\documentclass[10pt,reqno]{amsart}

%% GENERAL TYPESETTING

\setlength{\textwidth}{6 in}
\setlength{\textheight}{8.5 in}

\setlength{\oddsidemargin}{0.6 cm}
\setlength{\evensidemargin}{0.6 cm}
\setlength{\topmargin}{0 cm}

\renewcommand{\baselinestretch}{1.3}

%% HYPERLINKS

\newif\iffigures\figurestrue
\figuresfalse

\newif\ifhyper\IfFileExists{hyperref.sty}{\hypertrue}{\hyperfalse}

%\ifhyper\usepackage[hypertex]{hyperref}
\ifhyper\usepackage{hyperref}
\def\hitem#1#2{\item[\hypertarget{#1}{#2}]\expandafter\gdef\csname LBL#1ITM\endcsname{#2}}
\def\iref#1{\hyperlink{#1}{\csname LBL#1ITM\endcsname}}
\else
\def\hitem#1#2{\item[{#2}]\expandafter\gdef\csname LBL#1ITM\endcsname{#2}}
\def\iref#1{{\csname LBL#1ITM\endcsname}}
\fi

%% PACKAGES

\usepackage{amssymb,amsmath,mathrsfs}
\usepackage{wrapfig}
\usepackage{psfig}
\input epsf.sty
\input labelfig.tex
\usepackage{showlabels}


%% MACROS
%% DO NOT EDIT THE MACROS HERE, OR IF YOU DO, WARN ME.
%% IF YOU WANT SOME EXTRA DEFINITIONS, PUT THEM AFTER THE BEGINNING OF THE SECTION.

\def\lra{\leftrightarrow}
\def\llra{\longleftrightarrow}
\def\ora{\overrightarrow}

\def\Cl{\mathscr{C}}
\def\F{\mathscr{F}}
\def\A{{\mathcal A}}
\def\B{{\mathcal B}}
\def\XX{{\mathcal X}}
\def\TT{{\mathcal T}}

\def\text#1{\hbox{#1}}
\def\Z{{\mathbb Z}}
\def\R{{\mathbb R}}
\def\N{{\mathbb N}}
\def\T{{\mathbb T}}
\def\U{{\mathbb U}}
\def\C{{\mathbb C}}
\def\P{\mathbf P}
\def\E{\mathbf E}
\def\p{\partial}
\def\eps{\epsilon}
\def\Aut{{\rm Aut}}
\def\Re{{\mathrm {Re}}}
\def\Im{{\mathrm {Im}}}
\def\diam{{\rm diam}}
\def\length{{\rm length}}

\def\putfig{\vskip 0.5in \centerline{FIGURE}\vskip 0.5in} %if you think a figure is needed, but you don't have one 

\def\given{\,|\,} %the next few lines define nice Probaility and Expectation macros, with or without conditions.
\def\bgiven{\;\big|\;}
\def\Bgiven{\;\Big|\;}

\def\Bb#1#2{{\def\md{\bigm| }#1\bigl[\,#2\,\bigr]}}
\def\BB#1#2{{\def\md{\Bigm| }#1\Bigl[\,#2\,\Bigr]}}
\def\Bs#1#2{{\def\md{\mid}#1[\,#2\,]}}
\def\Pb{\Bb\P}
\def\Eb{\Bb\E}
\def\PB{\BB\P}
\def\EB{\BB\E}
\def\Ps{\Bs\P}
\def\Es{\Bs\E}

%% NUMBERING

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{exercise}{Exercise}[section]

\def\star{\hskip -0.05 in {\rm *} \hskip 0.05 in}
\def\starr{\hskip -0.05 in {\rm **} \hskip 0.05 in}
\def\starrr{\hskip -0.05 in {\rm ***} \hskip 0.05 in}


\theoremstyle{definition}
\numberwithin{equation}{section}

%% LET'S START

\begin{document}

\title{Probability and geometry on groups\\
(Lectures by G\'abor Pete)}

\author{NNN} %YOUR NAMES
\author{MMM}

\address{Department of Mathematics, University of Toronto, 
40 St George St, Toronto, M5S 2E4, Canada}
\email{} %YOUR EMAILS 
\email{} 

\date{\today}

\maketitle

\section{XXX}\label{s.X} %THE TITLE OF YOUR LECTURE, OR OF THE FIRST PART OF IT.

