff] (-0.9907695752874707,-2.9094520468273593) circle (2.5pt);
\draw [fill=qqqqff] (1.0484917714872686,-4.586908961109804) circle (2.5pt);
\draw [fill=qqqqff] (3.6140141109780695,-5.014496017691604) circle (2.5pt);
\draw [fill=qqqqff] (2.701554194561922,-2.578894969425186) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

\end{center}\pause

\end{multicols}

The homology (resp. cohomology) of the simplicial complex is $\mathrm{Ker}(\partial)/\mathrm{Im}(\partial)$ (resp. $\mathrm{Ker}(d)/\mathrm{Im}(d)$.\pause

\medskip

Simplicial chains and cochains can be equipped with $\ell^p$ norms.\pause

\medskip

In general, a \emph{normed chain complex} is a normed vector space $B$ equipped with a linear map $d:B\to B$ such that $d\circ d=0$.

\end{frame}

\begin{frame}

When $F:B_1\to B_2$ is a linear bijection, the robustness of the resolution of the equation
$$
Fx=y
$$
is governed by the \emph{conditioning number}
$$
\kappa(F)=|F||F^{-1}|.
$$\pause

For normed chain complexes, we first turn $d$ into a bijection $\bar d:B/\mathrm{Ker}(d)\to\mathrm{Im}(d)$, and set
$$
\kappa(B):=|\bar d||\bar d^{-1}|.
$$


\end{frame}

\subsection{Example}

\begin{frame}


\textbf{Example}. The $n$-stick satisfies $H^1=0$. The $1$-cochain $g$ equal to $\bar 1$ on the central edge and $0$ elsewhere can be written $df$ where
\begin{align*}
\|g\|_p=1,\quad \|f\|_p\sim n^{1/p}.
\end{align*}

\begin{center}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.3cm,y=0.3cm]
\clip(-2.1578568610390616,3.1283691165789764) rectangle (33.81224865175817,9.076181839167461);
\draw (0.,6.)-- (14.,6.);
\draw [color=ffqqqq](0.39120573435601774,7.498190708684802) node[anchor=north west] {$0$};
\draw [color=ffqqqq](2.4547325972948912,7.457729397646785) node[anchor=north west] {$0$};
\draw [color=ffqqqq](4.3564142160816965,7.417268086608768) node[anchor=north west] {$0$};
\draw [color=ffqqqq](6.41994107902057,7.417268086608768) node[anchor=north west] {$1$};
\draw [color=ffqqqq](8.483467941959445,7.417268086608768) node[anchor=north west] {$0$};
\draw [color=ffqqqq](10.425610871784265,7.376806775570751) node[anchor=north west] {$0$};
\draw [color=ffqqqq](12.408215112647104,7.376806775570751) node[anchor=north west] {$0$};
\draw [color=qqzzqq] (18.,7.)-- (20.,7.);
\draw [color=qqzzqq] (20.,7.)-- (22.,7.);
\draw [color=qqzzqq] (22.,7.)-- (24.,7.);
\draw [color=qqzzqq] (24.,7.)-- (26.,8.);
\draw [color=qqzzqq] (26.,8.)-- (28.,8.);
\draw [color=qqzzqq] (28.,8.)-- (30.,8.);
\draw [color=qqzzqq] (30.,8.)-- (32.,8.);
\draw [color=ffqqqq] (6.,8.)-- (8.,8.);
\draw [color=ffqqqq](6.703170256286691,5.596509089898007) node[anchor=north west] {$g$};
\draw [color=qqzzqq](24.667992357166295,5.596509089898007) node[anchor=north west] {$f$};
\begin{scriptsize}
\draw [fill=black] (0.,6.) circle (2.5pt);
\draw [fill=black] (14.,6.) circle (2.5pt);
\draw [fill=black] (2.,6.) circle (2.5pt);
\draw [fill=black] (4.,6.) circle (2.5pt);
\draw [fill=black] (6.,6.) circle (2.5pt);
\draw [fill=black] (8.,6.) circle (2.5pt);
\draw [fill=black] (10.,6.) circle (2.5pt);
\draw [fill=black] (12.,6.) circle (2.5pt);
\draw [fill=black] (18.,6.) circle (2.5pt);
\draw [fill=black] (20.,6.) circle (2.5pt);
\draw [fill=black] (22.,6.) circle (2.5pt);
\draw [fill=black] (24.,6.) circle (2.5pt);
\draw [fill=black] (26.,6.) circle (2.5pt);
\draw [fill=black] (28.,6.) circle (2.5pt);
\draw [fill=black] (30.,6.) circle (2.5pt);
\draw [fill=black] (32.,6.) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}\pause

When $n$ is large, solving $df=g$ is unstable. The computation of cohomology is ill-conditioned.

\begin{dfi}
The \textbf{conditionning number} of a graph $X$ is $\kappa(X,p,\k)=|\bar d||\bar d^{-1}|$ where $\bar d : C^0(X,\k)/\mathrm{Ker}(d)\to dC^0(X,\k)$. (It depends on $p$ and on the field $\k$).
\end{dfi}

\end{frame}

\subsection{Isoperimetry}

\begin{frame}[label=chee]

\begin{multicols}{2}

Isoperimetry = the art of cutting space apart.

