Finite Geometry

Topics

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

References

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

Prerequisites

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.

Exercises

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 Blokhuis-Mazzocca
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 Chen
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