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Born 9 March 1981. M.Sci. in Mathematics (First-class Hons), University of Glasgow 2002; Ph.D. in Mathematics (Thesis: Presentations for Subsemigroups of Groups), University of St Andrews 2005. Research Fellow at the University of St Andrews 2005¿–8, Centro de Álgebra da Universidade de Lisboa 2008–¿9, "Ciência 2008" Research Fellow at the Centro de Matemática da Universidade do Porto 2009–14; Senior Research Fellow at NOVA Math, Universidade Nova de Lisboa. (2014–23) Author or co-author of over fifty research articles in the fields of automata, semigroups, combinatorics, and philosophy of mathematics. Author of "Nine Chapters on the Semigroup Art" and "Form & Number: A History of Mathematical Beauty". Creator of an annotated edition of G.H. Hardy's "A Mathematician's Apology".
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Identificação

Identificação pessoal

Nome completo
Alan James Cain

Nomes de citação

  • Cain, Alan J.

Identificadores de autor

Ciência ID
3C13-398B-38A0
ORCID iD
0000-0002-0706-1354
Google Scholar ID
jSmVoLIAAAAJ
Researcher Id
G-7986-2014
Scopus Author Id
13608823100

Domínios de atuação

  • Ciências Exatas - Matemática - Matemática Pura
  • Humanidades - Filosofia, Ética e Religião - Filosofia

Idiomas

Idioma Conversação Leitura Escrita Compreensão Peer-review
Inglês (Idioma materno)
Português Utilizador elementar (A2) Utilizador independente (B1) Utilizador independente (B1) Utilizador independente (B1) Utilizador independente (B1)
Francês Utilizador elementar (A1) Utilizador independente (B1) Utilizador elementar (A2) Utilizador elementar (A1) Utilizador elementar (A2)
Japonês Utilizador independente (B1) Utilizador independente (B2) Utilizador independente (B1) Utilizador independente (B1) Utilizador elementar (A1)
Formação
Grau Classificação
2002/09/17 - 2005/09/20
Concluído
 Mathematics (Doctor of Philosophy)
University of St Andrews, Reino Unido
"Presentations for Subsemigroups of Groups" (TESE/DISSERTAÇÃO)
1998/09/17 - 2002/07/09
Concluído
 Mathematics (Master)
Especialização em Mathematics
University of Glasgow, Reino Unido
 First-class honours
Percurso profissional

Ciência

Categoria Profissional
Instituição de acolhimento 		Empregador
2019/01/12 - 2023/10/31 Investigador Contratado (Investigação) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
Universidade Nova Centro de Matemática e Aplicações, Portugal
2014/01/12 - 2019/01/11 Investigador Auxiliar (carreira) (Investigação) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
Universidade Nova Centro de Matemática e Aplicações, Portugal
2009/07/01 - 2014/01/10 Investigador Auxiliar (carreira) (Investigação) Universidade do Porto Faculdade de Ciências, Portugal
Universidade do Porto Faculdade de Ciências, Portugal
2008/09/01 - 2009/06/30 Pós-doutorado (Investigação) Universidade de Lisboa Faculdade de Ciências, Portugal
Universidade de Lisboa Faculdade de Ciências, Portugal
2005/09/01 - 2008/08/31 Investigador (Investigação) University of St Andrews, Reino Unido
University of St Andrews School of Mathematics and Statistics, Reino Unido
Projetos

Bolsa

Designação Financiadores
2014/07/01 - 2019/01/11 Exploratory Project (associated to "Investigador FCT" fellowship)
IF/01622/2013/CP1161/CT0001
Investigador responsável
Universidade Nova Centro de Matemática e Aplicações, Portugal
 Fundação para a Ciência e a Tecnologia
Concluído
2010/06/01 - 2014/05/31 Automata, Languages, Decidability in Algebra
EP/H011978/1
Outra
University of St Andrews School of Mathematics and Statistics, Reino Unido
 Engineering and Physical Sciences Research Council
Concluído
2005/09/01 - 2008/08/31 Critical Mass in Computational Algebra
EP/C523229/1
Bolseiro de Investigação
University of St Andrews School of Mathematics and Statistics, Reino Unido

