% The color scheme was originated by Joshua Fallon, an LSU Ph.D. alumnus, around March 2018.
% Updated by Bogdan Oporowski in March 2023.
%Updated by laura Kurtz in 2025 to add a slide counter and some other modern latex benefits.
\documentclass[xcolor=svgnames]{beamer}
\mode<presentation> {
  \usetheme{Warsaw} 
  \usecolortheme{beaver}
  \setbeamertemplate{navigation symbols}{} % Removes navigation buttons
  \setbeamertemplate{caption}[numbered] 
} 
\usepackage[T1]{fontenc}
\usepackage{tikz}
\usepackage{hyperref}
\usepackage{circuitikz}
\usefonttheme{professionalfonts}
\definecolor{LSUP}{RGB}{70,29,124}
\definecolor{LSUG}{RGB}{253,208,35}
\definecolor{LSULP}{RGB}{163,154,172}
\definecolor{LSULG}{RGB}{241,238,219}
\setbeamercolor{structure}{bg=LSUG,fg=LSUP}
\setbeamercolor{title}{bg=LSUP,fg=LSUG}
\setbeamercolor{title in head/foot}{bg=LSUP,fg=LSUG}
\setbeamercolor{frametitle}{bg=LSUP,fg=LSUG}
\setbeamercolor{author in head/foot}{bg=LSUG,fg=LSUP}
\setbeamercolor{date in head/foot}{bg=LSUG,fg=LSUP}
\setbeamercolor{section in head/foot}{bg=LSUG,fg=LSUP}
\setbeamercolor{subsection in head/foot}{bg=LSUP,fg=LSUG}
\setbeamercolor{block body}{bg=LSULG}

% ===== FOOTER WITH SLIDE COUNTER =====
\setbeamertemplate{footline}{%
  \leavevmode%
  \hbox{%
    % Left: Author
    \begin{beamercolorbox}[wd=0.333333\paperwidth,ht=2.25ex,dp=1ex,left]{author in head/foot}%
      \usebeamerfont{author in head/foot}\insertshortauthor
    \end{beamercolorbox}%
    % Center: Title
    \begin{beamercolorbox}[wd=0.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
      \usebeamerfont{title in head/foot}\insertshorttitle
    \end{beamercolorbox}%
    % Right: Slide counter
    \begin{beamercolorbox}[wd=0.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
      \usebeamerfont{date in head/foot}%
      Slide \insertframenumber{} of \inserttotalframenumber%
      \hspace*{2em}%
    \end{beamercolorbox}%
  }%
  \vskip0pt%
}

\newcommand{\gray}[1]{\color{LSULP}{#1}}
\newcommand{\Prob}{\mathbb{P}}
\newcommand{\Z}{\mathbb{Z}}
\usepackage{graphicx}
\usepackage{xcolor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%PREAMBLE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Short Title]{Title}
\author{Me}
\institute[LSU]{Louisiana State University \\ Mentored By someone}
\date{\today}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Frame %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Motivation}
some kind of intro
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Frame %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Motivation}
\begin{block}{Key Question}
What will you talk about?
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Frame %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Preliminaries}
\pause
\begin{block} {Definition}
A \textbf{Bernoulli random variable} is a random variable that takes only two values. (In our case, $-1$ and $1$.)
\end{block}
\pause
\begin{block} {Convention}
    When we talk about a collection of Bernoulli random variables, we often say they are independent and identically distributed, written "(i.i.d.)."
\end{block}
\pause
\begin{block}{Definition}
    \textbf{Identically distributed} means that they have the same distribution, meaning:
         \[
    \Prob(X \leq x) = \Prob(Y \leq x)
    \]
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Frame %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Intuition on higher dimensions}
  \begin{columns}
% First column - 1D case
\column{0.32\textwidth}
\begin{block}{1D - Narrow Valley}
\begin{itemize}
\item They have no choice but to turn around
\item Always returns to starting point
\item \textcolor{blue}{Recurrence: Certain}
\end{itemize}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[thick] (0,0) -- (3,0);
\draw[->] (1.5,0.5) -- (1.5,0);
\node[fill=red,circle] at (1.5,0) {};
\foreach \x in {0.5,1,2,2.5} {
    \draw[->] (\x,0.3) -- (\x+0.2,0.3);
    \draw[->] (\x+0.2,-0.3) -- (\x,-0.3);
}
\end{tikzpicture}
\end{block}
\pause
% Second column - 2D case
\column{0.32\textwidth}
\begin{block}{2D - City Grid}
\begin{itemize}
\item Might get lost temporarily
\item Still revisits intersections
\item \textcolor{orange}{Recurrence: Eventually}
\end{itemize}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[step=1,thin,gray] (0,0) grid (2,2);
\node[fill=red,circle] at (1,1) {};
\draw[->,red,thick] (1,1) -- (1.7,1);
\draw[->,red,thick] (1.7,1) -- (1.7,1.7);
\draw[->,red,thick] (1.7,1.7) -- (1,1.7);
\draw[->,red,thick] (1,1.7) -- (1,1);
\end{tikzpicture}
\end{block}
\pause
% Third column - 3D case
\column{0.32\textwidth}
\begin{block}{3D - Open Space}
\begin{itemize}
\item Too many possible directions
\item They become permanently lost
\item \textcolor{red}{Recurrence: Never}
\end{itemize}
\centering
\begin{tikzpicture}[scale=0.7]
\draw[->] (0,0) -- (2,0) node[right]{x};
\draw[->] (0,0) -- (0,2) node[above]{y};
\draw[->] (0,0) -- (-1,-1) node[below left]{z};
\node[fill=red,circle] at (0,0) {};
\draw[red,->,thick] (0,0) -- (0.5,0.3);
\draw[red,->,thick] (0.5,0.3) -- (0.2,1.2);
\draw[red,->,thick] (0.2,1.2) -- (-0.7,0.8);
\end{tikzpicture}
\end{block}
\end{columns}
\end{frame}
\begin{frame}
\begin{block}{Main Result}
  What did you do?
\end{block}
\end{frame}
\begin{frame}{References}
    \bibliographystyle{abbrv}
\bibliography{references}  % need to put bibtex references in references.bib 
 \nocite{*}
\end{frame}
\end{document}
