%\documentclass{beamer}  % {{{1
\documentclass[handout]{beamer}
\usepackage{amsfonts, amsmath, amssymb, amsthm, mathtools}
\usepackage{booktabs, enumitem, hyperref, graphicx, multicol, tikz}
\usepackage[final]{microtype}

% beamer settings
\usetheme{Madrid}
\setbeamercolor{alerted text}{fg=blue}
\beamertemplatenavigationsymbolsempty
\let\emph\textbf
\setitemize{
  itemsep=0.5em, topsep=0.5em,
  label=\usebeamerfont*{itemize item}
  \usebeamercolor[fg]{itemize item}
  \usebeamertemplate{itemize item}
}
\setbeamertemplate{items}[circle]
\setbeamertemplate{section in toc}[circle]
\newcommand{\slidefinish}{\hfill$\cdot$}
\tikzset{alert/.style = {color=blue, thick}}

% mathbb and mathcal symbols
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\renewcommand{\AA}{\mathbb{A}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\m}[1]{\mathbb{#1}}      % models
\newcommand{\alg}[1]{\m{#1}}         % algebras
\newcommand{\rel}[1]{\mathbf{#1}}    % relational structures
\newcommand{\cl}[1]{\mathcal{#1}}    % classes

% theorems and similar environments
\theoremstyle{plain}
  \newtheorem{thm}{Theorem}[section]  \newtheorem*{thm*}{Theorem}
  \newtheorem{claim}[thm]{Claim}      \newtheorem*{claim*}{Claim}
  \newtheorem{conj}[thm]{Conjecture}  \newtheorem*{conj*}{Conjecture}
  \newtheorem{cor}[thm]{Corollary}    \newtheorem*{cor*}{Corollary}
  \newtheorem{lem}[thm]{Lemma}        \newtheorem*{lem*}{Lemma}
  \newtheorem{prop}[thm]{Proposition} \newtheorem*{prop*}{Proposition}
\theoremstyle{definition}
  \newtheorem{defn}[thm]{Definition} \newtheorem*{defn*}{Definition}
  \newtheorem{ex}[thm]{Example}      \newtheorem*{ex*}{Example}
\theoremstyle{remark}
  \newtheorem{rk}[thm]{Remark}  \newtheorem*{rk*}{Remark}
\newcommand{\Case}[1]{\smallskip \textbf{Case #1:}}
\newenvironment{claimproof} {
  \begin{proof}[Proof of claim]
  \renewcommand{\qedsymbol}{\ensuremath{\circ}}
  } {
  \end{proof}
  }

% custom math commands
\DeclareMathOperator{\Cg}{Cg}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Rel}{Rel}
\DeclareMathOperator{\Sg}{Sg}
\DeclareMathOperator{\diag}{diag}
\newcommand{\bmat}[1]{ \begin{bmatrix} #1 \end{bmatrix} }
\newcommand{\Bmat}[1]{ \begin{Bmatrix} #1 \end{Bmatrix} }
\newcommand{\pmat}[1]{ \begin{pmatrix} #1 \end{pmatrix} }
\newcommand{\mat}[1]{ \begin{matrix} #1 \end{matrix} }
\newcommand{\vect}[1]{ \left< #1 \right> }
\newcommand{\ds}[1]{ \displaystyle{#1} }
\newcommand{\stack}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}

% tikz settings
\usetikzlibrary{calc, matrix, positioning}
\newcommand{\TikzCenterNode}{\node (center) at ($(current bounding box.north east)!0.5!(current bounding box.south west)$) {\phantom{$\cdot$}};}

% misc
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\TODO}[1]{\noindent\textbf{TODO: #1}}

% document specific stuff
\usepackage[thicklines]{cancel} \renewcommand{\CancelColor}{\color{blue}}

\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n} \newcommand{\Cong}[1]{\mathbbold{#1}}

\DeclareMathOperator{\M}{M}
\DeclareMathOperator{\C}{C}

% tikz
\usepackage{xparse}
\tikzset{myStyle/.style={baseline=(center.base), font=\small,
    every node/.style={inner sep=0.25em} }}

% Square commands. Syntax: 4 vertices, an optional scaling factor, an optional font.
\newcommand{\SquareUnwrapped}[4]{ % {{{
  \path (0,0)  node (nw) {$#1$}
      ++(1,0)  node (ne) {$#2$}
      ++(0,-1) node (se) {$#4$}
      ++(-1,0) node (sw) {$#3$};
  \draw (nw) -- (ne) -- (se) -- (sw) -- (nw);
  \TikzCenterNode
}  % }}}
\NewDocumentCommand{\Square}{ O{} O{} O{} O{} O{1} }{ % {{{
  \begin{tikzpicture}[myStyle, scale=#5*1]
    \SquareUnwrapped{#1}{#2}{#3}{#4}
  \end{tikzpicture}
}  % }}}
\NewDocumentCommand{\SquareScriptsize}{ O{} O{} O{} O{} O{1} }{ % {{{
  \begin{tikzpicture}[myStyle, scale=#5*0.9, font=\scriptsize]
    \SquareUnwrapped{#1}{#2}{#3}{#4}
  \end{tikzpicture}
}  % }}}

