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Dror Bar-Natan:
Talks:
# Two Talks at Carnegie Mellon University, April 3, 2015

## Commutators

### Undergraduate Lecture

**Abstract.** The commutator of two elements $x$ and $y$ in a
group $G$ is $xyx^{-1}y^{-1}$. That is, $x$ followed by $y$ followed
by the inverse of $x$ followed by the inverse of $y$. In my talk I will
tell you how commutators are related to the following four riddles:

- Can you send a secure message to a person you have never communicated
with before (neither privately nor publicly), using a messenger you
do not trust?
- Can you hang a picture on a string on the wall using $n$ nails,
so that if you remove any one of them, the picture will fall?
- Can you draw an $n$-component link (a knot made of $n$
non-intersecting circles) so that if you remove any one of those $n$
components, the remaining $n-1$ will fall apart?
- Can you solve the quintic in radicals? Is there a formula for the
zeros of a degree $5$ polynomial in terms of its coefficients, using
only the operations on a scientific calculator?

**Prerequisites.**

- The first week of any class on group theory.
- Knowing that every complex number other than to $0$ has exactly $n$ roots
of order $n$, and how to compute them.

**Handout:**
Commutators-Handout.pdf.
**Slides:**
Commutators-Slides.nb.
**Talk video:** .
**Sources:** pensieve.

## The Kashiwara-Vergne Problem and Topology

### Mathematics Colloquium

**Abstract.** I will describe a general machine, a close cousin of
Taylor's theorem, whose inputs are topics in topology and whose outputs
are problems in algebra. There are many inputs the machine can take, and
many outputs it produces, but I will concentrate on just one input/output
pair. When fed with a certain class of knotted 2-dimensional objects
in 4-dimensional space, it outputs the Kashiwara-Vergne Problem (1978,
solved Alekseev-Meinrenken 2006, elucidated Alekseev-Torossian 2008-2011),
a problem about convolutions on Lie groups and Lie algebras.

**A repeat of my November 2013 Bern
Colloquium,** see video there.

Links:
AM
AT
Bern
CS
Dal
F
inf
KV
mac
vX
WKO1
WKO2
X
ZD

**Handout:**
KVT.html,
KVT.pdf,
KVT.png.

See the papers: WKO1, WKO2.

**Sources:** pensieve.