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Week
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Date
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Topics
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Resources
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2
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Thu 22 Jan
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Introduction. Basic structure results. Connected component of the
identity and group of connected components. Barsotti-Chevalley
decomposition into affine normal subgroup and abelian variety
quotient. Basic algebraic geometry background. Existence of finite
dimensional linear representations for affine groups. Chevalley's
theorem on subgroups as stabilizers of lines in a finite dimensional
representation. Zariski closures.
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3
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Tue 27 Jan
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Nor'easter Juno!
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Thu 29 Jan
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Group objects in categories. Yoneda's Lemma. Representable
functors. The category of affine
schemes and affine group schemes. Hopf algebras. The coordinate algebra of an affine group scheme.
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Milne (AGS Ch. 1-2), Waterhouse (Ch. 1)
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4
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Tue 03 Feb
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Representability of the multiplicative group
on the category of all associative algebras (Zuckerman).
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5
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Tue 10 Feb
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Group scheme of units of an associative algebra. Theory of generic
norm and trace. Various example.
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Book of Involutions (Ex. 20.2), article by Skip
Garibaldi
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Thu 12 Feb
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Extension of scalars. Forms of linear algebraic groups. Subgroup schemes
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Book of Involutions (Sec. 20), Waterhouse (Ch. 2.1), Milne (AGS VII)
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6
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Tue 17 Feb
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Closed embeddings. Constant group schemes. Stabilizer subgroups. The
multiplicative group.
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Thu 19 Feb
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Etale algebras. Connected component.
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Book of Involutions (Prop. 20.10), Waterhouse (Ch. 6), Milne (AGS XIII)
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7
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Tue 24 Feb
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More connected components. Roots of unity. Orthogonal groups. Etale
groups and diagonalizable groups.
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Waterhouse (Ch. 2)
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Thu 26 Feb
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Galois descent of vector spaces and algebras. Grothendieck's Galois
theory. Etale algebras and continuous Galois representations to the
symmetric group.
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Book of Involutions (Sec. 18)
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8
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Tue 03 Mar
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Characters of algebraic groups. Group-like elements in Hopf
algebras. Diagonalizable groups.
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Book of Involutions (Prop. 20.17), Waterhouse (Ch. 7), Milne (AGS XIV)
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Thu 05 Mar
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Tori. Decomposition into split and anisotropic tori. Weil
restriction. Norm one tori.
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Waterhouse (Ch. 7.4)
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9
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Tue 10 Mar
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Spring Break!
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Thu 12 Mar
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Spring Break!
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10
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Tue 17 Mar
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Spring Break!
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Thu 19 Mar
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Spring Break!
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11
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Tue 24 Mar
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Galois cohomology. Profinite groups. Non-abelian Galois cohomology.
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Serre (Ch. 1 Sec. 1-2, 5), Book of Involutions (Sec. 28)
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Thu 26 Mar
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More non-abelian Galois cohomology. Torsors/principal homogeneous
spaces. Twisting. "Long" exact sequence. Example of degree 3
field extensions.
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Serre (Ch. I, Sec. 5)
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12
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Tue 31 Mar
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Sn and An extensions of fields and the discriminant.
Philosophy of forms. Automorphism groups of tensors.
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Serre (Ch. III, Sec. 1)
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Thu 02 Apr
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Hilbert Theorem 90 and Emmy Noether's version. Classification of
forms of tensors. Symmetric and
skew-symmetric forms.
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Serre (III, Sec. 1.4)
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13
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Tue 07 Apr
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Kummer theory. Classification of forms of SLn and
GLn, prelude. Automorphism group in terms of algebra
with involution. Central simple algebras. Artin-Wedderburn
theorem. Skolem-Noether theorem. Brauer group.
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Serre (III, Sec. 1.4), Kneser (Sec. 2.4)
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Thu 09 Apr
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Classification of forms of SLn and GLn, part I.
Inner and outer forms.
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Serre (III, Sec. 1.4), Kneser (Sec. 2.4)
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14
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Tue 14 Apr
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Classification of forms of SLn and GLn, part II.
Automorphism group in terms of algebra with involution.
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Kneser (Sec. 2.4), Book of Involutions (Sec. 26.A)
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Thu 16 Apr
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Classification of type groups of type Cn and
Bn. Orthogonal and symplectic involutions.
Classification of type groups of type Dn. Discriminant
of a quadratic form and an orthogonal involution. Glimpses of triality.
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Kneser (Sec. 2.4), Book of Involutions (Sec. 26.A)
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15
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Tue 21 Apr
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Central isogenies. Simply connected and adjoint groups. Automorphism groups.
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Book of Involutions (Sec. 25.B), Milne (iAG Ch. 23)
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Thu 23 Apr
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Classification of simply connected and adjoint types for all classical
groups. Description of all central isogenies. Clifford algebras
and Spin groups.
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16
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Tue 28 Apr
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Make-up class
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Tue 12 May
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Final Exam Due!
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Final Exam
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