|
Week
|
Date
|
Topics
|
Reading
|
Homework
|
|
1
|
Mon 14 Sep
|
History of quadratic forms. Bilinear forms.
|
|
|
|
Wed 16 Sep
|
Quadratic and symmetric bilinear forms. Space of bilinear forms. Radical.
|
Lam I.1-I.2, Kahn 1.1, Scharlau 1.1-1.3
|
Fri 18 Sep
|
Tensor products. Symmetric squares. Symmetric tensors.
|
Lam I.4, I.6-7, Kahn 1.2, Scharlau 1.5, Math3ma
|
|
2
|
Mon 21 Sep
|
Isometries. Orthogonal group. Orthogonal complements.
|
Lam I.4, I.6-7, Kahn 1.2, Scharlau 1.5
|
|
|
Wed 23 Sep
|
Orthogonal sum. Subform splitting lemma.
|
Lam I.4, I.6-7, Kahn 1.2, Scharlau 1.5
|
|
Fri 25 Sep
|
Diagonalization. Representing units. Reflections.
|
Lam I.4, I.6-7, Kahn 1.2, Scharlau 1.5
|
|
3
|
Mon 28 Sep
|
No class: Yom Kippur
|
|
|
Wed 30 Sep
|
Witt cancellation. Isotropy and hyperbolic spaces. Totally isotropic subspaces.
|
Lam I.3-I.4, Kahn 1.2, Scharlau 1.4-1.5
|
|
Fri 02 Oct
|
Square classes. Quadratic extensions and étale
algebras. Relations on diagonalizations.
|
Lam I.3-I.4, Kahn 1.2, Scharlau 1.4-1.5
|
|
4
|
Mon 05 Oct
|
Sylvester's law of inertia. Quadratic forms over finite fields.
|
Lam II.3, Scharlau 2.3-2.4
|
Problem Set #1
|
|
Wed 07 Oct
|
Binary forms. Witt decomposition. Witt group. Grothendieck-Witt group.
|
Lam II.1, Scharlau 2.1, Kahn 1.3
|
Fri 09 Oct
|
Tensor product of quadratic forms. Witt ring. Grothendieck-Witt
ring. Fundamental ideal. Reduced dimension and signed
discriminant. Fundamental filtration. Milnor conjecture.
|
Lam V.1-V.3, Scharlau 8.3, Kahn 6.2
|
|
5
|
Mon 12 Oct
|
Unital algebras. Tensor algebra. Graded rings and
homogeneous ideals. Clifford algebra.
|
Lam V.1-V.3, Scharlau 8.3, Kahn 6.2
|
|
Wed 14 Oct
|
Even Clifford algebra. Tensor product of algebras. Graded tensor
product of Z/2Z-algebras. Clifford algebra of an
orthogonal sum.
|
Lam V.1-V.3, Scharlau 8.3, Kahn 6.2
|
|
Fri 16 Oct
|
Center of Clifford algebra. Central simple algebras.
|
Lam V.1-V.3, Scharlau 8.3, Kahn 6.2
|
|
6
|
Mon 19 Oct
|
Division algebras. Artin-Wedderburn. Splitting of central simple
algebras. Brauer equivalence. Brauer group.
|
|
Problem Set #2
|
Wed 21 Oct
|
Quaternion algebras. Biquaternion algebras. Albert form.
Merkurjev's theorem the 2-torsion of the Brauer
group. A peek at cyclic algebras and
Merkurjev-Suslin. Clifford invariant and Hasse-Witt invariant.
|
|
|
Fri 23 Oct
|
Elementary invariants. Fundamental filtration of the Witt group.
|
|
|
7
|
Mon 26 Oct
|
Discrete valuations. Laurent series fields. Order of vanishing
on rational function fields. Ostrowski's theorem for rational
function fields.
|
Lam VI.1, Sharlau Ch. 3 Appendix, Dummit-Foote 16.2
|
|
Wed 28 Oct
|
Discrete valuation rings. p-adic valuation. Absolute values. Ostrowski's theorem.
|
Ostrowski's theorem Q
and F(t)
|
|
Fri 30 Oct
|
Metric topology induced by a valuation. Completions.
Inverse limits.
|
Lam VI.1, Sharlau Ch. 3 Appendix, Dummit-Foote 16.2
|
|
8
|
Mon 02 Nov
|
More completions. Hensel's Lemma.
|
Serre II.1
|
|
Wed 04 Nov
|
Hensel's Lemma. Square classes in complete discrete valuation fields.
|
Serre II.3.3
|
|
Fri 06 Nov
|
Newton's Method and Strong Hensel's Lemma. Multivariable Hensel's Lemma.
|
Serre II.2
|
|
9
|
Mon 09 Nov
|
No class: BATMOBYLE
|
|
Wed 11 Nov
|
Springer's theorem for complete discretely valued fields. Residue
maps on the Witt group. u-invariant. Tsen's theorem and
Tsen-Lang theory. Classification of unimodular quadratic forms over
complete DVRs.
|
Lam VI.1, Sharlau 6.2
|
|
Fri 13 Nov
|
Local fields. Brauer group of local fields. Classification of
quadratic forms over local fields.
|
Lam VI.2, Scharlau 5.6, Serre IV.2
|
|
10
|
Mon 16 Nov
|
Quadratic forms over global fields. Milnor exact sequence. Hasse-Minkowski theorem.
|
Lam VI.3, Scharlau 5.7, Serre IV.3
|
Final Exam
|