Assoc Prof Ian Wanless - Researcher Profile

Ian Wanless

Address

School of Mathematical Sciences
Building 28, Clayton

Contact Details

Tel: +61 3 990 54442

Email: Ian.Wanless@monash.edu


Biography

The puzzling mathematician

They say Sudoku puzzles require no mathematics to solve. Australian Mathematical Society 2009 medal winner Dr Ian Wanless disagrees. He views Sudoku as an example of a branch of mathematics called combinatorics. Its outstanding feature is Sunday-afternoon surface simplicity. But mathematical solutions to these puzzles hold surprising conceptual depth and applications in information and communication technology (ICT).

Dr Wanless has always loved solving puzzles and the field of combinatorics mathematics allows him to do just that. The problems he solves resemble those found in puzzle books, including anagrams and crosswords. They tend to involve permutations, combinations and rearrangement of symbols.

Much of the motivation for studying combinatorics comes from ICT and includes applications such as coding mobile phone signals or writing music on a CD. Ultimately, this is where a lot of his work finds application. Personally, he is not interested in applying his solutions.

“I like the problems for their elegance as an intellectual puzzle,” Ian says. The problems he finds most attractive are those that can be stated so simply that a child can understand them. Yet some deep mathematics are needed to arrive at a solution that works for every permutation.

The classic combinatorics example is the four-colour map problem. The problem seems simple. It asks how many colours are needed to colour regions on a map so that no two adjacent regions have the same colour. It was posed in the 19th century but took more than 100 years to solve.

“As mathematicians we are trying to find universal, absolute solutions, with the added bonus that nobody has done it before. You need an inescapable reason why whatever example of a puzzle you tried, the solution you propose would always work.

“A mathematical argument has to be general and cover every contingency,” he says. “That is the idea of proof and it is very important. Certain things that appear to be obvious can be extremely difficult to prove.”

His current favourite questions deal with Latin squares. A finished Sudoku puzzle is an example of a Latin square. These are problems in which symbols fit in a matrix and each symbol occurs once in each row and column.

“Latin squares arise in the design of statistical experiments in agriculture and psychology,” Ian says. “They are used to decide in which order to do experiments and they are useful in designing code for communication. I’m interested in basic properties like how many there are. Again, they are problems you can explain to a child yet some tricky math is needed to solve them.”

Contrary to stereotypes of the lone wolf intellectual, Ian enjoys collaborating. He has a dozen projects in the works, many involving colleagues in the US, Europe and Asia.

“I’m very much of the opinion that mathematics is a team sport,” he says. “I do write papers on my own. But it is more fun to collaborate. A lot of the best projects come from cross-fertilisation from different branches, as is the case in all science. You get surprising results when working between fields rather than in a field.”

Keywords

Latin squares, combinatorial designs, combinatorial enumeration, matrix permanents, Latin squares, graph theory

Qualifications

DOCTOR OF PHILOSOPHY IN MATHEMATICS
Institution: Australian National University
Year awarded: 1998
BACHELOR OF SCIENCE
Institution: Australian National University
Year awarded: 1992

Publications

Book Chapters

Wanless, I., 2011, Transversals in latin squares: a survey, in Surveys in Combinatorics 2011, eds Robin Chapman, Cambridge University Press, New York, pp. 403-437.

Colbourn, C.J., Dinitz, J.H., Wanless, I.M., 2007, Latin squares, in Handbook of Combinatorial Designs, eds C J Colbourn and J H Dinitz, Chapman & Hall, Boca Raton USA, pp. 135-152.

Wanless, I.M., 2007, Permanents, in Handbook of Linear Algebra, eds L Hogben, Chapman & Hall, Boca Raton USA, pp. 31.1-31.15.

Journal Articles

Stones, D.S., Wanless, I.M., 2012, A congruence connecting Latin rectangles and partial orthomorphisms, Annals of Combinatorics [P], vol 16, issue 2, Birkhauser Verlag AG, Basel Switzerland, pp. 349-365.

Barat, J., Wanless, I.M., 2012, A cube dismantling problem related to bootstrap percolation, Journal Of Statistical Physics [P], vol 149, issue 4, Springer, New York USA, pp. 754-770.

Vojtechovsky, P., Wanless, I.M., 2012, Closest multiplication tables of groups, Journal Of Algebra [P], vol 353, issue 1, Elsevier Science, San Diego USA, pp. 261-285.

