Math 604 Introduction to Quadratic Forms
The official syllabus in pdf form.
Syllabus
Updated September 13, 2016.
List of useful texts:
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Week
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Date
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Topics
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Resources
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1
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Wed 31 Aug
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Introduction. Conic sections. Symmetric bilinear forms.
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Lam I.1-I.2,
Kahn 1.1, Scharlau 1.1-1.3
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2
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Wed 07 Sep
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Quadratic and symmetric bilinear forms. Dimension. Orthogonal
complement. Radical.
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Lam I.1-I.2,
Kahn 1.1, Scharlau 1.1-1.3
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Fri 09 Sep
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Orthogonal sum. Tensor product. Subforms. Isometry. Orthogonal transformations. Diagonalization.
Representing units. Reflections.
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Lam I.4, I.6-7, Kahn 1.2, Scharlau 1.5
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3
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Fri 16 Sep
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Square classes. Quadratic extensions.
Witt cancellation. Isotropy and hyperbolic spaces.
Witt decomposition. Totally isotropic subspaces. Anisotropic forms. Norm forms.
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Lam I.3-I.4, Kahn 1.2, Scharlau 1.4-1.5
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4
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Fri 23 Sep
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Witt ring. Grothendieck-Witt ring. Generators and relations for the
Witt and Grothendieck-Witt ring.
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Lam II.1-II.5, Kahn 1.3--1.4, Scharlau 2.1-2.4
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5
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Fri 30 Sep
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Extension of scalars. Artin-Springer theorem. Springer's theorem.
Kernel of extension to quadratic fields.
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6
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Fri 07 Oct
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Classical invariant. Reduced dimension. Signed discriminant.
Quaternion algebras. Central simple algebras and the Brauer group.
Clifford algebra. Milnor K-theory.
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EKM II.11
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7
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Wed 12 Oct
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Fundamental filtration.
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Fri 14 Oct
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Pfister forms.
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8
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Fri 21 Oct
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Fall Break!
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9
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Fri 28 Oct
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10
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Fri 04 Nov
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Cancelled (AGNES)!
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11
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Fri 11 Nov
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12
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Fri 18 Nov
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13
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Fri 25 Nov
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Thanksgiving Break!
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14
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Fri 02 Dec
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15
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Fri 09 Dec
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