Geometry @ Ghent University

The research group Incidence Geometry is an interdepartemental group with members from the department of Mathematics: Algebra and Geometry (WE01) and the department of Mathematics: Analysis, Logic and Discrete Mathematics (WE16).

Members of our research group

Tenured

Postdocs

PhD students

Current research topics


Our main research interests are, amongst others, finite geometry and its applications; the theory of Tits-buildings, that is, the natural combinatorial objects of semisimple algebraic groups and their classical and mixed relatives.


Graph Theory

Spectral Graph Theory

Spectral graph theory associates a matrix to a graph and then seeks to deduce from the eigenvalues and eigenvectors of this matrix structural properties about the graph. In particular, one is interested in how far combinatorial properties are characterized by the spectrum. This gives rise to difficult but important problems that are also useful in other areas such as computer science, chemistry, and molecular biology.

Distance Regular Graphs and families of subspaces

For many geometrical and combinatorial structures there is a corresponding distance-regular graph. A lot of combinatorial information about these structures is encoded in these graphs, and hence they are a key tool in algebraic combinatorics. We investigate several families of subspaces in projective and polar spaces, such as t-intersecting sets (in particular Erdős-Ko-Rado sets) and Cameron-Liebler sets, that are interesting vertex sets in the corresponding distance-regular graph. E.g., an Erdős-Ko-Rado set in PG(n,q) is a set of subspaces such that no pair of the corresponding vertices is at maximal distance in the Grassmann graph.


Galois Geometry

Substructures in finite projective and polar spaces

We investigate families of point sets in finite projective and polar spaces that are defined through their intersection with lines or hyperplanes, or via collinearity. Classical examples are arcs, hyperovals, maximal arcs, caps, blocking sets, Korchmaros-Mazzocca arcs (KM-arcs), tight sets, ovoids, etc. We also investigate linear sets, which are an important tool for the construction of blocking sets and KM-arcs amongst others. The main goals are to construct new examples and to classify these objects (given a condition such as size or automorphism group).

Finite geometry and its relations to coding theory

Coding theory and finite geometry are two research areas which have many links. Many problems in coding theory have an equivalent geometrical problem. These links include: linear MDS codes and arcs, linear codes meeting the Griesmer bound and minihypers, covering radius of linear codes and saturating sets, codes arising from incidence matrices, and more recent links such as subspace codes, MRD codes and distributed storage. By studying the equivalent geometrical problems, new contributions to finite geometry and to coding theory are made.


Generalised polygons

Covers of quadrangles (and epimorphisms of polygons)

Generalized polygons are rank 2 buildings. We study covers of generalized quadrangles, in particular Kantor-Knuth generalized quadrangles. A key role is played by the ovoids of these generalized quadrangles. Also epimorphisms between generalized polygons turn up. In particular epimorphisms of thick generalized polygons onto thin poygons are investigated.

Infinite Moufang conditions

Moufang conditions arise naturally as fundamental group theoretical properties which live in higher dimensional buildings, but which have to be endowed on lower dimensional objects (because those objects can easily exist without Moufang properties). Besides global versions, research on local versions has the advantage that the properties occur in much more classes of examples, so that it could hand stronger tools to study global Moufang conditions. In the infinite case, such local versions are much harder to understand and control than in the finite case, and as such they present possibly very rewarding but hard problems to study.

Near polygons

We wish to obtain classification and characterization results for various classes of near polygons and generalized polygons. Also various substructures in these geometries are investigated, like hyperovals, m-ovoids, tight sets, distance-j-ovoids, hyperplanes, pseudo-hyperplanes and valuations.


Theory of buildings

We investigate geometric patterns in the Freudenthal-Tits magic square (FTMS). The latter is a 4x4 array comprising a remarkable set of (representations of) Lie algebras or buildings obtained using a pair of quadratic alternative algebras over an arbitrary field. The last row and column of the square contain exceptional spherical buildings of type F4, E6, E7 and E8. The FTMS builds up to these via classical groups/buildings since each entry is related to the one on its right and the one below by containment in some sense; and culminates in type E8.
Many of the following topics are linked, either directly or indirectly, to this square.

Geometric characterisation and constructions of the varieties of the FTMS

A vaste generalisation of Mazzocca and Melone's axioms for finite quadric Veronesean varieties seems to single out the varieties of the magic square among infinite families of possibly similar geometric objects. In a series of papers, we have established a geometric characterisation in this spirit for the second and third row, and we now aim at a characterisation for the fourth row.