Stones, D.S., Vojtechovsky, P., Wanless, I.M., 2012, Cycle structure of autotopisms of quasigroups and Latin squares, Journal of Combinatorial Designs [P], vol 20, issue 5, Wiley Periodicals, Malden MA USA, pp. 227-263.

Stones, D., Wanless, I., 2012, How not to prove the Alon-Tarsi conjecture, Nagoya Mathematical Journal [P], vol 205, Duke University Press, Durham NC USA, pp. 1-24.

Egan, J., Wanless, I., 2012, Latin squares with restricted transversals, Journal of Combinatorial Designs [P], vol 20, issue 2, Wiley Periodicals, USA, pp. 124-141.

Bryant, D.E., Cavenagh, N., Maenhaut, B.M., Pula, K., Wanless, I.M., 2012, Nonextendible Latin cuboids, Siam Journal On Discrete Mathematics [P], vol 26, issue 1, Siam Publications, Philadelphia PA USA, pp. 239-249.

Cheon, G., Wanless, I., 2012, Some results towards the Dittert conjecture on permanents, Linear Algebra and Its Applications [P], vol 436, issue 4, Elsevier Science, New York NY USA, pp. 791-801.

Brankovic, L., Wanless, I., 2011, Graceful labelling: State of the art, applications and future directions, Mathematics in Computer Science [E], vol 5, issue 1, Birkhaeuser Verlag AG, Switzerland, pp. 11-20.

Egan, J.A., Wanless, I.M., 2011, Indivisible partitions of latin squares, Journal of Statistical Planning and Inference [P], vol 141, issue 1, Elsvier BV, Netherlands, pp. 402-417.

Danziger, P., Wanless, I., Webb, B., 2011, Monogamous latin squares, Journal of Combinatorial Theory - Series A [P], vol 118, issue 3, Academic Press Inc Elsevier Science, USA, pp. 796-807.

Brouwer, A., Wanless, I., 2011, Universally noncommutative loops, Bulletin of the Institute of Combinatorics and its Applications [P], vol 61, The Institute of Combinatorics & Its Applications (I C A), Canada, pp. 113-115.

Stones, D.S., Wanless, I.M., 2010, Compound orthomorphisms of the cyclic group, Finite Fields And Their Applications [P], issue 4, Academic Press Inc Elsevier Science, San Diego CA USA, pp. 277-289.

Wanless, I.M., 2010, Counting matchings and tree-like walks in regular graphs, Combinatorics Probability & Computing [P], vol 19, issue 3, Cambridge University Press, New York USA, pp. 463-480.

Stones, D.S., Wanless, I.M., 2010, Divisors of the number of Latin rectangles, Journal of Combinatorial Theory - Series A [P], vol 117, Elsevier Science, San Diego CA USA, pp. 204-215.

Pula, K., Song, S.-., Wanless, I.M., 2010, Minimum permanents on two faces of the polytope of doubly stochastic matrices, Linear Algebra and Its Applications [P], vol 434, issue 1, Elsevier Science Inc, New York USA, pp. 232-238.

Cavenagh, N., Wanless, I.M., 2010, On the number of transversals in Cayley tables of cyclic groups, Discrete Applied Mathematics [P], vol 158, Elsevier BV, Netherlands, pp. 136-146.

Browning, J.M., Vojtechovsky, P., Wanless, I.M., 2010, Overlapping latin subsquares and full products, Commentationes Mathematicae Universitatis Carolinae [P], vol 51, issue 2, Univerzita Karlova v Praze, Praha Czech Republic, pp. 175-184.

Bryant, D.E., Egan, J.A., Maenhaut, B.M., Wanless, I.M., 2009, Indivisible plexes in latin squares, Designs Codes And Cryptography [P], vol 52, issue 1, Springer, Netherlands, pp. 93-105.

Cavenagh, N., Wanless, I.M., 2009, Latin trades in groups defined on planar triangulations, Journal of Algebraic Combinatorics, vol 30, Springer New York LLC, USA, pp. 323-347.

Lieby, P., McKay, B.D., McLeod, J.C., Wanless, I.M., 2009, Subgraphs of random k-edge-coloured k-regular graphs, Combinatorics Probability & Computing [P], vol 18, issue 4, Cambridge University Press, New York USA, pp. 533-549.

Bryant, D.E., Buchanan, M., Wanless, I.M., 2009, The spectrum for quasigroups with cyclic automorphisms and additional symmetries, Discrete Mathematics, vol 309, issue 4, Elsevier BV, Netherlands, pp. 821-833.