% Cube commands. Syntax: 8 vertices, an optional scaling factor, an optional font.
\newcommand{\CubeUnwrapped}[8]{ % {{{
  \path (0,0)  node (back_nw)    {$#1$}
    ++(1,0)  node (back_ne)      {$#2$}
    ++(0,-1) node (back_se)      {$#4$}
    ++(-1,0) node (back_sw)      {$#3$}
      (0.5,-0.5) node (front_nw) {$#5$}
    ++(1,0)  node (front_ne)     {$#6$}
    ++(0,-1) node (front_se)     {$#8$}
    ++(-1,0) node (front_sw)     {$#7$};
  \draw (front_nw) -- (front_ne) -- (front_se) -- (front_sw) -- (front_nw)
    (back_nw) -- (back_ne) (back_sw) -- (back_nw)
    (back_nw) -- (front_nw)
    (back_ne) -- (front_ne)
    (back_sw) -- (front_sw);
  \draw[densely dotted] (back_ne) -- (back_se) -- (back_sw)
    (back_se) -- (front_se);
  \TikzCenterNode
}  % }}}
\NewDocumentCommand{\Cube}{ O{} O{} O{} O{} O{} O{} O{} O{} O{1} }{  % {{{
  \begin{tikzpicture}[myStyle, scale=#9*1]
    \CubeUnwrapped{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}
  \end{tikzpicture}
}  % }}}
\NewDocumentCommand{\CubeScriptsize}{ O{} O{} O{} O{} O{} O{} O{} O{} O{1} }{  % {{{
  \begin{tikzpicture}[myStyle, scale=#9*0.9, font=\scriptsize]
    \CubeUnwrapped{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}
  \end{tikzpicture}
}  % }}}
%----------------------------------------------------------------------------}}}1

\begin{document}
% title page {{{1
\title{Supernilpotence Need Not Imply Nilpotence}
\author[Moore$^*$ (KU), Moorhead (Charles)]{
  Matthew Moore$^{*,\,1}$, 
  Andrew Moorhead$^2$
}
\institute[]{
  The University of Kansas$^1$,
  Charles University in Prague$^2$
}
\date{January 15, 2020}

\begin{frame}
\titlepage
\end{frame}
%----------------------------------------------------------------------------}}}1

\begin{frame} % notation {{{1
Let $\AA$ be an algebra and $\alg{R}\leq \AA^4$. \pause
\only<2|handout:0>{ \vspace{0.1em} }
\[
  \only<2|handout:0>{ \pmat{a \\ b \\ c \\ d} \in R }
  \only<3->{ \xcancel{\pmat{a \\ b \\ c \\ d} \in R} }
  \uncover<3->{
    \qquad \text{ instead, we will write } \qquad
    \Square[a][b][c][d] \in R
  }
\]

\pause \pause

Let $\alpha_1, \alpha_2$ be congruences of $\AA$. \pause
The \emph{$(\alpha_1, \alpha_2)$-squares} of $\AA$ are \pause
\[
  \M(\alpha_1, \alpha_2)
  \coloneqq \Sg_{\AA^{4}}\left\{ \Square[a][b][a][b], \Square[c][c][d][d] \mid 
    (a,b)\in \alpha_1,\ (c,d)\in \alpha_2 \right\}
\]

\pause

\emph{Observe}
\[
  \Square[a][b][c][d]\in \M(\alpha_1, \alpha_2)
  \pause
  \qquad\Longrightarrow\qquad
  \begin{tikzpicture}[myStyle]
    \SquareUnwrapped{a}{b}{c}{d}
    \draw (nw) to[out=60, in=180-60] node[above]{$\alpha_1$} (ne)
      (sw) to[out=-60, in=180+60] node[below]{$\alpha_1$} (se)
      (nw) to[out=180+30, in=180-30] node[auto, swap]{$\alpha_2$} (sw)
      (ne) to[out=-30, in=30] node[auto]{$\alpha_2$} (se);
  \end{tikzpicture}
\]
\end{frame}  % }}}1
\begin{frame} % 2-ary commutator {{{1
\[
  \M(\alpha_1, \alpha_2)
  \coloneqq
  \Sg_{\AA^{4}}\left\{ \Square[a][b][a][b], \Square[c][c][d][d] \mid 
    (a,b)\in \alpha_1,\ (c,d)\in \alpha_2 \right\}
\]

\vfill \pause

$\alpha_1$ \emph{centralizes} $\alpha_2$ modulo $\delta$ if \dots
\uncover<6->{ \hfill [ write $\C(\alpha_1, \alpha_2; \delta)$ ] }

\pause

\[
  \forall
  \Square[a][b][c][d]\in \M(\alpha_1, \alpha_2), \quad
  \pause
    \begin{tikzpicture}[myStyle]
      \SquareUnwrapped{a}{b}{c}{d}{e}
      \draw (nw) to[out=180+30, in=180-30] node[auto, swap]{$\delta$} (sw);
    \end{tikzpicture}
    \pause
    \qquad\Longrightarrow\qquad
    \begin{tikzpicture}[myStyle]
      \SquareUnwrapped{a}{b}{c}{d}
      \draw[densely dashed] (ne) to[out=-30, in=30] node[auto]{$\delta$} (se);
    \end{tikzpicture}
\]

\pause \pause

The \emph{commutator} of $\alpha_1$ and $\alpha_2$ is the smallest $\delta$
satisfying the above,

\pause

\[
  [ \alpha_1, \alpha_2] 
  \coloneqq \bigwedge_{ \C(\alpha_1, \alpha_2; \delta) } \delta
\]

\pause \bigskip

\emph{Can we generalize this to a higher arity commutator?}
\end{frame}  % }}}1

\begin{frame} % 3-dim notation {{{1
\only<1|handout:0>{Let $\AA$ be an algebra and $\alg{R}\leq \AA^4$.}
\only<1|handout:0>{ \[
  \xcancel{\pmat{a \\ b \\ c \\ d} \in R}
  \qquad \text{ instead, we will write } \qquad
  \Square[a][b][c][d] \in R
\] }
\only<2->{
  \uncover<3->{Let $\AA$ be an algebra and $\alg{R}\leq \AA^{2^3}$.}
  \uncover<4->{ \[
    \only<4|handout:0>{ \pmat{a \\ \vdots \\ h} \in R }
    \only<5->{ \xcancel{\pmat{a \\ \vdots \\ h} \in R} }
    \uncover<5->{
      \qquad \text{ instead, we will write } \qquad
      \Cube[a][b][c][d][e][f][g][h] \in R
    }
  \] }
}

\only<-5|handout:0>{
  Let $\alpha_1, \alpha_2$ be congruences of $\AA$.
  The \emph{$(\alpha_1, \alpha_2)$-squares} of $\AA$ are
  \[
    \M(\alpha_1, \alpha_2)
    \coloneqq \Sg_{\AA^{4}}\left\{ \Square[a][b][a][b], \Square[c][c][d][d] \mid 
      (a,b)\in \alpha_1,\ (c,d)\in \alpha_2 \right\}
  \]
  \vspace{1.1em}
}
\only<6->{
  \uncover<7->{Let $\alpha_1, \alpha_2, \alpha_3$ be congruences of $\AA$.}
  \uncover<8->{The \emph{$(\alpha_1, \alpha_2, \alpha_3)$-cubes} of $\AA$ are}
  \uncover<9->{ \small \[
    \M(\alpha_1, \alpha_2, \alpha_3)
    \coloneqq \Sg_{\AA^{2^3}}\left\{ 
      \Cube[a][b][a][b][a][b][a][b],
      \Cube[c][c][d][d][c][c][d][d],
      \Cube[e][e][e][e][f][f][f][f]
      \mid \begin{gathered}
        (a, b) \in \alpha_1, \\
        (c, d) \in \alpha_2, \\
        (e, f) \in \alpha_3
      \end{gathered} \right\}
  \] \normalsize }
}