One of our other targets is alternative, more geometric and explicit constructions for the buildings of type E6, E7, E8 and F4. Using the constructive nature of the FTMS, we developed such constructions and use these to solve some longer-standing open problems.

Parapolar spaces

Parapolar spaces, axiomatically defined point-line geometries equipped with polar spaces as convex substructures, were designed to capture the behaviour of the exceptional spherical buildings, and form a rich theory by themselves. We are looking for additional properties that are satisfied by the parapolar spaces related to buildings but not by too many others. Surprisingly, some of these partial classifications results again single out a large subset of the FTMS.

Automorphisms of (exceptional) buildings

We study automorphisms of (exceptional) buildings through the notion of an opposition diagram, which encodes the displacement properties of the given automorphism. We classify automorphisms with given opposition diagram. Again, there is a neat connection with the FTMS, as the linear analogues of the Galois descent corresponding with the Tits diagram of any cell have as opposition diagram precisely the Tits diagram of the cell lying symmetrically with respect to the main diagonal.

Twin buildings

We are involved in the classification of twin buildings, a rather extensive and deep project initiated and coordinated by Bernhard Mühlherr. The last cases are being resolved now, and one of them is de case of a twin building of type C2-tilde. The local structure versus the building at infinity provides an interesting interplay between exceptional and classical Moufang quadrangles. The existence part requires to write down explicit RGD-systems. All work is in progress here.

  • B. Mühlherr, H. Van Maldeghem, Affine twin buildings of type C2-tilde, in progress.

Lie incidence geometries and their embeddings

Projective representations and (pseudo-)hyperplanes

A projective embedding of a point-line geometry is a representation of the geometry as a (full) subgeometry of a projective space. We wish to obtain classification results for projective embeddings of certain geometries like dual polar spaces. In particular, we wish to find the universal embeddings of these geometries. Also the close relationship between projective embeddings and hyperplanes of geometries is under investigation, as well as some recent generalizations to pseudo-embeddings and pseudo-hyperplanes.

Full embeddings of Lie incidence geometries in each other

A Lie incidence geometry is a point-line geometry (a Grassmannian) associated to a building. They are instances of parapolar spaces. We study embeddings of ceertain such geometries in each other, in the sense that we assume that the point set of the smaller building is a subset of that of the larger building, and each line of the smaller building is a full line of the larger building. We have studied the full embeddings of F4,1 in E6,1, E6,2 and of F4,1, E6,1 and E6,2 in E7,1, and we now aim at the full embeddings of F4,1, E6,1, E6,2, E7,1 and E7,7 in E8,8.


Rank 1 geometries

Unitals

We study unitals, trying to isolate the classical ones amongst all translation unitals.

Moufang sets and Tits sets

A Moufang set is the permutation theoretical axiomatisation of a simple algebraic group G of relative isotropic rank 1. In 1995, Tits endowed certain Moufang sets with treads to obtain a rank 2 geometry, nowadays called a Tits web. The fundamental theorem of a Moufang set states that the automorphism group of a Tits web is essentially the corresponding group. Our programme consists in proving the fundamental theorem for the exceptional Moufang sets.


Tits sets are certain graphs that extend the notion of Moufang sets. We aim at characterisations of the exceptional polar Tits sets.


Incidence geometry in other fields

Absolute geometry and combinatorial motives

Especially due to the impetus of a foundational paper by Manin in the beginning of the 1990s about motives and regularized determinants, the geometric theory of the field with one element has evolved into a theory in its own right which combines elements of algebraic geometry, combinatorics, analysis and discrete geometry. The extensive work of Connes and Consani explained -- among many other results -- a connection between F1-theory and Singer actions on projective spaces, and this opened up a new way to look at old questions of -- finite and infinite -- incidence geometry. On the other side of the spectrum of F1-theory, extremely interesting questions about F1-zeta functions, Tate motives and combinatorial versions of motives (in a context of Grothendieck rings) arise.

Wootter's program

It has long been an understanding that many quantum information theoretic issues are deeply connected with certain problems in finite incidence geometry (often in combination with group theoretical aspects). Wootter's wrote some influential papers on this subject, and even conjectured that one of the most elusive open problems in quantum information theory -- the existence of maximal sets of mutually unbiased bases (MUBs) which attain the theoretical upper bound -- is directly connected to the prime power conjecture for projective planes in some way. Other work of e.g. Schumacher and Westmoreland, and Lev studied modal finite field quantum theories, and deepened Wootter's predictions. And even more recently Chang, Lewis, Minic and Takeuchi have studied F1-analogues of the approach Schumacher et al.