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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Ottawa-2306}
\def\title{Computing the Zombian of an Unfinished Columbarium}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to Ottawa!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/ott23}{http://drorbn.net/ott23}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
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%%%

\def\Confession{{{\bf Confession.} It's about 50\% of what I do.}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Apology.} It's a 20 minutes talk. Necessarily, it will be superficial.

{\red\bf Abstract.} The zombies need to compute a quantity, the zombian, that pertains to some
structure --- say, a columbarium. But unfortunately (for them), a part of that structure will only be
known in the future. What can they compute today with the parts they already have to hasten
tomorrow's computation?

That's a common quest, and I will illustrate it with a few examples from knot theory and with two
examples about matrices --- determinants and signatures. I will also mention two of my dreams (perhaps
delusions): that one day I will be able to reproduce, and extend, the Rolfsen table of knots using
code of the highest level of beauty.
}}}}

\def\freepik{{Image: \href{http://freepik.com}{Freepik.com}}}

\def\savings{{$2^{n/2}+2^{n/2}+2^{\sqrt{n}} \ll 2^n$}}

\def\Computing{{\raisebox{2mm}{\parbox[t]{2.87in}{
\parshape 1 0in 4in
{\red\bf Computing Zombians of Unfinished Columbaria.}
\begin{itemize}
\item Future zombies must be able to complete the computation.
\item Must be no slower than for finished ones.
\item Future zombies must not even know the size of the task that today's zombies were facing.
\item {\gray We must be able to extend to ZPUCs, Zombie Processed Unfinished Columbaria!}
\end{itemize}

\parshape 1 0in 2.87in
{\red\bf Exercise 1.} Compute the sum of 1,000 numbers, the last 50 of which are still unknown.

\parshape 2 0in 2.87in 0in 3.95in
{\red\bf Exercise 2.} Compute the determinant of a $1,000\times 1,000$ matrix in which $50$
entries are not yet given.

\parshape 1 0in 3.95in
{\red\bf Example 3.} Same, for signatures of matrices / quadratic forms.

}}}}

\def\Solution{{\raisebox{6pt}{\scalebox{0.25}{\parbox{3.2in}{
$\begin{CD}
  \begin{pmatrix} A & B \\ C & U \end{pmatrix}
  @>\det(A)>>
  \begin{pmatrix} I & A^{-1}B \\ C & U \end{pmatrix}
  @>1>>
\begin{pmatrix} I & A^{-1}B \\ 0 & U-CA^{-1}B \end{pmatrix}
\end{CD}$,
\newline
  so $\det\begin{pmatrix} A & B \\ C & U \end{pmatrix} = \det(A)\det(U-CA^{-1}B)$.
}}}}}

\def\Interp{{\raisebox{6pt}{\scalebox{0.25}{\parbox{2.5in}{
Roughly, $\det(A)$ is ``$\det$ on $\ker$'',
\newline $-CA^{-1}B$ is ``a pushforward of $\begin{pmatrix} A & B \\ C & U \end{pmatrix}$''.
\newline (what if $\not\exists A^{-1}$?)
}}}}}

\def\zfreepik{{Zombies: \href{http://freepik.com}{Freepik.com}}}

\def\mfreepik{{Image by macrovector on \href{http://freepik.com}{Freepik.com}}}

\def\WhyTangles{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Why Tangles?} $\bullet$ As common as knots!
\newline$\bullet$ Faster computations!
\newline$\bullet$ Conceptually clearer proofs of invariance
\newline\null\quad (and of skein relations).
\newline$\bullet$ Often fun and consequential:
\newline\null\ $\circ$ The Alexander polynomial $\leadsto$ Zombian $=\det$.
\newline\null\ $\circ$ Knot signatures $\leadsto$ Pushforwards of quadratic forms.
\newline\null\ $\circ$ The Jones Polynomial $\leadsto$ The Temperley-Lieb Algebra.
\newline\null\ $\circ$ Khovanov Homology $\leadsto$ ``Unfinished complexes'', complexes
\newline\null\qquad in a category.
\newline\null\ $\circ$ The Kontsevich Integral $\leadsto$ Drinfel'd Associators.\hfill$\cdots$\hfill\null
}}}}