\vskip.5cm

$|A|=5$, $|\partial A|=15$.

\begin{center}
\definecolor{qqzzqq}{rgb}{0.,0.6,0.}
\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.07cm,y=0.07cm]
\clip(-5.76000982177678,-18.30258608377214) rectangle (29.20345488179657,6.629028446151257);
\draw (-2.997139610017456,4.32663660301849)-- (-3.918096347270564,0.4125704696927845);
\draw (-3.918096347270564,0.4125704696927845)-- (1.377404891934806,0.5112444058270478);
\draw (1.377404891934806,0.5112444058270478)-- (-2.997139610017456,4.32663660301849);
\draw (-2.997139610017456,4.32663660301849)-- (2.758839997814468,2.9780928091835848);
\draw (2.758839997814468,2.9780928091835848)-- (1.377404891934806,0.5112444058270478);
\draw (1.377404891934806,0.5112444058270478)-- (5.094123152991991,0.6099183419613093);
\draw (2.758839997814468,2.9780928091835848)-- (5.094123152991991,0.6099183419613093);
\draw (-3.918096347270564,0.4125704696927845)-- (-4.740379148389411,-4.751365521333566);
\draw (-4.740379148389411,-4.751365521333566)-- (1.377404891934806,0.5112444058270478);
\draw [color=qqzzqq] (1.377404891934806,0.5112444058270478)-- (0.5222307787712068,-6.231474563347488);
\draw [color=qqzzqq] (0.5222307787712068,-6.231474563347488)-- (-4.740379148389411,-4.751365521333566);
\draw [color=qqzzqq] (0.5222307787712068,-6.231474563347488)-- (5.094123152991991,0.6099183419613093);
\draw (5.094123152991991,0.6099183419613093)-- (6.047971202289853,-2.2845171179770274);
\draw [color=qqzzqq] (6.047971202289853,-2.2845171179770274)-- (0.5222307787712068,-6.231474563347488);
\draw (2.758839997814468,2.9780928091835848)-- (5.192797089126253,5.116028092092583);
\draw (5.192797089126253,5.116028092092583)-- (5.094123152991991,0.6099183419613093);
\draw (5.094123152991991,0.6099183419613093)-- (9.567341591078517,2.747853624870308);
\draw (9.567341591078517,2.747853624870308)-- (5.192797089126253,5.116028092092583);
\draw (9.567341591078517,2.747853624870308)-- (6.047971202289853,-2.2845171179770274);
\draw (6.047971202289853,-2.2845171179770274)-- (11.507929001718995,-4.488235024975535);
\draw (11.507929001718995,-4.488235024975535)-- (9.567341591078517,2.747853624870308);
\draw (9.567341591078517,2.747853624870308)-- (16.079821375939783,5.839636957077166);
\draw (16.079821375939783,5.839636957077166)-- (11.507929001718995,-4.488235024975535);
\draw (11.507929001718995,-4.488235024975535)-- (21.210866054921386,1.4979837671696608);
\draw (16.079821375939783,5.839636957077166)-- (21.210866054921386,1.4979837671696608);
\draw (21.210866054921386,1.4979837671696608)-- (21.441105239234663,-5.376300450183889);
\draw (21.441105239234663,-5.376300450183889)-- (11.507929001718995,-4.488235024975535);
\draw (11.507929001718995,-4.488235024975535)-- (15.520669071178967,-9.882410200315162);
\draw (15.520669071178967,-9.882410200315162)-- (21.441105239234663,-5.376300450183889);
\draw [color=qqzzqq] (15.520669071178967,-9.882410200315162)-- (6.34399301069264,-7.349779172869118);
\draw [color=qqzzqq] (6.34399301069264,-7.349779172869118)-- (11.507929001718995,-4.488235024975535);
\draw [color=qqzzqq] (6.047971202289853,-2.2845171179770274)-- (6.34399301069264,-7.349779172869118);
\draw (6.34399301069264,-7.349779172869118)-- (0.5222307787712068,-6.231474563347488);
\draw [color=qqzzqq] (0.5222307787712068,-6.231474563347488)-- (-3.4247266665992564,-10.441562505075977);
\draw (-3.4247266665992564,-10.441562505075977)-- (-4.740379148389411,-4.751365521333566);
\draw [color=qqzzqq] (-3.4247266665992564,-10.441562505075977)-- (6.34399301069264,-7.349779172869118);
\draw (6.34399301069264,-7.349779172869118)-- (13.612972972583243,-13.105758780701038);
\draw [color=qqzzqq] (13.612972972583243,-13.105758780701038)-- (15.520669071178967,-9.882410200315162);
\draw (26.539258606171508,-12.217693355492685)-- (21.441105239234663,-5.376300450183889);
\draw (21.210866054921386,1.4979837671696608)-- (25.68408449300791,2.682071000780798);
\draw (25.68408449300791,2.682071000780798)-- (26.539258606171508,-12.217693355492685);
\draw (-3.4247266665992564,-10.441562505075977)-- (-4.773270460434165,-16.263324736997404);
\draw [color=qqzzqq] (-4.773270460434165,-16.263324736997404)-- (3.8771446073361004,-12.678171724119238);
\draw [color=qqzzqq] (3.8771446073361004,-12.678171724119238)-- (-3.4247266665992564,-10.441562505075977);
\draw (6.34399301069264,-7.349779172869118)-- (3.8771446073361004,-12.678171724119238);
\draw (3.8771446073361004,-12.678171724119238)-- (8.942406662228194,-14.256954702267421);
\draw (8.942406662228194,-14.256954702267421)-- (6.34399301069264,-7.349779172869118);
\draw (8.942406662228194,-14.256954702267421)-- (13.612972972583243,-13.105758780701038);
\draw [color=qqzzqq] (13.612972972583243,-13.105758780701038)-- (17.79016960226698,-17.217172786295265);
\draw (26.539258606171508,-12.217693355492685)-- (26.243236797768724,-17.151390162205757);
\draw (26.243236797768724,-17.151390162205757)-- (17.79016960226698,-17.217172786295265);
\draw (17.79016960226698,-17.217172786295265)-- (11.080341945137196,-17.381629346519034);
\draw [color=qqzzqq] (11.080341945137196,-17.381629346519034)-- (8.942406662228194,-14.256954702267421);
\draw (11.080341945137196,-17.381629346519034)-- (2.3312529412326692,-17.34873803447428);
\draw [color=qqzzqq] (2.3312529412326692,-17.34873803447428)-- (3.8771446073361004,-12.678171724119238);
\draw (2.3312529412326692,-17.34873803447428)-- (-4.773270460434165,-16.263324736997404);
\draw (21.441105239234663,-5.376300450183889)-- (17.79016960226698,-17.217172786295265);
\begin{scriptsize}
\draw [fill=qqqqff] (-2.997139610017456,4.32663660301849) circle (1.0pt);
\draw [fill=qqqqff] (-3.918096347270564,0.4125704696927845) circle (1.0pt);
\draw [fill=qqqqff] (1.377404891934806,0.5112444058270478) circle (1.0pt);
\draw [fill=qqqqff] (2.758839997814468,2.9780928091835848) circle (1.0pt);
\draw [fill=qqqqff] (5.094123152991991,0.6099183419613093) circle (1.0pt);
\draw [fill=qqqqff] (-4.740379148389411,-4.751365521333566) circle (1.0pt);
\draw [fill=ffqqqq] (0.5222307787712068,-6.231474563347488) circle (1.0pt);
\draw [fill=qqqqff] (6.047971202289853,-2.2845171179770274) circle (1.0pt);
\draw [fill=qqqqff] (5.192797089126253,5.116028092092583) circle (1.0pt);
\draw [fill=qqqqff] (9.567341591078517,2.747853624870308) circle (1.0pt);
\draw [fill=qqqqff] (11.507929001718995,-4.488235024975535) circle (1.0pt);
\draw [fill=qqqqff] (16.079821375939783,5.839636957077166) circle (1.0pt);
\draw [fill=qqqqff] (21.210866054921386,1.4979837671696608) circle (1.0pt);
\draw [fill=qqqqff] (21.441105239234663,-5.376300450183889) circle (1.0pt);
\draw [fill=qqqqff] (15.520669071178967,-9.882410200315162) circle (1.0pt);
\draw [fill=ffqqqq] (6.34399301069264,-7.349779172869118) circle (1.0pt);
\draw [fill=qqqqff] (-3.4247266665992564,-10.441562505075977) circle (1.0pt);
\draw [fill=ffqqqq] (13.612972972583243,-13.105758780701038) circle (1.0pt);
\draw [fill=qqqqff] (26.539258606171508,-12.217693355492685) circle (1.0pt);
\draw [fill=qqqqff] (25.68408449300791,2.682071000780798) circle (1.0pt);
\draw [fill=qqqqff] (-4.773270460434165,-16.263324736997404) circle (1.0pt);
\draw [fill=ffqqqq] (3.8771446073361004,-12.678171724119238) circle (1.0pt);
\draw [fill=ffqqqq] (8.942406662228194,-14.256954702267421) circle (1.0pt);
\draw [fill=qqqqff] (17.79016960226698,-17.217172786295265) circle (1.0pt);
\draw [fill=qqqqff] (26.243236797768724,-17.151390162205757) circle (1.0pt);
\draw [fill=qqqqff] (11.080341945137196,-17.381629346519034) circle (1.0pt);
\draw [fill=qqqqff] (2.3312529412326692,-17.34873803447428) circle (1.0pt);
\end{scriptsize}
\end{tikzpicture}
\end{center}\pause

\end{multicols}

\begin{dfi}
\textbf{Cheeger's constant} $h(X)$ of a graph $X$ is the largest $h$ such that for every set $A$ of vertices such that $|A|\le \frac{1}{2}|X|$,
\begin{align*}
|\partial A|\ge h\,|A|.
\end{align*}
Here, $\partial A$ is the set of edges connecting $A$ to its complement. 
\end{dfi}
\pause

\begin{pro}
$$
h(X)=\frac{2}{\kappa(X,1,\F)}=2(\|\bar d\|_{1\to 1}\|\bar d^{-1}\|_{1\to 1})^{-1}\text{ over }\F.%\qquad\hyperlink{hnc<1>}{\beamergotobutton{Démonstration}}
$$ 
\end{pro}

\end{frame}


\subsection{Laplacian}

\begin{frame}[label=Laplacian]

\begin{pro}
Let $\Delta$ be the self-adjoint operator corresponding to the quadratic form $f\mapsto \|df\|_2^2=\langle f,\Delta f \rangle$. Let $\lambda_1\le \lambda_2\le\cdots$ denote its eigenvalues. If the graph $X$ is connected, then $\lambda_1=0$ and
\begin{align*}
\lambda_2=(\|\bar d^{-1}\|_{2\to 2})^{-2}.%\qquad\qquad\hyperlink{lambda2<1>}{\beamergotobutton{Démonstration}}
\end{align*}
\end{pro}
\pause

In particular, 
$$
2\lambda_{2}^{-1}\le\kappa_0(X,2,\R)^2\leq 4\lambda_{2}^{-1}.
$$\pause

$\lambda_2$ is known as the \emph{spectral gap} of the graph.

