Finite Geometry

Summer 2019

FU Berlin, Mathematics and Informatics

Course Details


The lectures and exercises will be on Tuesdays from 14:15 to 15:45 and from 16:15 to 17:45 in Takustrasse 9 SR 006. The exercises will be held every two weeks from 14:15 to 15:45.

For announcements related to the course, please sign up here.


Anurag Bishnoi Tibor Szabó
Arnimallee 3, Room 206 Arnimallee 3, Room 211a
anurag.2357 at gmail dot com szabo at math dot fu-berlin dot de

Exams and Requirements

The grades will be based solely on the final exam. There will be oral exams offered in July, right after the end of the lectures, and in September/October. See here for full requirements of the course.


Over the course of this semester, we shall cover the following topics:

Finite affine and projective spaces: basic theory of abstract planes, latin squares, ovals and hyperovals, blocking sets, Kakeya and Nikodym sets
Spectral methods in finite geometry: strongly regular graphs, Moore graphs, eigenvalues of graphs, Hoffman's bound, expander mixing lemma
More finite geometries: generalized quadrangles, abstract polar spaces, quadrics, generalized polygons


The following will be the main references for the course:
  • Simeon Ball, Finite Geometry and Combinatorial Applications
  • Bart De Bruyn, An introduction to Incidence Geometry
  • Rey Casse, Projective Geometry, An Introduction
  • G Eric Moorhouse, Incidence Geometry
  • Chris Godsil and Gordon Royle, Algebraic Graph Theory

For further reading, you are encouraged to consult either of the texts below:
  • P. Dembowski, Finite Geometries
  • J. W. P. Hirschfeld and J. Thas, Projective Geometries over Finite Fields
  • A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications
  • L. M. Batten, Combinatorics of finite geometries
  • A. E. Brouwer and W. H. Haemers, Spectra of Graphs


Students should be familiar with basic graph theory and combinatorics, linear algebra, and finite fields. Here are some notes on the algebraic basics.

Lecture Notes

I will update the file containing the lecture notes (denoted by LN below) weekly and refer to it in the section on Course Progress. These notes also contain some additional exercises that are not in the weekly sheets; they are not for grading.


Exercise sheets will be posted below every second week, and should be submitted before 14:15 on Tuesday of the following week.

Assignment Due date Comments
Sheet 1 23/04/2019
Sheet 2 07/05/2019 Hints on the last page
Sheet 3 21/05/2019 3(b) corrected [9/05/2019]
Sheet 4 04/06/2019
Sheet 5 18/06/2019
Sheet 6 02/07/2019 4(b) corrected
Sheet 7 09/07/2019

Course Progress

As we make our way through the semester, we will provide below a brief review of the topics covered each week.

Week Topics References
1 finite affine planes, Mutually Orthogonal Latin Squares LN 1.1 and 1.2
2 projective planes, projective spaces, Desarguesian spaces, coordinates LN 1.3-1.6
Also see Casse Ch. 3 and 4
3 hypersurfaces in projective spaces, conics, ovals and hyperovals. LN 2.1-2.3
Also see Casse Ch. 7
4 dual ovals, blocking sets, Kakeya sets: introduction LN 2.3, 3.1, 3.2
5 Kakeya and Nikodym sets LN 3.1, Dvir Tao
6 Nikodym sets and minimal blocking sets, strongly regular graphs, eigenvalues of a graph LN 3.2 and 4.1-4.3
Godsil-Royle Ch 10
7 spectral methods, eigenvalues of strongly regular graphs, application to Moore graphs and Friendship theorem LN 4.3
8 polarities, Delsarte-Hoffman bound, two-intersection sets, generalized quadrangles LN 4.3-4.4, 5.1
Ball 5.4
9 generalized quadrangles, properties and examples LN 5.2-5.3
Godsil-Royle 5.4-5.5
10 chromatic number of ball packings, generalized polygons LN 5.4-5.5
Godsil-Royle 5.6
11 properties of generalized polygons LN 5.5
Godsil-Royle 5.6
12 abstract polar spaces, collinearity graph, polarities of projective spaces, quadrics and symmetric bilinear forms LN 6.1-6.3
Ball 3.6, 4.2
13 quadrics, Witt's theorem, counting totally singular subspaces LN 6.3-6.4
Ball 3.6, 4.2-4.4