From 28cb8dd720867c180cf4af12eaeac7c062456749 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 12:41:01 +0000
Subject: [PATCH] Update on Overleaf.
---
isospectral.tex | 4 ++--
prerequisites.tex | 4 ++--
2 files changed, 4 insertions(+), 4 deletions(-)
diff --git a/isospectral.tex b/isospectral.tex
index 7cab952..4270b12 100644
--- a/isospectral.tex
+++ b/isospectral.tex
@@ -7,10 +7,10 @@
\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 of this operator is a subset of $[0, \infty)$. We call it the \textcolor{blue}{\emph{spectrum}} of the manifold.
\pause
\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}.
+ \item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \textcolor{red}{\emph{isospectral pair}}.
\end{enumerate}
\end{frame}
diff --git a/prerequisites.tex b/prerequisites.tex
index beafa93..9411a5b 100644
--- a/prerequisites.tex
+++ b/prerequisites.tex
@@ -20,10 +20,10 @@
\begin{frame}{Quotients of the $3$-sphere}
\begin{itemize}
\item $\so 4$ is isomorphic to the isometry group of $\S^3$.
- \item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
+ \item Subgroups $\Gamma \leqslant \so 4$ define \textcolor{darkyellow}{equivalence classes} of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
\pause
\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
- \item This can be easily generalised to the $n$-sphere.
+ \item This can be easily generalized to the $n$-sphere.
\end{itemize}
\end{frame}