It governs the speed at which a random walk on the graph is mixing. In particular, the possibility of picking a vertex at random.\pause

\medskip

\textbf{Morality}. Normed chain complexes contain interesting information, beyond their mere homology. 

\end{frame}

\section{Distances}

\subsection{The discretization problem}


\begin{frame}

Given a metric space $X$, a finite subset $Y\subset X$ and $r>0$, the \emph{\v Cech simplicial complex} $Y_r$ has a simplex $(y_0,\ldots,y_k)$ each time $\bigcap_i B(y_i,r)\not=\emptyset$. Let $C^r_\cdot$ denote the simplicial chains of $Y_r$.

\vskip-0.5cm

\begin{center}

\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\definecolor{zzttqq}{rgb}{0.6,0.2,0.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.4cm,y=0.4cm]
\clip(-3.72,-2.02) rectangle (8.8,5.28);
\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (-1.88,-0.32) -- (-0.48,-0.82) -- (-0.58,0.76) -- cycle;
\fill[color=zzttqq,fill=zzttqq,fill opacity=0.1] (0.52,2.24) -- (1.62,2.86) -- (0.84,3.94) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (3.7,1.7) -- (5.12,1.3) -- (4.92,2.76) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.92,2.76) -- (4.5,1.74) -- (5.12,1.3) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.92,2.76) -- (4.5,1.74) -- (3.7,1.7) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (3.7,1.7) -- (4.5,1.74) -- (5.12,1.3) -- cycle;
\draw(-1.88,-0.32) circle (0.4cm);
\draw(0.52,2.24) circle (0.4cm);
\draw(-2.44,2.6) circle (0.4cm);
\draw(-0.58,0.76) circle (0.4cm);
\draw(4.92,2.76) circle (0.4cm);
\draw(3.7,1.7) circle (0.4cm);
\draw(5.12,1.3) circle (0.4cm);
\draw(4.5,1.74) circle (0.4cm);
\draw(0.84,3.94) circle (0.4cm);
\draw(1.62,2.86) circle (0.4cm);
\draw(-0.48,-0.82) circle (0.4cm);
\draw(7.5,4.1) circle (0.4cm);
\draw(6.22,3.52) circle (0.4cm);
\draw [color=zzttqq] (-1.88,-0.32)-- (-0.48,-0.82);
\draw [color=zzttqq] (-0.48,-0.82)-- (-0.58,0.76);
\draw [color=zzttqq] (-0.58,0.76)-- (-1.88,-0.32);
\draw [color=zzttqq] (0.52,2.24)-- (1.62,2.86);
\draw [color=zzttqq] (1.62,2.86)-- (0.84,3.94);
\draw [color=zzttqq] (0.84,3.94)-- (0.52,2.24);
\draw [color=qqqqff] (3.7,1.7)-- (5.12,1.3);
\draw [color=qqqqff] (5.12,1.3)-- (4.92,2.76);
\draw [color=qqqqff] (4.92,2.76)-- (3.7,1.7);
\draw [color=qqqqff] (4.92,2.76)-- (4.5,1.74);
\draw [color=qqqqff] (4.5,1.74)-- (5.12,1.3);
\draw [color=qqqqff] (5.12,1.3)-- (4.92,2.76);
\draw [color=qqqqff] (4.92,2.76)-- (4.5,1.74);
\draw [color=qqqqff] (4.5,1.74)-- (3.7,1.7);
\draw [color=qqqqff] (3.7,1.7)-- (4.92,2.76);
\draw [color=qqqqff] (3.7,1.7)-- (4.5,1.74);
\draw [color=qqqqff] (4.5,1.74)-- (5.12,1.3);
\draw [color=qqqqff] (5.12,1.3)-- (3.7,1.7);
\draw [color=ffqqqq] (-0.58,0.76)-- (0.52,2.24);
\draw [color=ffqqqq] (4.92,2.76)-- (6.22,3.52);
\draw [color=ffqqqq] (6.22,3.52)-- (7.5,4.1);
\begin{scriptsize}
\draw [fill=black] (-1.88,-0.32) circle (2.5pt);
\draw [fill=black] (0.52,2.24) circle (2.5pt);
\draw [fill=black] (-2.44,2.6) circle (2.5pt);
\draw [fill=black] (-0.58,0.76) circle (2.5pt);
\draw [fill=black] (4.92,2.76) circle (2.5pt);
\draw [fill=black] (3.7,1.7) circle (2.5pt);
\draw [fill=black] (5.12,1.3) circle (2.5pt);
\draw [fill=black] (4.5,1.74) circle (2.5pt);
\draw [fill=black] (0.84,3.94) circle (2.5pt);
\draw [fill=black] (1.62,2.86) circle (2.5pt);
\draw [fill=black] (-0.48,-0.82) circle (2.5pt);
\draw [fill=black] (7.5,4.1) circle (2.5pt);
\draw [fill=black] (6.22,3.52) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

\end{center}\pause

\vskip-0.5cm

\begin{thm}[Bobrowski-Weinberger 2017]
Fix $r<\frac12$ and $1\le k\le d$. Let $Y$ be an $n$-sample picked at random on the standard $d$-torus. Then, with high probability, the $k$-homology of $Y_r$ coincides with the homology of the torus as soon as
$$
\omega_d r^d \,n \gg \log n +k\log\log n,
$$ 
and this fails if $\omega_d r^d \,n \ll \log n +(k-2)\log\log n$. If $k=0$, the threshold is $2^{-d}\log n$.
\end{thm}\pause

\medskip

\textbf{Question}. Can one say that the chain complexes $C^r_\cdot$ converge to some chain complex attached to the torus?

\end{frame}

\begin{frame}

In order to define a distance between normed chain complexes, the first idea is to measure conditioning numbers of isomorphisms.

\begin{dfi}
\label{defBMdist}
Let $B_1\stackrel{d_1}{\to}B_1$ and $B_2\stackrel{d_2}{\to}B_2$ be normed chain complexes. The \textbf{Banach-Mazur distance} $\mathrm{BMDist}(B_1,B_2)$ is the infimum of $\log(|F||F^{-1}|)$ over all isomorphisms $F:B_1\to B_2$ duch that $Fd_1=d_2 F$.
\end{dfi}\pause

This is too restrictive: this implies $\mathrm{dim}(B_1)=\mathrm{dim}(B_2)$.

\end{frame}

\subsection{The homotopy distance}

\begin{frame}

The second idea is too measure the size of homotopies.

\begin{dfi}
\label{defhomdist}
Let $B_1\stackrel{d_1}{\to}B_1$ and $B_2\stackrel{d_2}{\to}B_2$ be normed chain complexes. Consider all bounded homotopies, i.e. 
\begin{itemize}
  \item bounded morphisms $F_1:B_1\to B_2$ and $F_2:B_2\to B_1$ such that 
  $$
  d_2F_1=F_1d_1,\quad d_1F_2=F_2d_2,
  $$
  \item bounded operators $Q_1:B_1\to B_1$ and $Q_2:B_2\to B_2$ such that
  $$
1-F_2F_1=d_1Q_1+Q_1d_1, \quad 1-F_1F_2=d_2Q_2+Q_2d_2.
  $$
\end{itemize}
Let $\displaystyle
q=\max\{|Q_1|,|Q_2|\}$, $f=\max\{1,|F_1||F_2|\}$. The \textbf{homotopy distance} $\mathrm{HomDist}(B_1,B_2)$ is the infimum over all homotopies of $\displaystyle \min\{\frac{q}{f}+\log f,\frac{f}{q}+\log q\}$.
\end{dfi}\pause

The weird expression guarantees a triangle inequality. %\hyperlink{trihomdist<1>}{\beamergotobutton{Détails}}

\end{frame}

\subsection{Continuous observables}

\begin{frame}

\begin{dfi}
Let $Null$ denote the set of null normed chain complexes (i.e. with $d=0$). Denote by\begin{align*}
\mathrm{ND}(B)=\mathrm{HomDist}(B,Null),\quad\mathrm{NH}(B)&=|\bar d^{-1}|.
\end{align*}
\end{dfi}\pause

\medskip

\textbf{Fact}. $\mathrm{ND}$ is continuous. $\mathrm{NH}$ is continuous on the complement of $Null$. \pause

\medskip

\textbf{Remark}. $\mathrm{ND}$ is a function of $\mathrm{NH}$ for prehilbertian complexes, but not in general.\pause

\medskip

\begin{dfi}
Let $B$ be a normed chain complex. Let $\bar B=B/\mathrm{Ker}(d)$ and $\bar d:\bar B\to \mathrm{Im}(d)$.

The \textbf{singular values} of $B$ are the numbers
\begin{align*}
\sigma_j=&\inf\{s\ge 0\,;\,\exists L\subset \bar B\text{ subvectorspace such that}\\
&\hskip2em\mathrm{dim}(L)\ge j \text{ and }\forall \bar x\in L,\,|\bar d\bar x| \le s|\bar x|\}. 
\end{align*}
\end{dfi}
\pause

\medskip

\textbf{Fact}. Each $\sigma_j$ is continuous in homotopy distance.

\end{frame}

\begin{frame}

\begin{dfi}
Say a normed chain complex $B$ is \textbf{precompact} if it is not null and belongs to the closure of finite dimensional normed chain complexes.
\end{dfi}
\pause

\medskip

\textbf{Example}. The (de Rham) complex of smooth differential forms on a smooth compact Riemannian manifold, in its $L^2$ norm, is precompact.\pause

\medskip

\textbf{Fact}. A prehilbertian chain complex is precompact $\iff$ its singular values form a finite sequence that tends to $+\infty$.

