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diff --git a/dissertation/dissertation.tex b/dissertation/dissertation.tex
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--- a/dissertation/dissertation.tex
+++ b/dissertation/dissertation.tex
@@ -2,6 +2,7 @@
 
 \input{preamble}
 
+\usepackage[firstpage]{draftwatermark}
 
 
 \begin{document}
@@ -78,18 +79,176 @@ Thank you!
 \chapter{Topological Data Analysis and Persistent Homology}%
 \label{cha:tda-ph}
 
+\section{Homology}%
+\label{sec:homology}
+
+Our goal is to understand the topological structure of a metric
+space. For this, we can use \emph{homology}, which consists in
+associating for a metric space $X$ and a dimension $i$ a vector space
+$H_i(X)$. The dimension of $H_i(X)$ will give us the number of
+$i$-dimensional components in $X$: the dimension of $H_0(X)$ is the
+number of path-connected components in $X$, the dimension of $H_1(X)$
+is the number of holes in $X$, and the dimension of $H_2(X)$ is the
+number of voids.
+
+Crucially, these vector spaces are robust to continuous deformation of
+the underlying metric space (they are \emph{homotopy
+  invariant}). However, computing the homology of an arbitrary metric
+space can be extremely difficult. It is necessary to approximate it in
+a structure that would be both combinatorial and topological in
+nature.
+
 \section{Simplicial Complexes}%
 \label{sec:simplicial-complexes}
 
-\section{Homology}%
-\label{sec:homology}
+In order to understand the topological structure of a metric space, we
+need a way to decompose it in smaller pieces which, when assembled,
+conserve the overall organisation of the space. For this, we use a
+structure called a \emph{simplicial complex}, which is a kind of
+higher-dimensional generalization of graphs.
+
+The building blocks of this representation will be \emph{simplices},
+which are simply the convex hull of an arbitrary set of
+points. Examples of simplices include single points, segments,
+triangles, and tetrahedrons (in dimensions 0, 1,, 2, and 3
+respectively).
+
+\begin{defn}[Simplex]
+  The \emph{$k$-dimensional simplex} $\sigma = [x_0,\ldots,x_k]$ is
+  the convex hull of the set $\{x_0,\ldots,x_k\} \in \mathbb{R}^d$,
+  where $x_0,\ldots,x_k$ are affinely independent. $x_0,\ldots,x_k$
+  are called the \emph{vertices} of $\sigma$, and the simplices
+  defined by the subsets of $\{x_0,\ldots,x_k\}$ are called the
+  \emph{faces} of $\sigma$.
+\end{defn}
+
+We then need a way to combine these basic building blocks meaningfully
+so that the resulting object can adequately reflect the topological
+structure of the metric space.
+
+\begin{defn}[Simplicial complex]
+  A \emph{simplicial complex} is a collection $K$ of simplices such
+  that:
+  \begin{itemize}
+  \item any face of a simplex of $K$ is a simplex of $K$
+  \item the intersection of two simplices of $K$ is either the empty
+    set or a common face or both.
+  \end{itemize}
+\end{defn}
+
+%% TODO figure with examples of simplicial complexes
+
+Using these definitions, we can define homology on simplicial
+complexes. %% TODO add reference for more details/do it myself?
 
 \section{Filtrations}%
 \label{sec:filtrations}
 
+If we consider that a simplicial complex is a kind of
+``discretization'' of a metric space, we realise that there must be an
+issue of \emph{scale}. For our analysis to be invariant under small
+perturbations in the data, we need a way to find the optimal scale
+parameter to capture the adequate topological structure, without
+taking into account some small perturbations, nor ignoring some
+important smaller features.
+
+%% TODO rewrite using the Cech filtration as an example?
+
+The ideal solution to these problems is to consider all scales at
+once: this is the objective of \emph{filtered simplical complexes}.
+
+\begin{defn}[Filtration]
+  A \emph{filtered simplicial complex}, or simply a \emph{filtration},
+  $K$ is a sequence ${(K_i)}_{i\in I}$ of simplicial complexes such
+  that:
+  \begin{itemize}
+  \item for any $i, j \in I$, if $i < j$ then $K_i \subseteq K_j$,
+  \item $\bigcup_{i\in I} K_i = K$.
+  \end{itemize}
+\end{defn}
+
+\section{Persistent Homology}%
+\label{sec:persistent-homology}
+
+We can now compute the homology for each step in a filtration. This
+leads to the notion of \emph{persistent homology}, which gives us all
+the information necessary to establish the topological structure of
+the metric space at multiple scales.
+
+\begin{defn}[Persistent homology]
+  The \emph{$p$-th persistent homology} of a simplicial complex
+  $K = {(K_i)}_{i\in I}$ is the pair
+  $(\{H_p(K_i)\}_{i\in I}, \{f_{i,j}\}_{i,j\in I, i\leq j})$, where
+  for all $i\leq j$, $f_{i,j} : H_p(K_i) \mapsto H_p(K_j)$ is induced
+  by the inclusion map $K_i \mapsto K_j$.
+\end{defn}
+
+The functions $f_{i,j}$ allow us to link generators in each successive
+homology space in the filtration. Since each generator correspond to a
+topological feature (connected component, hole, void, etc, depending
+on the dimension $p$), we can determine whether it survives in the
+next step of the filtration. We can now determine when each feature is
+born and when it dies (if it dies at all). This representation will be
+dependent on the choice of basis for each homology space
+$H_p(K_i)$. However, by the Fundamental Theorem of Persistent
+Homology, we can choose base vectors in each homology space such that
+the collection of half-open intervals is well-defined and unique. This
+construction is called a \emph{barcode}.
+%% TODO references for the Fundamental Theorem
+
 \section{Topological summaries: barcodes and persistence diagrams}%
 \label{sec:topol-summ}
 
