Alan James Cain  ·  Curriculum Vitæ Download as PDF ]
AddressCentro de Matemática e Aplicações
Faculdade de Ciência e Tecnologia
Universidade Nova de Lisboa
2829–516 Caparica
Portugal
Email addressDOT cain AT fct DOT unl DOT pt
Replace ‘AT’ with ‘@’ and ‘DOT’ with ‘.’
Web pagehttp://www-groups.st-and.ac.uk/~alanc
LanguagesEnglish (Native)
Japanese (Intermediate)
Portuguese (Basic)
French (Basic)
ORCiD0000-0002-0706-1354
ResearcherIDG-7986-2014
Scopus Author ID13608823100

Positions held

2014/01–
Present
‘Investigador FCT’ Senior Research Fellow
Centro de Matemática e Aplicações, Universidade Nova de Lisboa
2829–516 Caparica, Portugal
  • FCT-funded five-year personal research fellowship.
  • Awarded an associated grant of € 50000.
  • Taught courses ‘Semigroups, Automata, and Languages’, ‘Algebraic Theory of Automata’, ‘Rewriting Systems’, ‘Combinatorial Group Theory’.
  • Supervised two Ph.D. students (one unofficially, one officially) and one master’s student.
  • Organizer of the seminar of the Algebra & Logic group (2015–Present).
2009/07–
2014/01
‘Ciência 2008’ Research Fellow
Centro de Matemática da Universidade do Porto
Rua do Campo Alegre 687, 4169–007 Porto, Portugal
  • FCT-funded five-year personal research fellowship.
  • Developed and taught a master’s course ‘Semigroups’.
2008/09–
2009/06
Postdoctoral researcher
Centro de Álgebra da Universidade de Lisboa
Av. Prof. Gama Pinto 2, 1649–003 Lisboa, Portugal
  • Research within the project ‘Semigroups and Languages’, funded by FCT and PIDDAC (PTDC/MAT/69514/2006).
2005/09–
2008/08
Research Fellow
School of Mathematics & Statistics, University of St Andrews
St Andrews, Fife KY16 9AJ, United Kingdom
  • 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).
2002/09–
2005/05
Tutor & Computer laboratory demonstrator
School of Mathematics & Statistics, University of St Andrews
St Andrews, Fife KY16 9AJ, United Kingdom
  • Tutored the first-year mathematics courses MT1002 and MT1003, required for all students proceeding to a degree in mathematics.
  • Worked as a computer laboratory demonstrator, aiding students in using MAPLE to solve practical exercises related to their work in pre-honours lecture courses, and marking their work.

Education

2002/09–
2005/07
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. Nik Ruškuc
Examiners: Dr M. Quick (Internal), Prof. D.F. Holt (External, Univ. of Warwick)

1998/09–
2002/07
M.Sci. in Mathematics with First-class Honours
University of Glasgow
Glasgow G12 8QQ, United Kingdom
  • Cunninghame Prize in Mathematics; Class prizes in Mathematics in first and second year; Class prize in Computer Science in second year.
  • President of the Maclaurin Society, the University of Glasgow student society for mathematics and statistics.

Membership of Professional Societies

Teaching

2005–
2007
Fundamentals of Pure Mathematics
University of St Andrews
  • Lectured jointly with Prof. Nik Ruškuc
  • Completely revised the course to cover a number of foundational topics like constructing the real numbers, the fundamental theorem of algebra, and cardinality.
  • Student feedback was enthusiastically supportive, with 85% overall approval.
2012Semigroups
Universidade do Porto
  • Master’s-level course presenting an introduction to semigroup theory, including presentations, structure theory, the program of classification of finite semigroups via pseudovarieties, and the Eilenberg correspondence with varieties of rational languages.
  • Wrote the course from scratch and produced high-quality printed lecture notes for the students, including exercises, advanced problems, and solutions. These notes (see publication [C12d]) are distributed under a Creative Commons licence and have been used in courses at several other universities, including the University of Western Sydney and the University of Novi Sad.
2015–
Present
Semigroups, Automata, and Languages
Universidade Nova de Lisboa
  • Master’s-level course presenting an introduction to semigroup theory, including structure theory, and a brief introduction to pseudovarieties and the Eilenberg correspondence.
2016–
Present
Algebraic Theory of Automata
Universidade Nova de Lisboa
  • Course for mathematics and computer science students on finite and pushdown automata, Turing machines, and the Chomsky hierarchy of formal languages.
2016–
Present
Rewriting Systems
Universidade Nova de Lisboa
  • Master’s-level course on term rewriting.
2016–
Present
Combinatorial Group Theory
Universidade Nova de Lisboa
  • Doctoral-level course on combinatorial group theory, including Bass–Serre theory and hyperbolic groups.

