Research summary:

This position, an FCT-funded five-year personal research fellowship, made me responsible for choosing the direction of my research. So far, I have worked in several interrelated areas, both alone and with collaborators. One is hyperbolic semigroups, encompassing geometric, linguistic, and other generalizations of groups that are hyperbolic in the sense of Gromov. This has yielded results such as a polynomial-time algorithm for the word problem for hyperbolic monoids (improving on the previous-known exponential-time algorithm) and links between hyperbolicity and string-rewriting systems. Another is automatic presentations for relational structures. In joint work, I have classified various combinatorial structures that admit unary automatic presentations.

• Research within the project PTDC/MAT/69514/2006 ‘Semigroups and Languages’, funded by FCT and PIDDAC.

Research summary:

I studied automatic Clifford semigroups and also studied (with collaborators) how automatic structures and Malcev presentations interacted with the new notion of Green index subsemigroups.

Teaching:

Postgraduate-level short course on automatic groups and semigroups.

• I was offered this post before the completion of my Ph.D. based on my achievements during my doctoral studies.
• Member of Staff Council (March 2007–August 2008)

Research summary:

I worked as part of a large research team on the five-year interdisciplinary project `Critical Mass in Computational Algebra', funded by the EPSRC (EP/C523229/1). In particular, I researched decision problems for automatic semigroups: given an automatic structure for a semigroup, is it possible to decide certain algebraic properties of the semigroup? For some such properties, I developed algorithms; other cases, I proved that no such algorithm exists (for example, cancellativity and left-cancellativity are algorithmically undecidable). I also worked in related areas such as automaton semigroups (generalizing automaton groups) and automatic presentations. With collaborators, I classified all finitely generated cancellative semigroups that admit automatic presentations.

Teaching:

MT3600 ‘Fundamentals of Pure Mathematics’
(2005–06, 2006–07, 2007–08)
[with Prof. N. Ruškuc]

• Completely revised the course to concentrate on establishing the foundations of mathematics rigorously.
• Student feedback was enthusiastically supportive, with 85% overall approval.

Postgraduate-level short course on computing with automatic semigroups.

• I tutored the first-year mathematics courses MT1002 and MT1003. The former course (covering both pure and applied mathematics) is a prerequisite for further mathematical study at St Andrews, and therefore it was essential to help the students achieve a thorough understanding of the concepts involved.
• My work as a computer laboratory demonstrator consisted of aiding students to use MAPLE to solve various problems related to their work in the first year courses MT1002 and MT1003, and marking their work. The marks a student received from this laboratory work made up 10% of his or her total mark for the course.
• Undergraduates judged me to be an able teacher; this is one of the reasons I was asked to remain at St Andrews and to lecture in addition to my research.

Thesis: Presentations for Subsemigroups of Groups
Supervisors: Prof. E.F. Robertson & Prof. N. Ruškuc

Examiners: Dr M. Quick (Internal) & Prof. D.F. Holt (Univ.~of Warwick, External)

Summary of Ph.D. research:

My doctoral research spanned the boundary between mathematics, specifically algebra, and computer science, specifically the theory of algorithms. It focused on the study of combinatorial and computational properties of semigroups that are or can be embedded within groups. The most interesting semigroups are infinite. However, they sometimes have a finite description by means of a presentation or an automatic structure. My research studied both of these concepts and also Malcev presentations, a notion introduced in 1977 by J.-C. Spehner but neglected since then. In particular, I successfully linked the near-forgotten area of Malcev presentations with the new and active one of automatic semigroups. I then exploited this link in both directions, solving problems in the one area using the tools of the other.

• My thesis was accepted with no corrections being required.
• I won a prestigious Carnegie Doctoral Scholarship to fund my Ph.D. studies.

Prizes & awards:

Cunninghame Prize in Mathematics; Class prizes in Mathematics in first and second year; Class prize in Computer Science in second year.

Activities:

President of the Maclaurin Society, the University of Glasgow student society for mathematics and statistics.