LaTeX Beamer presentation template for my Talk

Here is the LaTeX Beamer presentation template for my talk:

A Modern Approach to Classical Mathematics: A Mathematical Feast

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\documentclass{beamer}

% Use Madrid theme as base
\usetheme{Madrid}

% Create custom navy blue and gold theme
\definecolor{navyblue}{RGB}{25, 40, 80} % Define navy blue
\definecolor{navydark}{RGB}{15, 25, 55} % Darker navy blue for contrast
\definecolor{navylight}{RGB}{65, 85, 135} % Lighter navy blue for backgrounds
\definecolor{gold}{RGB}{212, 175, 55} % Gold color
\definecolor{goldlight}{RGB}{232, 215, 155} % Light gold for backgrounds

% Set custom colors for various elements
\setbeamercolor{structure}{fg=gold} % Controls color of titles, sections, etc.
\setbeamercolor{palette primary}{fg=gold, bg=navyblue}
\setbeamercolor{palette secondary}{fg=gold, bg=navydark}
\setbeamercolor{palette tertiary}{fg=gold, bg=navydark!90!black}
\setbeamercolor{palette quaternary}{fg=gold, bg=navydark!95!black}

% Set colors for specific elements
\setbeamercolor{titlelike}{parent=palette primary}
\setbeamercolor{frametitle}{parent=palette primary}
\setbeamercolor{title}{parent=palette primary}
\setbeamercolor{item}{fg=gold}
\setbeamercolor{block title}{fg=gold, bg=navyblue}
\setbeamercolor{block body}{fg=black, bg=goldlight!40}

% Required packages
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{pifont}
\usepackage{tikz-cd}
\usepackage{tikz}
\DeclareGraphicsExtensions{.jpg}

% Define theorem-style environments with appropriate borders
\theoremstyle{plain}
\newtheorem{proposition}{Proposition}

% Custom commands for menu items
\newcommand{\menuitem}[1]{\textbf{\large #1}}
\newcommand{\menuitemdesc}[1]{\normalsize\textit{#1}}

% Title information - Replace with your own content
\title[Short Title]{Your Presentation Title Here}
\author[Your Name]{Your Name}
\institute[Institution]{Your Department\\Your Institution\\Your Address}
\date{\today}

\begin{document}

% Title slide
\begin{frame}
\titlepage
\end{frame}

% Creative menu-style table of contents
\begin{frame}
\frametitle{}
\begin{center}
{\fontsize{16}{18}\selectfont\textsc{Mathematical Menu}}
\vspace{0.1cm}
{\small\textit{An intellectual gastronomic experience}}
\vspace{0.4cm}
\begin{tikzpicture}
\draw[double, line width=0.6pt] (0,0) -- (9,0);
\end{tikzpicture}
\end{center}
\vspace{0.2cm}
\begin{description}
\item[\Large\textit{Apéritif}] 
    \begin{itemize}
    \item \small\textit{Your opening topic here}
    \item \small\textit{Another introductory concept}
    \end{itemize}
\vspace{0.2cm}
\item[\Large\textit{Entrée}]
    \begin{itemize}
    \item \small\textit{First main concept}
    \item \small\textit{Related theoretical foundation}
    \end{itemize}
\vspace{0.2cm}
\item[\Large\textit{Main Course}]
    \begin{itemize}
    \item \small\textit{Core topic of your presentation}
    \end{itemize}
\vspace{0.2cm}
\item[\Large\textit{Cheese Course}]
    \begin{itemize}
    \item \small\textit{Additional related topic}
    \item \small\textit{Secondary concept or application}
    \end{itemize}
\vspace{0.2cm}
\item[\Large\textit{Dessert}]
    \begin{itemize}
    \item \small\textit{Concluding topic or advanced application}
    \end{itemize}
\end{description}
\vspace{0.2cm}
\begin{center}
\begin{tikzpicture}
\draw[double, line width=0.6pt] (0,0) -- (9,0);
\end{tikzpicture}
\vspace{0.1cm}
{\small\textit{Bon appétit mathématique!}}
\end{center}
\end{frame}

% Traditional table of contents (alternative option)
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\frametitle{Outline}
\tableofcontents
\end{frame}

% Section 1
\section{Introduction}
\begin{frame}
\frametitle{Introduction}
\begin{itemize}
\item Your first main point here
\item Your second main point here
\item Your third main point here
\end{itemize}
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% Section 2
\section{Main Content}
\begin{frame}
\frametitle{Main Topic}
\begin{itemize}
\item Content point 1
\item Content point 2
\item Content point 3
\end{itemize}
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% Example of a frame with blocks
\begin{frame}
\frametitle{Using Blocks}
\begin{block}{Important Note}
This is an example of how blocks look with the navy and gold theme.
\end{block}

\begin{proposition}
This is how propositions appear in this template.
\end{proposition}
\end{frame}

% Example of a frame with mathematics
\begin{frame}
\frametitle{Mathematical Content}
Here's an example of mathematical notation:
\begin{align}
f(x) &= ax^2 + bx + c \\
\int_{-\infty}^{\infty} e^{-x^2} dx &= \sqrt{\pi}
\end{align}

You can also use inline math like $E = mc^2$.
\end{frame}

% Conclusion
\section{Conclusion}
\begin{frame}
\frametitle{Conclusion}
\begin{itemize}
\item Summary point 1
\item Summary point 2
\item Future work or next steps
\end{itemize}
\end{frame}

% Thank you slide
\begin{frame}
\frametitle{Thank You}
\centering
\Large Thank you for your attention!