\medskip

\begin{pro}
Let $B_i$ be precompact prehilbertian chain complexes. Then $B_i$ converges to $B$ $\iff$ for every $j$, $\sigma_j(B_i)$ tends to $\sigma_j(B)$. 
\end{pro}

\end{frame}

\section{Compactness}

\subsection{Analogy with Gromov-Hausdorff distance}

\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
{\color{red} Gromov-Hausdorff distance} & {\color{red} Homotopy distance}\\
\hline
Point & ?   \\
\hline
Bounded & ?   \\
\hline
Precompact & ?   \\
\hline
Compactness criterion (Gromov) & ? \\
\hline
\end{tabular}
\end{center}\pause

Let $X,Y$ be metric spaces.
\begin{align*}
GHDist(X,Y)&=\inf\{HDist_Z(i(X),j(Y))\,;\, Z\text{ metric space},\\
&\hskip3em  i:X\to Z,\,j:Y\to Z \text{ isometric embeddings}\}.
\end{align*}

\begin{center}

\definecolor{ffqqqq}{rgb}{1.,0.,0.}
\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.15cm,y=0.15cm]
\clip(-17.401640260380915,-21.243571081074414) rectangle (23.190799509570024,1.8251251385535558);
\fill[color=qqqqff,fill=qqqqff,fill opacity=1.0] (-2.,-4.) -- (2.,-6.) -- (-2.,-10.) -- (0.,-10.) -- (3.416666238709545,-6.659061619929286) -- (6.,-8.) -- (8.,-8.) -- (4.,-6.) -- (6.,-4.) -- (4.,-4.) -- (2.6272747496354523,-5.277626514049626) -- (0.,-4.) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=1.0] (-16.,-12.) -- (-12.,-16.) -- (-16.,-20.) -- (-14.,-20.) -- (-10.956837124847887,-17.019824914026742) -- (-8.,-20.) -- (-6.,-20.) -- (-10.,-16.) -- (-6.,-12.) -- (-8.,-12.) -- (-10.923945812803133,-14.914780943162496) -- (-14.,-12.) -- cycle;
\fill[color=ffqqqq,fill=ffqqqq,fill opacity=1.0] (12.,-12.) -- (16.,-16.) -- (12.,-20.) -- (14.,-20.) -- (22.,-12.) -- (20.,-12.) -- (17.000778113192887,-15.046346191341511) -- (14.,-12.) -- cycle;
\fill[color=ffqqqq,fill=ffqqqq,fill opacity=1.0] (-0.9578782632427166,-10.507345129165483) -- (0.7195786510397298,-10.540236441210237) -- (6.574232195005916,-4.488235024975533) -- (4.765210032544454,-4.554017649065041) -- (3.515340174843807,-6.00123537903421) -- (0.4893394667264529,-4.521126337020287) -- (-1.1223348234664858,-4.521126337020287) -- (2.8575139339487303,-6.659061619929286) -- cycle;
\draw [color=qqqqff] (-2.,-4.)-- (2.,-6.);
\draw [color=qqqqff] (2.,-6.)-- (-2.,-10.);
\draw [color=qqqqff] (-2.,-10.)-- (0.,-10.);
\draw [color=qqqqff] (0.,-10.)-- (3.416666238709545,-6.659061619929286);
\draw [color=qqqqff] (3.416666238709545,-6.659061619929286)-- (6.,-8.);
\draw [color=qqqqff] (6.,-8.)-- (8.,-8.);
\draw [color=qqqqff] (8.,-8.)-- (4.,-6.);
\draw [color=qqqqff] (4.,-6.)-- (6.,-4.);
\draw [color=qqqqff] (6.,-4.)-- (4.,-4.);
\draw [color=qqqqff] (4.,-4.)-- (2.6272747496354523,-5.277626514049626);
\draw [color=qqqqff] (2.6272747496354523,-5.277626514049626)-- (0.,-4.);
\draw [color=qqqqff] (0.,-4.)-- (-2.,-4.);
\draw [color=qqqqff] (-16.,-12.)-- (-12.,-16.);
\draw [color=qqqqff] (-12.,-16.)-- (-16.,-20.);
\draw [color=qqqqff] (-16.,-20.)-- (-14.,-20.);
\draw [color=qqqqff] (-14.,-20.)-- (-10.956837124847887,-17.019824914026742);
\draw [color=qqqqff] (-10.956837124847887,-17.019824914026742)-- (-8.,-20.);
\draw [color=qqqqff] (-8.,-20.)-- (-6.,-20.);
\draw [color=qqqqff] (-6.,-20.)-- (-10.,-16.);
\draw [color=qqqqff] (-10.,-16.)-- (-6.,-12.);
\draw [color=qqqqff] (-6.,-12.)-- (-8.,-12.);
\draw [color=qqqqff] (-8.,-12.)-- (-10.923945812803133,-14.914780943162496);
\draw [color=qqqqff] (-10.923945812803133,-14.914780943162496)-- (-14.,-12.);
\draw [color=qqqqff] (-14.,-12.)-- (-16.,-12.);
\draw [color=ffqqqq] (12.,-12.)-- (16.,-16.);
\draw [color=ffqqqq] (16.,-16.)-- (12.,-20.);
\draw [color=ffqqqq] (12.,-20.)-- (14.,-20.);
\draw [color=ffqqqq] (14.,-20.)-- (22.,-12.);
\draw [color=ffqqqq] (22.,-12.)-- (20.,-12.);
\draw [color=ffqqqq] (20.,-12.)-- (17.000778113192887,-15.046346191341511);
\draw [color=ffqqqq] (17.000778113192887,-15.046346191341511)-- (14.,-12.);
\draw [color=ffqqqq] (14.,-12.)-- (12.,-12.);
\draw [color=ffqqqq] (-0.9578782632427166,-10.507345129165483)-- (0.7195786510397298,-10.540236441210237);
\draw [color=ffqqqq] (0.7195786510397298,-10.540236441210237)-- (6.574232195005916,-4.488235024975533);
\draw [color=ffqqqq] (6.574232195005916,-4.488235024975533)-- (4.765210032544454,-4.554017649065041);
\draw [color=ffqqqq] (4.765210032544454,-4.554017649065041)-- (3.515340174843807,-6.00123537903421);
\draw [color=ffqqqq] (3.515340174843807,-6.00123537903421)-- (0.4893394667264529,-4.521126337020287);
\draw [color=ffqqqq] (0.4893394667264529,-4.521126337020287)-- (-1.1223348234664858,-4.521126337020287);
\draw [color=ffqqqq] (-1.1223348234664858,-4.521126337020287)-- (2.8575139339487303,-6.659061619929286);
\draw [color=ffqqqq] (2.8575139339487303,-6.659061619929286)-- (-0.9578782632427166,-10.507345129165483);
\draw (5.328432895472742,-0.587564190838727) node[anchor=north west] {$Z$};
\draw [->,color=ffqqqq] (12.94745183039575,-14.725077103418085) -- (3.8469569914599298,-8.460550423592503);
\draw [color=ffqqqq](9.095614479962453,-13.) node[anchor=north west] {$j$};
\draw [->,color=qqqqff] (-10.163572272204053,-11.592813763505294) -- (-0.3435034227477267,-7.360025466325846);
\draw [color=qqqqff](-12.,-9.) node[anchor=north west] {$i$};
\draw [rotate around={-1.135670666979445:(3.2966945128266016,-6.238336567573286)}] (3.2966945128266016,-6.238336567573286) ellipse (2.254302274062009cm and 1.1779434870219379cm);
\end{tikzpicture}

