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}