Professor Peter Higgins

Email
peteh@essex.ac.uk 
Telephone
+44 (0) 1206 873019

Location
6.317, Colchester Campus
Profile
Biography
I was born in Australia in 1956 and together with my English wife, Theresa, our son and three daughters we have lived in Colchester since 1990. I was promoted to Professor in 2000 andwasHead of the Dept of Mathematical Sciences from January 2005 until December 2010. My research is in algebraic semigroups which is the algebraic type that arises, quite literally, from the stringing together of words and the composition of functions. The former gives rise to connections with the theory of computing and automata while the latter leads to abstract algebra and sometimes to combinatorial problems, that is to say questions concerning counting and enumeration. Like most mathematicians, I try to maintain a wider interest in things mathematical. I have published over 50 research papers and seven books, the first of which was a specialist research monograph, Techniques of Semigroup Theory. I have fourpopularscience books with Oxford University Press, 'Matheamtics for the Curious', Mathematics for the Imagination','Nets, Puzzles and Postmen, an exploration of mathematical connections', (in hardback and paperback) and 'Numbers,A Very Short Introduction', which have been translated into Italian, Spanish, Ukrainian, Chinese, Japanese and Arabic. With Copernicus Press I have the hardback, 'Number Story, from Counting to Cryptography' now translated into Korean. In 2005 I invented 'Circular Sudoku', which now appears in magazines, newspapers and books around the world and also as a handheld computer game and an iPod Ap. The 'Official Book of Circular Sudoku' was published in 2006 by Plume Press in the States and now has translations in Japanese and Russian. In 2011 I was made a Fellow of the Institute of Mathematics and its Applications (F.I.M.A.) In 2013 I was awarded the Premio Peano for the best book on mathematics written in Italian for the translation of 'Nets, Puzzles and Postmen' entitled La matematica dei social network'. To keep uptodate with my mathematical activities, see my Web page: http://www.professorpeterhiggins.com/# PhD Research Students Previous: 1999. Paula Catarino, (Lecturer University of Trasos Montes, Portugal)  'The monoid of orientationpreserving mappings on a chain', external examiner Prof. J.M. Howie (St. Andrews). 1999. Alexei Vernitski (from Ekaterinburg, Russia, Lecturer, DMS)  'Classes of abstract semigroups closed under the formation of subsemigroups and finitary direct products', external examiner, Prof. J.M. Howie (St. Andrews). 2004 Chris Saker (Senior Lecturer, Essex), 'Unavoidable Sets and Aperiodic Unavoidable Sets', External Examiner, Dr Mark Lawson (Bangor) 2011, Imtiaz Ahmad, 'Bandwidth labellings of graphs and their associated semigroups', external examiner, Dr Robert Gray (Lisbon). 2014 Suhear Alwan 'Languages that require full scanning of words to determine membership' external examiner Dr Mark Lawson (HeriotWatt) Research interests: Linked with theSchool of Computer Science and Electronic Engineering. Member of theComputational Combinatorics Group. Member of theCrypto Group. An editor of the international journalCommunications in Algebra. You may also be interested inCircular Sudoku, a new adaptation of the popular Sudoku puzzles.
Qualifications

B.A. (ANU. Canberra, 1976), BSc (First Class, University of Tasmania, 1980), PhD, (Monash University, Melbourne, 1983).

Fellow of the Institute of Mathematics and its Applications, (2011  )
Research and professional activities
Research interests
Algebraic Semigroups (primary)
Formal Language Theory, Combinatorics and Graph Theory (secondary)
Conferences and presentations
Papers in Proceedings of International Conferences
'Semigroup epimorphisms', in P.R. Jones et al. (eds.), Proceedings of the Marquette Conference on Semigroups, 5164, September 1984.
'Random products in semigroups of mappings', in J. Almeida et al. (eds.), 89100, Lattices, Semigroups and Universa Algebra, Plenum Press, New York, 1990.
'On eventually regular semigroups', in J.M. Howie et al. (eds.), 170189, Semigroups with Applications, (Oberwolfach Conference) World Scientific, Singapore, 1992.
'Combinatorial Aspects of Semigroups of Orderpreserving and Decreasing Functions', 103110, Proceedings of the Milano International Conference on Semigroups, June 1992, World Scientific Publishers, 1993.
'Pseudovarieties generated by classes of transformation semigroups', Proc. St. Petersburgh Semigroup Conference Russian State Hydrometeorological Inst., 8594 (1999).
'The Converse of Lallement's Lemma', Proc. St. Andrews Semigroups Conference, World Scientific, 7886, (1998).
'On Special amalgamation Bases', (joint with Kunitaka Shoji, Shimane University, Japan). Proc. St. Andrews Semigroups Conference, World Scientific, 8797, (1998).
Deciding some embeddability problems for semigroups of mappings, (joint with D.H. Fremlin), Proc. Braga Semigroups Conference, World Scientific, 8795 (2000).
The map numbering problem, IEEE transactions, Proc CEEC 2010, 2nd Computer Science and Electronic Engineering Conference, joint with I. Ahmad. Winner of Prize for Best Paper.
Teaching and supervision

