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Dror Bar-Natan:
Talks:
### The Fields Institute, November 24-28, 2014

## On Maps, Machines and Roaches

#### High School Activities session, Tuesday November 25, 7-9PM.

**Abstract.** We try to map the configuration space of a simple machine,
a six-legged idealized roach, and find that good old cut-and-paste topology
can be a lot of fun.

Following ClassroomAdventures-1308.

**Talk video:**
@Fields

## Dessert: Hilbert's 13th Problem, in Full Colour

#### Expository Lecture, Wednesday November 26, 2:00-2:45PM

**Abstract.** To break a week of deep thinking with a nice colourful light
dessert, we will present the Kolmogorov-Arnold solution of Hilbert's 13th
problem with lots of computer-generated rainbow-painted 3D pictures.

In short, Hilbert asked if a certain specific function of three variables can
be written as a multiple (yet finite) composition of continuous functions of
just two variables. Kolmogorov and Arnold showed him silly (ok, it took about
60 years, so it was a bit tricky) by showing that ANY continuous function f of
any finite number of variables is a finite composition of continuous functions
of a single variable and several instances of the binary function "+"
(addition). For f(x,y)=xy, this may be xy=exp(log x + log y). For
f(x,y,z)=x^y/z, this may be exp(exp(log y + log log x) + (-log z)). What might
it be for (say) the real part of the Riemann zeta function?

The only original material in this talk will be the pictures; the math was
known since around 1957.

**Handout:**
H13.html,
H13.pdf,
H13.png.
**Sources:** pensieve.

**Talk video:**
@Fields

## Finite Type Invariants of Doodles

#### Research Lecture, Friday November 28, 9:45-10:15AM

**Abstract.** I will describe my former student's Jonathan Zung
work on finite type invariants of "doodles", plane curves modulo the
second Reidemeister move but not modulo the third. We use a definition
of "finite type" different from Arnold's and more along the lines of
Goussarov's "Interdependent Modifications", and come to a conjectural
combinatorial description of the set of all such invariants. We then
describe how to construct many such invariants (though perhaps not all)
using a certain class of 2-dimensional "configuration space integrals".

**Handout:**
Doodles.html,
Doodles.pdf,
Doodles.png.
**Sources:** pensieve.

**Talk video:** @Fields