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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{AGTW-2611}
\def\title{2D Intersection Theory and its Homological Model}
\def\subtitle{Why Am I in Sydney visiting Zsuzsi?}

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  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
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  Thanks for inviting me to the
  \href{https://sites.google.com/view/agtw/home}{Australian Geometric Topology Webinar}!
}}
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% From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes
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%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 8 0in 3.2in 0in 3.2in 0in 3.2in 0in 3.2in 0in 3.2in 0in 3.2in 0in 3.4in 0in 3.95in
{\red\bf Abstract.} When you intersect half-dimensional cycles in 4 or
higher dimensions, you get a count. In 2D, half-dimensional cycles can
be replaced by paths and they and their intersections carry homotopical
information. After inquiring why we should care, we ponder if this
intersection theory has a graded model, and an expansion, much as the
Taylor series is an expansion of smooth functions into their graded model,
the space of power series.

The tentative affirmative answer uses Drinfeld associators and/or
solutions of the KV equations, and makes drunken port calls at
``emergent'' knots in 3D which barely know knottedness, at truly knotted
knots in 4D, at a theory of convolutions on finite dimensional Lie groups
and algebras, and maybe in the theory of quatum groups. God was having
a good laugh when she threw all this at us without a user's manual.

{\red\bf Disclaimer.} Work in progress! This last month we worked out
the ideas and we believe them to be right. We dream that a paper will
come out in 2027 before the water level rises too high.
}}}}

\def\IntersectionTheory{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf 2D Intersection Theory} carries homotopy information and
not just counting! Let $D_p$ be the disk with $p$ punctures, let
$\pi\coloneqq\pi_1(D_p)=FG(x_1,\ldots,x_p)$, and let $|\pi|\coloneqq\pi/(g\sim h^{-1}gh)$
and $|\pi\times\pi|\coloneqq\pi\times\pi/((g_1,g_2)\sim(h^{-1}g_1h,h^{-1}g_2h))$.

The {\em extended Goldman bracket} $EG\colon\bbQ|\pi|\times\bbQ|\pi|\to\bbQ|\pi\times\pi|$ is:

\vskip 1cm
The {\em extended Turaev cobracket} $ET\colon\bbQ|\pi|\to\bbQ|\pi\times\pi|$ is:

\vskip 1.2cm
{\bf Applications.}
}}}}

\def\Conway{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf The Conway Quotient} is defined by $\doublepoint\to\upupsmoothing$, giving the {\em Goldman
bracket} $G\colon\bbQ|\pi|\times\bbQ|\pi|\to\bbQ|\pi|$ and the {\em Turaev
cobracket} $T\colon\bbQ|\pi|\to\bbQ|\pi|\times\bbQ|\pi|$ which together make the {\em Goldman-Turaev Lie
bialgebra}. Lie bialgebras are super cool! Yet,

{\bf Applications.} \hfill (though check~\cite{Chas:LieBialgebras})
}}}}

\def\AIStatement{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 1 0in 1.96in
{\red\bf AI Statement.} Since September 2025 I
am unable to draw a sharp line separating the intellectual ability of
humans from the intellectual ability of machines.

\parshape 4 0in 1.96in 0in 1.96in 0in 1.96in 0in 3.95in 
Seeing that the former is roughly constant and the latter is rapidly
rising, very soon the latter will vastly exceed the former, if it
hasn't yet. Thus soon biological brains and hence humans will no longer
be necessary for those who control the machines, be it a few powerful
humans or the machines themselves. Economic reality regrettably implies
that those who are unnecessary will shrink and starve. I believe this
is the gravest danger presently facing most humans.

I believe ``safe AI'' is wishful thinking, easily subverted.

Can AI and HI coexist? Like those jets that have bicycle pedals for
extra power? Wait, that's absurd.

Thus I believe that the only solution, unlikely and temporary as it
may be, is to pull the plug fully and completely. Yesterday is better
than today.

Yet while AI is here and while I still can, I will use it to my
advantage. I don't believe in sainthood and falling behind helps no one$^1$.

Students: Shamelessly use it when you can. Yet remember it's an enemy
and dream of its demise.

\rule{1in}{1pt}
\newline\footnotesize A classic prisoner's dilemma, isn't it? These are
more common than most people appreciate.
}}}}

\def\refs{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf References.}
{\scriptsize
%\par\vspace{-3mm}
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\begin{thebibliography}{}
\setlength{\parskip}{0pt}
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\bibitem[Ch]{Chas:LieBialgebras} M.~Chas,
  {\em Combinatorial Lie Bialgebras of Curves on Surfaces,}
  \arXiv{math/0105178}.

\end{thebibliography}}
}}}}

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