Another Look at Markoff's Work on Quadratic Forms

Host

Applied Mathematics

Speaker

Aaron Lauve
Loyola University-Chicago
http://webpages.math.luc.edu/~lauve/

Description

Abstract: During his study of binary quadratic forms ax^2 + bxy + cy^2 in the late 1800s, A.A. Markoff introduced necessary and sufficient conditions on a reduced form F so that its minimum is less than 3. He phrased his condition in terms of certain forbidden patterns of a bi-infinite sequence associated to F. From the Markoff condition, one gets a large class of binary sequences, or "words," that we call Markoff words.

In this talk, we will see that the Markoff words are palindromes and, indeed, are the important central words from the theory of Sturmian sequences. These words appear in a variety of other contexts as well, e.g., as "Euclidean rhythms" in music theory, and I will name a few.

What minimum values can such F attain? What if we relax the condition that it be less than 3? Here we run into Lagrange numbers, the famous Markov numbers, and the ubiquitous Fibonacci numbers, to boot.

(Based partly on joint work with Amy Glen and Franco Saliola.)