diff --git a/main.tex b/main.tex
index 2ff26cd1a83e093fffd6ddf1c03a418501563fcd..6c06bb69e5d6be4f524a938d7faa6966ef8b1fae 100644
--- a/main.tex
+++ b/main.tex
@@ -45,7 +45,6 @@ sorting=ynt
 \begin{document}
 
 
-
 \title{Tighter Bounds for Ranks of Tilt Semistabilizers on Picard Rank 1 Surfaces
 \\[1em] \large
 Practical Methods for Narrowing Down Possible Walls}
@@ -434,7 +433,7 @@ and rank zero cases.
 \end{definition}
 
 
-\subsection{Relevance of $V_v$}
+\subsection{Relevance of \texorpdfstring{$V_v$}{V_v}}
 \label{subsect:relevance-of-V_v}
 
 For the positive rank case, by definition of the first tilt $\firsttilt\beta$, objects of Chern character
@@ -455,7 +454,7 @@ $\firsttilt{\beta}$ for all $\beta$
 
 
 
-\subsection{Relevance of $\Theta_v$}
+\subsection{Relevance of \texorpdfstring{$\Theta_v$}{Θ_v}}
 
 Since $\chern_2^{\alpha, \beta}$ is the numerator of the tilt slope
 $\nu_{\alpha, \beta}$. The curve $\Theta_v$, where this is 0, firstly divides the
@@ -766,7 +765,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 
 \section{B.Schmidt's Solutions to the Problems}
 
-\subsection{Bound on $\chern_0(u)$ for Semistabilizers}
+\subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
 \label{subsect:loose-bound-on-r}
 
 The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
@@ -876,7 +875,8 @@ In section [ref], a different
 algorithm will be presented making use of the later theorems in this article,
 with the goal of cutting down the run time.
 
-\subsubsection{Finding possible $r$ and $c$}
+\subsubsection{Finding possible \texorpdfstring{$r$}{r} and
+\texorpdfstring{$c$}{c}}
 To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
 semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.
 
@@ -898,7 +898,8 @@ the Bogomolov inequalities and consequence 3 of lemma
 \ref{lem:pseudo_wall_numerical_tests}
 ($\chern_2^{\beta_{-}}(u)>0$).
 
-\subsubsection{Finding $d$ for fixed $r$ and $c$}
+\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
+and \texorpdfstring{$c$}{c}}
 
 $\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
 $\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
@@ -1000,7 +1001,7 @@ This section studies the numerical conditions that $u$ must satisfy as per
 lemma \ref{lem:num_test_prob1}
 (or corollary \ref{cor:num_test_prob2})
 
-\subsubsection{Size of pseudo-wall: $\chern_2^P(u)>0$ }
+\subsubsection{Size of pseudo-wall\texorpdfstring{: $\chern_2^P(u)>0$}{}}
 \label{subsect-d-bound-radiuscond}
 
 This condition refers to condition
@@ -1040,8 +1041,7 @@ Expressing this as a bound on $d$, then yields:
 \end{equation}
 
 
-\subsubsection{
-	Semistability of the Semistabilizer:
+\subsubsection{Semistability of the Semistabilizer:
 	\texorpdfstring{
 		$\Delta(u) \geq 0$
 	}{
@@ -1098,8 +1098,7 @@ Notice that in the context of problem \ref{problem:problem-statement-2}
 the constant and linear terms match up with the ones
 for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
 
-\subsubsection{
-	Semistability of the Quotient:
+\subsubsection{Semistability of the Quotient:
 	\texorpdfstring{
 		$\Delta(v-u) \geq 0$
 	}{
@@ -1147,8 +1146,8 @@ However, when specializing to problem \ref{problem:problem-statement-2} again
 And so in this context, the linear and constant terms do match up with the
 previous subsubsections.
 
-\subsubsection{All Bounds on $d$ Together for Problem
-\ref{problem:problem-statement-2}}
+\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
+\texorpdfstring{\ref{problem:problem-statement-2}}{2}}
 \label{subsubsect:all-bounds-on-d-prob2}
 %% RECAP ON INEQUALITIES TOGETHER
 
@@ -1272,8 +1271,8 @@ from plots_and_expressions import typical_bounds_on_d
 \label{fig:d_bounds_xmpl_gnrc_q}
 \end{figure}
 
-\subsubsection{All Bounds on $d$ Together for Problem
-\ref{problem:problem-statement-1}}
+\subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem
+\texorpdfstring{\ref{problem:problem-statement-1}}{1}}
 \label{subsubsect:all-bounds-on-d-prob1}
 
 Unlike for problem \ref{problem:problem-statement-2},
@@ -1874,7 +1873,7 @@ Goals:
 	\item calculate intersection of bounds?
 \end{itemize}
 
-\subsection{Irrational $\beta_{-}$}
+\subsection{Irrational \texorpdfstring{$\beta_{-}$}{êžµ_}}
 
 Goals:
 \begin{itemize}
@@ -1894,7 +1893,8 @@ above.
 The way it works, is by yielding solutions to the problem
 $u=(r,c\ell,\frac{e}{2}\ell^2)$ as follows.
 
-\subsection{Iterating Over Possible $q=\chern^{\beta_{-}}(u)$}
+\subsection{Iterating Over Possible
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
 
 Given a Chern character $v$, the domain of the problem are first verified: that
 $v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
@@ -1908,7 +1908,11 @@ $\chern_1^{\beta_{-}}(u)=q$ for one of the $q$ considered is equivalent to
 satisfying condition \ref{item:chern1bound:lem:num_test_prob2}
 in corollary \ref{cor:num_test_prob2}.
 
-\subsection{Iterating Over Possible $r=\chern_0(u)$ for Fixed $q=\chern^{\beta_{-}}(u)$}
+\subsection{Iterating Over Possible
+\texorpdfstring{$r=\chern_0(u)$}{r}
+for Fixed
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+}
 
 Let $q=\frac{b_q}{n}$ be one of the values of $\chern_1^{\beta_{-}}(u)$ that we
 have fixed. As mentioned before, the only values of $r$ which can
@@ -1947,8 +1951,13 @@ Iterate over such $r$ so that we are guarenteed to satisfy conditions
 in corollary
 \ref{cor:num_test_prob2}, and have a chance at satisfying the rest.
 
-\subsection{Iterating Over Possible $d=\chern_2(u)$ for Fixed $r=\chern_0(u)$
-and $q=\chern^{\beta_{-}}(u)$}
+\subsection{Iterating Over Possible
+\texorpdfstring{$d=\chern_2(u)$}{d}
+for Fixed
+\texorpdfstring{$r=\chern_0(u)$}{r}
+and
+\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
+}
 
 At this point we have fixed $\chern_0(u)=r$ and
 $\chern_1(u)=c=q+r\beta_{-}$.