Mathematical Skills (MA125)

Mathematics Careers and Employability (MA199)

Complex Variables and Applications (MA302)
Publications
Journals (55)
Higgins, PM. and Alwan, S., (2017). Languages with membership determined by single letter factors. Theoretical Computer Science. 680
Higgins, PM., (2017). Orthodox semigroups and permutation matchings. Semigroup Forum. 94 (1)
HIGGINS, PM., (2017). EMBEDDING IN A FINITE 2GENERATOR SEMIGROUP. Glasgow Mathematical Journal. 59 (01)
Higgins, PM., (2017). THE JOY OF QUIZ. TLSTHE TIMES LITERARY SUPPLEMENT (5966)
Higgins, PM., (2017). THE GCHQ PUZZLE BOOK. TLSTHE TIMES LITERARY SUPPLEMENT (5966)
Higgins, PM., (2017). CAN YOU SOLVE MY PROBLEMS? A casebook of ingenious, perplexing and totally satisfying puzzles. TLSTHE TIMES LITERARY SUPPLEMENT (5966)
Higgins, PM., (2016). THE PERFECT BET How science and maths are taking the luck out of gambling. TLSTHE TIMES LITERARY SUPPLEMENT (5923)
Catarino, P., Higgins, PM. and Levi, I., (2015). On inverse subsemigroups of the semigroup of orientationpreserving or orientationreversing transformations. Algebra and Discrete Mathematics. 19 (2)
Ahmad, I. and Higgins, PM., (2015). On the bandwidth of Mobius graphs. Bulletin of the Iranian Mathematical Society. 41 (3)
Higgins, PM., (2014). Permutations of a semigroup that map to inverses. Semigroup Forum. 89 (1)
Higgins, PM. and Alwan, S., (2013). Languages that require full scanning of words to determine membership. Journal of Automata Languages and Combinatorics. 18 (2)
Higgins, PM., (2012). Burrows–Wheeler transformations and de Bruijn words. Theoretical Computer Science. 457
Higgins, PM., (2012). The product of the idempotents and an $\mathcal{H}$ class of the finite full transformation semigroup. Semigroup Forum. 84 (2)
Higgins, PM., (2011). The Length Of Short Words In Unavoidable Sets. International Journal of Algebra and Computation. 21 (06)
HIGGINS, PM., (2011). THE LENGTH OF SHORT WORDS IN UNAVOIDABLE SETS. International Journal of Algebra and Computation. 21 (06)
Descalço, L. and Higgins, PM., (2010). Generalized Green'S Equivalences on the Subsemigroups of the Bicyclic Monoid. Communications in Algebra. 38 (12)
HIGGINS, PM., (2006). THE SEMIGROUP OF CONJUGATES OF A WORD. International Journal of Algebra and Computation. 16 (06)
Higgins, PM. and Saker, CJ., (2006). Unavoidable sets. Theoretical Computer Science. 359 (13)
HIGGINS, PM., MITCHELL, JD., MORAYNE, M. and RUŠKUC, N., (2006). RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS. Bulletin of the London Mathematical Society. 38 (02)
Higgins, PM., Howie, JM., Mitchell, JD. and Ruškuc, N., (2003). COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS. Proceedings of the Edinburgh Mathematical Society. 46 (3)
HIGGINS, PM., MITCHELL, JD. and RUŠKUC, N., (2003). GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS. Glasgow Mathematical Journal. 45 (3)
Higgins, PM., Howie, JM. and Ruškuc, N., (2003). Set products in transformation semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 133 (05)
Saker, C. and Higgins, P., (2002). Unavoidable Sets of Words of Uniform Length. Information and Computation. 173 (2)
CATARINO, PM. and HIGGINS, PM., (2002). THE PSEUDOVARIETY GENERATED BY ALL SEMIGROUPS OF ORIENTATIONPRESERVING TRANSFORMATIONS ON A FINITE CYCLE. International Journal of Algebra and Computation. 12 (03)
ALMEIDA, J., HIGGINS, PM. and VOLKOV, MV., (2001). THE GAP BETWEEN PARTIAL AND FULL: AN ADDENDUM. International Journal of Algebra and Computation. 11 (01)
Higgins, PM. and Margolis, SW., (2000). Finite aperiodic semigroups with commuting idempotents and generalizations. Israel Journal of Mathematics. 116 (1)
HIGGINS, PM., (2000). A NEW PROOF OF SCHUTZENBERGER'S THEOREM. International Journal of Algebra and Computation. 10 (02)
Higgins, PCP., (1999). The monoid of orientationpreserving mappings on a chain. Semigroup Forum. 58 (2)
