From 1a67366a27d2e6a92d07d060f28e5fea66fd2abb Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 12:57:16 +0000
Subject: [PATCH 1/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index d98b4df..6c8b73f 100644
--- a/main.tex
+++ b/main.tex
@@ -43,7 +43,7 @@ code-for-last-col = \color{blue}
}
-\title{Computing CMB temperature fluctuations for spherical spaces}
+\title{Computing CMB Temperature Fluctuations for Spherical Spaces}
\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
%\institute{Presenting: Javier, Juš}
From c8b7ccab94b3658d094b8dd3248201905f24545d Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 12:58:43 +0000
Subject: [PATCH 2/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 6c8b73f..480c094 100644
--- a/main.tex
+++ b/main.tex
@@ -45,7 +45,7 @@ code-for-last-col = \color{blue}
\title{Computing CMB Temperature Fluctuations for Spherical Spaces}
\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
-%\institute{Presenting: Javier, Juš}
+\institute{Presenting: Javier, Juš}
\date{March 24, 2025}
From fc67d22c14dfed5241b56c8dc26ff14d5573d4f7 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 14:01:05 +0000
Subject: [PATCH 3/9] Update on Overleaf.
---
isospectral.tex | 2 +-
main.tex | 7 ++++---
prerequisites.tex | 2 +-
3 files changed, 6 insertions(+), 5 deletions(-)
diff --git a/isospectral.tex b/isospectral.tex
index 034ad31..2259db0 100644
--- a/isospectral.tex
+++ b/isospectral.tex
@@ -44,7 +44,7 @@
\begin{frame}{The silver lining}
\begin{itemize}
- \item Isospectral pairs are intriguing in their own right, but they cannot occur in dimension $3$.
+ \item Isospectral manifolds are intriguing in their own right, but in dimension 3 all isospectral manifolds are isometric.
\item We can thus attempt to infer the shape of our universe based on its spectrum.
\end{itemize}
\end{frame}
diff --git a/main.tex b/main.tex
index 480c094..597aad4 100644
--- a/main.tex
+++ b/main.tex
@@ -45,7 +45,7 @@ code-for-last-col = \color{blue}
\title{Computing CMB Temperature Fluctuations for Spherical Spaces}
\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
-\institute{Presenting: Javier, Juš}
+\institute{University of Groningen}
\date{March 24, 2025}
@@ -96,7 +96,7 @@ code-for-last-col = \color{blue}
(\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0.
\end{align*}
\pause
- \item In fact $E_{\textcolor{red}{\beta}}^m = {\textcolor{red}{\beta}}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$. We call $\textcolor{red}{\beta}$ a wave number.
+ \item In fact $E_{\textcolor{red}{\beta}}^M = {\textcolor{red}{\beta}}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$. We call $\textcolor{red}{\beta}$ a wave number.
\pause
\item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
\pause
@@ -140,6 +140,7 @@ code-for-last-col = \color{blue}
\begin{frame}{Inhomogeneous spherical space}
\begin{itemize}
+ \pause
\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
\item \textcolor{red}{Multi-connected} space: it has non-contractable loops.
\item \textcolor{red}{Inhomogeneous} space: it does not look identical from every point in space.
@@ -226,7 +227,7 @@ code-for-last-col = \color{blue}
\textbf{Possible Explanations:}
\begin{itemize}
\item A real cosmological signal? → A finite universe or new physics.
- \item A systematic effect? → Foreground contamination or instrumental noise.
+ \item A systematic effect? → Technological or theory related limitations.
\end{itemize}
\end{columns}
\end{frame}
diff --git a/prerequisites.tex b/prerequisites.tex
index b53307a..822de1b 100644
--- a/prerequisites.tex
+++ b/prerequisites.tex
@@ -3,7 +3,7 @@
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
- \caption{The prototypical example of a manifold a mug [13].}
+ \caption{The prototypical example of a manifold -- a mug [13].}
\label{fig:mug-neighbourhoods}
\end{figure}
\pause
From df27bebc38daf2665d7a7ed31a2471afb2a7f897 Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:13 +0000
Subject: [PATCH 4/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 597aad4..565a372 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is isotropic, the \textcolor{red}{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
From 6d5864149711bebb1d33892c39b0c276921b7ddb Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:37 +0000
Subject: [PATCH 5/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 565a372..597aad4 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is isotropic, the \textcolor{red}{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
From addf05f02dfd5c844c02061ea140e73e518b5936 Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:49 +0000
Subject: [PATCH 6/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 597aad4..d86f896 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is \teisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
From 8cf31758282bd8043a3424246e80310d7f031105 Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:52 +0000
Subject: [PATCH 7/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index d86f896..c6bc6b0 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is \teisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is \textisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
From 66e1617cacbaa94cbc22ad0881ff59bbbfc6fba3 Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:54 +0000
Subject: [PATCH 8/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index c6bc6b0..e676bcb 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is \textisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is \textxisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
From 057c06368a5316a107b79ee480737f0c465508cf Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 14:08:59 +0000
Subject: [PATCH 9/9] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index e676bcb..1c10bd2 100644
--- a/main.tex
+++ b/main.tex
@@ -170,7 +170,7 @@ code-for-last-col = \color{blue}
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
- \item If the universe is \textxisotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \item If the universe is \textcolor{red}{isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]