\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{persistent homology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{persistent_homology}{}\section*{{Persistent homology}}\label{persistent_homology}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{more_detail}{More detail}\dotfill \pageref*{more_detail} \linebreak
\noindent\hyperlink{talks}{Talks}\dotfill \pageref*{talks} \linebreak
\noindent\hyperlink{software}{Software}\dotfill \pageref*{software} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Persistent homology is a [[homology theory]] adapted to a [[computation|computational]] context, for instance, in analysis of large data sets. It keeps track of homology classes which stay `persistent' when the approximate image of a space gets refined to higher resolutions.

\hypertarget{more_detail}{}\subsection*{{More detail}}\label{more_detail}

We suppose given a `data cloud of samples', $P\subset \mathbb{R}^m$, from some space $X$, yielding a [[simplicial complex]] $S_\rho(X)$ for each $\rho \gt 0$ via one of the family of simplicial complex approximation methods that are listed below (TO BE ADDED). For these, the important idea to retain is that if $\rho \lt \rho^\prime$, then

\begin{displaymath}
S_\rho(X) \hookrightarrow S_{\rho^\prime}(X),
\end{displaymath}
so we get a `filtration structure' on the complex.

The idea of \emph{persistent homology} is to look for features that persist for some range of parameter values. Typically a feature, such as a hole, will initially not be observed, then will appear, and after a range of values of the parameter it will disappear again. A typical feature will be a [[Betti number]] of the complex, $S_\rho(X)$, which then will vary with the parameter $\rho$.

\hypertarget{talks}{}\subsection*{{Talks}}\label{talks}

\begin{itemize}%
\item [[ACT-OIT.pdf|Slides:file]] for a talk by [[Tim Porter]] in 2008.

\end{itemize}
\hypertarget{software}{}\subsection*{{Software}}\label{software}

One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance \href{http://www.sas.upenn.edu/~vnanda/perseus}{Perseus}. The principal algorithm is based on bringing the filtered complex to its \textbf{canonical form} by upper-triangular matrices from (\href{https://www.researchgate.net/profile/Serguei_Barannikov/publication/267672645_The_Framed_Morse_complex_and_its_invariants/links/5970e947458515fa8de6e724/The-Framed-Morse-complex-and-its-invariants.pdf}{Barannikov1994, §2.1})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[persistence module]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A clear introduction to the use of persistent homology in data analysis is

\begin{itemize}%
\item [[Robert Ghrist]], \emph{Barcodes: The Persistent Topology of Data}, (\href{https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf}{pdf})

\end{itemize}
Bar-codes were discovered under the name of \emph{canonical forms invariants of filtered complexes} in

\begin{itemize}%
\item Serguei Barannikov, \emph{Framed Morse complex and its invariants}, \href{https://www.researchgate.net/profile/Serguei_Barannikov/publication/267672645_The_Framed_Morse_complex_and_its_invariants/links/5970e947458515fa8de6e724/The-Framed-Morse-complex-and-its-invariants.pdf}{pdf} Advances in Soviet Mathematics \textbf{21} 93–115 (1994)

\end{itemize}
Other references:

\begin{itemize}%
\item D. Le Peutrec, N. Nier, C. Viterbo, \emph{Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex}, Annales Henri Poincaré \textbf{14} (3): 567–610 (2013) \href{https://10.1007/s00023-012-0193-9}{doi}


\item [[Robert MacPherson]], Benjamin Schweinhart, \emph{Measuring shape with topology}, J. Math. Phys. \textbf{53}, 073516 (2012); \href{http://dx.doi.org/10.1063/1.4737391}{doi}


\item A. Zomorodian, [[Gunnar Carlsson]], \emph{Computing persistent homology}, Discrete Comput. Geom. \textbf{33}, 249--274 (2005)


\item Ulrich Bauer, Michael Kerber, Jan Reininghaus, \emph{Clear and compress: computing persistent homology in chunks}, \href{http://arxiv.org/abs/1303.0477}{arxiv/1303.0477}


\item Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, Shmuel Weinberger, \emph{Persistent homology for random fields and complexes} Institute of Mathematical Statistics Collections \textbf{6}:124--143, 2010 \href{http://arxiv.org/abs/1003.1001}{arxiv/1003.1001}


\item Robert J. Adler, Omer Bobrowski, Shmuel Weinberger, \emph{Crackle: the persistent homology of noise}, \href{http://arxiv.org/abs/1301.1466}{arxiv/1301.1466}


\item Pawe Dotko, Hubert Wagner, \emph{Computing homology and persistent homology using iterated Morse decomposition}, \href{http://arxiv.org/abs/1210.1429}{arxiv/1210.1429}


\item [[Gunnar Carlsson]], V. de Silva, \emph{Zigzag persistence}, \href{http://arxiv.org/abs/0812.0197}{arXiv:0812.0197}


\item [[Gunnar Carlsson]], \emph{Topology and data}, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308.


\item Francisco Belch\'i{} Guillam\'o{}n, Aniceto Murillo Mas, \emph{A-infinity persistence}, \href{http://arxiv.org/abs/1403.2395}{arxiv/1403.2395}


\item Jo\~a{}o Pita Costa, Mikael Vejdemo Johansson, Primo \v{S}kraba, \emph{Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology}, \href{http://arxiv.org/abs/1409.8613}{arxiv/1409.8613}


\item Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka, \emph{Persistence weighted Gaussian kernel for topological data analysis}, \href{http://arxiv.org/abs/1601.01741}{arxiv/1601.01741}


\item Heather A. Harrington, Nina Otter, Hal Schenck, [[Ulrike Tillmann]], \emph{Stratifying multiparameter persistent homology}, \href{https://arxiv.org/abs/1708.07390}{arxiv/1708.07390}


\item H. Edelsbrunner, D. Morozov, \emph{Persistent homology: theory and practice} \href{http://mrzv.org/publications/persistent-homology-theory-practice/ecm}{pdf}



\end{itemize}
The following paper uses persistent homology to single out features relevant for training neural networks,

\begin{itemize}%
\item Jean-Baptiste Bardin, Gard Spreemann, [[Kathryn Hess]], \emph{Topological exploration of artificial neuronal network dynamics}, \href{https://arxiv.org/abs/1810.01747}{arxiv/1810.01747}

\end{itemize}
\hypertarget{variants}{}\subsubsection*{{Variants}}\label{variants}

Discussion of persistent [[Cohomotopy]]:

\begin{itemize}%
\item [[Peter Franek]], [[Marek Krčál]], \emph{Persistence of Zero Sets}, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (\href{https://arxiv.org/abs/1507.04310}{arXiv:1507.04310}, \href{http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16}{doi:10.4310/HHA.2017.v19.n2.a16})


\item [[Peter Franek]], [[Marek Krčál]], \emph{Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory}, talk at \href{https://www2.ist.ac.at/acat}{ACAT meeting 2015} (\href{https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf}{pfd slides})


\item [[Peter Franek]], [[Marek Krčál]], Hubert Wagner, \emph{Solving equations and optimization problems with uncertainty}, J Appl. and Comput. Topology (2018) 1: 297 (\href{https://arxiv.org/abs/1607.06344}{arxiv:1607.06344}, \href{https://doi.org/10.1007/s41468-017-0009-6}{doi:10.1007/s41468-017-0009-6})



\end{itemize}
category: topology, applications

[[!redirects persistent homology]] [[!redirects persistent homology theory]]

[[!redirects persistence module]] [[!redirects persistence modules]]



\end{document}