Address	Centro de Matemática Universidade do Porto Rua do Campo Alegre 687 4169–007 Porto Portugal
Email address	ajcain AT fc DOT up DOT pt Replace ‘AT’ with ‘@’ and ‘DOT’ with ‘.’
Web page	www.fc.up.pt/pessoas/ajcain
Date of birth	9th March 1981
Nationality	British citizen
Languages	English (Native), French (Intermediate), Portuguese (Basic), Japanese (Basic).

Positions held

Jul. 2009 –Present	Research Fellow
	Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169–007 Porto, Portugal
	Research summary: This position, a five-year FCT Ciência 2008 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.

Sep. 2008 –Jun. 2009	Research Fellow
	Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649–003 Lisboa, Portugal
	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.

Sep. 2005 –Aug. 2008	Research Fellow
	School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, United Kingdom
	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.

Sep. 2002 –May 2005	Tutor & Computer laboratory demonstrator
	School of Mathematics & Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, United Kingdom
	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.

Education

Sep. 2002 –Jul. 2005	Ph.D. in Mathematics
	University of St Andrews, St Andrews, Fife, KY16 9AJ, United Kingdom
	Thesis: Presentations for Subsemigroups of Groups Supervisors: Prof. E.F. Robertson & Prof. N. Ruškuc Examiners: Dr M. Quick (Internal) & Prof. D.F. Holt (External) Submission: 11th Jul. 2005 Viva voce examination: 20th Sep. 2005 Graduation: 20th Jun. 2006 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 Carnegie Doctoral Scholarship to fund my Ph.D. studies.

Sep. 1998 –Jul. 2002	M.Sci. in Mathematics with First-class Honours
	University of Glasgow, Glasgow, G12 8QQ, United Kingdom
	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.

Membership of Professional Societies

London Mathematical Society

Elected: 17th Nov. 2006

Edinburgh Mathematical Society

Elected: 14th Jan. 2003

Publications

Presentations for Subsemigroups of Groups

Ph.D. Thesis, University of St Andrews, 2005.

‘Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness’

[with E. F. Robertson & N. Ruškuc]
Mathematical Proceedings of the Cambridge Philosophical Society 141 (2006), no. 1, pp. 57–66.

‘Subsemigroups of groups: presentations, Malcev presentations, and automatic structures’

[with E. F. Robertson & N. Ruškuc]
Journal of Group Theory 9 (2006), no. 3, pp. 397–426.

‘A group-embeddable non-automatic semigroup whose universal group is automatic’

Glasgow Mathematical Journal 48 (2006), no. 2, pp. 337–342.

‘Cancellativity is undecidable for automatic semigroups’

Quarterly Journal of Mathematics 57 (2006), no. 3, pp. 285–295.

‘Malcev presentations for subsemigroups of groups — a survey’

in C. M. Campbell, M. Quick, E. F. Robertson & G. C. Smith, editors, Groups St Andrews 2005 (Vol. 1), no. 339 in London Mathematical Society Lecture Note Series, Cambridge University Press, 2007, pp. 256–268.

‘Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions’

[with E. F. Robertson & N. Ruškuc]
Journal of the Australian Mathematical Society 84 (2008), no. 1, pp. 39–61.

‘Automatic presentations for cancellative semigroups’

[with G. Oliver, N. Ruškuc & R. M. Thomas]
in C. Martín-Vide, H. Fernau & F. Otto, editors, Language and Automata Theory and Applications: Second International Conference, Tarragona, Spain, March 13–19, 2008, no. 5196 in Lecture Notes in Computer Science, Springer, 2008, pp. 149–159.

‘Malcev presentations for subsemigroups of direct products of coherent groups’

Journal of Pure and Applied Algebra 213 (2009), no. 6, pp. 977–990.

‘Automaton semigroups’

Theoretical Computer Science 410 (2009), no. 47–49, pp. 5022–5038.

‘Automatic presentations for semigroups’

[with G. Oliver, N. Ruškuc & R. M. Thomas]
Information and Computation 207 (2009), no. 11, pp. 1156–1168.

‘Decision problems for finitely presented and one-relation semigroups and monoids’

[with V. Maltcev]
International Journal of Algebra and Computation 19 (2009), no. 6, pp. 747–770.

‘Monoids presented by rewriting systems and automatic structures for their submonoids’

International Journal of Algebra and Computation 19 (2009), no. 6, pp. 771–790.

‘Automatic semigroups and Bruck–Reilly extensions’

Acta Mathematica Hungarica 126 (2010), no. 1–2, pp. 1–15.