\only<-9|handout:0>{\emph{Observe}
  \[
    \Square[a][b][c][d]\in \M(\alpha_1, \alpha_2)
    \qquad\Longrightarrow\qquad
    \begin{tikzpicture}[myStyle]
      \SquareUnwrapped{a}{b}{c}{d}
      \draw (nw) to[out=60, in=180-60] node[above]{$\alpha_1$} (ne)
        (sw) to[out=-60, in=180+60] node[below]{$\alpha_1$} (se)
        (nw) to[out=180+30, in=180-30] node[auto, swap]{$\alpha_2$} (sw)
        (ne) to[out=-30, in=30] node[auto]{$\alpha_2$} (se);
    \end{tikzpicture}
  \]
}
\only<10->{ \uncover<11->{\emph{Observe}}
  \vspace{0.75em}
  \[
    \uncover<11->{\Cube[a][b][c][d][e][f][g][h]\in \M(\alpha_1, \alpha_2, \alpha_3)}
    \uncover<12->{
      \qquad\Longrightarrow\qquad
      \begin{tikzpicture}[myStyle]
        \draw (0,0) -- node[above]{$\alpha_1$} (1,0)
          (0,0) -- node[left]{$\alpha_2$} (0,1)
          (0,0) -- node[below left=-0.25em]{$\alpha_3$} (0.5,-0.5);
          \TikzCenterNode
          \node (center) at ($(center)+(0,0.05)$) {\phantom{$\cdot$}};
      \end{tikzpicture}
      \quad \text{in}\quad
      \Cube[a][b][c][d][e][f][g][h]
    }
  \]
  \vspace{0.75em}
}
\end{frame}  % }}}1
\begin{frame}[t] % 3-dim commutator {{{1
\only<1|handout:0>{ 
  \[
    \M(\alpha_1, \alpha_2)
    \coloneqq
    \Sg_{\AA^{4}}\left\{ \Square[a][b][a][b], \Square[c][c][d][d] \mid 
      (a,b)\in \alpha_1,\ (c,d)\in \alpha_2 \right\}
  \]
  \vspace{1.1em}
}
\only<2->{ \uncover<3->{
  { \small \[
    \M(\alpha_1, \alpha_2, \alpha_3)
    \coloneqq \Sg_{\AA^{2^3}}\left\{ 
      \Cube[a][b][a][b][a][b][a][b],
      \Cube[c][c][d][d][c][c][d][d],
      \Cube[e][e][e][e][f][f][f][f]
      \mid \begin{gathered}
        (a, b) \in \alpha_1, \\
        (c, d) \in \alpha_2, \\
        (e, f) \in \alpha_3
      \end{gathered} \right\}
  \] }
} }

\only<-3|handout:0>{
  $\alpha_1$ \emph{centralizes} $\alpha_2$ modulo $\delta$ if \dots
  \hspace{7.55em} [ write $\C(\alpha_1, \alpha_2; \delta)$ ]
}
\only<4->{ \uncover<5->{
  $\alpha_1, \alpha_2$ \emph{centralize} $\alpha_3$ modulo $\delta$ if \dots
  \uncover<10->{ \hspace{4.5em} [ write $\C(\alpha_1, \alpha_2, \alpha_3; \delta)$ ] }
} }
\only<-5|handout:0>{
  \vspace{0.57em}
  \[
    \forall
    \Square[a][b][c][d]\in \M(\alpha_1, \alpha_2), \qquad
      \begin{tikzpicture}[myStyle]
        \SquareUnwrapped{a}{b}{c}{d}{e}
        \draw (nw) to[out=180+30, in=180-30] node[auto, swap]{$\delta$} (sw);
      \end{tikzpicture}
      \qquad\Longrightarrow\qquad
      \begin{tikzpicture}[myStyle]
        \SquareUnwrapped{a}{b}{c}{d}
        \draw[densely dashed] (ne) to[out=-30, in=30] node[auto]{$\delta$} (se);
      \end{tikzpicture}
  \]
  \vspace{0.57em}
}
\only<6->{ \uncover<7->{ { \small \[
  \forall\ 
  \Cube[a][b][c][d][e][f][g][h]\in \M(\alpha_1, \alpha_2, \alpha_3), \qquad
  \uncover<8->{
    \begin{tikzpicture}[myStyle]
      \CubeUnwrapped{a}{b}{c}{d}{e}{f}{g}{h}
      \draw (back_sw) to[out=180+30,in=180] node[auto,swap]{$\delta$} (front_sw)
        (back_se) to[out=0,in=30] node[auto]{$\delta$} (front_se)
        (back_nw) to[out=180+30,in=180] node[auto,swap]{$\delta$} (front_nw);
      \TikzCenterNode
    \end{tikzpicture}
  }
  \uncover<9->{
    \qquad\Longrightarrow\qquad
    \begin{tikzpicture}[myStyle]
      \CubeUnwrapped{a}{b}{c}{d}{e}{f}{g}{h}
      \draw[dashed] (back_ne) to[out=0,in=30] node[auto]{$\delta$} (front_ne);
      \TikzCenterNode
    \end{tikzpicture}
  }
\] } } }

\only<-10|handout:0>{
  The \emph{commutator} of $\alpha_1$ and $\alpha_2$ is the smallest $\delta$
  satisfying the above,
  \[
    [ \alpha_1, \alpha_2 ] 
    \coloneqq \bigwedge_{ \C(\alpha_1, \alpha_2; \delta) } \delta
  \]
}
\only<11->{ \uncover<12->{ { \small
  The \emph{3-dim commutator} of $\alpha_1, \alpha_2, \alpha_3$ is the
  smallest $\delta$ satisfying the above,
  \[
    [ \alpha_1, \alpha_2, \alpha_3 ]
    \coloneqq \bigwedge_{ \C(\alpha_1, \alpha_2, \alpha_3; \delta) } \delta
  \]
  \vspace{0.5em} 
} } }

\emph{Can we generalize this to a higher arity commutator?}

\uncover<13->{ \hfill \alert{Yes, even for arbitrary $n$!} }
\end{frame}  % }}}1

\begin{frame} % abelian, solvable, nilpotent, supernilpotent {{{1
A general definition of the commutator yields general notions of
abelianness,  solvability, and nilpotence.

\begin{itemize}
  \item<2-> $\AA$ is \emph{abelian} if $[ \Cong{1}, \Cong{1} ] = \Cong{0}$.

