Lecture notes on the Web

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• Socrates Intensive Courses on Finite Geometry and Applications
• Lecture notes on permutation groups
• Individual sets of notes:
  • Ian Anderson and Iiro Honkala, A short course in combinatorial designs
  • R. A. Bailey, Association schemes and partially balanced designs
  • A. Betten, H. Fripertinger and A. Kerber, Algebraic combinatorics via finite group actions
  • Chris K. Caldwell, Graph theory tutorials
  • Peter J. Cameron, Classical groups
  • Peter J. Cameron, Projective and polar spaces
  • Peter J. Cameron, Polynomial aspects of codes, matroids and permutation groups
  • Bill Cherowitzo, Combinatorial structures
  • Queen Mary Combinatorics Study Group Papers
  • Stephen Donkin, Linear Algebra
  • Andrew Granville, Arithmetic properties of Binomial Coefficients
  • W. D. Joyner, The Mathematics of Rubik's Cube
  • M. Klin et al., Algebraic combinatorics in mathematical chemistry
  • László Lovász's Lecture Notes
  • T. W. Müller, Five lectures on generalized permutation representations
  • Steven R. Pagano, Matroids and signed graphs
  • John Preskill, Quantum information theory and quantum computing
  • D. R. Stinson, Combinatorial designs with selected applications
  • Herb Wilf, East side, west side
• Other sources of notes

• From Potenza 1999, available at
  http://dwispc8.vub.ac.be/Potenza/lectnotes.html,
  there are the following notes:
  • Peter J. Cameron, Finite geometry and coding theory
  • J. W. P. Hirschfeld, Semi-linear groups over finite fields
  • O. H. King, Classical groups
• From Ghent 2000, available at
  http://cage.rug.ac.be/~fdc/intensivecourse2/final.html,
  there are the following notes:
  • H. Van Maldeghem and J. A. Thas, Embeddings of point-line geometries in projective spaces
  • W. Haemers, Matrix techniques for strongly regular graphs and related geometries (with an addendum by Sara Cauchie and Elisabeth Kuijken)
  • F. Buekenhout, Prehistory and history of polar spaces and of generalized polygons
  • F. Mazzocca, Nuclei in finite projective planes (preliminary version of the notes in PostScript)
  • L. Storme and J. Eisfeld, Partial spreads in finite projective spaces (preliminary version of the notes in PostScript)
  • A. Bonisoli, On collineation groups of finite planes (with a first draft of an addendum by K. Thas)
  • M. Brown, (Hyper)ovals and ovoids in projective spaces

1. Systems of distinct representatives
2. 2-designs
3. t-designs and Steiner systems
4. Codes and designs

1. Definitions of association scheme
2. Adjacency matrices
3. Some special association schemes
4. The Bose-Mesner algebra
5. Character tables
6. Techniques
7. Strongly regular graphs
8. Block designs
9. Partially balanced block designs
10. A little statistics
11. Efficiency
12. Cyclic designs
13. Families of partitions
14. Orthogonal block structures

1. Actions (Actions of groups; Bilateral classes, symmetry classes of mappings; Finite symmetric groups; Complete monomial groups; Enumeration of symmetry classes; The involution principle; Special symmetry classes)
2. Weights (Enumeration by weight; Cycle indicator polynomials; Sums of cycle indicators, recursive methods; A generalization; The Decomposition Theorem; Species)
3. Marks
4. Constructions (Orbit evaluation; Transversals of symmetry classes; Orbits of centralizers; Recursion and orderly generation; Generating orbit representatives; Symmetry adapted bases)
5. Index

1. Introduction
2. The basics
3. Some classics
4. Machinery
5. Primitive ovals in projective planes of odd order
6. An excursion into graphs
7. The uniqueness of the dual Lüneburg plane of order 64
8. Recent results, open problems

1. Notation, definitions, and some exercises
2. Ovals, ovoids, and a theorem concerning the fixed element structure of a collineation of a finite projective plane
3. Translation nets, translation planes, Moufang planes, and ( p,L)-transitivity
4. Some easy remarks on the proofs of some propositions
5. On Proposition 4.3
6. On Section 5
7. Moufang sets and collineations of projective planes
8. Collineations of finite projective planes and the Petersen graph
9. References

1. Introduction
2. (Hyper)ovals in PG(2,q)
3. Ovoids in PG(3,q)
4. References

1. Introduction
2. Projective spaces
3. Duality and polarity
4. The classical groups, Jordan 1870 and Dickson 1901
5. Tits 1956
6. Veldkamp 1959
7. Tits 1959, Feit-Higman 1964
8. Tits 1961-1974
9. Buekenhout-Shult 1974
10. To be done
11. References

1. Introduction to graph theory
2. Euler circuits and paths
3. Coloring problems

1. Codes
2. Symplectic and quadratic forms
3. Reed-Muller codes
4. Self-dual codes
5. Bent functions
6. Kerdock codes
7. Some resolved designs
8. Extraspecial 2-groups
9. Quantum computing
10. Quantum codes
11. Z₄ codes
12. Bibliography

1. Fields and vector spaces
2. Linear and projective groups
3. Polarities and forms
4. Symplectic groups
5. Unitary groups
6. Orthogonal groups
7. The Klein correspondence and triality
8. Further topics
9. A short bibliography on classical groups

1. Projective spaces
2. Projective planes
3. Coordinatisation of projective spaces
4. Various topics
5. Buekenhout geometries
6. Polar spaces
7. Axioms for polar spaces
8. The Klein quadric and triality
9. The geometry of the Mathieu groups
10. Exterior powers and Clifford algebras
11. References and index

