Jeremy Nicholson, Pure Mathematics, University of Waterloo
"The Frobenius Problem and Combinatorics on Words"
Given a set of positive integers $S=\{m_1, m_2,...,m_k\}$, what is the largest integer (if it exists) that cannot be written as a non-negative integer linear combination of the elements in $S$? The problem above is known as the Frobenius problem and was first introduced by German mathematician Ferdinand Frobenius in the late 1800s. The focus of my talk will be about generalizing the Frobenius problem to a combinatorics on words setting. Combinatorics on words is a relatively new field of mathematics focusing on the study of words (sequences of symbols called letters) over a given alphabet. Several results in the realm of combinatorics on words have been applied to other mathematical fields including logic, linear algebra and group theory. Most notably, I will show how to use the existence of an infinite ternary square-free word to construct a counterexample to the Burnside problem for semigroups.