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# Talks at the "Legacy of Vladimir Arnold" Conference

## 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.

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