slides-nis2023.tex

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\title{Constructive mathematics for mathematical phantoms: a report on the
emerging synthetic account of algebraic geometry}

\author{Ingo Blechschmidt}
\date{June 27th, 2023}

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  Constructive mathematics for mathematical phantoms: \\
  \hil{synthetic algebraic geometry}
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  \scriptsize
  \textit{-- an invitation --}

  \bigskip

  CM:FP 2023 in Niš \\
  June 27th, 2023

  Ingo Blechschmidt
  \par
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\section{Not in this talk}

\begin{frame}{Not in this talk}
  ``Can we salvage the result if we require the function to be uniformly continuous?''
  \bigskip

  ``Can we weaken dependent choice to countable choice?''
  \bigskip

  ``Can we weaken the decidability assumption?''
  \bigskip

  ``Can pointfree topology help?''

  \vspace*{0.2cm}

  \centering
  \qquad\quad\includegraphics[width=0.3\textwidth]{confusion}
\end{frame}

\begin{frame}[t]{Mathematical phantoms}
  \begin{columns}[T]
    \begin{column}{0.25\textwidth}
      \centering
      \includegraphics[width=\textwidth]{wraith-portrait} \\
      \scriptsize
      Gavin Wraith
    \end{column}

    \begin{column}{0.7\textwidth}
      \emph{One of the recurring themes of mathematics,
      and one that I have always found seductive,
      is that of \\\medskip
      \triang{} the nonexistent entity which ought to be there \\
      \phantom{\triang{}} but apparently is not; \\\medskip
      \triang{} which nevertheless obtrudes its effects so
      convincingly that \\
      \phantom{\triang{}} one is forced to concede
      a broader notion of existence.}
    \end{column}
  \end{columns}

  \bigskip

  \mbox{\begin{minipage}{0.15\textwidth}
    \centering\small
    \includegraphics[height=5em]{zeta-function} \\
    $\mathbb{C}$
  \end{minipage}\quad
  \begin{minipage}{0.20\textwidth}
    \centering\small
    \includegraphics[height=5em]{3-adic-numbers} \\
    $\mathbb{Q}_p$
  \end{minipage}\quad
  \begin{minipage}{0.20\textwidth}
    \centering\small
    \includegraphics[height=5em]{bruyn-pope} \\
    $\mathbb{F}_1$
  \end{minipage}\quad
  \begin{minipage}{0.42\textwidth}
    \centering\small
    \includegraphics[height=5em]{hilbert-hotel} \\
    $\infty$
  \end{minipage}}
\end{frame}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{5.95cm}\includegraphics[width=\paperwidth]{fr1}\end{minipage}}
\begin{frame}{A glimpse of algebraic geometry}
  Algebraic geometry studies \hil{solution sets} of polynomial systems of equations, \\
  and spaces obtained by \hil{gluing} such sets:

  \begin{center}
    \includegraphics[angle=90,width=0.5\textwidth]{algebraic-surfaces}

    \scriptsize C. Stussak, P. Schenzel. Interactive visualisation of algebraic surfaces as a tool for shape creation. \\ Int. J. Arts Technol. 4:2 (2011), pp. 216--218
  \end{center}

  Concrete results such as Fermat's Last Theorem: For~$n \geq 3$, no positive integers satisfy
  \[ a^n + b^n = c^n. \]
\end{frame}}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{5.95cm}\includegraphics[width=\paperwidth]{fr1}\end{minipage}}
\begin{frame}{Mathematics vs. algebraic geometry}
  Let~$k$ be a base field,
  e.g. $\QQ$, $\FF_p$, \ldots
  \bigskip

  \subhead{Functions}
  Which functions~$k^2 \to k$ are there?
  \vspace*{-1em}

  \begin{columns}[c]
    \begin{column}{0.4\textwidth}
      \centering
      \[ (x,y) \mapsto x^3 + xy^2 - y^4 \]

      \ \\
      \cmark{} polynomial
    \end{column}
    \begin{column}{0.4\textwidth}
      \centering
      \[ (x,y) \mapsto \begin{cases}
	1, & \text{if $x = 0$,} \\
	0, & \text{else.}
      \end{cases} \]

      \xmark{} non-polynomial
    \end{column}
  \end{columns}
  \pause
  \bigskip

  \subhead{Implications}
  \begin{columns}[c]
    \begin{column}{0.4\textwidth}
      \centering
      \[
	\left.\begin{array}{@{}r@{\ }c@{\ }l@{}}
	  1+x^2 &=& 0 \\
	  1+x^2+x^4 &=& 0
	\end{array}\right\} \Rightarrow 1 = 0
      \]
      \visible<3->{\cmark{} algebraic certificate:
      \vspace*{-1em}
      \[ 1 = (-x^2)\cdot(1+x^2) + 1\cdot(1+x^2+x^4)\]}
    \end{column}
    \begin{column}{0.4\textwidth}
      \centering
      \[ \left.\begin{array}{@{}l@{}}\phantom{x^2}\\\phantom{x^2}\end{array}\right.\!\!\!\!
      x^2 = 0 \Rightarrow x = 0 \]

