diff --git a/isospectral.tex b/isospectral.tex index 3492287..ab9f9ff 100644 --- a/isospectral.tex +++ b/isospectral.tex @@ -8,6 +8,7 @@ \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. + \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}. \end{enumerate} @@ -32,6 +33,12 @@ \end{comment} \begin{frame}{The silver lining} + \begin{tikzpicture} + \clip (0,0) rectangle (\paperwidth,\paperheight); + \fill[color=orange] (\paperwidth-10pt,0) rectangle (\paperwidth,\paperheight); + % Added + \fill[color=orange](0,0) rectangle (10pt,\paperheight); + \end{tikzpicture} \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. diff --git a/main.pdf b/main.pdf index 98d316f..3355227 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/prerequisites.tex b/prerequisites.tex index bc486a8..d309d29 100644 --- a/prerequisites.tex +++ b/prerequisites.tex @@ -6,6 +6,7 @@ \caption{The prototypical example of a manifold a mug [13].} \label{fig:mug-neighbourhoods} \end{figure} + \pause \begin{figure}[H] \centering @@ -20,6 +21,7 @@ \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 $M \in \Gamma$. + \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. \end{itemize} @@ -49,5 +51,6 @@ \end{align*} \end{definition} + \pause In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb Z$. \end{frame}