\def\QuadraticForms{{\raisebox{0mm}{\parbox[t]{3.96in}{
A {\red quadratic form} on a v.s.\ $V$ over $\bbC$ is a quadratic $Q\colon
V\to\bbC$, or a sesquilinear Hermitian $\langle\cdot,\cdot\rangle$
on $V\times V$ (so
$\langle x,y\rangle = \overline{\langle y,x\rangle}$ and
$Q(y)=\langle y,y\rangle$), or given a basis $\eta_i$ of
$V^\ast$, a matrix $A=(a_{ij})$ with $A=\bar{A}^T$ and $Q =
\sum a_{ij}\bar\eta_i\eta_j$. The {\red signature} $\sigma$ of
$Q$ is $\sigma_+-\sigma_-$, where for some $P$, $\bar{P}^TAP =
\text{diag}(1,\stackrel{\sigma_+}{\cdots},1,-1,\stackrel{\sigma_-}{\cdots},-1,0,\ldots)$.
}}}}

\def\PQ{{\raisebox{0mm}{\parbox[t]{3.96in}{
A {\red\em Partial Quadratic (PQ)} on $V$ is a quadratic $Q$
defined only on a subspace $\calD_Q\subset V$. We add PQs with
$\calD_{Q_1+Q_2}\coloneqq\calD_{Q_1}\cap\calD_{Q_2}$. Given a linear $\psi\colon V\to W$ and a PQ $Q$
on $W$, there is an obvious {\em pullback} $\psi^*Q$, a PQ on $V$.

\parshape 1 0in 3.33in
{\bf\red Theorem 1} (with Jessica Liu).
Given a linear $\phi\colon V\to W$ and a PQ $Q$ on $V$, there is a unique {\em
pushforward} PQ $\phi_*Q$ on $W$ such that for every $PQ$ $U$ on $W$,
\[ \sigma_V(Q+\phi^*U) = \sigma_{\ker\phi}(Q|_{\ker\phi}) + \sigma_W(U+\phi_*Q). \]
}}}}

\def\Gist{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Gist of the Proof.}
}}}}

\def\BT{$\bar{B}^T$}
\def\CT{$\bar{C}^T$}
\def\sigker{$\sigma(Q|_{\ker\phi})$}

\def\NowD{{\raisebox{-1mm}{\parbox[t]{3.96in}{
\ldots and the quadratic $F\eqqcolon\phi_*Q$ is well-defined only on $D\coloneqq\ker C$.
\newline\footnotesize\null\hfill (more at \web{icerm}.)
}}}}

\def\Embarrassments{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf One more story is left to tell, of knot tabulation.}

\vskip 1mm
\parshape 1 0in 4in {\tiny\begin{spacing}{0.5}
Two slides from R.~Jason Parsley's \web{history}:
\newline\includegraphics[width=2in]{Parsley-1.pdf}\includegraphics[width=2in]{Parsley-2.pdf}
\newline There's also Burton's tabulation to 19 crossings \web{Burton}, and Khesin's K250, \arXiv{1705.10319}.
\end{spacing}}

\vskip 1mm
{\red\bf Embarrassment 1 (personal).} I don't know how to reproduce the
Rolfsen table of knots! Many others can, yet I still take it on faith,
contradicting one of the tenets of our practice, ``thou shalt not use
what thou canst not prove''.

It's harder than it seems! Producing all knot diagrams is a mess,
identifying all available  Reidemeister moves is a mess, and you sometimes
have to go up in crossing number before you can go down again.

{\red\bf Embarrassment 2 (communal).} There isn't anywhere a tabulation
of tangles! When you want to test your new discoveries, where do you go?
}}}}

\def\Dream{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Dream.} Conquer both embarrassments at once. Reproduce the
Rolfsen table, and extend it to tangles, using code of the highest
level of beauty. The algorithm should be so clear and simple that anyone
should be able to easily implement it in an afternoon without messing
with any technicalities.
}}}}

\def\BenefitA{{\raisebox{0mm}{\parbox[t]{1.25in}{\footnotesize
We don't even need to look at all knot diagrams!
}}}}

\def\BenefitB{{\raisebox{0mm}{\parbox[t]{1.75in}{\footnotesize
The dreaded slide moves, which go up in crossing number, are parametrized by tangles!
}}}}

\def\BenefitC{{\raisebox{0mm}{\parbox[t]{0.52in}{\footnotesize
R-moves are tangle equalities!
}}}}

\def\Acknowledgement{{\raisebox{3mm}{\parbox[t]{3.95in}{
{\red\bf Acknowledgement.} This work was partially supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).
}}}}

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