McKay, B.D., Wanless, I.M., 2008, A census of small latin hypercubes, Siam Journal on Discrete Mathematics, vol 22, issue 2, Society for Industrial and Applied Mathematics, USA, pp. 719-735.

Egan, J.A., Wanless, I.M., 2008, Latin squares with no small odd plexes, Journal of Combinatorial Designs, vol 16, issue 6, John Wiley & Sons Inc, USA, pp. 477-492.

Wanless, I.M., 2008, On the Brualdi-Liu conjecture for the even permanent, Electronic Journal of Linear Algebra, vol 17, issue 1, International Linear Algebra Society, USA, pp. 284-286.

Cavenagh, N., Greenhill, C.S., Wanless, I.M., 2008, The cycle structure of two rows in a random latin square, Random Structures and Algorithms, vol 33, issue 3, John Wiley & Sons Inc, USA, pp. 286-309.

Wanless, I.M., 2007, A computer enumeration of small latin trades, Australasian Journal of Combinatorics, vol 39, Centre for Discrete Mathematics & Computing, Qld Australia, pp. 247-258.

Cheon, G., Wanless, I.M., 2007, An interpretation of the Dittert conjecture in terms of semi-matchings, Discrete Mathematics, vol 307, issue 21, Elsevier BV, Netherlands, pp. 2501-2507.

Wanless, I.M., 2007, On Minc's sixth conjecture, Linear & Multilinear Algebra, vol 55, issue 1, Taylor & Francis Ltd, Oxon UK, pp. 57-63.

Maenhaut, B., Wanless, I.M., Webb, B.S., 2007, Subsquare-free latin squares of odd order, European Journal of Combinatorics, vol 28, Academic Press, UK, pp. 322-336.

Wanless, I.M., 2007, Transversals in latin squares, Quasigroups and Related Systems, vol 15, issue 1, Academia de Stiinte a Moldovej, Moldova Poland, pp. 169-190.

Wanless, I.M., 2006, Addendum to Schrijver's work on minimum permanents, Combinatorica, vol 26, Springer, Germany, pp. 743-745.

Taylor, S., Wanless, I.M., Boland, N.L., 2006, Distance domination and amplifier placement problems, Australasian Journal of Combinatorics, vol 34, Centre for Discrete Mathematics & Computing, Australia, pp. 117-136.

Bryant, D.E., Maenhaut, B., Wanless, I.M., 2006, New families of atomic latin squares and perfect 1-factorisations, Journal of Combinatorial Theory Series A, vol 113, issue 4, Academic Press Inc Elsevier Science, San Diego USA, pp. 608-624.

Wanless, I.M., Webb, B.S., 2006, The existence of latin squares without orthogonal mates, Designs Codes and Cryptography, vol 40, issue 1, Springer, Dordrecht Netherland, pp. 131-135.

McKay, B.D., McLeod, J.C., Wanless, I.M., 2006, The number of transversals in a Latin square, Designs Codes and Cryptography, vol 40, issue 3, Springer, Dordrecht Netherland, pp. 269-284.

Cheon, G., Wanless, I.M., 2005, An update on Minc's survey of open problems involving permanents, Linear Algebra and its Applications, vol 403, Elsevier, USA, pp. 314-342.

Wanless, I.M., 2005, Atomic Latin Squares based on Cyclotomic Orthomorphisms, Electronic Journal Of Combinatorics, vol 12, issue R22, Electronic Journal of Combinatorics, USA, pp. 1-23.

Cameron, P.J., Wanless, I.M., 2005, Covering radius for sets of permutations, Discrete Mathmatics, vol 293, Elsevier B.V., Netherlands, pp. 91-109.

McKay, B.D., Wanless, I.M., 2005, On the number of Latin squares, Annals of Combinatorics, vol 9, pp. 335-344.

Wanless, I.M., 2005, Permanents of matrices of signed ones, Linear and Multilinear Algebra, vol 53, issue 6, Taylor & Francis Group Ltd, UK, pp. 427-433.

Wanless, I.M., Ihrig, E.C., 2005, Symmetries that Latin squares inherit from 1-factorizations, Journal of Combinatorial Designs, vol 13, issue 1, John Wiley & Sons Inc, USA, pp. 157-172.

Wanless, I.M., 2004, A partial latin squares problem posed by Blackburn, Bulletin of the ICA, vol 42, The Institute of Combinatorics and its Applications, Canada, pp. 76-80.

