main.tex

\documentclass[aspectratio=169]{beamer}

\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{sagetex}

\usepackage{ulem}
\usepackage{xcolor}

\usetheme{edmaths}
\renewcommand\mathfamilydefault{\rmdefault}

\title{Bridgeland Stabilities and Finding Walls}
\subtitle{}
\author{Luke Naylor}
\institute{Hodge Club}
\date{March 2023}

\newcommand\RR{\mathbb{R}}
\newcommand\CC{\mathbb{C}}
\newcommand\centralcharge{\mathcal{Z}}
\newcommand\coh{\operatorname{Coh}}
\newcommand\rank{\operatorname{rk}}
\newcommand\degree{\operatorname{deg}}
\newcommand\realpart{\mathfrak{Re}}
\newcommand\imagpart{\mathfrak{Im}}
\newcommand\bigO{\mathcal{O}}

\begin{document}

\begin{sagesilent}
	from sagetexscripts import *
\end{sagesilent}

\begin{frame}
	\titlepage
\end{frame}

\section{Transitioning to Stab on Triangulated Categories}

\begin{frame}{Central Charge}
\end{frame}

\begin{frame}{Central Charge for Mumford Stability}
	\begin{align*}
		&\centralcharge \colon \coh(X) \to \CC \\
		&\centralcharge (E) = - \degree(E) + i \rank(E)
	\end{align*}

	\vfill

	\resizebox{1\hsize}{!}{
		\sageplot{fig1.plot()}
	}

	\begin{columns}[T] % align columns
	\begin{column}{.48\linewidth}
		\[
			\centralcharge (E) = r(E) e^{i\pi \varphi(E)}
		\]
		\begin{center}
			\begin{large}
			$\varphi$ called "phase"
			\end{large}
		\end{center}
	\end{column}%
	%\hfill%
	\begin{column}{.48\linewidth}
		\[
			\mu(E) =
			\frac{
				- \realpart(\centralcharge(E))
			}{
				\imagpart(\centralcharge(E))
			}
			\quad
		\]
		\begin{center}
			\begin{large}
				(allow for $+\infty$)
			\end{large}
		\end{center}
	\end{column}%
	\end{columns}

\end{frame}


\begin{frame}
	\resizebox{0.5\hsize}{!}{
		\sageplot{fig2.plot()}
	}
\end{frame}

\begin{frame}
	\resizebox{0.5\hsize}{!}{
		\sageplot{fig3.plot()}
	}
\end{frame}

\begin{frame}
	\resizebox{0.5\hsize}{!}{
		\sageplot{fig4.plot()}
	}
\end{frame}

\begin{frame}{title}
\begin{columns}[t,onlytextwidth] % align columns
	\begin{column}{.33\linewidth}%
	\resizebox{1.1\hsize}{!}{
		\sageplot{fig2.plot()}
	}
	\end{column}%
	%\hfill%
	\begin{column}{.33\linewidth}%
	\resizebox{1.1\hsize}{!}{
		\sageplot{fig3.plot()}
	}
	\end{column}%
	%\hfill%
	\begin{column}{.33\linewidth}%
	\resizebox{1.1\hsize}{!}{
		\sageplot{fig4.plot()}
	}
	\end{column}%
	\end{columns}

	\vfill

	\begin{itemize}
		\item $\centralcharge(\bigO_X) = \centralcharge(\bigO_X[2])$
		\item But $\phi(\bigO_X) \not= \phi(\bigO_X[2])$
		\item Stability decided among $E \in D^b(X) \colon \phi(E) \in (0, \pi]$
	\end{itemize}
\end{frame}

\section{Section 1}

\section{Section 2}

\subsection{Subsection 2.1}


\end{document}