\documentclass[11pt]{amsart}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ... 
%\geometry{landscape}                % Activate for for rotated page geometry
%\usepackage[parfill]{parskip}    % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\newtheorem{theorem}{Theorem}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conj}[theorem]{Conjecture}

\newcommand{\todo}[1]{\vspace{5 mm}\par \noindent
\marginpar{\textsc{ToDo}}
\framebox{\begin{minipage}[c]{0.95 \textwidth}
\tt #1 \end{minipage}}\vspace{5 mm}\par}

%remove the comment from the following line to remove all the
% 'to do' remarks:
%\renewcommand{\todo}[1]{} 

\newcommand{\idiot}[1]{\vspace{5 mm}\par \noindent
\marginpar{\textsc{Note}}
\framebox{\begin{minipage}[c]{0.95 \textwidth}
#1 \end{minipage}}\vspace{5 mm}\par}

%remove the comment from the following line to remove all the
% extra proofs:
%\renewcommand{\idiot}[1]{}


\def\la{{\lambda}}
\def\shuf{{\sqcup\!\sqcup}}
\def\QQ{{\mathbb Q}}
\def\RR{{\mathcal R}}
\def\NN{{\mathbb N}}
\def\CC{{\mathbb C}}
\def\F{{\mathcal F}}

\newcommand{\pchoose}[2]{\begin{pmatrix}#1\\ #2\end{pmatrix}}
\newcommand{\bchoose}[2]{\begin{bmatrix}#1\\ #2\end{bmatrix}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Young diagrams, grace a Francois Bergeron

\newdimen\squaresize \squaresize=12pt
\newdimen\thickness \thickness=0.4pt


\def\square#1{\hbox{\vrule width \thickness
     \vbox to \squaresize{\hrule height \thickness\vss
        \hbox to \squaresize{\hss#1\hss}
     \vss\hrule height\thickness}
\unskip\vrule width \thickness}
\kern-\thickness}

\def\vsquare#1{\vbox{\square{$#1$}}\kern-\thickness}
\def\blk{\omit\hskip\squaresize}
\def\noir{\vrule height\Squaresize width\Squaresize}%

\def\young#1{
\vbox{\smallskip\offinterlineskip
\halign{&\vsquare{##}\cr #1}}}

\def\thisbox#1{\kern-.09ex\fbox{#1}}
\def\downbox#1{\lower1.200em\hbox{#1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\title{Sample file}
\author{Your name here}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
\maketitle
%\section{}
%\subsection{}

You can delete all of this text and just start
typing, but I am going to include a few examples.
Paragraphs are separated by a single blank line.
If
you
type
some
words
they are
arranged
on the page
in a paragraph
form.

Mostly, if you want to do something that you haven't done before
in LaTeX, then you can easily find how to do this in Google.  Just
ask a question such as `{\it how do I insert a picture in latex?}'
or `{\bf what is the command for the symbol theta in latex?}' 
or `{\tiny how do I make a list in latex?}'

Here is a good reference for the answer to the last question
\begin{center}
{\tt https://www.sharelatex.com/learn/Lists}
\end{center}

An equation surrounded by a \$ on either side such as 
{\tt\$\string\frac\{n(n+1)\}\{2\}\$} produces
the equation $\frac{n(n+1)}{2}$
as part of the text (this is called an `in-line equation').
If you want an equation of the text or a formula on a separate line like
$$\sqrt{\sqrt{2} + \frac{\sqrt{3} \sqrt{5}}{2}}
$$
then you need to surround it by a double dollar sign.

But beware. Just because the formula looks pretty does not mean that it is correct.
If you would like to number your equations you can wrap your equation in 
a {\tt\string\begin\{equation\}}
 and {\tt\string\end\{equation\}}
and a command {\tt\string\label\{myequationname\}}
doesn't cause the equation to be labeled, it just
makes LaTeX remember what the equation number is.
\begin{equation}\label{myequationname}
1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n(n+1)^2(n+2)}{12}
\end{equation}
That way if you want to refer to the equation later you just write 
{\tt\string\eqref\{myequationname\}} and then you can point out that
when $n=2$ the left hand side of equation
\eqref{myequationname} is equal to $9$ while the
right hand side of equation \eqref{myequationname} is equal to $6$ (and
hence it is not true in general).

Instead the equation should read
\begin{equation}\label{anotherequation}
1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}~.
\end{equation}
Just make sure you
give a different name to each equation.  At least for $n=2$ the right hand
side of equation \eqref{anotherequation} is also equal to $9$.

\end{document}