I created this video for the “Dance Your PhD” competition sponsored by Science and TEDxBrussels, which was a perfect excuse for me to combine two of my great passions: mathematics and dance.
My area of research is complex dynamics.
Now most of you probably don’t know what this is. Or maybe complex dynamics means something different to you than to me. So let me explain what this means in the math world.
First of all, what are the dynamics of a map? Well, consider the map x → x^2 on the real line. The dynamics are how points on the real line behave upon repeated application of this map. For instance:
0 → 0^2 = 0 → . . . → 0
½ → ¼ → 1/16 → . . . moves towards 0
±1 → 1 → . . . → 1
2 → 4 → 16 → . . . moves towards infinity
From this we can see that under repeated application of the map:
(1) 0 is a fixed point
(2) points on the line between (but not including) -1 and 1 become closer to 0
(3) 1 is a fixed point
(4) points with an absolute value greater than 1 move toward infinity
This describes the basic dynamics of the map x → x^2.
Secondly, what makes the dynamics complex? The inputs and outputs of the map are complex numbers. So what are complex numbers? Well, real numbers are all the numbers that you normally think of, such as -6, sqrt(2) and pi. The complex numbers are combinations of real numbers and imaginary numbers, such as sqrt(-1)=i. For example, ½ and 2+3i are both complex numbers, but only the first one is a real number. So the set of complex numbers is twice the "size" (i.e. dimension) of the set of real numbers. Visually, 1 copy of the complex numbers is a plane.
This video and most of my thesis focus on a particular type of map that sends two copies of the complex numbers (think 4 real dimensions) to itself AND fixes a point. Both the video and most of my thesis focus on the dynamics of this map.
The conclusion of this work:
If f satisfies a few properties (such as it has a fixed point and only one direction along which points can be attracted to that fixed point), then f has an attracting domain in which all points are attracted to the fixed point along a particular direction. Furthermore, there exists a map q such that q○f○(q inverse) acts on the attracting domain by sending (z,w) to (z+1,w).
By making different assumptions on the map f, other mathematicians have shown that it is sometimes, but not always, possible to get the same conclusion.
If you would like to see the math behind this video, look for my paper on the arXiv:
“Attracting domains of maps tangent to the identity whose only characteristic direction is non-degenerate."