  \item<4-> Define the \emph{derived series},
    \[
      [ \alpha ]_0 \coloneqq \alpha
      \qquad\qquad
      [ \alpha ]_{n+1} \coloneqq \big[ [\alpha]_n, [\alpha]_n \big].
    \]
    \uncover<5->{ 
      $\AA$ is \emph{$m$-solvable} if $[ \Cong{1} ]_m = \Cong{0}$ for some $m$.
    }

  \item<6-> define the \emph{lower central series},
    \[
      ( \alpha ]_0 \coloneqq \alpha
      \qquad\qquad
      ( \alpha ]_{n+1} \coloneqq \big[ \alpha, (\alpha]_n \big].
    \]
    \uncover<7->{
      $\AA$ is (left) \emph{$m$-nilpotent} if $( \Cong{1} ]_m = \Cong{0}$ for
      some $m$.
    }

  \item<8-> The higher dimensional commutator has its own series,
    \[
      \alpha 
      \geq [\alpha, \alpha] 
      \geq [\alpha, \alpha, \alpha] 
      \geq \dots
      \geq [ \alpha, \dots, \alpha ]
      \geq \dots
    \]
    \uncover<9->{
      $\AA$ is \emph{$(m+1)$-supernilpotent} if $[\Cong{1}, \dots, \Cong{1}] =
      \Cong{0}$ ($m$-ary)
    }
\end{itemize}

\uncover<3->{
  ($\Cong{1}$ is the universal congruence and $\Cong{0}$ is the trivial
  congruence)
}
\end{frame}  % }}}1
\begin{frame} % does supernilt imply nilp? {{{1
For an algebra $\AA$ we have general notions of
\begin{multicols}{2}
\begin{itemize}
  \pause\item abelianness
  \pause\item solvability
  \pause\item nilpotence
  \pause\item supernilpotence
\end{itemize}
\end{multicols}

\pause

\textbf{How are these related?}

\pause \bigskip

\textbf{How are nilpotence and supernilpotence related?}

\pause \bigskip

\textbf{Does supernilpotence imply nilpotence?}

\pause \bigskip

\begin{thm}[Moorhead]
If $\AA$ satisfies a non-trivial idempotent equational condition, then
``\emph{Yes}''.
\end{thm}

\pause

\begin{thm}[Kearnes, Szendrei]
If $\AA$ is finite, then ``\emph{Yes}''.
\end{thm}

\pause

\begin{thm}[M+M]
Supernilpotence does not imply nilpotence in general.
\end{thm}
\end{frame}  % }}}1

\begin{frame} % proof strategy {{{1
\emph{Idea:} \pause Construct $\AA$ as simply as possible.
\begin{itemize}
  \pause\item Make $\AA$ 2-supernilpotent: $[ \Cong{1}, \Cong{1}, \Cong{1} ] = \Cong{0}$.
  \pause\item When $[\cdot, \cdot]$ is not symmetric, there are many different
    notions of nilpotence. \pause All of them imply solvability.
  \pause\item Make $\AA$ not solvable. 
\end{itemize}

\pause

\emph{What does $[ \Cong{1}, \Cong{1}, \Cong{1} ] = \Cong{0}$ mean?}

\pause

\[
  \M(\Cong{1}, \Cong{1}, \Cong{1})
  \coloneqq \Sg_{\AA^{2^3}}\left\{ 
    \Cube[a][b][a][b][a][b][a][b],
    \Cube[a][a][b][b][a][a][b][b],
    \Cube[a][a][a][a][b][b][b][b]
    \mid a, b\in A \right\}
\]

