News
3 November 2025: Schläfli lecture by Kathlén Kohn.

Teaching

Discrete Structures

Course material

Summary of the topics treated thus far in the course

Week 1

Introduction to fields and vector spaces and finite projective planes. Wedderburn's theorem.

Week 2

Collineations. The fundamental theorem of projective geometry. The Desargues configuration.

Week 3

Counting subspaces and determining group orders. Coordinatisation of projective spaces of order 2.

Week 4

Affine spaces. Transitivity of parallelism. (Sections 3.5 and 3.6 of the course material). Spreads and translation planes (Section 4.1). Segre's theorem (Section 4.3).

Week 5

The projective line and cross-ratio (Section 4.5); application to 3-designs. Baer subplanes, and Aiden Bruen's theorem. Blokhuis's theorem (without proof).

Week 6

Dualities and polarities of projective spaces and sesquilinear forms (Section 6.1 and 6.2).

Week 7

Polarities and related forms and their geometry (remaider chapter 6).

Week 8

Polar spaces, an axiomatic approach (chapter 7)

Subjects

Further reading

Projective Geometry by Albrecht Beutelspacher and Ute Rosenbaum. Cambridge University Press, 1998.

A slow-paced elementary treatment for readers with little mathematical background.

Designs, Graphs, Codes, and their Links by Peter J. Cameron and J.H. van Lint. London Math. Soc. Student Texts.

For people interested in design theory, partial geometries, and their relation to finite projective planes.

Miniquaternion Geometry by T.G. Room and P.B. Kirkpatrick. Cambridge University Press, 1971.

An introduction to the study of projective planes with emphasis on the four different planes of order 9, their coordinatisation, and the study of substructures such as ovals, Hermitian sets and subplanes.

Finite Geometries by Peter Dembowski. Springer Verlag, 1968 (reprinted 1979).

The classic treatment of the combinatorics of finite projective and affine planes and their classification in terms of groups acting on them.

Combinatorics---topics, techniques, algorithms by Peter J. Cameron. Cambridge University Press.

Contains a very readable introductory chapter on finite geometry.

Projective geometries over finite fields by J.W.P. Hirschfeld. Oxford University Press.

A very useful collection of results on combinatorial structures in projective planes such as arcs, ovals, cubic curves, blocking sets; and a special chapter on the planes of orders 2,3,...,13.

Projective Planes by Daniel R. Hughes and Fred C. Piper. Springer Verlag.

Elaborate study of the algebraic structures that are used to coordinatise the different kinds of finite projective planes.

Eindige Meetkunde by J.J. Seidel.

Lecture notes in Dutch. Contains all the basics and an elaborate treatment of quadratic and hermitian forms, as well as an introduction to polar spaces.