‘Automatic presentations and semigroup constructions’

[with G. Oliver, N. Ruškuc & R. M. Thomas]
Theory of Computing Systems 47 (2010), no. 2, pp. 568–592.

‘Deus ex machina and the aesthetics of proof’

Mathematical Intelligencer 32 (2010), no. 3, pp. 7–11.

An annotated translation of Yves Marie André’s Essay on Beauty (1741)

Ebook, 2010. vii+136 pp.

‘Unary FA-presentable semigroups’

[with N. Ruškuc & R. M. Thomas]
International Journal of Algebra and Computation, 2012. Forthcoming.

‘Automatic structures for subsemigroups of Baumslag–Solitar semigroups’

Submitted.

‘Automatic Clifford semigroups’

Submitted.

‘Green index in semigroup theory: generators, presentations, and automatic structures’

[with R. Gray & N. Ruškuc]
Submitted.

‘Hyperbolicity of monoids presented by confluent monadic rewriting systems’

Submitted.

‘Markov semigroups, monoids, and groups’

[with V. Maltcev]
Submitted.

‘Context-free rewriting systems and word-hyperbolic structures with uniqueness’

[with V. Maltcev]
Submitted.

‘Unary FA-presentable algebraic and relational structures’

[with N. Ruškuc]
In preparation.

‘Decision problems for word-hyperbolic monoids’

In preparation.

‘Notions of hyperbolicity in monoids’

[with V. Maltcev]
In preparation.

‘On weak automatic presentations’

In preparation.

Seminars & Conference talks

‘Certain properties of subsemigroups of free groups’

Seminar: University of St Andrews, 13th August 2003.

‘Automatic semigroups & Malcev presentations I, II’

Informal seminars: Friday Group, University of St Andrews, 26th March, 7th May 2004.

‘Malcev presentations’

Seminar: University of St Andrews, 17th June 2004.

‘Malcev presentations’

Conference talk: Postgraduate Pure Mathematics in the North East 2004, University of Newcastle, 28th June 2004.

‘Malcev presentations for subsemigroups of groups’

Invited seminar: University of Glasgow, 13th October 2004.

‘A survey of Malcev presentations for subsemigroups of groups’

Conference talk: Groups St Andrews 2005, University of St Andrews, 31st July 2005.

‘Decidability and undecidability for automatic semigroups’

Invited seminar: University of Edinburgh & Heriot–Watt University (Joint), 15th November 2005.

‘An introduction to decidability for automatic semigroups’

Short course: University of St Andrews, 28th November–12th December 2005.

‘Cancellative and Malcev presentations for subsemigroups and extensions’

Conference talk: Semigroup Day ’06, Heriot–Watt University, 5th July 2006.

‘Decidability for automatic semigroups’

Conference talk: International Workshop on Computational and Algorithmic Aspects of Semigroup Theory, University of St Andrews, 6th September 2006.

‘Automatic presentations for cancellative semigroups’

Conference talk: 2nd International Conference on Language and Automata Theory and Applications, Universitat Rovira i Virgili, Tarragona, Spain, 18th March 2008.

‘Automatic presentations for semigroups’

Conference talk: 2008 British Mathematical Colloquium, University of York, 27th March 2008.

‘Automaton semigroups’

Seminar: Pure Mathematics Colloquium, University of St Andrews, 29th May 2008.

‘Automatic semigroups’

Short course: Centro de Álgebra da Universidade de Lisboa, 22nd–29th October 2008.

‘Malcev presentations for subsemigroups of groups’

Seminar: Centro de Álgebra da Universidade de Lisboa, 31st October 2008.

‘Automaton semigroups’

Invited conference talk: North Britain Semigroups and Applications Network, University of St Andrews, 16th April 2009.

‘Automatic presentations and semigroups’

Seminar: Centro de Matemática da Universidade do Porto, 25th September 2009.

‘Malcev presentations for subsemigroups of groups’

Seminar: Centro de Matemática da Universidade do Porto, 30th October 2009.

‘Computing with automatic semigroups’

Conference talk: Centro de Matemática da Universidade do Porto, 15th June 2010.

‘Automatic presentations and semigroups’

Invited seminar: Centro de Álgebra da Universidade de Lisboa, 10th September 2010.

‘Hyperbolic and word-hyperbolic semigroups’

Invited conference talk: North Britain Semigroups and Applications Network, University of St Andrews, 19th May 2011.