**Day 1.** "The Kashiwara-Vergne Problem and Topology" following my
future

Bern-131104
talk (Zurich-time handout at

pensieve),
then "Finite Type Invariants of Ribbon Knotted Balloons and Hoops"
following my

Geneva-131024
talk. Videos:

hour
1,

hour 2.

**Day 2.** The "Bonn Sequence" following my

Bonn-0908 talk.
Videos:

hour
1,

hour 2.

**Day 3.** Quickly "18 Conjectures" following my

Chicago-1009 talk
2. Then "Trees and Wheels and Balloons and Hoops" in the spirit
of my

Zurich-130919
talk, but hopefully with some further details. Videos:

hour 1,

hour 2.

© | << < ? > >> |
Dror Bar-Natan:
Talks:

# Informal Talks on the Topology, Combinatorics, and Low and High Algebra of
w-Knots

#### University
of Zurich, Tuesday October 29 3-5, Wednesday October 30 10-12, and
Thursday October 31 1-3, 2013

**Abstract.** Taylor's theorem maps smooth functions to power series. In
other words, it maps the smooth to the combinatorial and algebraic, which is
susceptible to an inductive degree-by-degree study. Surprisingly, there is a
notion of "expansions" for topological things, which shares the spirit of the
original Taylor expansion while having nothing to do with approximations of
smooth functions.

"w-Knots", or more generally "w-knotted objects", are knotted 2-dimensional
objects in 4-dimensional space (some restrictions apply). They have a rich
theory of "expansions" which takes topology into combinatorics. That
combinatorics, in itself, turns out to be the combinatorics of formulas that
can be written universally in arbitrary finite-dimensional Lie algebras ("low
algebra"). Taylor's theorem for a certain class of w-knotted objects turns out
to be equivalent to some global statements about Lie algebras and Lie groups
("Kashiwara-Vergne", "high algebra"). I will do my best to talk about all
these things.

"w-Knotted objects" contain the usual "u-knotted objects" (braids, knots,
links, tangles, knotted graphs, etc.) and are quotients of the more general
"v-knotted objects". To within reason I will also speak about the relationship
of "w" with "u" and "v", where the key words are "associators" and "Lie
bi-algebras", respectively.

Anna asked me to talk for up to 6 hours, and that's more than I can prepare in
detail in advance. Hence the adjective "informal": I have a general idea of
what I want to say and much of it I've said many times before. Beyond that
things will flow, if they won't stand still, chaotically and randomly.

----

See the papers: **WKO**, **KBH**!