\vspace{1cm}
Questions?
\end{frame}

\end{document}

 

Functions with Compact Support and Algebraic Structures

Functions with Compact SupportRing‐valued functionsModule‐valued functionsExample: infinite coproduct

 

Functions with Compact Support

Let $G$$G$ be a group and $X$$X$ a topological space. Then ${\mathrm{Hom}}_{\mathbf{Set}}(X,G)$$\mathrm{Hom}_{\mathbf{Set}}(X,G)$ is a group under pointwise multiplication:

$$(fg)(x)=f(x)g(x),f^{-1}(x)=f(x{)}^{-1},e(x)=e.$$

Define

$$D(f)=\{x\in X:f(x)\ne e\},\mathrm{supp}(f)=\overline{D(f)}.$$

Lemma 1. $\overline{U\cup V}=\bar{U}\cup \bar{V}.$$\overline{U\cup V} = \overline{U}\,\cup\,\overline{V}.$

Proof. It is easy to see that

$$\bar{U}\subseteq K⟺U\subseteq i(K)$$

The closure operator $\overline{(-)}$$\overline{(-)}$ is left adjoint to the inclusion of closed sets into all subsets, hence preserves unions (colimits). $◻$$\square$

It follows that

$$\mathrm{supp}(fg)=\overline{D(fg)}\subseteq \overline{D(f)\cup D(g)}=\mathrm{supp}(f)\cup \mathrm{supp}(g).$$

---

Lemma 2. The union of two compact subsets of $X$$X$ is compact.

Proof. If $\{U_i{\}}_{i\in I}$$\{U_i\}_{i\in I}$ covers $U\cup V$$U\cup V$, it covers each of $U$$U$ and $V$$V$. By compactness extract finite subcovers for $U$$U$ and $V$$V$, then take their union.
$◻$$\square$

Lemma 3. A closed subset of a compact space is compact.

Proof. If $\{U_i\}$$\{U_i\}$ covers $F\subseteq K$$F\subseteq K$, then $\{X\setminus F\}\cup \{U_i\}$$\{X\setminus F\}\cup\{U_i\}$ covers $K$$K$. Extract a finite subcover and discard $X\setminus F$$X\setminus F$.
$◻$$\square$

Corollary. If $f,g$$f,g$ have compact support then so does $fg$$f g$.

Proposition.

$$K=\{f\in {\mathrm{Hom}}_{\mathbf{Set}}(X,G):\mathrm{supp}(f)\text{ is compact}\}$$

is a subgroup.

Proof. The constant map $e$$e$ has empty (hence compact) support. Closure under products follows from the corollary. For inverses note $\mathrm{supp}(f^{-1})=\mathrm{supp}(f)$$\mathrm{supp}(f^{-1})=\mathrm{supp}(f)$.
$◻$$\square$

---

Ring‐valued functions

If $R$$R$ is a ring, pointwise addition and multiplication make ${\mathrm{Hom}}_{\mathbf{Set}}(X,R)$$\mathrm{Hom}_{\mathbf{Set}}(X,R)$ into a ring. Defining $D(f)$$D(f)$ and $\mathrm{supp}(f)$$\mathrm{supp}(f)$ as above, one shows:

• $\mathrm{supp}(f+g)\subseteq \mathrm{supp}(f)\cup \mathrm{supp}(g)$$\mathrm{supp}(f+g)\subseteq \mathrm{supp}(f)\cup\mathrm{supp}(g)$.

• $\mathrm{supp}(fg)\subseteq \mathrm{supp}(f)\cap \mathrm{supp}(g)$$\mathrm{supp}(f g)\subseteq \mathrm{supp}(f)\cap\mathrm{supp}(g)$.

Hence

$$K=\{f:X\to R:\mathrm{supp}(f)\text{ is compact}\}$$

is an ideal, and if $X$$X$ itself is compact, a subring.

---

Module‐valued functions

If $M$$M$ is an $R$$R$‐module then ${\mathrm{Hom}}_{\mathbf{Set}}(X,M)$$\mathrm{Hom}_{\mathbf{Set}}(X,M)$ is an $R$$R$‐module pointwise, and

$$\mathrm{supp}(f+g)\subseteq \mathrm{supp}(f)\cup \mathrm{supp}(g),\mathrm{supp}(rf)\subseteq \mathrm{supp}(f).$$

Thus

$$K=\{f:X\to M:\mathrm{supp}(f)\text{ is compact}\}$$

is an $R$$R$‐submodule.

---

Example: infinite coproduct

Let $I$$I$ be discrete. Then the only compact subsets of $I$$I$ are the finite ones, and we could construct the coproduct as:

$$\underset{i\in I}{⨁}{M}_i:=\{(x_i{)}_{i\in I}\in \prod _{i\in I}{M}_i|(x_i{)}_{i\in I}\text{ has compact support}\}$$