Dr Amy Glen

Room: S&C/3.027

Projects

Combinatorics on Words

"Combinatorics on words" is a relatively new area of research within Discrete Mathematics which has flourished during the last few decades. The motivation comes not only from different modern, as well as classical, fields of mathematics, but also from computer science, physics, and biology. In fact, many fundamental results of the theory have been discovered, or rediscovered, when using words as tools for other sciences.

The following diagram illustrates the main connections between word combinatorics and other scientific fields.

Although there have been important contributions on "words" starting from the very beginning of last century, they were scattered and typically needed as tools to achieve some other goals in mathematics. A noteworthy exception is combinatorial group theory, which studies combinatorial problems on words as representing group elements. Now, and even more so after the appearance of Lothaire's book -- Combinatorics on Words -- in 1983, the topic has become a challenging research field in its own right.

The central notion is a "word", i.e., a finite or infinite sequence of symbols (called "letters") taken from a finite non-empty set A (called an "alphabet"). It follows immediately that the mathematical research of words exploits two features: discreteness and non-commutativity. The latter aspect is what makes this field challenging: many easily formulated problems are often difficult to solve, mainly because of the limited availability of mathematical tools to deal with non-commutative structures compared to commutative ones.

There is a wealth of interesting open questions and problems in word combinatorics, many of which would make excellent Honours projects, leading to original research results and publication. Such projects would afford students the opportunity to meet and apply tools and techniques from a variety of mathematical fields, particularly combinatorics, algebra, geometry, and number theory.

Possible specific topics could involve: Sturmian and episturmian words; Fibonacci and k-bonacci words; Thue-Morse word and its generalizations; lexicographic orderings; Lyndon words; return words; quasiperiodicity; palindromic and periodic words; balanced words; factor complexity; repetitions in words; avoidable and unavoidable patterns; property-preserving morphisms; infinite words generated by morphisms; automatic sequences; theory of codes; invertible substitutions; numeration systems; continued fractions; transcendental numbers.

There are no prerequisites other than a sound background in undergraduate mathematics. Computer programming skills could also be useful, depending on the chosen project.

I welcome enquiries from prospective Honours and PhD students. Feel free to e-mail me or, if you're in Perth, drop by my office to discuss possible topics. I am happy to supervise Honours projects and PhD studies in the general areas of Combinatorics, Discrete Mathematics, and Number Theory (especially combinatorics on words). To find out more about my research interests, check out my web-site at http://wwwstaff.murdoch.edu.au/~aglen

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