"Axiomatic and Transformational Geometries Class Notes

Axiomatic and Transformational Geometry Class Notes - Finite Geometry
Finite Geometry and Combinatorial Applications,
by Simeon Ball,
London Mathematical Society Student Texts 82 (2015). Ball's Finite Geometry and Combinatorial Applications book

Ball's Finite Geometry and Combinatorial Applications book

These notes would constitute part of the material of "Axiomatic and Transformational Geometry" (MATH 5330). The catalog description in the 2014-15 ETSU Graduate Catalog was: "Axiomatic and finite geometries. Euclidean geometry (synthetic/analytic), transformational geometries, non-Euclidean and projective geometries." The course was removed from the catalog in 2015. The prerequisites are Calculus 2 (MATH 1920), Linear Algebra (2010), and Mathematical Reasoning (MATH 3000). This course was previously titled "Vector Geometry" and the description in the 1988-90 ETSU Graduate Catalogue was: "Projective geometry, affine geometry and affine transformation, Euclidean geometry, non-Euclidean geometries." It seems that "Vector Geometry" was split into "Introduction to Modern Geometry" (MATH 4157/5157) and "Axiomatic and Transformational Geometry" (MATH 5330) sometime in the 1990s. Honestly, the material in these notes is not at the level of a graduate-only course, and is more appropriate for a cross-listed undergraduate-graduate level class... perhaps it can become "Introduction to Modern Geometry 2" (MATH 4167/5167) in the future.

These notes constitute the finite geometry part of Axiomatic and Transformational Geometry. The transformational geometry part is covered in my online transformational geometry class notes. Online notes for axiomatic geometry are also available, though they are primarily meant for use in Introduction to Modern Geometry.

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

Chapter 1. Fields.

Section 1.1. Rings and Fields.
Section 1.2. Finite Automorphisms.
Section 1.3. The Multiplicative Group of a Finite Field

Chapter 2. Vector Spaces.

Section 2.1. Vector Spaces and Subspaces.
Section 2.2. Linear Maps and Linear Forms.
Section 2.3. Determinants.
Section 2.4. Quotient Spaces.

Chapter 3. Forms.

Section 3.1. σ-Sesquilinear Forms (partial). Section 3.1 notes

Supplement. Proofs from Section 3.1. Beamer file of Section 3.1 proofs
Supplement. Printouts of Proofs from Section 3.1. Print file of Section 3.1 proof

Section 3.2. Classification of Reflexive Forms.
Section 3.3. Alternating Forms.
Section 3.4. Hermetian Forms.
Section 3.5. Symmetric Forms.
Section 3.6. Quadratic Forms.

Chapter 4. Geometries.

Section 4.1. Projective Spaces (partial). Section 4.1 notes
Section 4.2. Polar Spaces.
Section 4.3. Quotient Geometries.
Section 4.4. Counting Subspaces.
Section 4.5. Generalised Polygons.
Section 4.6. Plücker Coordinates.
Section 4.7. Polarities.
Section 4.8. Ovoids.

Chapter 5. Combinatorial Applications.

Section 5.1. Groups.
Section 5.2. Finite Analogues of Structures in Real Space.
Section 5.3. Codes.
Section 5.4. Graphs.
Section 5.5. Designs.
Section 5.6. Permutation Polynomials.

Chapter 6. The Forbidden Subgraph Problem.

Section 6.1. The Erdős-Stone Theorem.
Section 6.2. Even Cycles.
Section 6.3. Complete Bipartite Graphs.
Section 6.4. Graphs Containing no K_2,s .
Section 6.5. A Probabilistic Construction of Graphs Containing no K_t,s .
Section 6.6. Graphs Containing no K_3,3 .
Section 6.7. The Norm Graph.
Section 6.8. Graphs Containing no K_5,5 .

Chapter 7. MDS Codes.

Section 7.1. Singleton Bound.
Section 7.2. Linear MDS Codes.
Section 7.3. Dual MDS Codes.
Section 7.4. The MDS Conjecture.
Section 7.5. Polynomial Interpolation.
Section 7.6. The A-Functions.
Section 7.7. Lemma of Tangents.
Section 7.8. Combining Interpolation with the Lemma of Tangents.
Section 7.9. A Proof of the MDS Conjecture for k ≤ p.
Section 7.10. More Examples of MDS Codes of Length q + 1
Section 7.11. Classification of Linear MDS Codes of length q + 1 for k ≤ p.
Section 7.12. The Set of Linear Forms Associated with a Linear MDS Code.
Section 7.13. Lemma of Tangents in the Deal Space.
Section 7.14. The Algebraic Hypersurface Associated with a Linear MDS Code.
Section 7.15. Extendability of Linear MDS Codes.
Section 7.16. Classification of Linear MDS Codes of Length q + 1 for k < c√q.
Section 7.17. A Proof of the MDS Conjecture for k < c√q.

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