slides-filmat2018.tex

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\title{Exploring mathematical objects from custom-tailored mathematical universes}
\author{Ingo Blechschmidt}
\date{May 26th, 2018}

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\begin{document}

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{\title{}
\begin{frame}[c]
  \centering
  \medskip
  \includegraphics[height=0.4\textheight]{acquaviva-platani-street}%
  \includegraphics[height=0.4\textheight]{mussomeli-sheep}%
  \includegraphics[height=0.4\textheight]{mussomeli-performance}%
  \includegraphics[height=0.4\textheight]{mussomeli-castle}%
  \medskip

  \hil{Exploring mathematical objects from \\ custom-tailored mathematical universes} \\

  \emph{-- an invitation --}

  \bigskip

  \scriptsize
  Ingo Blechschmidt \\
  University of Augsburg / \\
  Max Planck Institute for Mathematics in the Sciences, Leipzig \\
 
  \medskip

  Third international conference of the \\
  \emph{Italian Network for the Philosophy of Mathematics} \\
  in Mussomeli
  \medskip

  May 26th, 2018
  \par
\end{frame}}


\section[Toposophic landscape]{A glimpse of the toposophic landscape}

\begin{frame}[fragile]{A glimpse of the toposophic landscape}
  \tikzstyle{topos} = [draw=mypurple, very thick, rectangle, rounded corners, inner sep=5pt, inner ysep=10pt]
  \tikzstyle{title} = [fill=mypurple, text=white]

  \input{images/primes.tex}
  %\renewcommand{\sieve}[2]{SIEVE}
  %\renewcommand{\fakesieve}[2]{SIEVE}

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    \node[topos, #4] [fit = #3] (#1) {};
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      \only<2->{\fewstuff}
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    \node (prop-set1) [below of=objs-set1, align=left] {
      \only<1->{%
        The usual laws \\
        of logic hold.
      }
    };

    \node (prop-eff1) [below of=objs-eff1, align=left] {
      \only<2->{%
        Every function \\
        is computable.
      }
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        The axiom of \\
        choice fails.
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\note{\justifying
  The concept of a platonic heaven, where the
  objects live which mathematicians study, can be mathematically realized by
  the topos~$\Set$. As a category, it consists of all sets and all maps between
  these sets.

  But besides~$\Set$, there are many more toposes. Each such topos can be
  thought of as a mathematical universe, containing their own versions of the
  natural numbers,
}


\section[Internal universe]{The internal universe of a topos}

\begin{frame}{The internal universe of a topos}
  For any topos~$\E$ and any statement~$\varphi$, we define the meaning of
  \vspace*{-0.5em}
  \[
    \text{``$\E \models \varphi$''} \quad
    \text{(``$\varphi$ holds in the internal universe of~$\E$'')}
  \]

  \vspace*{-1.0em}
  using the \hil{Kripke--Joyal semantics}.

  \vspace*{-1em}
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      $\Set \models \varphi$ \\
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      $\Sh(X) \models \varphi$ \\
      ``$\varphi$ holds continuously.''
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      \centering
      $\Eff \models \varphi$ \\
      ``$\varphi$ holds computably.''
    \usebeamertemplate{block end}\end{column}
  \end{columns}
  \medskip

  \pause
  Any topos supports \hil{mathematical reasoning}:

  \vspace*{-1em}
  \begin{hilblock}
    If~$\E \models \varphi$ and if~$\varphi \vdash \psi$
    \pointthis{<3>}{intuitionistically}{%
      no $\varphi \vee \neg\varphi$,\ \
      no $\neg\neg\varphi \Rightarrow \varphi$,\ \
      no axiom of choice},
    then~$\E \models \psi$.
  \end{hilblock}
\end{frame}


\section[First steps]{First steps in alternate universes}

\newcommand{\intex}[3]{#1 \quad #2\par #3\bigskip\medskip}

\begin{frame}{First steps in alternate universes}
  \begin{changemargin}{-1.1em}{0em}
    \fontsize{10pt}{12pt}\selectfont
    \begin{itemize}
      \item \intex{
        $\Eff \models \text{``Any number is prime or is not prime.''}$
      }{\textcolor{green!90}{\cmark}}{
        Meaning: There is a \hil{Turing machine} which determines of
        any given number whether it is prime or not.
      }

      \item \intex{
        $\Eff \models \text{``There are infinitely many prime numbers.''}$
      }{\textcolor{green!90}{\cmark}}{
        Meaning: There is a \hil{Turing machine} producing arbitrarily many
        primes.\\[0em]
      }

      \item \intex{
        $\Eff \models \text{``Any function~$\NN \to \NN$ is the zero function or not.''}$
      }{\textcolor{red!80}{\xmark}}{
        Meaning: There is a \hil{Turing machine} which, given a Turing
        machine computing a function~$f : \NN \to \NN$, determines whether~$f$
        is zero or not.
      }

