Reformat twisted chern and loose bound on r into theorem envs

main.tex

-%% Write basic article template
+%% Write  basic article template
 
 \documentclass[12pt]{article}
 \usepackage{amsmath}
@@ -25,6 +25,7 @@
 
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemmadfn}{Lemma/Definition}[section]
+\newtheorem{dfn}{Definition}[section]
 
 \begin{document}
 
@@ -84,12 +85,15 @@ bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 \section{Characteristic Curves of Stability Conditions Associated to Chern
 Characters}
 
-\section{Twisted Chern Characters of Pseudo Destabilizers}
-\label{sec:twisted-chern}
-
-For a given $\beta$, we can define a twisted Chern character
-$\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
+\section{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
+$\beta\in\QQ$}
 
+\begin{dfn}[Twisted Chern Character]
+\label{sec:twisted-chern}
+For a given $\beta$, define the twisted Chern character as follows.
+\[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
+\noindent
+Component-wise, this is:
 \begin{align*}
 	\chern^\beta_0(E) &= \chern_0(E)
 \\
@@ -99,6 +103,7 @@ $\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
 \end{align*}
 
 % TODO I think this^ needs adjusting for general Surface with $\ell$
+\end{dfn}
 
 $\chern^\beta_1(E)$ is the imaginary component of the central charge
 $\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
@@ -120,6 +125,18 @@ normal one. So $0 \leq \Delta(E)$ yields:
 	2\chern^\beta_0(E) \chern^\beta_2(E) \leq \chern^\beta_1(E)^2
 \end{equation}
 
+\begin{theorem}[Bound on $r$ - Benjamin Schmidt]
+Given a Chern character $v$ such that $\beta_{-}(v)\in\QQ$, the rank $r$ of
+any semistabilizer $E$ of some $F \in \firsttilt\beta$ with $\chern(F)=v$ is
+bounded above by:
+
+\begin{equation*}
+	r \leq \frac{mn^2 \chern^\beta_1(v)^2}{\gcd(m,2n^2)}
+\end{equation*}
+\end{theorem}
+
+\begin{proof}
+
 The restrictions on $\chern^\beta_0(E)$ and $\chern^\beta_2(E)$
 is best seen with the following graph:
 
@@ -142,6 +159,8 @@ for the rank of $E$:
 	&\leq \frac{mn^2 \chern^\beta_1(F)^2}{\gcd(m,2n^2)}
 \end{align}
 
+\end{proof}
+
 \section{B.Schmidt's Method}
 
 \section{Limitations}