Michel Lavrauw


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RESEARCH

My research is focussed on the areas of combinatorics, finite geometry, incidence geometry, finite fields and coding theory. More specifically, finite semifields (non-associative division algebras), projective planes, translation generalised quadrangles (equivalent to the theory of eggs in projective spaces), polynomial techniques in finite geometry, blocking sets, linear sets and the links between incidence geometry and coding theory.



PUBLICATIONS |
FinInG | 


PROJECTS

`publications`

Preprints are available for most of my published research articles. These preprints may differ from the published version (for which I have signed a copyright agreement). In case you want a reprint of a published article (e.g. for references), you can download the publication from the journal website, or contact me.

PhD thesis

Scattered spaces with respect to spreads and eggs in finite projective spaces. pdf

research articles

Linear (q+1)-fold blocking sets in PG(2,q⁴), with Simeon Ball and Aart Blokhuis, Finite Fields Appl. 6 (2000), no. 4, 294--301. preprint
Scattered spaces with respect to a spread in PG(n,q), with Aart Blokhuis, Geom. Dedicata 81 (2000), no. 1-3, 231--243. preprint
On eggs and translation generalised quadrangles, with Tim Penttila, J. Combin. Theory Ser. A 96 (2001), no. 2, 303--315. preprint
On two-intersection sets with respect to hyperplanes in projective spaces, with Aart Blokhuis, J. Combin. Theory Ser. A 99 (2002), no. 2, 377--382. preprint
Commutative semifields of rank 2 over their middle nucleus, with Simeon Ball, Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Mullen, G. L. (ed) et al, Springer-Verlag, 1--21, 2002. preprint
On the classification of semifield flocks, with Simeon Ball and Aart Blokhuis, Adv. Math. 180 (2003) 104--111. preprint
Eggs in PG(4n-1,q), q even, containing a pseudo-conic, with M. R. Brown, Bull. London Math. Soc. 36 (2004), no. 5, 633--639. preprint
Symplectic Spreads, with S. Ball, J. Bamberg and T. Penttila, Des. Codes Cryptogr. 33 (2004) 9--14. preprint
Eggs in PG(4n-1,q), q even, containing a pseudo pointed conic, with M. R. Brown, European J. Combin. 26 (2005), no.1, 117--128. preprint
Good Eggs and Veronese Varieties, with G. Lunardon, Discrete Math. 294 (2005), no. 1-2, 119--122. preprint
Semifield flocks, eggs, and ovoids of Q(4,q), Adv. Geom. 5 (2005), no. 3, 333--345. preprint
BLT-sets admitting the symmetric group S_4, with M. Law, Australasian J. Combinatorics, 33 (2005), 307--316. preprint
Characterizations and properties of good eggs in PG(4n-1,q), q odd, Discrete Math. 301, (2005), 106--116. preprint
The two sets of three semifields associated with a semifield flock. Innov. Incidence Geom. 2 (2005), 101--107. preprint
Sublines of prime order contained in the set of internal points of a conic. Des. Codes Cryptogr. 38 (2006), no. 1, 113--123. preprint
On the graph of a function in two variables over a finite field, with S. Ball, J. Algebraic Combin. 23 (2006) 243--253. preprint
How to use Redei polynomials in higher dimensional spaces, with S. Ball, Le Matematiche (Catania), 59 (2004), 39--52 (2006). preprint
A geometric construction of finite semifields. J. Algebra 311 (2007), no. 1, 117--129 (with G. Ebert and S. Ball). preprint
On the Hughes-Kleinfeld and Knuth's semifields two dimensional over a weak nucleus. Des. Codes Cryptogr. 44 (2007), no. 1-3, 63--67 (with S. Ball). preprint
On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual. Des. Codes Cryptogr. 48 (2008) 231--245 (with L. Storme and G. Van de Voorde). preprint
On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual. Finite Fields and Appl. 14 (2008) 1020--1038 (with L. Storme and G. Van de Voorde). preprint
On the isotopism classes of finite semifields. Finite Fields Appl. 14 (2008) 897--910. preprint
An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n,q). J. Combin. Theory Ser. A (available online 29 January 2009) (with L. Storme, P. Sziklai and G. Van de Voorde). preprint
A proof of the linearity conjecture for k-blockings sets in PG(n,q^3), q prime (with L. Storme and G. Van de Voorde). To appear in J. Combin. Theory Ser. A preprint
On linear sets on a projective line (with G. Van de Voorde). To appear in Des. Codes Cryptogr. preprint
Finite semifields with a large nucleus and higher secant varieties to Segre varieties. To appear in Adv. Geom. preprint
Codes from projective spaces (with G. Van de Voorde en L. Storme). To appear in Proceedings of the Conference on Error-Correcting Codes, Cryptography and Finite Geometries, AMS book seriess. preprint
F_q-pseudoreguli of PG(3,q^3) and scattered semifields of order q^6 (with G. Marino, O. Polverino and R. Trombetti). To appear in Finite Fields Appl. preprint
Semifields from skew polynomial rings (with J. Sheekey). preprint
On the elation group of the dual of the H(4,q^2) generalised quadrangle (with J. Bamberg and T . Penttila). In preparation.

