% Template for reports
% of the seminar ``Simplicial Topology''
% WS 2015/16
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% General advice
% - Try to keep the LaTeX code as transparent as possible; (La)TeX's
%   error messages can seem cryptic -- in well-structured source code, 
%   debugging is much easier than in chaotic source code!
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Every LaTeX document needs a \documentclass declaration; this will take
%   care of the general layout
\documentclass[a4paper,twoside,DIV8,10pt]{scrartcl}
  
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % including further packages
  % \usepackage{german}    % fuer die deutschen Trennmuster
  % \usepackage{ngerman} % entsprechend fuer die neue Rechtschreibung
  \usepackage[latin1]{inputenc} % useful for Umlauts
  \usepackage{amsmath}   % useful math macros
  \usepackage{amsthm}    % for theorems, definitions, proofs, etc.
  \usepackage{amsfonts}  % special AMS math fonts
  \usepackage{tikz}      % for easy generation of simple graphics
  \usepackage[headinclude]{scrpage2}  % for page headings

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % some math macros
  \newcommand{\N}{\ensuremath{\mathbb{N}}}   % natural numbers
  \newcommand{\Z}{\ensuremath{\mathbb{Z}}}   % integers
  \newcommand{\Q}{\ensuremath{\mathbb{Q}}}   % rational numbers
  \newcommand{\R}{\ensuremath{\mathbb{R}}}   % real numbers

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % declaration of theorem/definition/etc environments, based on amsthm
  \newtheorem{thm}{Theorem}[section]
  \newtheorem{prop}[thm]{Proposition}
  \newtheorem{lemma}[thm]{Lemma}
  \newtheorem{cor}[thm]{Corollary}
  \theoremstyle{definition}
  \newtheorem{definition}[thm]{Definition}
  \newtheorem{rem}[thm]{Remark}
  \newtheorem{exercise}[thm]{Exercise}
  \newtheorem{example}[thm]{Example}
  %% German version:
  %\newtheorem{satz}{Satz}[section]
  %\newtheorem{lemma}[satz]{Lemma}
  %\newtheorem{korollar}[satz]{Korollar}
  %\theoremstyle{definition}
  %\newtheorem{definition}[satz]{Definition}
  %\newtheorem{bemerkung}[satz]{Bemerkung}
  %\newtheorem{aufgabe}[satz]{Aufgabe}
  %\newenvironment{beweis}%
  %  {\begin{proof}[Beweis]}
  %  {\end{proof}}
  %\newtheorem{beispiel}[satz]{Beispiel}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % further macros
  \makeatletter                    % footnotes without labels
    \def\blfootnote{\xdef\@thefnmark{}\@footnotetext}
  \renewcommand{\sectfont}{\normalfont} % changes section font

  % page layout
  \renewcommand{\headfont}{\sffamily}  % sans serif headings
  \renewcommand{\pnumfont}{\sffamily}  % sans serif page numbers
  \automark[section]{section}          % determines content of \headmark
  \deftripstyle{myheadings}{\headmark}{}{\pagemark}% head
                           {}{}{}% foot
  \pagestyle{myheadings}               % activating the new page style


  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  % title of report
  \author{N.~Imeta (\textsf{mail@spam.blorx})}
  \date{February 30, 2010}
  \title{House of Santa Claus%
    \blfootnote{Seminar ``Simplicial Topology'', 
      WS~2015/16, Universit\"at Regensburg}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% start of the actual document
\begin{document}

  % title of the report
  \maketitle

  \begin{abstract}
    \noindent
    Here is a short summary or overview of the talk -- approximately
    one to ten lines. You should describe briefly what the main goal 
    of the talk is and how this is accomplished.
  \end{abstract}

  \section{Basics}

  \begin{definition}[House of Santa Claus]
    \label{def:housesanta}
    The \emph{House of Santa Claus} is the graph~$(V,E)$, defined as follows: 
    \begin{align*}
      V & := \{1, \dots, 5\}
      \\
      E & := \bigl\{ \{1,2\}
                   , \{1,5\}
                   , \{2,3\}
                   , \{2,4\}
                   , \{2,5\}
                   , \{3,4\}
                   , \{3,5\}
                   , \{4,5\}
             \bigr\}
    \end{align*}
  \end{definition}

  One can illustrate the House of Santa Claus as in
  Figure~\ref{fig:housesanta}; more information on Ti\textit{k}Z can
  be found in the documentation~\cite{tantau}.

