Add section about isospectral pairs

.gitignore

@@ -4,6 +4,10 @@
 *.fdb_latexmk
 *.fls
 *.log
+*.nav
+*.snm
+*.toc
+*.vrb
 *.out
 *.synctex
 *.synctex(busy)

isospectral.tex

@@ -0,0 +1,37 @@
+\section{ The spectral geometry of hyperbolic and spherical spaces}
+\begin{frame}{The Laplace-Beltrami operator}
+	\begin{enumerate}
+		\begin{definition}[Laplace-Beltrami operator]
+			a generalization of the Laplace operator to more general spaces
+			\begin{align*}
+				\Delta f \coloneq \divergence \nabla f
+			\end{align*}
+		\end{definition}
+		\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \emph{spectrum} of the manifold.
+		\item The spectrum determines many things about the space (like its volume).
+		\item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \emph{isospectral pair}.
+	\end{enumerate}
+\end{frame}
+
+\begin{frame}{Can isospectral pairs approximate your mom}
+	\begin{itemize}
+		\item One of the problems the paper studies is finding volume-maximizing isospectral pairs of spherical forms (i.e. the spaces we are interested in).
+		\item Isospectral pairs share the order of their homotopy groups.
+		\item The paper classifies spherical forms with homotopy group of order $<24$.
+		\item It follows that any such isospectral pairs (with group order $<24$) are lens spaces.
+	\end{itemize}
+\end{frame}
+
+\begin{frame}{Combining lenses into larger spaces}
+	\begin{itemize}
+		\item We can combine lens spaces by concatenating their lists of indices.
+		\item Together with a computational search, the paper builds a table of the solutions for a bunch of dimensions. The general case is still an open question.
+	\end{itemize}
+\end{frame}
+
+\begin{frame}{The silver lining}
+	\begin{itemize}
+		\item Isospectral pairs are cool and all, but they cannot occur in dimension $3$.
+		\item We can thus attempt to infer the shape of our universe based on its spectrum.
+	\end{itemize}
+\end{frame}

main.tex

@@ -12,11 +12,17 @@
 \usepackage{tikz}
 \usepackage{pgfplots}
 \usepackage{verbatim}
+\usepackage{mathtools}
 \pgfplotsset{compat = newest}
 \usetikzlibrary{matrix}
 \usepackage[dvipsnames]{xcolor}
 \usetikzlibrary{perspective}
 
+\DeclareMathOperator{\divergence}{div}
+\DeclareMathOperator{\lensop}{L}
+\DeclareMathOperator{\rotmat}{R}
+\newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
+
 % cool color
 
 \usepackage{xcolor}
@@ -43,6 +49,8 @@ code-for-last-col = \color{blue}
 
 \maketitle
 
+\section{Introduction & preliminaries}
+
 \begin{frame}{Outline}
 	%Should we make an outline?
 	\begin{enumerate}%[<+->]
@@ -71,6 +79,8 @@ code-for-last-col = \color{blue}
 	ddgagagagaga
 \end{frame}
 
+\include{isospectral}
+
 \begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
 	\begin{itemize}
 		\item Manifolds $M := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite