From acc7e17c1758481a36286f422113fa22f11fc13e Mon Sep 17 00:00:00 2001
From: prescientmoon <git@moonythm.dev>
Date: Thu, 20 Mar 2025 16:46:52 +0100
Subject: [PATCH] Lens spaces and whatnot
---
isospectral.tex | 2 ++
main.tex | 17 ++++++++++-------
prerequisites.tex | 42 +++++++++++++++++++++++++++++++++++++-----
3 files changed, 49 insertions(+), 12 deletions(-)
diff --git a/isospectral.tex b/isospectral.tex
index 1d705e2..3492287 100644
--- a/isospectral.tex
+++ b/isospectral.tex
@@ -13,6 +13,7 @@
\end{enumerate}
\end{frame}
+\begin{comment}
\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).
@@ -28,6 +29,7 @@
\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}
+\end{comment}
\begin{frame}{The silver lining}
\begin{itemize}
diff --git a/main.tex b/main.tex
index 61b063c..7e47211 100644
--- a/main.tex
+++ b/main.tex
@@ -2,8 +2,6 @@
\usetheme{Warsaw}
\usecolortheme{lily}
-
-
% boadilla seems to be the only one with enough space for stuff.compile it now it is very toxic but works
\usepackage{graphicx} % Required for inserting images
@@ -20,8 +18,12 @@
\DeclareMathOperator{\divergence}{div}
\DeclareMathOperator{\lensop}{L}
-\DeclareMathOperator{\rotmat}{R}
+\DeclareMathOperator{\rotmatop}{R}
+\DeclareMathOperator{\soop}{SO}
\newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
+\newcommand*{\so}[1]{\soop\left(#1\right)}
+\newcommand*{\rotmat}[1]{\rotmatop\left(#1\right)}
+\renewcommand{\S}{\mathbb{S}}
% cool color
@@ -49,9 +51,12 @@ code-for-last-col = \color{blue}
\setbeamertemplate{headline}{%
\leavevmode%
\begin{beamercolorbox}[wd=\paperwidth,ht=2.5ex,dp=1.5ex]{section in head/foot}%
- \hspace{.5em}\strut\insertsectionhead\hfill\mbox{}%
+ \mbox{}\hspace{.5em}\strut\insertsectionhead\hfill%
\end{beamercolorbox}%
}
+
+\setbeamercolor{block title}{fg=white, bg=purple!50!black}
+\setbeamercolor{block body}{fg=black, bg=pink!20}
\begin{document}
@@ -76,10 +81,10 @@ code-for-last-col = \color{blue}
\item 1
\item 2
\end{itemize}
-
\end{frame}
\include{prerequisites}
+
\include{isospectral}
\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
@@ -123,7 +128,6 @@ code-for-last-col = \color{blue}
\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}
@@ -158,7 +162,6 @@ code-for-last-col = \color{blue}
\centering
\includegraphics[width=0.6\linewidth]{DSE-Test Graph}
\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
- \label{fig:Decorrelated Statistical Spectrum}
\end{figure}
\end{frame}
diff --git a/prerequisites.tex b/prerequisites.tex
index 0f7edf6..920cbd1 100644
--- a/prerequisites.tex
+++ b/prerequisites.tex
@@ -1,9 +1,9 @@
\section{Prerequisites}
-\begin{frame}{Introduction}
+\begin{frame}{Manifolds \& Homotopy groups}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
- \caption{The prototypical example of a manifold a mug. Image source: [13]}
+ \caption{The prototypical example of a manifold a mug [13].}
\label{fig:mug-neighbourhoods}
\end{figure}
@@ -11,11 +11,43 @@
\centering
%$ \captionsetup{width=.75\linewidth}
\includegraphics[width=0.2\linewidth]{Contractible loops.png}
- \caption{Diagram showing two double tori with (non)-contractible paths. Image source[7]}
+ \caption{Two double tori with (non)-contractible paths [7].}
\label{fig:CoLoop}
\end{figure}
\end{frame}
-\begin{frame}{Preliminaries- quotient groups (Put actual title later)}
- ddgagagagaga
+\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 $\R^3$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
+ \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}
+\end{frame}
+
+\begin{frame}{Lens spaces}
+ \begin{itemize}
+ \item Lens spaces are obtained by taking the quotient of some $n$-sphere by a cyclic group.
+ \item They cannot be distinguished by their homotopy group alone.
+ \end{itemize}
+\end{frame}
+\begin{frame}{Lens spaces — the explicit construction}
+ \begin{definition}[Lens space]
+ Given $q \in \mathbb N$ and $s \in \mathbb Z ^n$ each coprime with $q$
+ \begin{align*}
+ \lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
+ \end{align*}
+ where $\langle M \rangle$ is the group generated by
+ \begin{align*}
+ M_{q,s} \coloneq
+ \begin{pmatrix}
+ \rotmat{2 \pi s_1 / q} & & & \\
+ & \rotmat{2 \pi s_2 / q} & & \\
+ & & \ddots & \\
+ & & & \rotmat{2 \pi s_n / q}
+ \end{pmatrix}
+ \end{align*}
+ \end{definition}
+
+ In the $\S^3$ case, we denote $\lens p q \coloneq \lens p {(1, q)}$ for coprime $p, q \in \mathbb \mathbb Z$.
\end{frame}