Research funding

2002/10–
2005/09
Carnegie Doctoral Scholarship
Carnegie Trust for the Universities of Scotland
  • Ph.D. scholarship covering tuition fees and a stipend. These scholarships are prestigious and highly competitive, with fewer than 10% of applications funded.
2009/07–
2014/01
‘Ciência 2008’ fellowship
Fundação para a Ciência e a Tecnologia (FCT) [National science agency]
  • Five-year fellowship to pursue research at a university in Portugal. Funding for 90% of the salary, with the remaining 10% paid by the host institution.
2010/06–
2014/05
Project ‘Automata, Languages, Decidability in Algebra’
EP/H011978/1
Engineering and Physical Sciences Research Council (EPSRC)
  • Prof. Nik Ruškuc, Dr Martyn Quick, and I wrote the application for this grant, which was funded to a value of £ 348646. The original plan was for me to be employed as a researcher on this project at the University of St Andrews, but I declined in favour of the ‘Ciência 2008’ fellowship. I remained an active participant in the project and made frequent research visits to St Andrews.
2014/01–
Present
‘Investigador FCT’ fellowship
Fundação para a Ciência e a Tecnologia (FCT) [National science agency]
  • Five-year advanced fellowship to pursue research at a university in Portugal. This programme is a much more selective than the ‘Ciência’ scheme. The evaluation panel rated my application at the highest grade 9, defined as ‘exceptionally strong with essentially no weaknesses’.
2014/07–
Present
Exploratory Project (Principal investigator)
IF/01622/2013/CP1161/CT0001
Fundação para a Ciência e a Tecnologia (FCT) [National science agency]
  • Special grant scheme, only open to ‘Investigador FCT’ fellows. Value € 50000, with freedom to allocate this to travel, visitors, equipment, etc., and with the usual deductions for administrative overheads being waived.
2015/04‘Research in Pairs’ grant
London Mathematical Society
  • Small grant (value £ 1000) to visit the University of St Andrews to work with Dr Markus Pfeiffer and speak at a conference.
2016/06–
Present
Project ‘Hilbert’s 24th Problem’ (Subproject leader)
PTDC/MHC-FIL/2583/2014
Fundação para a Ciência e a Tecnologia (FCT) [National science agency]
  • Value € 199902. I am leader of one of the three subprojects, and a member of another subprojects.

Research visits

University of St Andrews, United Kingdom
2009/05; 2010/08; 2011/05; 2011/08; 2012/06; 2013/05; 2015/04; 2015/09.
Centro de Álgebra da Universidade de Lisboa, Portugal
2010/09; 2012/05; 2012/10; 2013/11.
Sultan Qaboos University, Muscat, Oman
2013/04.
University of York, United Kingdom
2013/11.
Uniwersytet Warszawski, Poland
2016/10.
Université Paris-Diderot, France
2017/04.

