Spring 2018

During Spring 2018 MEGL ran a 6 project program with 19 participants (faculty, graduate, and undergraduate students). The five research/visualization groups were titled: Visualizing Continued Fractions, Asymptotic Dynamics on Arithmetic Curves, Applications of Diffusion Maps, Riemann Hilbert Correspondence in Categorical Language, and Equivariant K-theory of Flag Varieties. Additionally, there was one public engagement group which we refer to below as Outreach. The research/visualization groups engaged in experimental explorations involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, to make conjectures from data, and to work on theory resulting from conjectures. The outreach group involved faculty, graduate students, and undergraduates to develop and implement activities for elementary and high school students that were presented at local schools and public libraries. We concluded with an end of term symposium and a poster session (scroll down for pictures).

It is standard to write numbers as a sequence of decimals, such as $\sqrt{2}=1.41421...$ Another way to write a number is to specify a sequence of descending continued fractions that represent it. In this way, we can write $\sqrt{2}=1\frac{1}{2+\frac{1}{2+\frac{1}{...}}}$. This idea can then be extended to complex numbers as well. The goal of this project was to visualize continued fractions and explore the relationship between two natural equivalence relations on continued fractions: tail equivalence and matrix equivalence.

Two complex numbers $x$ and $y$ are said to be matrix equivalent if there exist Gaussian integers $a$, $b$, $c$, and $d$ such that $x=\frac{ay+b}{cy+d}$ and $ad-bc=\pm 1$. Two complex numbers $x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{...}}}$ and $y=b_0+\frac{1}{b_1+\frac{1}{b_2+\frac{1}{...}}}$ are said to be tail equivalent if there exist positive integers $r$ and $s$ such that $a_{n+r}=b_{n+s}$ for all positive integers $n$. In other words, $x$ and $y$ are tail equivalent if there is a way to align their continued fraction digits so that their tails match. It can be shown that two real numbers are matrix equivalent if and only if they are tail equivalent. For complex numbers, however, tail equivalence implies matrix equivalence, but there exist exceptions to the converse. Previously, the only known example of matrix equivalent numbers which are not tail equivalent was given by Richard Lakein (1974). In this project, we successfully found more such examples.