Alan James Cain  ·  Curriculum Vitæ
 Address Centro de Matemática e Aplicações Faculdade de Ciência e Tecnologia Universidade Nova de Lisboa 2829–516 Caparica Portugal Email address a DOT cain AT fct DOT unl DOT ptReplace ‘AT’ with ‘@’ and ‘DOT’ with ‘.’ Web page http://www-groups.st-and.ac.uk/~alanc Languages English (Native)Japanese (Intermediate)Portuguese (Basic)French (Basic)
 ORCiD 0000-0002-0706-1354 ResearcherID G-7986-2014 Scopus Author ID 13608823100

Positions held

 2014/01–Present ‘Investigador FCT’ Senior Research Fellow Centro de Matemática e Aplicações, Universidade Nova de Lisboa2829–516 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 PortoRua 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 LisboaAv. 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 AndrewsSt 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 AndrewsSt 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 AndrewsSt 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) Funded by a prestigious Carnegie Doctoral Scholarship. Thesis accepted without corrections. 1998/09–2002/07 M.Sci. in Mathematics with First-class Honours University of GlasgowGlasgow 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

• London Mathematical Society (Elected 2006)
• Edinburgh Mathematical Society (Elected 2003)

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/H011978/1Engineering 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/CT0001Fundação 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 to speak at a conference. 2016/06–Present Project ‘Hilbert’s 24th Problem’ (Subproject leader) PTDC/MHC-FIL/2583/2014Fundação 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 subproject.

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; 2017/04.
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: 1336.20062.
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. MR: 3443762. ZBL: 1363.20044.
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. MR: 3537824. ZBL: 06621117.
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. MR: 3604065. ZBL: 1359.05135.
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. MR: 3621637. ZBL: 06714329.
Automaton semigroups: new constructions results and examples of non-automaton semigroups
[with T. Brough]
Theoretical Computer Science, 674 (2017), pp. 1–15.
DOI: 10.1016/j.tcs.2017.02.003. MR: 3634703. ZBL: 06715025.
On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids
[with R. D. Gray & A. Malheiro]
Information and Computation, 255, no. 1 (August 2017), pp. 68–93.
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, and related monoids
[with A. Malheiro]
In S. Brlek, F. Dolce, C. Reutenauer, & É. Vandomme, eds, Combinatorics on Words: 11th International Conference, Montréal, Canada, September 11–15, 2017, vol. 10432 of Lecture Notes in Computer Science, pp. 190–202 (Springer, 2017).
Crystals and trees: quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences
[with A. Malheiro]
Journal of Algebra, 502 (May 2018), pp. 347–381.
The monoids of the patience sorting algorithm
[with A. Malheiro & F. Silva]
International Journal of Algebra and Computation, Forthcoming.
arXiv: 1706.06884.
Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids
[with A. Malheiro]
Electronic Journal of Combinatorics, Forthcoming.
arXiv: 1611.04151.
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.
A note on identities in plactic monoids and monoids of upper-triangular tropical matrices
[with G. Klein, Ł. Kubat, A. Malheiro & J. Okniński]
Submitted.
arXiv: 1705.04596.
Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids
[with A. Malheiro]
Submitted.
arXiv: 1709.03974.
Combinatorics of patience sorting monoids
[with A. Malheiro & F. Silva]
Submitted.
arXiv: 1801.05591.
Visual Thinking and Simplicity in Proof
Submitted.
arXiv: 1803.00038.
Conjugacy in patience sorting monoids
[with A. Malheiro & F. Silva]
Submitted.
arXiv: 1803.00361.
Two applications of monoid actions to cross-sections
[with T. Brough & V. Maltcev]
Submitted.
arXiv: 1803.10747.

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.
‘Simplicity and the aesthetics of mathematics’
Invited conference talk: Third Lisbon International Conference on Philosophy of Science, 14th December 2016.
‘Combinatorics of cyclic shifts in the plactic, hypoplactic, sylvester, and related monoids’
Invited seminar: Uniwersytet Warszawski, Poland, 20th April 2017.
‘Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids’
Invited conference talk: 7th Combinatorics Day, Universidade de Évora, 26th May 2017.