University of St Andrews School of Computer Science, Reino Unido
Engineering and Physical Sciences Research Council
Concluído

Projeto

Designação Financiadores
2018/10/01 - 2021/09/30 Semigroups: Conjugacy, Computation, Crystals and Combinatorics
PTDC/MAT-PUR/31174/2017
Associação para a Inovação e Desenvolvimento da FCT, Portugal

Universidade Aberta, Portugal
Fundação para a Ciência e a Tecnologia
Em curso
2016/06/01 - 2019/11/30 Hilbert's 24th Problem
PTDC/MHC-FIL/2583/2014
Universidade Nova Centro de Matemática e Aplicações, Portugal

Associação para a Inovação e Desenvolvimento da FCT, Portugal
Fundação para a Ciência e a Tecnologia
Concluído
Produções

Publicações

Artigo em revista
  1. Alan J. Cain; António Malheiro; Duarte Ribeiro. "Identities and bases in the sylvester and Baxter monoids". Journal of Algebraic Combinatorics (2023): https://doi.org/10.1007/s10801-022-01202-6.
    10.1007/s10801-022-01202-6
  2. Alan J. Cain; Marianne Johnson; Mark Kambites; António Malheiro. "Representations and identities of plactic-like monoids". Journal of Algebra (2022): https://doi.org/10.1016/j.jalgebra.2022.04.033.
    10.1016/j.jalgebra.2022.04.033
  3. Brough, Tara; Cain, Alan J.. "A language hierarchy of binary relations". Information and Computation 275 (2020): 104607. http://dx.doi.org/10.1016/j.ic.2020.104607.
    Publicado • 10.1016/j.ic.2020.104607
  4. Tara Brough; Alan J. Cain; Victor Maltcev. "Two applications of monoid actions to cross-sections". Communications in Algebra (2020): 1-10. https://doi.org/10.1080/00927872.2019.1710160.
    10.1080/00927872.2019.1710160
  5. Alan J. Cain. "Visual thinking and simplicity of proof". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377 2140 (2019): 20180032-20180032. https://doi.org/10.1098/rsta.2018.0032.
    10.1098/rsta.2018.0032
  6. Alan J. Cain; Robert D. Gray; António Malheiro. "Crystal monoids & crystal bases: Rewriting systems and biautomatic structures for plactic monoids of types A, B, C, D, and G2". Journal of Combinatorial Theory, Series A 162 (2019): 406-466. https://doi.org/10.1016/j.jcta.2018.11.010.
    10.1016/j.jcta.2018.11.010
  7. Cain, Alan J.; Malheiro, António; CMA - Centro de Matemática e Aplicações; DM - Departamento de Matemática. "Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids". (2019): http://www.scopus.com/inward/record.url?scp=85068467706&partnerID=8YFLogxK.
    https://doi.org/10.1016/j.jalgebra.2019.06.025
  8. Cain, Alan J.; CMA - Centro de Matemática e Aplicações. "Visual thinking and simplicity of proof". (2019): http://www.scopus.com/inward/record.url?scp=85061334936&partnerID=8YFLogxK.
    https://doi.org/10.1098/rsta.2018.0032
  9. Cain, Alan J.; Malheiro, António; Silva, Fábio M.; CMA - Centro de Matemática e Aplicações; DM - Departamento de Matemática. "Combinatorics of patience sorting monoids". (2019): http://www.scopus.com/inward/record.url?scp=85066807002&partnerID=8YFLogxK.
    https://doi.org/10.1016/j.disc.2019.05.022
  10. Alan J. Cain; António Malheiro. "Identities in Plactic, Hypoplactic, Sylvester, Baxter, and Related Monoids". The Electronic Journal of Combinatorics (2018): https://doi.org/10.37236/6873.
    10.37236/6873
  11. Alan J. Cain; António Malheiro. "Crystals and trees: Quasi-Kashiwara operators, monoids of binary trees, and Robinson–Schensted-type correspondences". Journal of Algebra 502 (2018): 347-381. https://doi.org/10.1016/j.jalgebra.2018.01.036.
    10.1016/j.jalgebra.2018.01.036
  12. Cain, Alan J.; Malheiro, António; DM - Departamento de Matemática; CMA - Centro de Matemática e Aplicações. "Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids". (2018): http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p30.