\end{center}

\end{frame}

\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
Gromov-Hausdorff distance & Homotopy distance\\
\hline
{\color{red} Point} & {\color{red} Null complex (i.e. $d=0$)}   \\
\hline
Bounded & ?  \\
\hline
Precompact & ?  \\
\hline
Compactness criterion (Gromov) & ? \\
\hline
\end{tabular}
\end{center}

\vskip5cm

\end{frame}

\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
Gromov-Hausdorff distance & Homotopy distance\\
\hline
Point & Null complex (i.e. $d=0$)   \\
\hline
{\color{red} Bounded} & {\color{red} Homotopic to a null complex}   \\
\hline
Precompact & ?  \\
\hline
Compactness criterion (Gromov) & ? \\
\hline
\end{tabular}
\end{center}

$B$ is homotopic to a null complex $\iff$ $\mathrm{ND}(B)<\infty$.

\medskip

One can think of $\mathrm{ND}(B)=\mathrm{HomDist}(B,Null)$ as an analogue of diameter.

\vskip4cm

\end{frame}

\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
Gromov-Hausdorff distance & Homotopy distance\\
\hline
Point & Null complex (i.e. $d=0$)   \\
\hline
Bounded & Homotopic to a null complex   \\
\hline
{\color{red} Precompact} & {\color{red} In the closure of finite dim. complexes}  \\
\hline
Compactness criterion (Gromov) & ? \\
\hline
\end{tabular}
\end{center}

$B$ is precompact $\implies$ $B$ has a finite sequence of singular values that tends to $+\infty$ ($\iff$ if $B$ is prehilbertian).

\vskip4cm

\end{frame}

\subsection{Compactness criterion}


\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
Gromov-Hausdorff distance & Homotopy distance\\
\hline
Point & Null complex (i.e. $d=0$)   \\
\hline
Bounded & Homotopic to a null complex   \\
\hline
Precompact & In the closure of finite dim. complexes  \\
\hline
{\color{red} Compactness criterion (Gromov)} & {\color{red} ????} \\
\hline
\end{tabular}
\end{center}

\begin{dfi}
$X$ precompact metric space, $\epsilon>0$. The \textbf{covering number} $N(X,\epsilon)$ is the minimal number of $\epsilon$-balls that can cover $X$.
\end{dfi}\pause

\begin{thm}[Gromov's compactness criterion]
A collection $\mathcal{T}$ of precompact metric spaces is precompact in Gromov-Hausdorff distance if and only if there is a function $\nu$ which serves as a covering number for all spaces in $\mathcal{T}$, i.e. 
$$
\forall\epsilon>0,\quad \forall X\in \mathcal{T},\quad N(X,\epsilon)\le \nu(\epsilon).
$$
\end{thm}

\end{frame}


\begin{frame}

Analogy between normed chain complexes and metric spaces.

