Algebra and Geometry
What used to be 2WF02 is now 2WF05.
Old: trimester course - New: semester course
Time and place: Thu 13:45-15:30, Auditorium 11
Starting April 26: HG 6.96.
Exercises
Send answers/solutions to aeb@cwi.nl.
Results.
Some notes:
Lattices (dvi,
pdf),
Spectra of some small graphs (dvi,
pdf).
Spectra of graphs (pdf).
Old Notes:
Golay (dvi,
pdf),
Much more detail on Witt designs, Golay codes, Mathieu groups
(dvi, pdf),
Leech (dvi,
pdf),
Graph spectra and Perron-Frobenius theorem
(html).
Much more detail on graph spectra
(pdf).
Chevalley-Warning, quadrics (dvi,
pdf),
Ovals and conics (dvi,
pdf),
Generalized quadrangles (dvi,
pdf),
Representations of finite groups (dvi,
pdf).
Coherent configurations (dvi,
pdf),
Groups and graphs (dvi,
pdf),
Buekenhout-Tits geometries (dvi,
pdf).
Very old notes:
Gröbner bases (dvi,
pdf),
p-adic numbers (dvi,
pdf),
Nullstellensatz (dvi,
pdf),
Bezout's theorem (dvi,
pdf),
Resultant, discriminant (dvi,
pdf),
Varieties (first definitions) (dvi,
pdf),
Dimension (definition) dvi,
pdf),
Hilbert function (dvi,
pdf),
Zeta function (dvi,
pdf),
Blowup (dvi,
pdf),
Elliptic functions (dvi,
pdf),
Mordell (dvi,
pdf).
Genus of a plane curve (dvi,
pdf),
Regular differential forms (dvi,
pdf),
Riemann-Roch (dvi, pdf).
Topics:
- Spectrum of graphs. Adjacency, Laplace, Seidel matrix.
- Friendship theorem. Spectrum of strongly regular graphs.
- Paley graphs.
- Perron-Frobenius theorem.
- Classification of graphs with largest eigenvalue at most 2.
- Interlacing.
- Very first intro to lattices and modular forms.
- Steiner triple systems and Latin squares.
- Steiner systems, Golay codes, Leech lattice, sporadic simple groups.
- Classification of root lattices.
- Chevalley-Warning.
- Projective geometry. Polar spaces. Generalized quadrangles.
- (Hyper)ovals in a projective plane, number of absolute points of a polarity
- Quadratic forms, Witt index.
- Intro symplectic, orthogonal and unitary geometry.
- Association schemes. Delsarte LP.
- Bounds on the size of (co)cliques and chromatic number.
- Krein parameters and absolute bound for association schemes.
- Restrictions on automorphisms of association schemes.
- Sign patterns in the eigenmatrices of a P- or Q-polynomial association scheme.
- Linear representations of finite groups.
- Buekenhout-Tits geometries.
- Two-weight codes, projective two-intersection sets, and strongly regular graphs.
- Equiangular lines.
- Majorization order, theorems of Schur, Courant-Weil, Ky Fan, Gale-Ryser.
- Grone-Merris conjecture, Hua Bai theorem.
Literature
A.E. Brouwer, A.M. Cohen & A. Neumaier,
Distance-regular graphs,
Springer, 1989.
J.H. Conway & N.J.A. Sloane,
Sphere Packings, Lattices and Groups,
Springer, 1988.
Jean-Pierre Serre,
A Course in Arithmetic,
Springer, 1973.
A.E. Brouwer & W.H. Haemers,
Spectra of Graphs,
Springer, 2012.