+In order to interpret the results of the persistent homology
+computation, we need to compare the output for a particular data set
+to a suitable null model. For this, we need some kind of a similarity
+measure between barcodes and a way to evaluate the statistical
+significance of the results.
+
+One possible approach for this is to define a space in which we can
+project barcodes and study their geometric
+properties. \emph{Persistence diagrams} are an example of such a
+space.
+
+\begin{defn}[Persistence diagrams]
+  A \emph{persistence diagram} is the union of a finite multiset of
+  points in $\bar{\mathbb{R}}^2$ zith the diagonal
+  $\Delta = \{(x,x) \;|\; x\in\mathbb{R}^2\}$, where every point of
+  $\Delta$ has infinite multiplicity.
+\end{defn}
+
+The diagonal $\Delta$ is added to facilitate comparisons between
+diagrams, as points near the diagonal correspond to short-lived
+topological feature, thus likely to be caused by small perturbations
+in the data.
+
+We can now define several distances on the space of persistence
+diagrams.
+
+\begin{defn}[Wasserstein distance]
+  The \emph{$p$-th Wasserstein distance} between two diagrams $X$ and
+  $Y$ is
+  \[ W_p[d](X, Y) = \inf_{\phi:X\mapsto Y} \left[\sum_{x\in X} {d\left(x, \phi(x)\right)}^p\right] \]
+  for $p\in [1,\infty)$, and
+  \[ W_\infty[d](X, Y) = \inf_{\phi:X\mapsto Y} \sup_{x\in X} d\left(x,
+      \phi(x)\right) \] for $p = \infty$, where $d$ is a distance on
+  $\mathbb{R}^2$ and $\phi$ ranges over all bijections from $X$ to
+  $Y$.
+\end{defn}
+
+\begin{defn}[Bottleneck distance]
+  The \emph{bottleneck distance} is defined as the infinite
+  Wasserstein distance with $d$ the uniform norm:
+  $d_B = W_\infty[L_\infty]$.
+\end{defn}
+
+Since the bottleneck distance is by far the most commonly used, we
+will focus on it in the following. It is symmetric, non-negative, and
+satisfies the triangle inequality. However, it is not a true distance,
+as it is fairly straightforward to come up with two distinct diagrams
+at bottleneck distance zero, even on multisets not touching the
+diagonal $\Delta$.
+
 \section{Stability}%
 \label{sec:stability}
 
@@ -97,10 +256,6 @@ Thank you!
 
 \chapter{Temporal Networks}%
 \label{cha:temporal-networks}
-
-
-
-
 \backmatter%
 
 \nocite{*}
diff --git a/dissertation/preamble.tex b/dissertation/preamble.tex
index df6c935..2054723 100644
--- a/dissertation/preamble.tex
+++ b/dissertation/preamble.tex
@@ -16,6 +16,20 @@
 \usepackage{lettrine}
 
 \usepackage{amssymb, amsmath}
+\usepackage{amsthm}
+
+\theoremstyle{plain}
+\newtheorem{thm}{Theorem}[chapter]
+\newtheorem{lem}[thm]{Lemma}
+\newtheorem{cor}[thm]{Corollary}
+\newtheorem{prop}[thm]{Proposition}
+\theoremstyle{definition}
+\newtheorem{defn}{Definition}[chapter]
+\newtheorem{expl}{Example}[chapter]
+\theoremstyle{remark}
+\newtheorem*{rem}{Remark}
+\newtheorem*{note}{Note}
+\newtheorem*{notation}{Notation}
 
 \usepackage{pdfpages}