Publications

Presentations for Subsemigroups of Groups
Ph.D. thesis, University of St Andrews, 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, no. 1 (2006), pp. 57–66.
DOI: 10.1017/s0305004106009236. MR: 2238642. ZBL: 1115.20043.
Subsemigroups of groups: presentations, Malcev presentations, and automatic structures
[with E. F. Robertson & N. Ruškuc]
Journal of Group Theory, 9, no. 3 (2006), pp. 397–426.
DOI: 10.1515/jgt.2006.027. MR: 2226621. ZBL: 1151.20044.
A group-embeddable non-automatic semigroup whose universal group is automatic
Glasgow Mathematical Journal, 48, no. 2 (2006), pp. 337–342.
DOI: 10.1017/s0017089506003107. MR: 2256982. ZBL: 1108.20056.
Cancellativity is undecidable for automatic semigroups
Quarterly Journal of Mathematics, 57, no. 3 (2006), pp. 285–295.
DOI: 10.1093/qmath/hai023. MR: 2253587. ZBL: 1126.20039.
Malcev presentations for subsemigroups of groups — a survey
In C. M. Campbell, M. Quick, E. F. Robertson, & G. C. Smith, eds, Groups St Andrews 2005 (Vol. 1), no. 339 in London Mathematical Society Lecture Note Series, pp. 256–268 (Cambridge: Cambridge University Press, 2007).
DOI: 10.1017/CBO9780511721212.018. MR: 2328165.
Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions
[with E. F. Robertson & N. Ruškuc]
Journal of the Australian Mathematical Society, 84, no. 1 (2008), pp. 39–61.
DOI: 10.1017/s1446788708000086. MR: 2469266. ZBL: 1156.20048.
Automatic presentations for cancellative semigroups
[with G. Oliver, N. Ruškuc & R. M. Thomas]
In C. Martín-Vide, H. Fernau, & F. Otto, eds, Language and Automata Theory and Applications: Second International Conference, Tarragona, Spain, March 13–19, 2008, no. 5196 in Lecture Notes in Computer Science, pp. 149–159 (Springer, 2008).
DOI: 10.1007/978-3-540-88282-4_15. MR: 2540320. ZBL: 1157.20332.
Malcev presentations for subsemigroups of direct products of coherent groups
Journal of Pure and Applied Algebra, 213, no. 6 (2009), pp. 977–990.
DOI: 10.1016/j.jpaa.2008.10.006. MR: 2498789. ZBL: 1178.20050.
Automaton semigroups
Theoretical Computer Science, 410, no. 47–49 (2009), pp. 5022–5038.
DOI: 10.1016/j.tcs.2009.07.054. MR: 2583696. ZBL: 1194.68133.
Automatic presentations for semigroups
[with G. Oliver, N. Ruškuc & R. M. Thomas]
Information and Computation, 207, no. 11 (2009), pp. 1156–1168.
DOI: 10.1016/j.ic.2009.02.005. MR: 2566948. ZBL: 1192.20040.
Decision problems for finitely presented and one-relation semigroups and monoids
[with V. Maltcev]
International Journal of Algebra and Computation, 19, no. 6 (2009), pp. 747–770.
DOI: 10.1142/s0218196709005366. MR: 2572873. ZBL: 1201.20055.
Monoids presented by rewriting systems and automatic structures for their submonoids
International Journal of Algebra and Computation, 19, no. 6 (2009), pp. 771–790.
DOI: 10.1142/s0218196709005317. MR: 2572874. ZBL: 1201.20054.
Automatic semigroups and Bruck–Reilly extensions
Acta Mathematica Hungarica, 126, no. 1–2 (2010), pp. 1–15.
DOI: 10.1007/s10474-009-8063-8. MR: 2593314. ZBL: 1259.20061.
Automatic presentations and semigroup constructions
[with G. Oliver, N. Ruškuc & R. M. Thomas]
Theory of Computing Systems, 47, no. 2 (2010), pp. 568–592.
DOI: 10.1007/s00224-009-9216-4. MR: 2652030. ZBL: 1204.68118.
Deus ex machina and the aesthetics of proof
Mathematical Intelligencer, 32, no. 3 (September 2010), pp. 7–11.
DOI: 10.1007/s00283-010-9141-z. MR: 2721302. ZBL: 1247.00009.
An annotated translation of Yves Marie André’s Essay on Beauty (1741)
(Ebook, 2010).
Unary FA-presentable semigroups
[with N. Ruškuc & R. M. Thomas]
International Journal of Algebra and Computation, 22, no. 4 (2012).
DOI: 10.1142/S0218196712500385. MR: 2946303. ZBL: 1285.03048.
Green index in semigroup theory: generators, presentations, and automatic structures
[with R. Gray & N. Ruškuc]
Semigroup Forum, 85, no. 3 (2012), pp. 448–476.
DOI: 10.1007/s00233-012-9406-2. MR: 3001595. ZBL: 1270.20059.
Context-free rewriting systems and word-hyperbolic structures with uniqueness
[with V. Maltcev]
