\documentclass[12pt]{article}
\usepackage{amssymb} % adds some useful symbols
\usepackage{amsmath}
\usepackage{amsrefs} % this is the package I'm suggesting for your
                     % bibliography
\usepackage{amsthm}
\usepackage{array}
\usepackage{rotating}
\usepackage{verbatim}
\usepackage{enumitem} %\begin{enumerate}[resume] to continue old numbers
\usepackage{showlabels} % this allows you to print drafts showing the
                        % names you have given to equations, theorems,
                        % and so on.
\showlabels{cite}

\newenvironment{proof_sketch}{\noindent{\it Sketch of
    Proof.}\hspace*{1em}}{\qed\bigskip} 
\theoremstyle{plain}
\newtheorem{theorem}[equation]{Theorem} % this uses the same counter
                                % to number both theorems and equations
\newtheorem{corollary}[equation]{Corollary}
\newtheorem{lemma}[equation]{Lemma}
\newtheorem{conjecture}[equation]{Conjecture}
\newtheorem{proposition}[equation]{Proposition}

\theoremstyle{definition}
\newtheorem{algorithm}[equation]{Algorithm}
\theoremstyle{remark}

\newtheorem{definition}[equation]{Definition}
\newtheorem{example}[equation]{Example}
\newtheorem{remark}[equation]{Remark}

\newcommand{\ZZ}{\mathbb{Z}} % this provides convenient abbreviations
                             % for some common symbols. Don't make too
                             % many of these: they can make your LaTeX
                             % harder to read. 
\newcommand{\NN}{\mathbb{N}}                                                    
\newcommand{\RR}{\mathbb{R}}

\DeclareMathOperator\Hom{Hom}
\DeclareMathOperator\Span{span}
\DeclareMathOperator\Range{Range}
\DeclareMathOperator\Id{Id}
\DeclareMathOperator\End{End}

\renewcommand{\theequation}{\thesection.\arabic{equation}} % this makes equation
% numbers (x.y) for equation number y in section number x.
%\newcommand{\Cite}[1]{\hspace{-.7ex}\cite{#1}} 
\newcounter{saveenumi}

\begin{document}
\title{Here is a captivating yet pithy description of our project}
\author{First Author
\and Second Author \and Third Author}
\date{\today}

\maketitle

\begin{abstract}
This document is a template for the real papers you will write
for 18.821.
% An abstract is a SHORT summary of the main points in the project.
% It is meant to be understandable in isolation from your paper: it
% should not refer to your bibliography or to specifics of the paper
% itself. (For instance, don't say ``In Theorem 3 we prove the Riemann
% hypothesis.'')
\end{abstract}

\section{Introduction}\label{sec:intro}
\setcounter{equation}{0} % point of this is to restart equation
                         % numbering in each section.
This paper offers a tour of the
exciting geometry of local rings, leading to our mastery of the
pronunciation of the name of the journal in which \cite{LocalRings} is
published. The main tool we use is \cite{ODE}*{Chapter 1}: we write
% notice that the argument of \cite is the short name of the
% bibliographic reference that you choose for your convenience; it
% does not appear in the LaTeX output. The following ``*{detailed
% location}'' should be present more often than not: if your reader
% needs to look at a reference, your reader needs help to find the
% relevant location!
down a simple differential equation
\begin{equation}\label{eq:downhill}
  \frac {dU}{dt} = -U(t), \qquad U(0)=.01
\end{equation}
describing the evolution of our understanding $U$ of local rings. The
meaning of this equation will become clearer in Section
\ref{sec:local} below. For the moment, notice only that the variable
$t$ takes values in $\RR$, the real numbers.

An unexpected benefit of our investigations was our discovery of the
open source software \cite{atlas}. One of the scripts included in the
software distribution is \verb|sudoku.ax|, which will solve sudokus
for you instantly. (Typesetting here used the package \verb?verbatim?
which is loaded at the top of this file. Great for email and web
addresses, computer code, and quotations from
Hemingway.\footnote{{\tt In order to write about life first you must live it.}} \footnote{Fortunately this is less true of mathematics.}
% turns out the \verb command doesn't work inside \footnote. Oh well.
\section{The role of local rings in mathematics}\label{sec:local}
\label{sec:setup}
\setcounter{equation}{0}
In the body of the paper, you will begin to fulfill the promises you
made in Section \ref{sec:intro}. Because this is only a template, and
because of certain gender deficiencies, the
author has chosen to hear what he wants to hear and disregard the
rest. To put it another way, you will not learn here (does the
repetition of the sound {\em hear} have a Hemingwayesque character?)
what a local ring is. We will not even solve \eqref{eq:downhill} for you.

As small compensation, here is an aligned equation for you:
\begin{equation}\label{eq:alignedExample}
\begin{aligned}
  \ZZ/2\ZZ &= \{0,1\}\\
  \ZZ/3\ZZ &= \{0,1,2\}\\
  \ZZ/5\ZZ &= \{0,1,2,3,4\}.\\
\end{aligned}
\end{equation}
If I just put these in one after the other, I'd get something quite
different from \eqref{eq:alignedExample}:
\begin{equation}
  \ZZ/2\ZZ = \{0,1\}
\end{equation}
\begin{equation}
  \ZZ/3\ZZ = \{0,1,2\}
\end{equation}
\begin{equation}
  \ZZ/5\ZZ = \{0,1,2,3,4\}.
\end{equation}
You might sometimes prefer
\begin{equation}\label{eq:casesExample}
  \\Z/p\ZZ = \begin{cases} \{0,1\} & \text{(if $p=2$)}\\
    \{0,1,2\} & \text{(if $p=3$)}\\
    \{0,1,2,3,4\} & \text{(if $p=4$)}\\
    \qquad \vdots & \qquad \vdots \end{cases}
\end{equation}

Amazingly enough, Section \ref{sec:local} now provides three examples
of local rings! 
\begin{bibdiv}
\begin{biblist}[\normalsize]

  \bib{LocalRings}{article}{ % this is copied straight from
                             % MathSciNet; I just changed ``MR0268188'' to
                             % ``LocalRings''
   author={Artin, M.},
   title={Algebraic approximation of structures over complete local rings},
   journal={Inst. Hautes \'Etudes Sci. Publ. Math.}, % MathSciNet
                                % knows how to spell ``\'Etudes, and
                                % knows the correct abbreviation of
                                % the journal name!
   number={36},
   date={1969},
   pages={23--58},
   issn={0073-8301},
 %  review={\MR{0268188}},
  }
  
\bib{ODE}{book}{ % the information copied from MathSciNet had
                 % MR0069338 instead of ODE.
   author={Coddington, Earl A.},
   author={Levinson, Norman}, % I picked this book because it's an MIT author
   title={Theory of ordinary differential equations},
   publisher={McGraw-Hill Book Company, Inc., New York-Toronto-London},
   date={1955},
   pages={xii+429},
%   review={\MR{0069338}},
}

\bib{TW}{article}{ % this is a do-it-yourself reference to a preprint
                   % at arxiv.org
author={Casian, Luis}, 
author={Kodama, Yuji},
title={On the cohomology of real Grassmann manifolds},
eprint={arxiv:1309.5520}% [Version 1: Sat, 21 Sep 2013 19:20:43 GMT]}
}

\bib{atlas}{misc}{ % this is a do-it-yourself reference to some open
                   % source software
title={Atlas of Lie Groups and Representations software},
note={\url{https://github.com/jeffreyadams/atlasofliegroups}},
year={2016}
}






\end{biblist}
\end{bibdiv}

\end{document}