  13. "Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids". Electronic Journal of Combinatorics (2018): http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p30.
  14. Cain, Alan J.. "Growths of endomorphisms of finitely generated semigroups". Journal of the Australian Mathematical Society (2017): 1-22. https://doi.org/10.1017%2Fs1446788716000264.
    10.1017/S1446788716000264
  15. Tara Brough; Alan J. Cain. "Automaton semigroups: New constructions results and examples of non-automaton semigroups". Theoretical Computer Science (2017):
    10.1016/j.tcs.2017.02.003
  16. Alan J. Cain; António Malheiro. "Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson–Schensted–Knuth-type correspondence for quasi-ribbon tableaux". Journal of Algebraic Combinatorics (2017): https://doi.org/10.1007%2Fs10801-016-0714-6.
    10.1007/s10801-016-0714-6
  17. Cain, Alan J.; Malheiro, António; CMA - Centro de Matemática e Aplicações; DM - Departamento de Matemática. "Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson-Schensted-Knuth-type correspondence for quasi-ribbon tableaux". (2017): http://www.scopus.com/inward/record.url?scp=84988709956&partnerID=8YFLogxK.
    https://doi.org/10.1007/s10801-016-0714-6
  18. Brough, Tara; Cain, Alan J.; CMA - Centro de Matemática e Aplicações. "Automaton semigroups: New constructions results and examples of non-automaton semigroups". (2017): http://www.scopus.com/inward/record.url?scp=85015758537&partnerID=8YFLogxK.
    https://doi.org/10.1016/j.tcs.2017.02.003
  19. Cain, Alan J.; Maltcev, Victor; CMA - Centro de Matemática e Aplicações. "Growths of endomorphisms of finitely generated semigroups". (2017): http://www.scopus.com/inward/record.url?scp=84978044464&partnerID=8YFLogxK.
    https://doi.org/10.1017/S1446788716000264
  20. Cain, A.J.; Gray, R.D.; Malheiro, A.. "On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids". Information and Computation 255 (2017): 68-93. http://www.scopus.com/inward/record.url?eid=2-s2.0-85020090084&partnerID=MN8TOARS.
    10.1016/j.ic.2017.05.003
  21. Cain, A.J.; Pfeiffer, M.. "Decision problems for word-hyperbolic semigroups". Journal of Algebra 465 (2016): 287-321. http://www.scopus.com/inward/record.url?eid=2-s2.0-84989918507&partnerID=MN8TOARS.
    10.1016/j.jalgebra.2016.07.007
  22. Al-Kharousi, Fatma; Cain, Alan J.; Maltcev, Victor; Umar, Abdullahi; CMA - Centro de Matemática e Aplicações. "A countable family of finitely presented infinite congruence-free monoids". (2015): http://www.scopus.com/inward/record.url?scp=84953876715&partnerID=8YFLogxK.
    https://doi.org/10.14232/actasm-013-028-z
  23. Cain, A.J.; Malheiro, A.. "Deciding conjugacy in sylvester monoids and other homogeneous monoids". International Journal of Algebra and Computation 25 5 (2015): 899-915. http://www.scopus.com/inward/record.url?eid=2-s2.0-84939567901&partnerID=MN8TOARS.
    10.1142/S0218196715500241
  24. Cain, Alan J.. "Finite Gröbner-Shirshov bases for plactic algebras and biautomatic structures for plactic monoids". Journal of Algebra 423 (2015): 37-53. http://www.scopus.com/inward/record.url?eid=2-s2.0-84908322877&partnerID=MN8TOARS.
    10.1016/j.jalgebra.2014.09.037
  25. Cain, A.J.; Gray, R.D.; Malheiro, A.. "Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids". International Journal of Algebra and Computation 25 1-2 (2015): 51-80. http://www.scopus.com/inward/record.url?eid=2-s2.0-84928485399&partnerID=MN8TOARS.
    10.1142/S0218196715400044
  26. Cain, A.J.; Maltcev, V.. "Finitely presented monoids with linear Dehn function need not have regular cross-sections". Semigroup Forum 88 2 (2014): 300-315. http://www.scopus.com/inward/record.url?eid=2-s2.0-84897989172&partnerID=MN8TOARS.
