52 lines
2.1 KiB
TeX
52 lines
2.1 KiB
TeX
\section{The spectral geometry of hyperbolic and spherical spaces}
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\begin{frame}{The Laplace-Beltrami operator}
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\begin{enumerate}
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\begin{definition}[Laplace-Beltrami operator]
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a generalization of the Laplace operator to more general spaces
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\begin{align*}
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\Delta f \coloneq \divergence \nabla f.
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\end{align*}
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\end{definition}
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\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \textcolor{blue}{\emph{spectrum}} of the manifold.
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\pause
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\item The spectrum determines many things about the space (like its volume).
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\item When two distinct (up to isomorphism) manifolds share a spectrum, they for an \textcolor{red}{\emph{isospectral pair}}.
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\end{enumerate}
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\end{frame}
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\begin{comment}
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\begin{frame}{Can isospectral pairs approximate your mom}
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\begin{itemize}
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\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).
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\item Isospectral pairs share the order of their homotopy groups.
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\item The paper classifies spherical forms with homotopy group of order $<24$.
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\item It follows that any such isospectral pairs (with group order $<24$) are lens spaces.
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\end{itemize}
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\end{frame}
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\begin{frame}{Combining lenses into larger spaces}
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\begin{itemize}
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\item We can combine lens spaces by concatenating their lists of indices.
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\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.
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\end{itemize}
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\end{frame}
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\end{comment}
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{
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\usebackgroundtemplate{
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\begin{tikzpicture}
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\clip (0,0) rectangle (\paperwidth,\paperheight);
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\fill[color=silver] (\paperwidth-10pt,0) rectangle (\paperwidth,\paperheight);
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% Added
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\fill[color=silver](0,0) rectangle (10pt,\paperheight);
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\end{tikzpicture}
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}
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\begin{frame}{The silver lining}
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\begin{itemize}
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\item Isospectral manifolds are intriguing in their own right, but in dimension 3 all isospectral manifolds are isometric.
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\item We can thus attempt to infer the shape of our universe based on its spectrum.
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\end{itemize}
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\end{frame}
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}
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