\pause

\[
  \begin{tikzpicture}[myStyle]
    \only<-10|handout:0>{
      \CubeUnwrapped{a}{b}{c}{d}{e}{f}{g}{h}
    }
    \only<11->{
      \CubeUnwrapped{a}{\alert{b}}{c}{d}{a}{\alert{f}}{c}{d}
    }
    \only<-9|handout:0>{
      \draw (back_sw) to[out=180+30,in=180] node[auto,swap]{$\delta$} (front_sw)
        (back_se) to[out=0,in=30] node[auto]{$\delta$} (front_se)
        (back_nw) to[out=180+30,in=180] node[auto,swap]{$\delta$} (front_nw);
    }
    \only<10-:|handout:0>{ \uncover<10>{
      \draw (back_sw) to[out=180+30,in=180] node[auto,swap]{$\Cong{0}$} (front_sw)
        (back_se) to[out=0,in=30] node[auto]{$\Cong{0}$} (front_se)
        (back_nw) to[out=180+30,in=180] node[auto,swap]{$\Cong{0}$} (front_nw);
    } }
    \TikzCenterNode
  \end{tikzpicture} \in \M(\Cong{1}, \Cong{1}, \Cong{1})
  \qquad\Longrightarrow\qquad
  \begin{tikzpicture}[myStyle]
    \uncover<-10|handout:0>{
      \CubeUnwrapped{a}{b}{c}{d}{e}{f}{g}{h}
    }
    \only<-9|handout:0>{
      \draw[dashed] (back_ne) to[out=0,in=30] node[auto]{$\delta$} (front_ne);
    }
    \only<10-|handout:0>{ \uncover<10>{
      \draw[dashed] (back_ne) to[out=0,in=30] node[auto]{$\Cong{0}$} (front_ne);
    } }
    \TikzCenterNode
    \only<11->{ \node at (center) {$\alert{b} = \alert{f}$}; }
  \end{tikzpicture}
\]

\uncover<12->{
  \emph{We will need to carefully analyze the generation of $\M(\Cong{1}, \Cong{1}, \Cong{1})$.}
}
\end{frame}  % }}}1
\begin{frame} % A_2 motivation {{{1
\small
\[
  \M(\Cong{1}, \Cong{1}, \Cong{1})
  \coloneqq \Sg_{\AA^{2^3}}\left\{ 
    \CubeScriptsize[a][b][a][b][a][b][a][b],
    \CubeScriptsize[a][a][b][b][a][a][b][b],
    \CubeScriptsize[a][a][a][a][b][b][b][b]
    \mid a, b\in A \right\}
\]

\pause

\[ \tag{$\dagger$}
  [ \Cong{1}, \Cong{1}, \Cong{1} ] = \Cong{0}
  \qquad\text{iff}\qquad
  \left[
    \CubeScriptsize[a][\alert{b}][c][d][a][\alert{f}][c][d] \in \M(\Cong{1}, \Cong{1}, \Cong{1})
    \quad\Longrightarrow\quad
    \alert{b} = \alert{f}
  \right]
\]

\pause

\emph{What is the simplest way to ensure $[ \Cong{1}, \Cong{1}, \Cong{1} ] = \Cong{0}$?}
\begin{itemize}
  \pause\item consider $\AA = \big< A \;;\; t(x,y) \big>$ where $t(x,y)$ is
    injective (infinite $\AA$)

  \pause\item generating $\M(\Cong{1}, \Cong{1}, \Cong{1})$ in layers, look for
    first failure of ($\dagger$)
  \[
    \only<-9|handout:0>{ \uncover<7->{
      t\left( \CubeScriptsize[\alert<9>{a_1}][b_1][c_1][d_1][\alert<9>{e_1}][f_1][g_1][h_1][1.05],
              \CubeScriptsize[\alert<9>{a_2}][b_2][c_2][d_2][\alert<9>{e_2}][f_2][g_2][h_2][1.05] \right)
    } }
    \only<10|handout:0>{
      t\left( \CubeScriptsize[\alert<10>{a_1}][b_1][c_1][d_1][\alert<10>{\mathbf{a_1}}][f_1][g_1][h_1][1.05],
              \CubeScriptsize[\alert<10>{a_2}][b_2][c_2][d_2][\alert<10>{\mathbf{a_2}}][f_2][g_2][h_2][1.05] \right)
    }
    \only<11|handout:0>{
      t\left( \CubeScriptsize[a_1][b_1][\alert<11>{c_1}][\alert<11>{d_1}][a_1][f_1][\alert<11>{\mathbf{c_1}}][\alert<11>{\mathbf{d_1}}][1.05],
              \CubeScriptsize[a_2][b_2][\alert<11>{c_2}][\alert<11>{d_2}][a_2][f_2][\alert<11>{\mathbf{c_2}}][\alert<11>{\mathbf{d_2}}][1.05] \right)
    }
    \only<12-13|handout:1>{
      t\left( \CubeScriptsize[a_1][\alert<13>{b_1}][c_1][d_1][a_1][\alert<13>{f_1}][c_1][d_1][1.05],
              \CubeScriptsize[a_2][\alert<13>{b_2}][c_2][d_2][a_2][\alert<13>{f_2}][c_2][d_2][1.05] \right)
    }
    \only<14|handout:2>{
      t\left( \CubeScriptsize[a_1][\alert<14>{b_1}][c_1][d_1][a_1][\alert<14>{\mathbf{b_1}}][c_1][d_1][1.05],
              \CubeScriptsize[a_2][\alert<14>{b_2}][c_2][d_2][a_2][\alert<14>{\mathbf{b_2}}][c_2][d_2][1.05] \right)
    }
    \only<15-|handout:0>{
      t\left( \CubeScriptsize[a_1][\alert<15->{b_1}][c_1][d_1][a_1][\alert<15->{b_1}][c_1][d_1][1.05],
              \CubeScriptsize[a_2][\alert<15->{b_2}][c_2][d_2][a_2][\alert<15->{b_2}][c_2][d_2][1.05] \right)
    }
    \uncover<7->{ = }
    \only<6-15|handout:1>{
      \CubeScriptsize[\alert<8-10>{a}][\alert<15>{b}][\alert<11>{c}][\alert<11>{d}][\alert<8-10>{a}][\alert<15>{f}][\alert<11>{c}][\alert<11>{d}]
    }
    \only<16-|handout:2>{
      \CubeScriptsize[a][\alert<16->{b}][c][d][a][\alert<16->{\mathbf{b}}][c][d]
    }
    \uncover<6->{
      \hspace{4em} \big( \text{all in } \M(\Cong{1}, \Cong{1}, \Cong{1}) \big)
    }
  \]