    10.1007/s00233-013-9531-6
  27. Cain, A.J.; Ruškuc, N.. "Subalgebras of FA-presentable algebras". Algebra Universalis 72 2 (2014): 101-123. http://www.scopus.com/inward/record.url?eid=2-s2.0-84908147785&partnerID=MN8TOARS.
    10.1007/s00012-014-0293-0
  28. Cain, A.J.; Maltcev, V.. "Hopfian and co-hopfian subsemigroups and extensions". Demonstratio Mathematica 47 4 (2014): 791-804. http://www.scopus.com/inward/record.url?eid=2-s2.0-84919414091&partnerID=MN8TOARS.
    10.2478/dema-2014-0064
  29. Cain, A.J.; Maltcev, V.; de Luca, A.. "Markov semigroups, monoids and groups". International Journal of Algebra and Computation 2 (2014): http://www.scopus.com/inward/record.url?eid=2-s2.0-84904274960&partnerID=MN8TOARS.
    10.1142/S021819671450026X
  30. Cain, A.J.; Maltcev, V.. "A simple non-bisimple congruence-free finitely presented monoid". Semigroup Forum (2014): http://www.scopus.com/inward/record.url?eid=2-s2.0-84901888757&partnerID=MN8TOARS.
    10.1007/s00233-014-9607-y
  31. Brough, T.; Cain, A.J.. "Automaton semigroup constructions". Semigroup Forum (2014): http://www.scopus.com/inward/record.url?eid=2-s2.0-84908009618&partnerID=MN8TOARS.
    10.1007/s00233-014-9632-x
  32. Cain, A.J.. "Hyperbolicity of monoids presented by confluent monadic rewriting systems". Beitrage zur Algebra und Geometrie 54 2 (2013): 593-608. http://www.scopus.com/inward/record.url?eid=2-s2.0-84883784128&partnerID=MN8TOARS.
    10.1007/s13366-012-0116-4
  33. Cain, A.J.. "Automatic structures for subsemigroups of Baumslag-Solitar semigroups". Semigroup Forum 87 3 (2013): 537-552. http://www.scopus.com/inward/record.url?eid=2-s2.0-84887989321&partnerID=MN8TOARS.
    10.1007/s00233-013-9490-y
  34. Cain, A.J.; Maltcev, V.. "Context-free rewriting systems and word-hyperbolic structures with uniqueness". International Journal of Algebra and Computation 22 7 (2012): http://www.scopus.com/inward/record.url?eid=2-s2.0-84870620241&partnerID=MN8TOARS.
    10.1142/S0218196712500610
  35. Cain, A.J.; Ruškuc, N.; Thomas, R.M.. "Unary FA-presentable semigroups". International Journal of Algebra and Computation 22 4 (2012): http://www.scopus.com/inward/record.url?eid=2-s2.0-84862547988&partnerID=MN8TOARS.
    10.1142/S0218196712500385
  36. Cain, A.J.; Gray, R.; Ruškuc, N.. "Green index in semigroups: Generators, presentations, and automatic structures". Semigroup Forum 85 3 (2012): 448-476. http://www.scopus.com/inward/record.url?eid=2-s2.0-84871329719&partnerID=MN8TOARS.
    10.1007/s00233-012-9406-2
  37. Cain, A.J.. "Automatic semigroups and Bruck-Reilly extensions". Acta Mathematica Hungarica 126 1 (2010): 1-15. http://www.scopus.com/inward/record.url?eid=2-s2.0-77349105736&partnerID=MN8TOARS.
    10.1007/s10474-009-8063-8
  38. Cain, A.J.; Oliver, G.; Ruškuc, N.; Thomas, R.M.. "Automatic presentations and semigroup constructions". Theory of Computing Systems 47 2 (2010): 568-592. http://www.scopus.com/inward/record.url?eid=2-s2.0-77952952975&partnerID=MN8TOARS.
    10.1007/s00224-009-9216-4
  39. Cain, A.J.. "Deus ex Machina and the Aesthetics of Proof". Mathematical Intelligencer 32 3 (2010): 7-11. http://www.scopus.com/inward/record.url?eid=2-s2.0-77956678650&partnerID=MN8TOARS.
    10.1007/s00283-010-9141-z
  40. Cain, A.J.; Oliver, G.; Ruškuc, N.; Thomas, R.M.. "Automatic presentations for semigroups". Information and Computation 207 11 (2009): 1156-1168. http://www.scopus.com/inward/record.url?eid=2-s2.0-70349754454&partnerID=MN8TOARS.
    10.1016/j.ic.2009.02.005
  41. Cain, A.J.; Maltcev, V.. "Decision problems for finitely presented and one-relation semigroups and monoids". International Journal of Algebra and Computation 19 6 (2009): 747-770. http://www.scopus.com/inward/record.url?eid=2-s2.0-70350306574&partnerID=MN8TOARS.
