diff --git a/main.tex b/main.tex
index 09ba6f52d0a86fb18d21a33977a136c1fbb32768..e64982bd5be5277d9ba16d2cd5b3888fb686b36d 100644
--- a/main.tex
+++ b/main.tex
@@ -77,6 +77,31 @@ Practical Methods for Narrowing Down Possible Walls}
 \section{Introduction}
 \label{sec:intro}
 
+The theory of Bridgeland stability conditions \cite{BridgelandTom2007SCoT} on
+complexes of sheaves was developed as a generalisation of stability for vector
+bundles. The definition is most analoguous to Mumford stability, but is more
+aware of the features that sheaves can have on spaces of dimension greater
+than 1. Whilst also asymptotically matching up with Gieseker stability.
+For K3 surfaces, explicit stability conditions were defined in
+\cite{Bridgeland_StabK3}, and later shown to also be valid on other surfaces.
+
+The moduli spaces of stable objects of some fixed Chern character $v$ is
+studied, as well as how they change as we vary the Bridgeland stability
+condition. They in fact do not change over whole regions of the stability
+space (called chambers), but do undergo changes as we cross `walls' in the
+stability space. These are where there is some stable object $F$ of $v$ which
+has a subobject who's slope overtakes the slope of $v$, making $F$ unstable
+after crossing the wall.
+
+% NOTE: SURFACE SPACIALIZATION
+% (come back to these when adjusting to general Picard rank 1)
+In this document we concentrate on two surfaces: Principally polarized abelian
+surfaces and the projective surface $\PP^2$. Although this can be generalised
+for Picard rank 1 surfaces, the formulae will need adjusting.
+The Bridgeland stability conditions (defined in \cite{Bridgeland_StabK3}) are
+given by two parameters $\alpha \in \RR_{>0}$, $\beta \in \RR$, which will be
+illustrated throughout this article with diagrams of the upper half plane.
+
 It is well known that for any rational $\beta_0$,
 the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
 intersects finitely many walls
@@ -103,11 +128,11 @@ conditions only admit finitely many solutions (despite the fact that the same
 conditions admit infinitely many solutions when $\beta_{-}$ is irrational).
 
 
-For a destabilizing sequence
+For a semistabilizing sequence
 $E \hookrightarrow F \twoheadrightarrow G$ in $\mathcal{B}^\beta$
 we have the following conditions.
-There are some Bogomolov-Gieseker type inequalities:
-$0 \leq \Delta(E), \Delta(G)$ and $\Delta(E) + \Delta(G) \leq \Delta(F)$.
+There are some Bogomolov-Gieseker inequalities:
+$0 \leq \Delta(E), \Delta(G)$.
 We also have a condition relating to the tilt category $\firsttilt\beta$:
 $0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)$.
 Finally, there is a condition ensuring that the radius of the circular wall is
@@ -663,7 +688,7 @@ stability and the stability condition corresponding to $P$.
 \end{problem}
 
 \begin{problem}[all `left' pseudo-walls]
-\label{problem:problem-statement-1}
+\label{problem:problem-statement-2}
 
 Fix a Chern character $v$ with positive rank, $\Delta(v) \geq 0$,
 and $\beta_{-}(v) \in \QQ$.
@@ -856,7 +881,7 @@ Finally, $r>0$ as per the statement of the problem, so the right-hand-side
 of equation \ref{eqn:bgmlv1-pt1} is always greater than, or equal, to zero.
 And so, when $P\coloneqq(\beta_{-},0)$, this condition $\Delta(u,v-u) \geq 0$ is
 always satisfied when $2r \geq R$, provided that the other conditions of the
-problem statement (\ref{subsect:problem-statement}) hold.
+problem statement (\ref{subsect:problem-statement-2}) hold.
 
 However, when $2r<R$, this condition does add potentially independent condition
 of the others:
@@ -1397,7 +1422,7 @@ no possible solutions for $d$. At that point, there are no Chern characters
 $(r,c,d)$ that satisfy all inequalities to give a pseudowall.
 
 
-\subsubsection{All Semistabilizers Left of Vertical Wall for Rational Beta min}
+\subsubsection{All Semistabilizers Left of $V_v$ for Rational beta}
 
 
 The strategy here is similar to what was shown in (sect
diff --git a/references.bib b/references.bib
index 78f44248d3c9cae41f7e1de6e92b2cc699887378..6fbe7e3f91ed2f46da8042c24245ac354f2917d6 100644
--- a/references.bib
+++ b/references.bib
@@ -133,4 +133,31 @@
   publisher = {Wiley},
   title = {An introduction to the theory of numbers / Ivan Niven, Herbert S. Zuckerman.},
   year = {1966},
-}
\ No newline at end of file
+}
+@article{Bridgeland_StabK3,
+  author = {Tom Bridgeland},
+  title = {{Stability conditions on $K3$ surfaces}},
+  volume = {141},
+  journal = {Duke Mathematical Journal},
+  number = {2},
+  publisher = {Duke University Press},
+  pages = {241 -- 291},
+  year = {2008},
+  doi = {10.1215/S0012-7094-08-14122-5},
+  URL = {https://doi.org/10.1215/S0012-7094-08-14122-5}
+}
+@article{BridgelandTom2007SCoT,
+  author = {Bridgeland, Tom},
+  address = {PRINCETON},
+  copyright = {Copyright 2007 Princeton University (Mathematics Department)},
+  issn = {0003-486X},
+  journal = {Annals of mathematics},
+  keywords = {Algebra ; Exact sciences and technology ; General mathematics ; General, history and biography ; Homomorphisms ; Mathematics ; Morphisms ; Nitration ; Physical Sciences ; Quotients ; Real numbers ; Science & Technology ; Sciences and techniques of general use ; String theory ; Topological spaces ; Topology},
+  language = {eng},
+  number = {2},
+  pages = {317-345},
+  publisher = {Princeton University Press},
+  title = {Stability Conditions on Triangulated Categories},
+  volume = {166},
+  year = {2007},
+}