book

Scattered Spaces and Eggs in Finite Projective Spaces, Topics in Finite Geometry
VDM Verlag Dr. Muller Aktiengesellschaft & Co. KG (2008) ISBN-10: 363906397X, ISBN-13: 978-3 639063974 (Available from amazon)

chapter in book

Finite Semifields (with Olga Polverino) preprint
Chapter to appear in Current Research Topics in Galois Geometries (NOVA collected works)

`FinInG`

FinInG is a GAP package that provides users with the basic tools to work in various areas of finite incidence geometry, from the realms of projective spaces to the flat lands of generalised polygons. It uses the algebraic power of GAP, particularly its implementation of matrix and permutation groups. If you would like to try the package, don't hesitate to contact me, or any other member of the development team. Examples and documentation can be found here.

`projects`

Here is a list of fellowships and grants I have been awarded:

Current projects:

My research project (postdoctoraal onderzoeksmandaat), funded by FWO: Finite Semifields.

The project is situated in two domains. On the one hand, in the theory of non-associative algebras, and on the other hand in finite geometry. A finite semifield (also called division algebra or distributive quasifield in the earlier literature) is a finite dimensional, distributive, algebra (associativity is not assumed) over a finite field with a unit and without zero-divisors. From now on, the term semifield will refer to a finite semifield. If the existence of a unit is not required, then we speak of presemifields.

The study of semifields was initiated about a century ago by Dickson (1905-1906), shortly after the classification of finite fields in 1893, [Dickson, Leonard Eugene Linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc. 7 (1906), no. 3, 370--390] and [Dickson, L. E.; On Linear Algebras. Amer. Math. Monthly 13 (1906), no. 11, 201--205]. It might have been Dickson's intention to prove that they must be fields, in the light of Wedderburn's theorem for finite skew-fields. But instead of proving a result in that direction, Dickson constructed the first examples of semifields that are not fields, and the next characterisation result for finite fields is the Artin-Zorn theorem, which says that finite alternative semifields must be fields (this is in fact a special case of the Bruck-Kleinfeld-Skornyakov theorem). Dickson's student A. A. Albert extended this result to power-associative semifields. The proofs of the above results are all from a purely algebraic point of view.

By 1960, the relevance of semifields in the theory of translation planes was well known (e.g. [Dembowski1968]) and Albert proved that isomorphic planes correspond to isotopic semifields . A good survey of the state of the art on semifields up to 1965 is Knuth's paper: Finite semifields and projective planes. Since then, it has been shown that semifields are related to a large number of interesting geometric structures besides translation planes, such as flocks of a quadratic cone, spreads, ovoids and BLT sets of polar spaces, blocking sets, translation generalised quadrangles and eggs in projective spaces; they play a key role in finite geometry. By now, the theory of semifields is a vast area in mathematics and, as shown by the number of publications in the area and the diversity of the journals in which these appear, there seems to be an ever-increasing interest in the subject. This is no doubt stimulated by the numerous connections with structures in finite geometry, as well as the applications in the theory of finite fields and cryptology.

It is this interdisciplinary aspect of semifield theory, connecting algebra and geometry, that we want to exploit to achieve significant progress in the classification of finite semifields. The overall aim is to obtain constructions, characterisations and classification results with regard to finite semifields and related structures in finite geometry. We also want to provide a classification system for finite semifields, including a set of new invariants of the isotopism class of a semifield, and design deterministic methods in order to ascertain whether two semifields are isotopic or not. More ambitiously, we seek to find a canonic representative (complete invariant) for the isotopy class of a semifield. Further goals of the research project involve structures in finite geometry related to semifields, with an emphasis on translation generalised quadrangles (TGQ) and BLT sets of polar spaces.

Travel grant from NWO (for a period of six months) to visit the University of Western Australia (December 1999 - June 2000)
Marie Curie Fellowship withing the 5th framework, (for the period of two years) with as host institution Naples University (awarded July 2001)
Veni grant (for a period of three years) from NWO, with Eindhoven University of Technology as a host institution (awarded July 28 2004).
BOF research grant (for a period of three years) from Ghent University (awarded June 2007).
Travel grant from FWO (for a period of 6 weeks) to visit Naples University (January 19, 2008 - March 1, 2008)
PhD grant (aspirantenmandaat) for Geertrui Van de Voorde together with Leo Storme, supported by FWO: Blocking sets in finite projective spaces and in coding theory (PhD awarded April 2010).