  \item<17-> $\AA$ is $2$-supernilpotent! 
    \uncover<18->{$\AA$ \emph{is also nilpotent!}} 
\end{itemize}
\end{frame}  % }}}1
\begin{frame} % A_2 defn {{{1
\emph{Idea:}
\begin{itemize}
  \pause\item $\AA = \left< A \;;\; t(x,y) \right>$, injective $t(x,y)$
  \pause\item carefully redefine $t(x,y)$ to sabotage injectivity and nilpotence
  \pause\item be clever so that $t(x,y)$ is injective enough to ensure 
    $[ \Cong{1}, \Cong{1}, \Cong{1} ] = \Cong{0}$
\end{itemize}

\pause

Define \pause
\[
  O \coloneqq \big\{ o_i^j \mid i,j\in \NN \big\},
  \qquad\quad
  \pause
  R \coloneqq \big\{ r_i^j \mid i,j\in \NN \big\},
  \pause
  \qquad\quad
  A \coloneqq O \cup R \cup \NN.
\]

\pause

Fix an injection $s: A^2 \to \NN$. 
\pause Define
\[
  t\left( \Square[r_{4i}^j][r_{4i+2}^j][r_{4i}^j][r_{4i+2}^j],
          \Square[r_{4i}^j][r_{4i}^j][r_{4i+2}^j][r_{4i+2}^j] \right)
  \coloneqq \Square[o_i^j][r_i^{j+1}][o_i^j][r_{i+1}^{j+1}]
\]

\pause

Otherwise, define $t(x,y) \coloneqq s(x,y)$. 
\pause Let $\AA = \left< A \;;\; t(x,y) \right>$.

\bigskip

\pause 
\emph{Observe:} $t(x,y)$ is injective except when the output is in $O$.
\end{frame}  % }}}1
\begin{frame}[t] % A_2 is not solvable {{{1
Define $A = O \cup R \cup \NN$ and
\[
  t\left( \SquareScriptsize[r_{4i}^j][r_{4i+2}^j][r_{4i}^j][r_{4i+2}^j],
          \SquareScriptsize[r_{4i}^j][r_{4i}^j][r_{4i+2}^j][r_{4i+2}^j] \right)
  \coloneqq \SquareScriptsize[o_i^j][r_i^{j+1}][o_i^j][r_{i+1}^{j+1}],
\]
otherwise define $t(x,y) \coloneqq s(x,y)$ for some injection $s: A^2 \to \NN$.

\vfill

\pause $\AA$ \emph{is not solvable}
\begin{itemize}
  \pause\item Dervived series: \quad $[\alpha]_0 \coloneqq \alpha$, \qquad\qquad 
    $[\alpha]_{n+1} \coloneqq \big[ [\alpha]_n, [\alpha]_n \big]$.

  \pause\item $R^j \coloneqq \big\{ r_i^j \mid i,j\in \NN \big\}$ is contained
    in a $[\Cong{1}]_j$-class (induction on $j$).

