Koç University, Mathematics Seminar

Date & Time: Tuesday, December 11, 17:30-18:30

Place: SCI-103

Speaker: Michel Lavrauw (University Padova)

Title: A geometric construction of finite non-assciative division algebras

Abstract: Shortly after the classification of finite fields by E. H. Moore, announced in 1893 at the International Mathematical Congres in Chicago, mathematicians turned their attention to algebraic systems (K,+,•), satisfying a slightly weaker set of axioms compared to a finite field. For instance, L. E. Dickson and J. Wedderburn studied finite domains, omitting the axiom: (K*,•) is commutative. In 1905 they proved that there is no such thing as a non-commutative finite domain. This is now known as Wedderburn's little theorem. Similarly, Dickson studied algebraic systems (K,+,•) omitting the axiom of associativity (K,+,•) as well: non-associative division algebras. In this case however there do exist proper examples, i.e. which are not fields. This study was continued by his student A. A. Albert, and picked up shortly after by many other mathematicians, including E. Artin, M. A. Zorn, L. A. Skornyakov, R. H. Bruck, D. E. Knuth, E. Kleinfield, I. Kaplansky, to name but a few. Nowadays these algebras are called semifields (a term introduced by D. E. Knuth in 1965), and semifields are studied up to isotopisms instead of isomorphisms. We will explain the BEL-configuration from [1] and report on the structural consequences on the set of isotopism classes of semifields from [2].

[1] S. Ball, G. Ebert, M. Lavrauw. A geometric construction of finite semifields. J. Algebra (2007)

[2] M. Lavrauw and J. Sheekey. On BEL-configurations and finite semifields. arXiv1402.2486

About the Speaker: Michel Lavrauw (Izegem, Belgium, 1974) obtained his degree in Mathematics from Ghent University, Belgium, with a master thesis on Knot Theory, supervised by Wim Mielants. He worked on a PhD in the field of Combinatorics, more specific Finite Geometry, at Eindhoven University of Technology, supervised by Aart Blokhuis and Andries E. Brouwer. In 2000, he spent six months at the University of Western Australia on a research visit working on the theory of translation generalised quadrangles in collaboration with Tim Penttila. After obtaining his PhD in 2001, he was awarded a two-year Marie Curie Individual Fellowship from the European Commission, at Naples University, with Guglielmo Lunardon as scientist in charge. After this position, he was employed by Barcelona's Technical University of Catalonia (UPC) to collaborate with the research group at the Department of Applied Mathematics within a European Commission-funded Research Training Network: Combinatorial Structure of Intractable Problems (COMBSTRU). In 2004, he was awarded a VENI grant, funded by NWO’s Innovational Research Incentives Scheme (Netherlands Organisation for Scientific Research) for his project on Semifields and related structures in finite geometry, with Eindhoven University of Technology as host institution. He moved to Ghent in September 2006 to collaborate on a project on Incidence Geometry at the Department of Mathematics at Ghent University. In 2008 he got awarded a postdoctoral research fellowship from FWO (Organisation for Scientific Research - Flanders) with a project on Finite Semifields, affiliated with Ghent University and Vrije Universiteit Brussel. He is currently working as an associate professor at the University of Padova, Italy. From September 2014 until March 2015 he is a visiting professor with Tübitak, affiliated with Sabanci University, Istanbul. His research is focussed on Finite Geometry and Algebra. He is one of the authors of the computer algebra package FinInG, a GAP package for Finite Incidence Geometry (http://cage.ugent.be/fining). Information on relevant conferences and events, as well as a download link for his publications can be found from his webpage: <http://cage.ugent.be/~ml>.

Michel Lavrauw's visit to Sabanci University is supported by TÜBİTAK 2221 - Fellowships for Visiting Scientists Program.