McKay, B.D., Oggier, F.E., Royle, G.F., Sloane, N.J.A., Wanless, I.M., Wilf, H.S., 2004, Acyclic Digraphs and Eigenvalues of (o, 1)-Matrices, Journal of Integer Sequences, vol 7, University of Waterloo, Canada, pp. 1-5.

Maenhaut, B.M., Wanless, I.M., 2004, Atomic Latin Squares of Order Eleven, Journal of Combinatorial Designs, vol 12, John Wiley & Sons, Hoboken New Jersey USA, pp. 12-33.

Wanless, I.M., 2004, Cycle switches in Latin squares, Graphs and Combinatorics [P], vol 20, Springer-Verlag, Japan, pp. 545-570.

Wanless, I.M., 2004, Diagonally cyclic latin squares, European Journal of Combinatorics, vol 25, Elsevier Ltd, UK, pp. 393-413.

Wanless, I.M., 2003, A lower bound on the maximum permanent in Lambda (K,n), Linear Algebra and its Applications, vol 373, Elsevier Inc, USA, pp. 153-167.

Wanless, I.M., Vaughan-Lee, M., 2003, Latin squares and the Hall-Paige conjecture, London Mathematical Society Bulletin, vol 35, Oxford University Press, UK, pp. 191-195.

Bryant, D.E., Maenhaut, B.M., Wanless, I.M., 2002, A family of perfect factorisations of complete bipartite graphs, Journal of Combinatorial Theory Series A, vol 98, issue 1, Elsevier, USA, pp. 328-342.

Wanless, I.M., 2002, A generalisation of transversals for Latin Squares, Electronic Journal Of Combinatorics, vol 9, issue R12, Electronic Journal of Combinatorics, USA, pp. 1-15.

McKay, B.D., Wanless, I.M., Wormald, N.C., 2002, Asymptotic enumeration of graphs with a given upper bound on the maximum degree, Combinatorics, Probability and Computing, vol 11, issue 1, Cambridge University Press, UK, pp. 373-392.

Wanless, I.M., 2001, Answers to Questions by Denes on Latin power Sets, European Journal of Combinatorics, vol 22, Academic Press, UK, pp. 1009-1020.

Gao, Z.J., Wanless, I.M., Wormald, N.C., 2001, Counting 5-connected planar triangulations, Journal of Graph Theory, vol 38, John Wiley & Sons, Inc, pp. 18-35.

Wanless, I.M., 2001, Latin squares with one subsquare, Journal of Combinatorial Designs, vol 9, John Wiley & Sons, Hoboken New Jersey USA, pp. 128-145.

Wanless, I.M., 2001, On McLeish's construction for latin squares without intercalates, Ars Combinatoria, vol 58, Charles Babbage Research Centre, Winnipeg Canada, pp. 313-317.

Wanless, I.M., 2001, Path achievement games, Australasian Journal of Combinatorics, vol 23, Centre for Discrete Mathematics & Computing, Australia, pp. 9-18.

Wanless, I.M., Wormald, N.C., 2001, Regular Graphs with No Homomorphisms onto Cycles, Journal of Combinatorial Theory Series B, vol 82, Academic Press, USA, pp. 155-166.

Postgraduate Research Supervisions

Current Supervision

Program of Study:
(DOCTORATE BY RESEARCH).
Thesis Title:
Adding numbers to RAMs.
Supervisors:
Farr, G (Main), Wanless, I (Associate).
Program of Study:
(DOCTORATE BY RESEARCH).
Thesis Title:
Autoparatopisms of Quasigroups.
Supervisors:
Wanless, I (Main), Horsley, D (Associate).
Program of Study:
(DOCTORATE BY RESEARCH).
Thesis Title:
Cyclotomic Orthomorphisms.
Supervisors:
Wanless, I (Main), Horsley, D (Associate).
Program of Study:
(DOCTORATE BY RESEARCH).
Thesis Title:
The search for connected regular integral graphs.
Supervisors:
Wanless, I (Main), Horsley, D (Associate).

Completed Supervision

Student:
Egan, J.
Program of Study:
Transversals, indivisible plexes and partitions of latin squares. (PHD) 2010.
Supervisors:
Wanless, I (Main), Farr, G (Associate).
Student:
Smith, L.
Program of Study:
Controlling Selmer groups over Affinoid algebras. (Masters) 2012.
Supervisors:
Delbourgo, D (Main), Wanless, I (Associate).
Student:
Stones, D.
Program of Study:
On the number of Latin rectangles. (PHD) 2009.
Supervisors:
Wanless, I (Main), Farr, G (Associate).