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.gitignore vendored
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*.fdb_latexmk
*.fls
*.log
*.nav
*.snm
*.toc
*.vrb
*.out
*.synctex
*.synctex(busy)

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isospectral.tex Normal file
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\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}

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main.tex
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@ -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}
@ -43,77 +49,81 @@ code-for-last-col = \color{blue}
\maketitle
\section{Introduction & preliminaries}
\begin{frame}{Outline}
%Should we make an outline?
\begin{enumerate}%[<+->]
\item Motivation
\item Part 1
\item Part 2
\item Part 3
\item etc
\end{enumerate}
%Should we make an outline?
\begin{enumerate}%[<+->]
\item Motivation
\item Part 1
\item Part 2
\item Part 3
\item etc
\end{enumerate}
\end{frame}
% should we make is so that these bullet points appear one after another? Yes
\begin{frame}{Abstracting Independence - Motivation}
\begin{itemize}
\item 1
\item 2
\end{itemize}
\begin{itemize}
\item 1
\item 2
\end{itemize}
\end{frame}
\begin{frame}{introduction}
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121fdaaeegag
\end{frame}
\begin{frame}{Preliminaries- quotient groups (Put actual title later)}
ddgagagagaga
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
\begin{itemize}
\item Manifolds $M := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
\item Helmholtz equation on $M$ given by
$$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
\item Helmholtz equation on $M$ given by
$$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
\item In fact $E_\beta^m = \beta^2-1$ for $\beta \in \mathbb{N}$ we call $\beta$ a wave number
\item In fact $E_\beta^m = \beta^2-1$ for $\beta \in \mathbb{N}$ we call $\beta$ a wave number
\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$
\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$
\item Solutions to the Helmholtz equation are calculated for each $H$ and CMB anisotropies $\frac{\delta T}{T}$ computed
\end{itemize}
\item Solutions to the Helmholtz equation are calculated for each $H$ and CMB anisotropies $\frac{\delta T}{T}$ computed
\end{itemize}
\end{frame}
\begin{comment}
\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
\begin{itemize}
\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
\item If $H = \mathbb Z/m\mathbb Z$ for odd $m$ the set of wave numbers is $$\{1,3,\cdots, m\}\cup \{m+1, m+2, m+3, \cdots \}.$$
\item If $H = O^*$, the binary tetrahedral group --- a subgroup of $SO(4)$ of size 24, the set of wave numbers is $$\{ 1,7,9 \} \cup \{13,15,17,\cdots \}.$$
\end{itemize}
\begin{itemize}
\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
\item If $H = \mathbb Z/m\mathbb Z$ for odd $m$ the set of wave numbers is $$\{1,3,\cdots, m\}\cup \{m+1, m+2, m+3, \cdots \}.$$
\item If $H = O^*$, the binary tetrahedral group --- a subgroup of $SO(4)$ of size 24, the set of wave numbers is $$\{ 1,7,9 \} \cup \{13,15,17,\cdots \}.$$
\end{itemize}
\end{frame}
\end{comment}
\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces - Conclusion}
\begin{itemize}
\item For the majority of groups $H$ the anisotropies $\frac{\delta T}{T}$ do not coincide with observations
\item The only for which it does are $H = O^*$ and $H = I^*.$ the \textit{binary octahedral} and \textit{binary icosahedral} groups of order 48 and 120
\begin{itemize}
\item For the majority of groups $H$ the anisotropies $\frac{\delta T}{T}$ do not coincide with observations
\item The only for which it does are $H = O^*$ and $H = I^*.$ the \textit{binary octahedral} and \textit{binary icosahedral} groups of order 48 and 120
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
\caption{Graphical representation of the binary tetrahedral group
[5]}
\label{fig:binary-octahedron}
\end{figure}
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
\caption{Graphical representation of the binary tetrahedral group
[5]}
\label{fig:binary-octahedron}
\end{figure}
\end{frame}
@ -121,71 +131,71 @@ $$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
\begin{frame}[fragile]{Article 2}
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vsgaegaagjadhvakva
\end{frame}
\begin{frame}{CMB radiation in an inhomogeneous spherical space}
Manifolds
Manifolds
\end{frame}
\begin{frame}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
\textit{statistical isotropy}, meaning that the statistical properties of the CMB should be the same in all directions. Testing this assumption is essential, as deviations from isotropy could indicate alternative topologies for the universe.
\textit{statistical isotropy}, meaning that the statistical properties of the CMB should be the same in all directions. Testing this assumption is essential, as deviations from isotropy could indicate alternative topologies for the universe.
\end{frame}
\begin{frame}{Conclusion}
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akbbfbsKJBKJBLJs
\end{frame}
\begin{frame}{To summerize}
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\begin{itemize}
\item 1
\item 2
\item 3
\end{itemize}
egqgaaf
\begin{itemize}
\item 1
\item 2
\item 3
\end{itemize}
\end{frame}
\begin{frame}{References}
\begin{itemize}
\item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
\item R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
\item R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
\item P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
\item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
\end{itemize}
\begin{itemize}
\item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
\item R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
\item R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
\item P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
\item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
\end{itemize}
\end{frame}
\begin{frame}{References}
\begin{itemize}
\begin{itemize}
\item N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
\item M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
\item A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
\item D. Kashino, K. Ichiki, and T. Takeuchi. \textit{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}. Physics Review, 2012.
\item P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
\end{itemize}
\item N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
\item M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
\item A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
\item D. Kashino, K. Ichiki, and T. Takeuchi. \textit{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}. Physics Review, 2012.
\item P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
\end{itemize}
\end{frame}
\begin{frame}{References}
\begin{itemize}
\item E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
\item R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
\item M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
\item B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
\end{itemize}
\begin{itemize}
\item E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
\item R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
\item M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
\item B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
\end{itemize}
\end{frame}
\begin{frame}{}
\begin{center}
\huge Thank You!
\end{center}
\begin{center}
\huge Thank You!
\end{center}
\end{frame}
\end{document}