%% The beamer package is used to make presentations
\documentclass[12pt]{beamer}
%% The handout option can be used to compile a version without
%% pauses that can be printed convienently
%%\documentclass[11pt,handout]{beamer}

% %% Use a specific theme
\usetheme{Warsaw}

% %% And a specific color theme
\usecolortheme{orchid}

%% Use a serif font theme
\usefonttheme{serif}

%%% See for instance https://deic.uab.cat/~iblanes/beamer_gallery/
%%% for samples of all the combinations.

%% Use also various symbols from the AMS
\usepackage{amsmath,amssymb}

%% A font with slightly different calligraphic letters
\usepackage{mathrsfs}

%% The basic package to include graphics
\usepackage{graphicx}

%% For commutative diagrams
\usepackage{tikz-cd}
\tikzset{commutative diagrams/arrow style=math font}

%% Beamer already defined theorem, lemma and corollary.
\theoremstyle{plain}
% \newtheorem{theorem}{Theorem}[section]
% \newtheorem*{theorem*}{Theorem}
% \newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
% \newtheorem{corollary}[theorem]{Corollary}

%% Style for remarks
\theoremstyle{remark}
\newtheorem{remark}{Remark}

%% Style for definitions
%% Beamer already has definition.
\theoremstyle{definition}
% \newtheorem{definition}[theorem]{Definition}

%% Define a new command to typeset SL as a function
\DeclareMathOperator{\SL}{SL}

%% Some other operators
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\St}{St}

%% An operator where the subscripts/superscripts appear under/over the
%% symbol
\DeclareMathOperator*{\expect}{\mathbf{E}}

%% Allow to put some text over an arrow: this is a command with one
%% argument
\newcommand{\fleche}[1]{\stackrel{{#1}}{\longrightarrow}}

%% Use the AMS versions of the "less than or equal to" and "larger
%% than or equal to" signs
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}

%% A command "boldmath" to type bold mathematical formulas in such a
%% way that small letters remain slanted instead of being straight
\newcommand{\bm}[1]{\text{\boldmath${#1}$}}

%% Modulo with parentheses but smaller horizontal space than \pmod
\newcommand{\mods}[1]{\,(\mathrm{mod}\,{#1})}

%% The basic number sets in the bold style
\newcommand{\C}{\mathbf{C}}
\newcommand{\R}{\mathbf{R}}
\newcommand{\F}{\mathbf{F}}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Pp}{\mathbf{P}}

%% A symbol for disjoint union; mathrel indicates that the spacing
%% around the symbol should as for other relation symbols.
\newcommand{\dunion}{\mathrel{\sqcup}}

%% Prefer the varepsilon for epsilon, and give it a shorter name
\newcommand{\eps}{\varepsilon}

%%%% Some beamer configuration options

%% Remove the navigation symbols at the bottom of the slides
\setbeamertemplate{navigation symbols}{}

    

%% Begin the text
\begin{document}

%%% The basic information, like for a paper

%% Title
\title{File for 25.10.2023}

%% Author
\author{E. Kowalski and Someone Else}

%% Physical address
\institute{ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland}

%5 Each slide appears in a "frame" environment
\begin{frame}

  \titlepage
  
\end{frame}

\begin{frame}

  \frametitle{\phantom{Title}}
%%  \frametitle{The Riemann--Roch Theorem}

  \begin{theorem}[Riemann, Roch]
    Let $C$ be a smooth projective curve over an algebraically closed
    field $k$, and $g\geq 0$ its genus. For any divisor $D$ on $C$, we
    have the formula
    $$
    \ell(D)-\ell(K-D)=\deg(D)+1-g,
    $$
    where $K$ denotes a canonical divisor on $C$.
  \end{theorem}
  
\end{frame}

\begin{frame}
  \frametitle{A commutative diagram}

  $$
  %% tikz-cd doesn't always work well with beamer
  %% The ampersand replacement helps
  \begin{tikzcd}[ampersand replacement=\&]
    1\arrow[r] \& A\arrow[d,"f"]\arrow[r] \& B \arrow[r]\arrow[d] \& C\arrow[d]\arrow[r]\& 0
    \\
    1\arrow[r] \& A'\arrow[r] \& B'\arrow[r]\& C'\arrow[r]\& 0
\end{tikzcd}
$$

\end{frame}

\begin{frame}
  \frametitle{Pausing}

  Here is a first sentence.
  \pause Which continues.

  \pause

  And another one.
  
\end{frame}

\begin{frame}
  \frametitle{Lists}

  \begin{itemize}
  \item<1-3> First item.
  \item<2-> This item is the second, and will stay there.
  \item<3> This item is the third, and will now disappear.
  \item<1,4> This item was there at the beginning, and comes back.
  \item<5-> This is the last.
  \end{itemize}
\end{frame}



\begin{frame}
  \frametitle{Including pictures}
  
   \begin{center}
   \begin{columns}
     \begin{column}{2.5in}
       \framebox{\includegraphics[width=2in]{notes-rhb-60.pdf}}
     \end{column}
     \begin{column}{1.5in}
       %%\vspace{3cm}
       \framebox{\includegraphics[width=1.2in]{savant.jpg}}
     \end{column}
   \end{columns}
 \end{center}
 
\end{frame}



\begin{frame}
  \frametitle{Moving pictures}

  %% Divide in multiple columns
  \begin{columns}
    %% Width of the first column
    \column{3cm}
    \begin{itemize}
    \item<1-> First item.
    \item<2-> Second item.
    \item<3> Third item.
    \end{itemize}
    %% Second column
    \column{7cm}
    %% "Fixes" the positions of the items so that they do not move
    %% vertically when passing from one to the other.
    \begin{overlayarea}{\textwidth}{7cm}
      %% Some vertical space
      \vskip0.5cm
      %% This will appear with the first item Other combinations are
      %% allowed like <1,2>, <2->, etc.  In this case it would not be
      %% useful since the pictures would take too much space
      \only<1>{
        \framebox{\includegraphics[width=6.5cm]{birch-10007.pdf}}}
      \only<2>{
        \framebox{\includegraphics[width=6.5cm]{kl-10007.pdf}}}
      \only<3>{
        \framebox{\includegraphics[width=6.5cm]{buffon.jpg}}}
    \end{overlayarea}
  \end{columns}
\end{frame}



\end{document}