International Journal of Algebra and Computation, 22, no. 7 (2012).
DOI: 10.1142/S0218196712500610. MR: 2999367. ZBL: 1284.68320.
Nine Chapters on the Semigroup Art
(Lecture notes, 2012).
Hyperbolicity of monoids presented by confluent monadic rewriting systems
Beiträge zur Algebra und Geometrie, 54, no. 2 (October 2013), pp. 593–608.
DOI: 10.1007/s13366-012-0116-4. MR: 3095744. ZBL: 1326.20056.
Automatic structures for subsemigroups of Baumslag–Solitar semigroups
Semigroup Forum, 87, no. 3 (2013), pp. 537–552.
DOI: 10.1007/s00233-013-9490-y. MR: 3128706. ZBL: 1326.20057.
Finitely presented monoids with linear Dehn function need not have regular cross-sections
[with V. Maltcev]
Semigroup Forum, 88, no. 2 (2014), pp. 300–315.
DOI: 10.1007/s00233-013-9531-6. MR: 3189098. ZBL: 1300.20057.
Markov semigroups, monoids, and groups
[with V. Maltcev]
International Journal of Algebra and Computation, 24, no. 5 (August 2014).
DOI: 10.1142/S021819671450026X. MR: 3254716. ZBL: 1325.20055.
Subalgebras of FA-presentable algebras
[with N. Ruškuc]
Algebra Universalis, 72, no. 2 (October 2014), pp. 101–123.
DOI: 10.1007/s00012-014-0293-0. MR: 3257650. ZBL: 1321.08001.
Hopfian and co-hopfian subsemigroups and extensions
[with V. Maltcev]
Demonstratio Mathematica, 47, no. 4 (2014), pp. 791–804.
DOI: 10.2478/dema-2014-0064. MR: 3290386. ZBL: 1312.20051.
Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
[with R. D. Gray & A. Malheiro]
Journal of Algebra, 423 (Febuary 2015), pp. 37–53.
DOI: 10.1016/j.jalgebra.2014.09.037. MR: 3283708. ZBL: 1311.20055.
A simple non-bisimple congruence-free finitely presented monoid
[with V. Maltcev]
Semigroup Forum, 90, no. 1 (Febuary 2015), pp. 184–188.
DOI: 10.1007/s00233-014-9607-y. MR: 3297818. ZBL: 1317.20049.
Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids
[with R. D. Gray & A. Malheiro]
International Journal of Algebra and Computation, 25, no. 1-2 (2015).
DOI: 10.1142/S0218196715400044. MR: 3325877. ZBL: 1326.20058.
Automaton semigroup constructions
[with T. Brough]
Semigroup Forum, 90, no. 3 (June 2015), pp. 763–774.
DOI: 10.1007/s00233-014-9632-x. MR: 3345953. ZBL: 06493123.
Deciding conjugacy in sylvester monoids and other homogeneous monoids
[with A. Malheiro]
International Journal of Algebra and Computation, 25, no. 5 (August 2015).
DOI: 10.1142/S0218196715500241. MR: 3384086. ZBL: 06481131.
A countable family of congruence-free finitely presented monoids
[with F. Al-Kharousi, V. Maltcev & A. Umar]
Acta Universitatis Szegediensis: Acta Scientiarum Mathematicarum, 81, no. 3–4 (2015), pp. 437–445.
DOI: 10.14232/actasm-013-028-z.
Decision problems for word-hyperbolic semigroups
[with M. Pfeiffer]
Journal of Algebra, 465 (November 2016), pp. 287–321.
DOI: 10.1016/j.jalgebra.2016.07.007.
Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson–Schensted–Knuth-type correspondence for quasi-ribbon tableaux
[with A. Malheiro]
Journal of Algebraic Combinatorics, 45, no. 2 (March 2017), pp. 475–524.
DOI: 10.1007/s10801-016-0714-6.
Growths of endomorphisms of finitely generated semigroups
[with V. Maltcev]
Journal of the Australian Mathematical Society, 102, no. 2 (April 2017), pp. 163–184.
DOI: 10.1017/S1446788716000264.
Automaton semigroups: new constructions results and examples of non-automaton semigroups
[with T. Brough]
Theoretical Computer Science, 674 (2017), p. 1–15.
DOI: 10.1016/j.tcs.2017.02.003.
On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids
[with R. D. Gray & A. Malheiro]
Submitted.
arXiv: 1407.7428.
Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types ${A}_n$, ${B}_n$, ${C}_n$, ${D}_n$, and ${G}_2$
[with R. D. Gray & A. Malheiro]
Submitted.
arXiv: 1412.7040.
Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids
[with A. Malheiro]
Submitted.
arXiv: 1611.04151.
Combinatorics of cyclic shifts in sylvester monoids
[with A. Malheiro]
Submitted.
arXiv: 1611.04152.
Crystals & trees: quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences
[with A. Malheiro]
Submitted.
arXiv: 1702.02998.