    10.1142/S0218196709005366
  42. Cain, A.J.. "Automaton semigroups". Theoretical Computer Science 410 47-49 (2009): 5022-5038. http://www.scopus.com/inward/record.url?eid=2-s2.0-78649243772&partnerID=MN8TOARS.
    10.1016/j.tcs.2009.07.054
  43. Cain, A.J.. "Monoids presented by rewriting systems and automatic structures for their submonoids". International Journal of Algebra and Computation 19 6 (2009): 771-790. http://www.scopus.com/inward/record.url?eid=2-s2.0-70350327375&partnerID=MN8TOARS.
    10.1142/S0218196709005317
  44. Cain, A.J.. "Malcev presentations for subsemigroups of direct products of coherent groups". Journal of Pure and Applied Algebra 213 6 (2009): 977-990. http://www.scopus.com/inward/record.url?eid=2-s2.0-60449083962&partnerID=MN8TOARS.
    10.1016/j.jpaa.2008.10.006
  45. Cain, Alan J.. "Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions". Journal of the Australian Mathematical Society 84 1 (2008): 39-61. http://www.scopus.com/inward/record.url?eid=2-s2.0-48349100626&partnerID=MN8TOARS.
    10.1017/S1446788708000086
  46. Cain, A.J.. "A group-embeddable non-automatic semigroup whose universal group is automatic". Glasgow Mathematical Journal 48 2 (2006): 337-342. http://www.scopus.com/inward/record.url?eid=2-s2.0-33747631021&partnerID=MN8TOARS.
    10.1017/S0017089506003107
  47. Cain, A.J.; Robertson, E.F.; Ruškuc, N.. "Subsemigroups of groups: Presentations, Malcev presentations, and automatic structures". Journal of Group Theory 9 3 (2006): 397-426. http://www.scopus.com/inward/record.url?eid=2-s2.0-33646771687&partnerID=MN8TOARS.
    10.1515/JGT.2006.027
  48. Cain, A.J.. "Cancellativity is undecidable for automatic semigroups". Quarterly Journal of Mathematics 57 3 (2006): 285-295. http://www.scopus.com/inward/record.url?eid=2-s2.0-33748769509&partnerID=MN8TOARS.
    10.1093/qmath/hai023
  49. Cain, A.J.; Robertson, E.F.; Ruškuc, N.. "Subsemigroups of virtually free groups: Finite Malcev presentations and testing for freeness". Mathematical Proceedings of the Cambridge Philosophical Society 141 1 (2006): 57-66. http://www.scopus.com/inward/record.url?eid=2-s2.0-33745714195&partnerID=MN8TOARS.
    10.1017/S0305004106009236
Capítulo de livro
  1. Tara Brough; Alan J. Cain; Markus Pfeiffer. "Context-Free Word Problem Semigroups". 2019.
    10.1007/978-3-030-24886-4_22
  2. Cain, Alan J.; Malheiro, António. "Combinatorics of Cyclic Shifts in Plactic, Hypoplactic, Sylvester, and Related Monoids". In Combinatorics on Words: 11th International Conference, Montréal, Canada, September 11–15, 2017,, 190-202. Springer, 2017.
    Publicado • 10.1007/978-3-319-66396-8_18
  3. Cain, Alan J.; Oliver, Graham; Ruškuc, Nik; Thomas, Richard M.. "Automatic Presentations for Cancellative Semigroups". In Language and Automata Theory and Applications, 149-159. Springer, 2008.
    Publicado • 10.1007/978-3-540-88282-4_15
Recurso online
  1. Cain, Alan J.. Autor correspondente: Cain, Alan J.. Form & Number: A History of Mathematical Beauty. 2024. https://archive.org/details/cain_formandnumber_ebook_large.
  2. Cain, Alan J.. An annotated edition, with commentary, of G.H. Hardy's "A Mathematician's Apology". 2019. https://www.archive.org/details/hardy_annotated.
  3. Cain, Alan J.. Nine Chapters on the Semigroup Art. 2012. http://www-groups.mcs.st-and.ac.uk/~alanc/pub/c_semigroups/.
Tese / Dissertação
  1. Cain, Alan J.. "Presentations for Subsemigroups of Groups". Doutoramento, University of St Andrews, 2005. http://www-groups.mcs.st-andrews.ac.uk/~alanc/pub/c_phdthesis.pdf.
Tradução
  1. Cain, Alan J.. An annotated translation of Yves-Marie André's "Essay on Beauty". 2010.
    Publicado
Atividades