This is the way to refer to a section: see Section~\ref{s.X} or~\ref{s.Y}. Don't worry if a reference to something in a different lecture comes out just as ??, still use the label from that previous lecture, as shown on the margin. I'm typically using labels {s.something} for sections, {t.something} for theorems {l.something} for lemmas, {e.something} for equations. {\bf Again:} never refer to anything manually! Writing ``by Theorem 1.1'' is forbidden, you have to write ``by Theorem~\ref{t.Polya}''. Citing a reference: see~\cite{evolving}.

The labels chosen should have a serious chance to be unique across the entire course. So, don't call you first section {s.first}, and your second lemma {l.second}; rather, give descriptive names. It's also easier for you to remember when you want to use them.

\begin{theorem}\label{t.Polya}
My way of writing multiline formulas:
\begin{equation}
\label{e.d/2}
\begin{aligned}
\Pb{ X_n=y \md X_0=x} &= \sum_{k} etc\\
& \asymp n^{-d/2}\,.
\end{aligned}
\end{equation}
Note that one label refers here to the entire multiline formula.
\end{theorem}

Or, if you want separate labels for the lines,
\begin{align}
\label{e.etc}
\EB{ \big| \{\text{some set}\}\big|  \md \text{some condition}} &= \sum_k etc \\
\nonumber
& \asymp \text{something}\\
\label{e.else}
& = \text{something else}\,.
\end{align}

If you don't want labels on some equation at all, use double dollar signs for single-line equations, or the starred equation* or align* commands. For instance,
\begin{align*}
\EB{ \big| \{\text{some set}\}\big|  \md \text{some condition}} &= \sum_k etc \\
& \asymp \text{something}\\
& = \text{something else}\,.
\end{align*}

It is good to label lemmas and theorems always, but use only as many labels on the equations as you actually need. Another request regarding equations is that please use commas and full stops when the equations form part of a sentence. See the examples above.

You should also use the preset commands and labels for exercises:

\begin{exercise}\label{ex.AlphaCentauri}
\starr This is the first double-starred exercise.
\end{exercise}

Some TeX tricks: $\langle \, a,b \mid [a,b] \, \rangle$ looks infinitely better than $< a,b | [a,b] >$. Furthermore, $l$ is hard to distinguish from $1$ $t$ $($ $|$, so it's better to use $\ell$.

Here is how to include a figure: 

%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\centerline{
\epsfysize=2 true in \epsffile{F2Cayley.eps}
}
\caption{A Cayley graph.}
\label{f.F2}
\end{figure}
%%%%%%%%%%%%%%%%%

Moreover, you can write inside it:

%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\SetLabels
(.15*.9) a letter $\xi$ here\\
(-0.05*.2) and another there\\
(.4*.4) $\xi$\\
\endSetLabels
%\ShowGrid%comment this command out after you have set the labels at the right places
\centerline{
\AffixLabels{\epsfysize=2 true in \epsffile{F2Cayley.eps}}
}
\caption{A Cayley graph.}
\label{f.F2again}
\end{figure}
%%%%%%%%%%%%%%%%%




\begin{thebibliography}{ABBP06} %ADD HERE THE REFERENCES, IN A STYLE SHOWN BELOW

\bibitem[Gri99]{Grimm}
G. Grimmett. {\it Percolation. Second edition}. Grundlehren der
Mathematischen Wissenschaften, 321. Springer-Verlag, Berlin, 1999.

\bibitem[MoP05]{evolving} B. Morris and Y. Peres. 
Evolving sets, mixing and heat kernel bounds. 
{\it Prob. Th. Rel. Fields} {\bf 133} (2005), no. 2, 245--266. 
{\tt arXiv:math.PR/0305349}

\end{thebibliography}
\end{document}
