CHAPTER 1

An Introduction to Incidence Geometry

Francis BUEKENHOUT

CP 216, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

Contents

1. Incidence geometry within mathematics and geometry

1.1. The subject briefly explained

1.2. On mathematics

1.3. On geometry

1.4. Incidence, related subjects and the foundations of geometry

1.5. Is incidence geometry useful?

2. Incidence geometry as rooted in division rings and in dimension

2.1. The reason for division rings (or skew fields)

2.2. The role of dimension and vector spaces

2.3. Projective spaces and affine spaces

2.4. Topology, order, metric

2.5. Linear spaces

2.6. Linear spaces with dimension, matroids, geometric lattices

2.7. Designs

2.8. Finite Geometry

3. Excursion on the hill of the Handbook

3.1. On this side of the hill: linear algebra and its zone of influence

3.2. The other side of the hill: buildings and groups

4. Incidence geometry and groups

4.1. The role of symmetry

4.2. Klein's Erlangen Program (1872)

4.3. Flag-transitivity and coset geometries

4.4. Classical groups and Lie groups. The reappearance of and

4.5. The meeting of projective geometry and Lie groups

4.6. The rise of buildings

4.7. Around buildings

References