ScheduleThe 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 RequirementsThe 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
ReferencesThe 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
PrerequisitesStudents should be familiar with basic graph theory and combinatorics, linear algebra, and finite fields. Here are some notes on the algebraic basics.
Lecture NotesI 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.
|Sheet 2||07/05/2019||Hints on the last page|
|Sheet 3||21/05/2019||3(b) corrected [9/05/2019]|
|Sheet 6||02/07/2019||4(b) corrected|
As we make our way through the semester, we will provide below a brief review of the topics covered each week.
|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, 5.1Ball 5.4|
|9||generalized quadrangles, properties and examples||LN 5.2-5.3Godsil-Royle 5.4-5.5|
|10||chromatic number of ball packings, generalized polygons||LN 5.4-5.5Godsil-Royle 5.6Chen|
|11||properties of generalized polygons||LN 5.5Godsil-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|