Invited seminars & conference talks

‘Malcev presentations for subsemigroups of groups’
Invited seminar: University of Glasgow, 13th October 2004.
‘Decidability and undecidability for automatic semigroups’
Invited seminar: University of Edinburgh & Heriot–Watt University (Joint), 15th November 2005.
‘Automaton semigroups’
Invited conference talk: North Britain Semigroups and Applications Network, University of St Andrews, 16th April 2009.
‘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 2010.
‘Hyperbolicity for semigroups’
Invited seminar: Centro de Álgebra da Universidade de Lisboa, 11th May 2012.
‘Automatic presentations for algebraic and combinatorial structures’
Invited seminar: Centro de Matemática da Universidade de Coimbra, 20th March 2013.
‘Plactic monoids and rewriting systems’
Invited seminar: Sultan Qaboos University, 23rd April 2013.
‘Unary FA-presentable algebraic and combinatorial structures’
Invited conference talk: Workshop on Algebraic Structures and Semigroups, Centro de Álgebra da Universidade de Lisboa, 5th July 2013.
‘Endomorphisms of semigroups: growth and interactions with subsemigroups’
Invited conference talk: North Britain Semigroups and Applications Network, University of York, 20th November 2013.
‘Automaticity, finite complete rewriting systems, and finite derivation type for homogeneous monoids’
Invited seminar: Centro de Matemática da Universidade do Porto, 17th January 2014.
‘Computing with automatic and word-hyperbolic semigroups’
Invited conference talk: Workshop on Computational Algebra, Centro de Álgebra da Universidade de Lisboa, 22nd July 2014.
‘Endomorphisms of semigroups: growth and interactions with subsemigroups’
Invited seminar: Centro de Álgebra da Universidade de Lisboa, 11th November 2014.
‘Plactic, hypoplactic, and sylvester monoids, and other homogeneous monoids’
Invited seminar: Centro de Matemática da Universidade de Coimbra, 25th February 2015.
‘Computation and conjugacy in hypoplactic and sylvester monoids and other homogenous monoids’
Invited conference talk: North Britain Semigroups and Applications Network, University of St Andrews, 25th April 2015.
‘Crystal bases, finite convergent presentations, and automaticity for plactic monoids’
Invited conference talk: International meeting of the American Mathematical Society, European Mathematical Society, and Sociedade Portuguesa de Matemática: Special Session on Algebraic Theory of Semigroups and Applications, University of Porto, 11th June 2015.
‘Quasi-ribbons & quasi-crystals’
Invited seminar: Pure Mathematics Colloquium, University of St Andrews, 24th September 2015.
‘Combinatorial and computational properties of the sylvester monoid’
Invited conference talk: 6th Iberian Mathematical Meeting, University of Santiago de Compostela, 6th October 2016.
‘Crystals, tableaux, and the (hypo)plactic monoid’
Invited seminar: Uniwersytet Warszawski, Poland, 27th October 2016.
‘Combinatorics of cyclic shifts in the plactic, hypoplactic, sylvester, and related monoids’
Invited seminar: Uniwersytet Warszawski, Poland, 20th April 2017.