      \item \intex{
        $\Eff \models \text{``Any function~$\NN \to \NN$ is computable.''}$
      }{\textcolor{green!90}{\cmark}}{}

      \item \intex{
        $\Sh(X) \models \text{``Any cont. function with opposite signs has
        a zero.''}$
      }{\textcolor{red!80}{\xmark}}{
        Meaning: Zeros can locally be picked \hil{continuously} in
        continuous families of continuous functions.
        \textcolor{red!80}{(\href{https://rawgit.com/iblech/internal-methods/master/images/zeros-in-families.mp4}{video} for counterexample)}
      }
    \end{itemize}
  \end{changemargin}
\end{frame}


\section[Applications]{Applications in commutative algebra}

\begin{frame}{Applications in commutative algebra}
  Let~$A$ be a reduced commutative ring.

  \hil{For instance:}\ \ $\ZZ$,\ \ $\ZZ[X]$,\ \ $\ZZ[X,Y,Z]/(X^n+Y^n-Z^n)$,\ \ $\QQ$,\ \ $\RR$

  \mbox{The \hil{little Zariski topos} of~$A$ contains a \hil{mirror image} of~$A$:
  $A^\sim$.}

  \setlength\columnsep{0pt}
  \begin{multicols*}{2}
    \begin{enumerate}
      \item $A^\sim$ is always a \hil{field}. \columnbreak
      \item $A^\sim$ is still \hil{very close} to~$A$.
    \end{enumerate}
  \end{multicols*}
  % \item Studying~$A^\sim$ is the same as studying~$A$ \hil{from a different
  % point of view}.

  \vspace*{-2em}
  \begin{columns}[t]
    \begin{column}[t]{0.47\textwidth}
      \centering

      \begin{varblock}{\textwidth}{A baby application}
        \justifying
        Let~$M$ be a surjective matrix with more rows than columns over a ring~$A$.
        Then~$A = 0$.
      \end{varblock}

      \scalebox{0.8}{$\begin{pmatrix}
        \cdot & \cdot \\
        \cdot & \cdot \\
        \cdot & \cdot
      \end{pmatrix}$}
    \end{column}

    \begin{column}[t]{0.47\textwidth}
      \centering

      \begin{varblock}{\textwidth}{Generic freeness\phantom{p}}
        \justifying
        Generically, any finitely generated module over a reduced ring is free.
      \end{varblock}
      \vspace*{-0.5em}

      \includegraphics[width=0.6\textwidth]{generic-freeness}
    \end{column}
  \end{columns}
\end{frame}

{\renewcommand{\insertframenumber}{4a}
\begin{frame}{The little Zariski topos in more detail}
  Recall~$A[f^{-1}] = \bigl\{ \frac{u}{f^n} \,|\, u \in A, n \in \NN \bigr\}$.

  \begin{itemize}
    \item $\Sh(\Spec(A)) \models \text{``For all~$x \in A^\sim$, \ldots''}$

          Meaning: For all~$f \in A$ and all~$x \in A[f^{-1}]$, \ldots
          \medskip

    \item $\Sh(\Spec(A)) \models \text{``There is~$x \in A^\sim$ such that \ldots''}$

          \mbox{Meaning: There is a partition of unity,~$1 = f_1 + \cdots + f_n \in A$,}
          such that for each~$i$, there exists~$x_i \in A[f_i^{-1}]$
          with~\ldots
          \medskip

    \item $\Sh(\Spec(A)) \models \text{``$\varphi$ implies $\psi$''}$

          Meaning: For all~$f \in A$, if~$\varphi$ on stage~$f$, then~$\psi$ on
          stage~$f$.
  \end{itemize}
\end{frame}
\addtocounter{framenumber}{-1}}


\appendix
\section{Conclusion}

\begin{frame}{Topos theory \ldots}
  \vspace*{-0.5em}
  \begin{itemize}
    \item enriches the platonism debate,
    %\item yields a new view on ``continuous dependence on parameters'',
    \item uncovers further relations between objects,
    \item allows to study objects from a different point of view,
    \item has applications in mathematical practice.
  \end{itemize}

  \centering
  \href{https://www.oliviacaramello.com/Papers/Papers.htm}{\includegraphics[height=0.45\textheight]{olivia-tst}}
  \href{http://math.andrej.com/2014/03/04/intuitionistic-mathematics-and-realizability-in-the-physical-world/}{\includegraphics[height=0.45\textheight]{zenil-computable-universe}}
  \href{https://pizzaseminar.speicherleck.de/skript2/zariski-topos-klein.pdf}{\includegraphics[height=0.45\textheight]{fun-with-the-zariski-topos}}
  \href{https://rawgit.com/iblech/internal-methods/master/notes.pdf}{\includegraphics[height=0.45\textheight]{phd-cover}}
\end{frame}

\addtocounter{framenumber}{-1}

\end{document}