1. Codes
2. Codes over Z₄
3. Matroids
4. Matroids and codes
5. Permutation groups
6. Cycle index
7. Codes and permutation groups
8. IBIS groups

1. Latin squares
2. Introduction to finite fields
3. Hadamard matrices
4. Block designs
5. Finite geometries

1. Fibonacci notes, by Peter Cameron and Dima Fon-Dear-Flaass
2. Notes on quantum error correction, by Harriet Pollatsek and Keldon Drudge
3. Problems from the First Anglo-Hungarian Meeting on Groups and Geometries
4. Five lectures on generalized permutation representations, by Thomas Müller
5. Borcherds' proof of the moonshine conjecture, after V. Nikulin
6. Partially ordered sets, by Thomas Britz and Peter Cameron
7. Causal set glossary and bibliography, by Rafael D. Sorkin
8. A Markov chain for Steiner triple systems, working notes by Peter Cameron
9. Primitive lambda-roots, by Peter Cameron and Donald Preece
10. Decoding the Mathieu group M₁₂, by Robert F. Bailey

1. Matrices
2. Inversion, elementary row operations
3. Determinants
4. Vector spaces
5. Linear maps
6. The rank of a matrix

1. t-spreads of projective spaces
2. t-covers and partial t-spreads in PG(N,q), where t+1 does not divide N+1
3. Extendability of spreads in PG(3,q)
4. Partial spreads in PG(N,q), where t+1 divides N+1
5. Minimal t-covers in PG(N,q), where t+1 divides N+1
6. General covering and blocking problems in projective spaces
7. References

1. Introduction
2. Elementary Number Theory
3. Generalization of Lucas' Theorem
4. Congruences for sums of binomial coefficients
5. Computing binomial coefficients modulo prime powers
6. Recognizing the primes
7. Pascal's triangle via cellular automata
8. Studying binomial coefficients through their generating function
9. Bernoulli numbers and polynomials
10. Theorems of Morley and Emma Lehmer and their generalizations
11. Some useful p-adic numbers
12. Congruences modulo higher powers of primes
13. Concluding remarks
14. References

1. Strongly regular graphs
2. Association schemes
3. Matrix tools
4. The (81, 20, 1, 6) strongly regular graph
5. References

1. Eigenvalues of a regular graph
2. The friendship property; polarities in projective planes
3. The line graph of a graph
4. (Partial) linear spaces and their point and line graphs
5. Pseudo-geometric graphs
6. Spreads
7. References

1. Polarities
2. Groups on the line
3. Orders and isomorphisms among the semi-linear groups
4. Bibliography

1. Logic and sets
2. Functions, matrices, relations and counting
3. Permutations
4. Permutation puzzles
5. Groups, I
6. Graphs and "God's Algorithm"
7. Symmetry groups of the Platonic solids
8. Groups, II
9. The Rubik's cube and the word problem
10. The 2 × 2 and 3 × 3 cube groups
11. Other Rubik-like puzzle groups
12. Interesting subgroups of the cube group
13. Crossing the Rubicon
14. Appendix: some solution strategies

1. Introduction
2. The subject of algebraic combinatorics
3. Problems related to the perception of the symmetry of chemical graphs
4. Fundamentals of permutation group theory
5. Centralizer algebras of permutation groups
6. Cellular algebras
7. Galois correspondence between permutation groups and cellular algebras
8. S-rings over cyclic groups
9. Automorphism groups of certain chemical graphs
10. Concluding remarks

1. Forms and groups
2. Isomorphisms between classical groups
3. Aschbacher's Theorem
4. Bibliography

1. Some combinatorial aspects of permutation representations
2. Generalizing permutation representations
3. Some examples and a formula for the exterior function
4. Explicit formulae for abelian groups and computations in Quillen complexes
5. Asymptotics of Hom(G, H wr S_n) and subgroup growth

1. Semidefinite optimization
2. Topological methods in combinatorics
3. Complexity of algorithms
4. (with K. Vesztergombi) Discrete Mathematics

1. Definition and examples
2. Some results
3. The Segre-Korchmáros lemma and its applications
4. Sets with the maximal number of nuclei
5. Quasi-odd sets
6. Generalizations
7. Further generalizations
8. References

1. Introduction and some notes
2. What is a matroid?
3. Some common examples of matroids
4. Circuits, bases, rank, closure
5. Duality
6. Minors
7. Representability
8. Connectivity
9. What the heck is a signed graph?
10. What are you trying to find out?
11. Have you found anything interesting yet?
12. Any good references for all this stuff?

1. Introduction and overview
2. Foundations of quantum theory I: States and ensembles
3. Foundations of quantum theory II: Measurements and ensembles
4. Quantum entanglement
5. Quantum information theory
6. Quantum computation
7. Quantum error correction

1. Basic definitions and properties of BIBDs
2. Symmetric BIBDs
3. Resolvable BIBDs
4. Steiner triple systems
5. Orthogonal Latin squares
6. Authentication codes
7. Threshold schemes
8. Visual cryptography
9. Group testing
10. Two-point sampling
11. References

1. Definitions
2. Some important finite point-line geometries
3. Embeddings of generalized quadrangles
4. Embeddings of polar spaces
5. Embeddings of partial geometries
6. Embeddings of the flag geometries of projective planes
7. Embeddings of generalized hexagons
8. Polarized, flat and lax embeddings of generalized polygons
9. Open cases and conjectures
10. References

1. Introduction
2. About programming in Maple
3. Sets and subsets
4. Permutations and their cycles
5. Set partitions
6. Integer partitions
7. ... and all around the town
8. The EastWest Maple package
9. Program notes

• Paderborn
• MathArchive
• World Combinatorial Exchange

Peter J. Cameron
20 May 2003