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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Banff-1911}
\def\title{The Mysteries of BF}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to \magenta Banff!}}
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\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Abstract.} The BF quantum field theory on one hand, and prior
art regarding finite type invariants of w-knots on the other hand,
suggest that there should be a ``Kontsevich Integral'' for general
2-knots in 4-space (of not necessarily the ``simple'' or ``welded'' or
``w'' type). Where is it? Why don't we know about it?  Why aren't we
studying it?

This will be a ``large structures'' talk. I will explain how several
large structures fit together nicely in 3D and several other large
structures show potential in 4D. Personally I prefer ``every detail
shown'' talks. Sorry.

{\footnotesize {\red\bf Confession.}
I haven't touched this material for a few years, and I barely know what I'm talking about.
}
}}}}

\def\Abstracts{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Recent Abstracts.} $\bullet$ {\red\em Everything around $sl_{2+}^\epsilon$ is DoPeGDO. So what?}
UCLA, Nov 2019, \web{ucla}: I'll explain what "everything around" means: classical and quantum $m$,
$\Delta$, $S$, $tr$, $R$, $C$, and $\theta$, as well as $P$, $\Phi$, $J$, ${\mathbb D}$, and more, and all
of their compositions. What DoPeGDO means: the category of Docile Perturbed Gaussian Differential
Operators. And what $sl_{2+}^\epsilon$ means: a solvable approximation of the semi-simple Lie algebra
$sl_2$.

Knot theorists should rejoice because all this leads to very powerful and well-behaved poly-time-computable
knot invariants. Quantum algebraists should rejoice because it's a realistic playground for testing
complicated equations and theories.

$\bullet$ {\red\em Some Feynman Diagrams in Algebra.} Sydney, Oct 2019, \web{syd2}: I will explain how the
computation of compositions of maps of a certain natural class, from one polynomial ring into another,
naturally leads to a certain composition operation of quadratics and to Feynman diagrams.

$\bullet$ {\red\em Algebraic Knot Theory.} Sydney, Sep 2019, \web{syd1}: This will be a very ``light'' talk:
I will explain why about 13 years ago, in order to have a say on some problems in knot theory, I've set out
to find tangle invariants with some nice compositional properties. In later talks in different seminars
here in Sydney I will explain how such invariants were found --- though they are yet to be explored and
utilized.

$\bullet$ {\red\em The Dogma is Wrong.} Les Diablerets, Aug 2017, \web{ld}: It has long been known that
there are knot invariants associated to semi-simple Lie algebras, and there has long been a dogma as for
how to extract them: ``quantize and use representation theory''. We present an alternative and better
procedure: ``centrally extend, approximate by solvable, and learn how to re-order exponentials in a
universal enveloping algebra''. While equivalent to the old invariants via a complicated process, our
invariants are in practice stronger, faster to compute (poly-time vs. exp-time), and clearly carry
topological information.

$\bullet$ {\red\em On Elves and Invariants,} Knots in Washington, Dec 2016, \web{kiw}:Whether or not you
like the formulas on this page, they describe the strongest truly computable knot invariant we know.
}}}}

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%{\red\bf References.}
{\footnotesize
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\bibitem[Za]{Zagier:Dilogarithm} D.~Zagier,
  {\em The Dilogarithm Function,}
  in Cartier, Moussa, Julia, and Vanhove (eds) {\em Frontiers in Number
  Theory, Physics, and Geometry II.} Springer, Berlin, Heidelberg,
  and \web{Za}.

\end{thebibliography}}
}}}}

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