  \begin{figure}[h]
    \begin{center}
      \begin{tikzpicture}
         \begin{scope}[color=black!20]
           \draw[fill] (0,0) circle (2pt);
           \draw[fill] (1,0) circle (2pt);
           \draw[fill] (0,1) circle (2pt);
           \draw[fill] (0.5,1.5) circle (2pt);
           \draw[fill] (1,1) circle (2pt);
         \end{scope}
         \begin{scope}[color=blue]
           \draw (0.5,1.5) node[above] {$1$};
           \draw (1,1) node[right] {$2$};
           \draw (1,0) node[right] {$3$};
           \draw (0,0) node[left] {$4$};
           \draw (0,1) node[left] {$5$};
         \end{scope}
         \draw[->,rounded corners=0.1cm]
            (0,0) -- (1,0) -- (0,1) -- (1,1) -- (0,0)
         -- (0,1) -- (0.5,1.5) -- (1,1) -- (1,0.2);
      \end{tikzpicture}
    \end{center}
    \caption{House of Santa Claus}
    \label{fig:housesanta}
  \end{figure}

  \section{Properties of the House of Santa Claus}

  \begin{thm}[House of Santa Claus]
    The House of Santa Claus is not complete.
  \end{thm}
  \begin{proof}
    We use the notation from Definition~\ref{def:housesanta}.  The
    House of Santa Claus is not a complete graph because the
    edge~$\{1,3\}$ is not contained in the House of Santa Claus.
  \end{proof}

  \section{Examples}

  \begin{example}
    \hfil
    \begin{itemize}
      \item Here is an example
      \item \dots and another one
      \item \dots and another one
    \end{itemize}
  \end{example}

  \begin{exercise}
    Please do not forget to insert a few exercises -- so that the 
    participants can test their understanding of the topic.
  \end{exercise}

  \begin{example}
    \hfil
    \begin{enumerate}
      \item An example \dots
      \item \dots using numbers.
    \end{enumerate}
  \end{example}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Bibliography
% - For simplicity, the references from the seminar schedule are already
%   included
% - Please remove all items that you do not cite
% - Conversely, you shoul add all the literature necessary for your talk
%   here, www.ams.org/mathscinet can help to compile the necessary 
%   bibliographical data.
  \begin{thebibliography}{99}

    \bibitem{beutelspacher} A.~Beutelspacher. \emph{Das ist o.B.d.A.\
        trivial!}, 9th edition, Vie\-weg$+$Teub\-ner, 2009.

    \bibitem{chua-leong-lim}
      C.K.~Chua, K.F.~Leong, C.S.~Lim.
      \emph{Rapid Prototyping: Principles and Applications}, 
      World Scientific Publishing, 2010.

    \bibitem{edelsbrunner-harer}
      H.~Edelsbrunner, J.L.~Harer.
      \emph{Computational topology. 
        An introduction}, \
      American Mathematical Society, 2010.

    \bibitem{ghrist}
      R.~Ghrist.
      Barcodes: the persistent topology of data,
      \emph{Bull.\ Amer.\ Math.\ Soc.},~45, S.~61--75, 2008. 
 
    \bibitem{kmm}
      T.~Kaczynski, K.~Mischaikow, M.~Mrozek, Marian.
      \emph{Computational homology}, 
      Applied Mathematical Sciences, 157. Springer, 2004. 

    \bibitem{lurie}
      J.~Lurie.
      \emph{Higher Topos Theory},
      Annals of Mathematics Studies, 170, 
      Princeton University Press, 2009.

    \bibitem{maclane}
        S.~Mac~Lane.
        \emph{Categories for the Working Mathematician},
        second edition, Springer, 1998.

    \bibitem{massey} W.S.~Massey.
      \emph{Algebraic Topology: An Introduction}, 
      Graduate Texts in Mathematics, 56, Springer, 1989.

    \bibitem{may}
      J.P.~May.
      \emph{Simplicial Objects in Algebraic Topology},
      University of Chicago Press, 1993.

    \bibitem{mohar-thomassen}
      B.~Mohar,C.~Thomassen.
      \emph{Graphs on surfaces}, 
      Johns Hopkins Studies in the Mathematical Sciences, 
      Johns Hopkins University Press, 2001.

    \bibitem{munkresalg}
      J.R.~Munkres. \emph{Elements of algebraic topology},
      Addison-Wesley, 1984.

    \bibitem{companion} F.~Mittelbach, M.~Goossens, J.~Braams,
      D.~Carlisle, C.~Rowley. \emph{The \LaTeX\ Companion}, zweite
      Auf\-lage, Addison-Wesley, 2004.

    \bibitem{rotman} J.J.~Rotman.
      \emph{An Introduction to the Theory of Groups}, fourth edition, 
      Graduate Texts in Mathematics, 148, Springer, 1999. 

    \bibitem{tantau} 
        T.~Tantau. \emph{The {\normalfont Ti\textit{k}Z} and
          {\normalfont PGF} Packages}, 
        \\
        \textsf{http://www.ctan.org/tex-archive/graphics/pgf/base/doc/generic/pgf/pgfmanual.pdf} 

    \bibitem{weibel}
      C.A.~Weibel.
      \emph{An introduction to homological algebra},
      Cambridge Studies in Advanced Mathematics, 38, 
      Cambridge University Press, 1994. 

  \end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% end of document 
% everything below this line will be ignored by LaTeX
\end{document}