      \visible<4->{\xmark{} no algebraic certificate:
      \vspace*{-1em}
      \[ x = \ldots x^2 \ldots?! \]}
    \end{column}
  \end{columns}

\end{frame}}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{3.95cm}\includegraphics[width=\paperwidth]{staircase}\end{minipage}}
\begin{frame}{Transfinite methods?}
  The standard road to algebraic geometry:
  \begin{enumerate}
    \item Invent topological spaces.
    \item Put the Zariski topology on~$k^n$.
    \item Add non-maximal prime ideals to soberify the space.
    \item Invent sheaves.
    \item Construct the structure sheaf.
  \end{enumerate}
  \pause
  This requires \ldots
  \begin{multicols}{2}
    \begin{itemize}
      \item large structures
      \item powersets
      \item law of excluded middle
      \item axiom of choice
    \end{itemize}
  \end{multicols}
  \pause
  \vspace*{-0.5em}
  despite:
  \begin{itemize}
    \item concrete subject matter
    \item practical computer algebra systems for computations
    %\item well-established constructive algebra
    \item high-level proofs often constructive
  \end{itemize}
\end{frame}}

{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{3.23cm}\includegraphics[width=\paperwidth]{topos-horses-lighter-lighter}\end{minipage}}
\begin{frame}{Synthetic algebraic geometry}
  \hil{Postulate.}
  %Algebraic geometry can alternatively be developed from just
  %three postulates:
  We have a~$k$-algebra~$R$ such that
  \begin{enumerate}
    \item $R$ is local%
    \only<2->{: \emph{if a sum is invertible, then so is one of the summands}},

    \item $R$ is \hil{quasicoherent}%
    \only<3->{: \emph{the map~$A \to R^{\Spec(A)}$ is bijective for every f.p.~$R$-algebra $A$}},
  \end{enumerate}
  and such that we have
  \begin{enumerate}
    \addtocounter{enumi}{2}
    \item \hil{Zariski-choice}.
  \end{enumerate}
  \medskip

  \pause
  \pause
  \pause
  \textbf{Def.} $\Spec(A) \defeq \Hom_R(A,R) = \{ \varphi : A \to R \,|\, \text{$\varphi$ is an~$R$-algebra homomorphism} \}$.
  \pause

  \textbf{Ex.} $\Spec(R[X_1,\ldots,X_n]) \cong R^n$\pause,\quad
  $\Spec(R) \cong R^0 = \{ () \}$\pause,\quad
  $\Spec(R/(x)) \cong \{ () \,|\, x = 0 \}$.
  \pause

  \textbf{Prop.} $R^{(R^n)} \cong R[X_1,\ldots,X_n]$.
  \pause

  \textbf{Prop.} (``Nullstellensatz'') If~$\Spec(A) = \emptyset$, then~$A = 0$.
  \pause

  \textbf{Cor.} For all~$x \in R$, if~$x \neq 0$, then~$x$ is invertible.
  \pause

  \textbf{Prop.} It is \emph{not the case} that for every~$\varepsilon \in R$,
  if~$\varepsilon^2 = 0$ then~$\varepsilon = 0$.
  \pause

  \textbf{Cor.} It is \emph{not the case} that for every~$x \in R$,
  either~$x = 0$ or~$x \neq 0$.
\end{frame}}


%\begin{frame}{Formalizing algebraic geometry}
%  \bigskip
%  \bigskip
%  \bigskip
%  \includegraphics[width=\textwidth]{schemes-in-lean2}
%
%  \vspace*{-22em}
%  \includegraphics[frame=0.1em,bb=0 170 610 625,width=0.3\textwidth]{schemes-in-coq}\hfill
%  \includegraphics[frame=0.1em,bb=0 170 610 625,width=0.3\textwidth]{schemes-in-lean}
%\end{frame}

\end{document}
  The \hil{space} associated to an~$R$~\hil{algebra}~$A$ is
  \[ \Spec(A) \defeq \Hom_R(A,R) = \{ \varphi : A \to R \,|\, \text{$\varphi$ is an~$R$-algebra homomorphism} \}. \]
  Every element~$f \in A$ gives rise to a function~$\mathrm{ev}_f$ on~$\Spec(A)$:
  \[ \mathrm{ev}_f : \Spec(A) \to R,\ \varphi \mapsto \varphi(f). \]