  \pause\item Thus $[\Cong{1}]_j \neq \Cong{0}$ for any $j$, so $\AA$ is not
    solvable.
\end{itemize}

\vfill
\end{frame}  % }}}1
\begin{frame} % A_2 is supernilpotent {{{1
\small
Define $A = O \cup R \cup \NN$ and
\[
  t\left( \SquareScriptsize[r_{4i}^j][r_{4i+2}^j][r_{4i}^j][r_{4i+2}^j][0.9],
          \SquareScriptsize[r_{4i}^j][r_{4i}^j][r_{4i+2}^j][r_{4i+2}^j][0.9] \right)
  \coloneqq \SquareScriptsize[o_i^j][r_i^{j+1}][o_i^j][r_{i+1}^{j+1}][0.9],
\]
otherwise define $t(x,y) \coloneqq s(x,y)$ for some injection $s: A^2 \to \NN$.

\bigskip

\uncover<2->{ 
  Generating $\M(\Cong{1}, \Cong{1}, \Cong{1})$ in layers, look for first
  instance of
}
\[
  \only<-8|handout:1>{ \uncover<4->{
    t\left( \CubeScriptsize[\alert<8>{a_1}][b_1][\alert<8>{c_1}][\alert<8>{d_1}][\alert<8>{e_1}][f_1][\alert<8>{g_1}][\alert<8>{h_1}][1.05],
            \CubeScriptsize[a_2][b_2][c_2][d_2][e_2][f_2][g_2][h_2][1.05] \right)
  } }
  \only<9-10|handout:0>{
    t\left( \CubeScriptsize[\alert{a_1}][b_1][\alert{c_1}][\alert{d_1}][\alert{\mathbf{a_1}}][f_1][\alert{\mathbf{c_1}}][\alert{\mathbf{d_1}}][1.05],
            \CubeScriptsize[a_2][b_2][c_2][d_2][e_2][f_2][g_2][h_2][1.05] \right)
  }
  \only<11|handout:0>{
    t\left( \CubeScriptsize[a_1][\alert<11>{b_1}][c_1][d_1][a_1][\alert<11>{f_1}][c_1][d_1][1.05],
            \CubeScriptsize[a_2][b_2][c_2][d_2][e_2][f_2][g_2][h_2][1.05] \right)
  }
  \only<12|handout:0>{
    t\left( \CubeScriptsize[a_1][\alert<12>{b_1}][c_1][d_1][a_1][\alert<12>{\mathbf{b_1}}][c_1][d_1][1.05],
            \CubeScriptsize[a_2][b_2][c_2][d_2][e_2][f_2][g_2][h_2][1.05] \right)
  }
  \only<13-|handout:2>{
    t\left( \CubeScriptsize[a_1][\alert<14->{b_1}][c_1][d_1][a_1][\alert<14->{b_1}][c_1][d_1][1.05],
            \CubeScriptsize[a_2][\alert<14->{b_2}][c_2][d_2][e_2][\alert<14->{f_2}][g_2][h_2][1.05] \right)
  }
  \uncover<4->{ = }
  \uncover<3->{
    \CubeScriptsize[\alert<7-8>{a}][\alert<14->{b}][\alert<7-8>{c}][\alert<7-8>{d}][\alert<7-8>{a}][\alert<14->{f}][\alert<7-8>{c}][\alert<7-8>{d}],
    \hspace{3em} b\neq f
  }
\]


\uncover<5->{\ \emph{Observations}}
\begin{itemize}
  \item<6-> If $t(a,b) = t(c,d)$ then $a = c$.

  \item<10-> Argument squares: if 3 back-to-front edges are equal, then the
    4th is too.

  \item<14-> $b_2 \neq f_2$, otherwise $b=f$.
    \uncover<15->{At most 2 back-to-front edges in the second argument cube are equal.}
\end{itemize}

\uncover<16->{\ The rest of the argument is technical \dots}
\end{frame}  % }}}1

\begin{frame} % results {{{1
\begin{thm}[M+M]
The algebra $\AA$ is 3-supernilpotent but not solvable.
\end{thm}

\bigskip

\begin{itemize}[leftmargin=*, itemsep=1em]
  \pause\item nilpotence and solvability can be generalized using the $n$-ary
    commutator

  \pause\item call the generalized notions $n$-dimensional nilpotence and
    solvability

  \pause\item a similar construction yields 
    $\AA_n \coloneqq \left< A \;;\; t(x_1,\dots,x_n) \right>$.
\end{itemize}

\bigskip \pause

\begin{thm}[M+M]
The algebra $\AA_n$ is $(n+1)$-supernilpotent but not $n$-dimensional solvable.
\end{thm}
\end{frame}  % }}}1
\begin{frame}[t] \frametitle{\inserttitle} % thanks {{{1

\pause

\begin{thm}[M+M]
The algebra $\AA$ is 3-supernilpotent but not solvable.
\end{thm}

\bigskip\pause

\begin{thm}[M+M]
The algebra $\AA_n$ is $(n+1)$-supernilpotent but not $n$-dimensional solvable.
\end{thm}

\pause\vspace{4em}

\centering Thank you for your attention.
\end{frame}  % }}}1
\end{document}