Orientação

Título / Tema
Papel desempenhado 		Curso (Tipo)
Instituição / Organização
2021/09/01 - Atual [Undecided; topic is commuting graphs of semigroups and related topics]
Coorientador
 Mathematics (Doutoramento)
Universidade Nova de Lisboa Departamento de Matemática, Portugal
2016/09/01 - 2022/05/20 Algebraic Theory of Quasi-Crystals: A Generalization of the Hypoplactic Monoid and a Littelmann Path Model
Coorientador
 Mathematics (Doutoramento)
Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2017/09/01 - 2022/02/11 Identities and bases in plactic, hypoplactic, sylvester, and related monoids
Coorientador
 Mathematics (Doutoramento)
Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal

Júri de grau académico

Tema
Tipo de participação 		Nome do candidato (Tipo de grau)
Instituição / Organização
2020/12/18 Automaton Structures -- Decision Problems and Structure Theory
Arguente
 Jan Phillipp Wächter (Doutoramento)
Universität Stuttgart Fakultät 5 für Informatik Elektrotechnik und Informationstechnik, Alemanha
2017/07/21 Thesis entitled "Coherent presentation for the hypoplactic monoid of rank n"
Vogal
 Duarte Chambel Ribeiro (Mestrado)
Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal

Curso / Disciplina lecionado

Disciplina Curso (Tipo) Instituição / Organização
2022/01/01 - 2022/06/30 Topics in Semigroups Mathematics (Doutoramento) Universidade Nova de Lisboa Departamento de Matemática, Portugal
2015/09/01 - 2020/01/31 Semigroups, Automata, and Languages Mathematics (Mestrado) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2017/03/01 - 2018/07/31 Error-correcting Codes Information theory (Pós-Graduação) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2016/03/01 - 2016/07/31 Algebraic Theory of Automata Mathematics (Mestrado) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2016/03/01 - 2016/07/31 Rewriting Systems Mathematics (Mestrado) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2016/03/01 - 2016/07/31 Combinatorial Group Theory Mathematics (Curso de doutoramento (conclusão de unidades curriculares)) Universidade Nova de Lisboa Faculdade de Ciências e Tecnologia, Portugal
2012/09/01 - 2013/01/31 Semigroups Mathematics (Mestrado) Universidade do Porto Faculdade de Ciências, Portugal
2005/10/01 - 2007/01/31 Fundamentals of Pure Mathematics Mathematics (Bachelor) University of St Andrews, Reino Unido