Jeroen Schillewaert

Characterizations of Hermitian varieties by intersection numbers

In this paper, we give characterizations of the classical generalized quadrangles H(3, q2) and H(4, q2), embedded in PG(3, q2) and PG(4, q2), respectively. The intersection numbers with lines and planes characterize H(3, q2), and H(4, q2) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.

Characterizations of finite classical polar spaces by intersection numbers with hyperplanes and spaces of codimension 2

In this article we show that non-singular quadrics and non-singular Hermitian varieties are completely characterized by their intersection numbers with respect to hyperplanes and spaces of codimension 2. This strongly generalizes a result by Ferri and Tallini [5] and also provides necessary and sufficient conditions for quasi-quadrics (respectively their Hermitian analogues) to be non-singular quadrics (respectively Hermitian varieties).

Hermitian Veronesean Caps

In [J. Schillewaert and H. Van Maldeghem, Quadric Veronesean caps, Discrete Mathematics], a characterization theorem for Veronesean varieties in \(\mathsf{PG}(N, \mathbb{K})\), with \(\mathbb{K}\) a skewfield, is proved. This result extends the theorem for the finite case proved in [J. A. Thas and H. Van Maldeghem, Quart. J. Math. 55 (2004), 99–113]. In this paper, we prove analogous results for Hermitian varieties, extending the results obtained in the finite case in [B. Cooperstein, J. A. Thas and H. Van Maldeghem, Forum Math. 16 (2004), 365–381] in a non-trivial way.

The 2-Transitive Transplantable Isospectral Drums

In this paper we investigate pairs of Euclidean TI-domains which are isospectral but not congruent. For Riemannian manifolds there are several examples which are isospectral but not isometric, see e.g. J. Milnor [2]. The method we use dates back to T. Sunada [3] considering the problem as a geometric analogue of a method in number theory which uses Dedekind zeta functions. Counter examples to M. Kac’s conjecture so-far can all be constructed by a certain tiling method (“transplantability”) using special linear operator groups which act 2-transitively on certain associated modules. In this talk we show that if any such operator group acts 2-transitively on the associated module, no new counter examples can occur.

A flat Laguerre plane of Kleinewillinghöfer type V

The Kleinewillinghofer types of Laguerre planes reflect the transitivity properties of certain groups of central automorphisms. Polster and Steinke have shown that some of the conceivable types for flat Laguerre planes cannot exist and given models for most of the other types. The existence of only a few types is still in doubt. One of these is type V.A.1, whose existence we prove here. In order to construct our model, we make systematic use of the restrictions imposed by the group. We conjecture that our example belongs to a one-parameter family of planes all of type V.A.1.

Construction and Comparison of Authentication Codes

One of the main problems we want to address is the fact that in the construction theory of authentication codes, the expression “new code” is used all the time, while it is almost never shown that the obtained codes indeed are new, or even what the notion “new” means to begin with. In this paper, we formally define authentication codes and introduce a comparison method for them, using the notion of isomorphism and the invariant isomorphism group. We then define operations in this framework which enable us to “multiply” and “add” arbitrary authentication codes, regardless of how they are initially constructed. We obtain (nonisomorphic) codes with easily calculated parameters, a large number of which are new.

A combinatorial characterization of the Lagrangian Grassmannian LG(3,6)()

We provide a combinatorial characterization of LG(3, 6)(K) using an axiom set which is the natural continuation of the Mazzocca-Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields, Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal-Tits magic square.

Probabilistic constructions in generalized quadrangles

We study random constructions in incidence structures, and illustrate our techniques by means of a well-studied example from finite geometry. A maximal partial ovoid of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical lower bounds on the size of a maximal partial ovoid in a quadrangle of order ( s , t ) are linear in s . In this paper, in a wide class of quadrangles of order ( s , t ) we give a construction of a maximal partial ovoid of size at most s ? polylog ( s ) , which is within a polylogarithmic factor of theoretical lower bounds. The construction substantially improves previous quadratic upper bounds in quadrangles of order ( s , s 2 ) , in particular in the well-studied case of the elliptic quadrics Q - ( 5 , s ) .