\begin{center}
\begin{tabular}{|c|c|}
\hline
Metric space  & Normed chain complex  \\
\hline
\hline
Gromov-Hausdorff distance & Homotopy distance\\
\hline
Point & Null complex (i.e. $d=0$)   \\
\hline
Bounded & Homotopic to a null complex   \\
\hline
Precompact & In the closure of finite dim. complexes  \\
\hline
{\color{red} Compactness criterion (Gromov)} & {\color{red} ????} \\
\hline
\end{tabular}
\end{center}

\begin{dfi}
Let $(B,d)$ be a normed chain complex that belongs to the closure of finite dimensional normed complexes. Its \textbf{profile} is the smallest function $\pi=(\pi_d,\pi_c):(0,+\infty)\to(0,+\infty)^2$ with the following property. For every $\epsilon>0$, there exists a finite-dimensional normed complex $(B',d')$ such that
$$
HomDist(B,B')<\epsilon, \quad \mathrm{dim}(B')\le \pi_d(\epsilon),\quad \kappa(B',d')\le \pi_c(\epsilon).
$$
\end{dfi}\pause

\begin{thm}
A collection of nonnull normed chain complexes is precompact if and only if a same profile serves for all and the distances to null complexes are bounded below.
\end{thm}

\end{frame}

\subsection{Discretizing Riemannian manifolds}

\begin{frame}

\begin{lem}
Let $B$ be a prehilbertian chain complex. Then the profile of $B$ is determined by the asymptotics of eigenvalues,
$$
\pi_d(\epsilon)\leq \mathrm{Card}\{\lambda\in\mathrm{spectrum}(d^*d)\,;\,\lambda < \frac{1}{\epsilon^2}\},\quad \pi_c(\epsilon)\le \frac{1}{\epsilon\sqrt{\lambda_2}}.
$$
\end{lem}\pause

\begin{exa}
Let $M$ be a smooth compact Riemannian manifold. Consider the (de Rham) complex of smooth differential forms on $M$ in its $L^2$ norm. Its profile satisfies $\pi_d(\epsilon)\le C\,\epsilon^{-N}$, where $N=\mathrm{dim}(M)$.
\end{exa}
\pause

\medskip

\begin{coj}
Consider finer and finer triangulations of a fixed compact manifold. The corresponding complexes of simplicial cochains in their weighted $\ell^p$ norms form a precompact family.
\end{coj}
Here, the weight of a simplex is a function of its volume.

\end{frame}


\begin{frame}

Let $Y$ be a finite metric space. The \emph{complete simplicial complex} $\Delta_Y$ on $Y$ takes as simplices all tuples of points of $Y$. Pick a function of the diameter as a weight. Use weighted $\ell^p$ norms on cochains. This gives a normed chain complex $C^\cdot(Y)$.\pause

%\begin{multicols}{2}

\begin{center}

\definecolor{qqqqff}{rgb}{0.,0.,1.}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.7cm,y=0.7cm]
\clip(3.646905423022828,-2.9423433588721153) rectangle (7.593862868393299,1.1690706467221121);
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (6.738688755229687,-1.297777756634422) -- (5.598266370781756,0.5467288138167195) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (5.598266370781756,0.5467288138167195) -- (6.146645138424117,-1.8898213734399907) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (5.598266370781756,0.5467288138167195) -- (6.146645138424117,-1.8898213734399907) -- (6.738688755229687,-1.297777756634422) -- cycle;
\fill[color=qqqqff,fill=qqqqff,fill opacity=0.1] (4.403405600052162,-1.3964516927686834) -- (6.738688755229687,-1.297777756634422) -- (6.146645138424117,-1.8898213734399907) -- cycle;
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (4.403405600052162,-1.3964516927686834);
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (4.403405600052162,-1.3964516927686834);
\draw [color=qqqqff] (5.598266370781756,0.5467288138167195)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (5.598266370781756,0.5467288138167195);
\draw [color=qqqqff] (4.403405600052162,-1.3964516927686834)-- (6.738688755229687,-1.297777756634422);
\draw [color=qqqqff] (6.738688755229687,-1.297777756634422)-- (6.146645138424117,-1.8898213734399907);
\draw [color=qqqqff] (6.146645138424117,-1.8898213734399907)-- (4.403405600052162,-1.3964516927686834);
\begin{scriptsize}
\draw [fill=qqqqff] (4.403405600052162,-1.3964516927686834) circle (2.5pt);
\draw [fill=qqqqff] (5.598266370781756,0.5467288138167195) circle (2.5pt);
\draw [fill=qqqqff] (6.738688755229687,-1.297777756634422) circle (2.5pt);
\draw [fill=qqqqff] (6.146645138424117,-1.8898213734399907) circle (2.5pt);
\end{scriptsize}
\end{tikzpicture}

\vskip-.5cm

The complete simplicial complex on 4 points.

\end{center}\pause

%\end{multicols}


Let $(X,\mu)$ be a metric measure space. Same construction with the same weight $w$ and $L^p(\mu^{\otimes\cdot})$ norms yields a normed chain complex $C^\cdot(X)$.

\textbf{Example}. $1$-cochains are functions $c$ on $X\times X$. The squared weighted $L^2$ norm is
$$
\int_{X\times X}w(|x-x'|)|c(x,x')|^2\,d\mu(x)\,d\mu(x').
$$
\pause

\medskip

\textbf{Question}. Given a metric measure space $(X,\mu)$ and a finite sample $Y\subset X$. Does $C^\cdot(Y)$ converge to $C^\cdot(X)$?

\end{frame}

\end{document}
