Course Details
Schedule
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.
Instructors
Anurag Bishnoi  Tibor Szabó  
Arnimallee 3, Room 206  Arnimallee 3, Room 211a  
anurag.2357 at gmail dot com  szabo at math dot fuberlin 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.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 1.31.6 Also see Casse Ch. 3 and 4 

3  hypersurfaces in projective spaces, conics, ovals and hyperovals.  LN 2.12.3 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 BlokhuisMazzocca 

6  Nikodym sets and minimal blocking sets, strongly regular graphs, eigenvalues of a graph  LN 3.2 and 4.14.3 GodsilRoyle Ch 10 

7  spectral methods, eigenvalues of strongly regular graphs, application to Moore graphs and Friendship theorem  LN 4.3  
8  polarities, DelsarteHoffman bound, twointersection sets, generalized quadrangles  LN 4.34.4, 5.1 Ball 5.4 

9  generalized quadrangles, properties and examples  LN 5.25.3 GodsilRoyle 5.45.5  
10  chromatic number of ball packings, generalized polygons  LN 5.45.5 GodsilRoyle 5.6 Chen 

11  properties of generalized polygons  LN 5.5 GodsilRoyle 5.6 

12  abstract polar spaces, collinearity graph, polarities of projective spaces, quadrics and symmetric bilinear forms  LN 6.16.3 Ball 3.6, 4.2 

13  quadrics, Witt's theorem, counting totally singular subspaces  LN 6.36.4 Ball 3.6, 4.24.4 