\documentclass[11pt,oneside]{amsart}
\usepackage{amssymb, amsmath, amsthm}

\usepackage{epsfig}
\usepackage{epstopdf}
\usepackage{graphicx}
\usepackage{color}
%\usepackage[shortalphabetic]{amsrefs}
\usepackage{mathptmx}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
%\usepackage{url}

%\usepackage{multicols}

%\usepackage{pinlabel}

%Define theorem styles
     \theoremstyle{definition}
      \newtheorem{theorem}{Theorem}[section]
      \newtheorem*{theorem*}{Theorem}
      \newtheorem{definition}[theorem]{Definition}
      \newtheorem{exercise}[theorem]{Exercise}
      \newtheorem{example}[theorem]{Example}
      \newtheorem{conj}{Conjecture}
      \newtheorem{proposition}[theorem]{Proposition}
      \newtheorem{corollary}[theorem]{Corollary}
      \newtheorem{claim}{Claim}
      \newtheorem{lemma}[theorem]{Lemma}
    
   % Define new commands

      \newcommand{\R}{{\mathbb R}}
      \newcommand{\C}{\mathbb C}
      \newcommand{\Z}{\mathbb Z}
	\newcommand{\N}{\mathbb N}
	\newcommand{\Q}{\mathbb Q}
	\newcommand{\Hy}{\mathbb H}
	\newcommand{\E}{\mathbb E}
      \newcommand{\nil}{\varnothing}
	\newcommand{\til}{\tilde}
	\newcommand{\boundary}{\partial}
	\newcommand{\m}{\backslash}
	\newcommand{\mc}[1]{\mathcal{#1}}
	\newcommand{\vect}[1]{\mathbf{#1}}
	\newcommand{\prob}[1]{\textbf{Problem {#1}: }}
	\newcommand{\defn}[1]{\textbf{#1}}
	\newcommand{\ob}[1]{\overline{#1}}
	%\newcommand{\ker}{\operatorname{ker}}
	\newcommand{\im}{\operatorname{im}}
	\newcommand{\tr}{\operatorname{tr}}
	\newcommand{\rank}{\operatorname{rank}}
	\newcommand{\rref}{\operatorname{rref}}
	\newcommand{\hyp}{\text{hyp}}
	
	\newcommand{\diver}{\operatorname{div}}
	\newcommand{\curl}{\operatorname{curl}}
	\newcommand{\grad}{\operatorname{grad}}

	\newcommand{\lpartial}[1]{\frac{\partial}{\partial #1}}
	\newcommand{\llpartial}[2]{\frac{\partial #1}{\partial #2}}
	\newcommand{\co}{\mskip0.5mu\colon\thinspace} 
	\newcommand{\spacing}{\parskip 6.6pt
\parindent 0pt}

	\newcommand{\custthm}[2]{
			\framebox[0.95\linewidth]{
			\begin{minipage}{0.9\linewidth}

			\begin{theorem}\label{#1}
			#2	
			\end{theorem}
			\end{minipage}
			}
	}
	
	
	
	\newenvironment{enumer}
		{\begin{enumerate}
		\parskip 6.6pt
		\parindent 0pt
		\renewcommand{\labelenumi}{(\alph{enumi})}
		
		}
		{\end{enumerate}}


	  
\parskip 6.6pt
\parindent 0pt

%\renewcommand{\labelenumi}{(\alph{enumi})}


	
   % The following mysterious maneuver gets rid of AMS junk at the top
   % and bottom of the first page.
   
      \makeatletter
      \def\@setcopyright{}
      \def\serieslogo@{}
      \makeatother
   
% This ends the preamble.  We now proceed to the document itself.

\begin{document}


   % title

   \title[short title]{long title}
   \author{}
   % Note that the short title for running heads goes in square
   % brackets.  This is optional.  The long title goes in curly
   % braces.  In the long title, line breaks are indicated by \\.


%Here are two sets of commands for including images.
  
%\begin{center}
%\begin{figure}[h]
%\includegraphics[scale=0.4]{FinalExam-prob10.pdf}
%\end{figure}
%\end{center}

%\begin{figure}[ht] 
%\scalebox{0.5}{\input{Pen.eps_t}}
%\caption{A pox on both your houses!}
%\end{figure}

   

   % today's date, or fill in whatever date you prefer
   \date{\today}

\framebox{
\raisebox{0in}[.5in][.2in]{
\textbf{MA 314 Homework 14: Surfacing for air} 
}
}



\section{Reading}

\begin{enumerate}\spacing
\item Read Bonahon Sections 4.1 and 4.2 and Schwartz Sections 3.1, 3.2, and 3.3.
\item Suggest some sources which you'll use as the basis for your blog essay.

\end{enumerate}

\section{Mortar}

\begin{enumerate}\spacing
\item Bonahon defines the length of a discrete walk and then defines distance in the the quotient space as the infimum of lengths of discret walks. Schwartz defines chains and pathifications. Articulate how these definitions do or don't do the same thing. Are they equivalent concepts?
\end{enumerate}

\section{Bricks}

\begin{enumerate}\spacing
\item Prove that the result of removing an open disc from the projective plane is a M\"obius band. If it helps, the projective plane is pictured below:

\begin{center}
\includegraphics[scale=0.5]{RP2}
\end{center}


\item Let $X$ be the strip in $\R^2$ bounded by the lines $x=-1$ and $x = +1$. For a fixed $r \geq 0$, let $\ob{X}_r$ be the result gluing the points $(-1, y)$ to $(1, y + r)$ for each $y \in \R$. We can give $\ob{X}_r$ a path metric inherited from $X$ as discussed in class for the torus. For the following problems, you may informally. Recall that a geodesic is a locally length-minimizing path. The geodesics in $\ob{X}_r$ are all unions of straight line segments in $X$.
\begin{enumerate}\spacing
\item For $r = 0$, sketch a picture of a geodesic loop in $\ob{X}_0$. What is its length? Are there any other geodesic loops?
\item For $r = 1$, sketch a picture of a geodesic loop in $\ob{X}_1$. What is its length? Are there any other geodesesic loops?
\item Prove that if $r,s \geq 0$ with $r \neq s$, then $\ob{X}_r$ is not isometric to $\ob{X}_s$.
\item Describe the isometries of $\ob{X}_r$ for different values of $r$. Do the $\ob{X}_r$ for different values of $r$ all have similar types of isometries?
\end{enumerate}

\end{enumerate}


\end{document}