Higgins, PM., Howie, JM. and Ruškuc, N., (1998). Generators and factorisations of transformation semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 128 (06)
Higgins, PM., Howie, JM. and Ruškuc, N., (1998). Generators and factorisations of transformation semigroups. Royal Society of Edinburgh  Proceedings A. 128 (6)
Howie, JM., Ruškuc, N. and Higgins, P., (1998). On relative ranks of full transformation semigroups. Communications in Algebra. 26 (3)
Higgins, PM., (1997). A proof of Simon's theorem on piecewise testable languages. Theoretical Computer Science. 178 (12)
Almeida, J. and Higgins, PM., (1997). Monoids RespectingnChains of Intervals. Journal of Algebra. 187 (1)
Higgins, PM., (1996). Epis from locally inverse semigroups are onto. Semigroup Forum. 52 (1)
Higgins, PM., (1995). Embedding semigroups in semibands of depth two. Semigroup Forum. 51 (1)
Higgins, PM., (1994). The Mitsch order on a semigroup. Semigroup Forum. 49 (1)
Higgins, PM., (1994). Idempotent depth in semigroups of orderpreserving mappings. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 124 (05)
Higgins, PM., (1993). A Class of Eventually Regular Semigroups Determined by PseudoRandom Sets. Journal of the London Mathematical Society. s248 (1)
Higgins, PM., (1993). Combinatorial results for semigroups of orderpreserving mappings. Mathematical Proceedings of the Cambridge Philosophical Society. 113 (02)
Higgins, P., (1990). A short proof of Isbell’s zigzag theorem. Pacific Journal of Mathematics. 144 (1)
Higgins, PM., (1990). An algebraic proof that pseudovarieties are defined by pseudoidentities. Algebra Universalis. 27 (4)
Higgins, PM., (1990). Embedding in bisimple semigroups. Semigroup Forum. 40 (1)
Brown, B. and Higgins, PM., (1988). Finite full transformation semigroups as collections of random functions. Glasgow Mathematical Journal. 30 (02)
Higgins, PM., (1988). Digraphs and the semigroup of all functions on a finite set. Glasgow Mathematical Journal. 30 (01)
Higgins, PM., (1988). The range order of a product of i transformations from a finite full transformation semigroup. Semigroup Forum. 37 (1)
Higgins, PM., (1986). Completely semisimple semigroups and epimorphisms. Proceedings of the American Mathematical Society. 96 (3)
Higgins, PM., (1986). Dense subsets of some common classes of semigroup. Semigroup Forum. 34 (1)
Higgins, PM., (1985). Epimorphisms, dominions and semigroups. Algebra Universalis. 21 (23)
Higgins, P., (1985). A new determination of the permutation identities which ensure that a semigroup variety is finitely based. Journal of Pure and Applied Algebra. 38 (1)
Higgins, PM., (1984). Saturated and epimorphically closed varieties of semigroups. Journal of the Australian Mathematical Society. 36 (02)
Higgins, PM., (1984). Epimorphisms, permutation identities and finite semigroups. Semigroup Forum. 29 (1)
Higgins, PM., (1983). A semigroup with an epimorphically embedded subband. Bulletin of the Australian Mathematical Society. 27 (02)
Higgins, PM., (1983). The varieties of commutative semigroups for which epis are onto. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 94 (12)
Higgins, PM., (1983). The determination of all varieties consisting of absolutely closed semigroups. Proceedings of the American Mathematical Society. 87 (3)
Higgins, PM., (1981). Epis are onto for generalised inverse semigroups. Semigroup Forum. 23 (1)
Books (2)
Higgins, PM., (2008).Nets, puzzles and postmen, an exploration of mathematical connections. Oxford University Press. 9780199218424
Higgins, PM., (2008).Number Story, from Counting to Cryptography. Copernicus Press, SpringerVerlag. 9781848000001
Conferences (2)
Higgins, PM., Howie, JM. and Ruškuc, N., (2003). Set products in transformation semigroups
Higgins, PM., (1996). Ramsey's theorem in algebraic semigroups