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Course Criteria
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3.00 Credits
Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.
Prerequisite:
Prereq: 18.702
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3.00 Credits
Wedderburn theory, Morita equivalence, localization and Goldie's theorem, central simple algebras and the Brauer group, maximal orders, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.
Prerequisite:
Prereq: 18.705
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3.00 Credits
Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.
Prerequisite:
Prereq: 18.702 or 18.703
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3.00 Credits
Introduction to affine and projective algebraic geometry. Emphasis on basic examples of complex algebraic varieties, including algebraic curves.
Prerequisite:
Prereq: 18.702, 18.901
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3.00 Credits
No course description available.
Prerequisite:
Prereq: None. Coreq: 18.705
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3.00 Credits
Continuation of the introduction to algebraic geometry given in 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.
Prerequisite:
Prereq: 18.725
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3.00 Credits
Topics vary from year to year.
Prerequisite:
Prereq: 18.725
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3.00 Credits
Topics vary from year to year.
Prerequisite:
Prereq: 18.705
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3.00 Credits
Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.
Prerequisite:
Prereq: 18.705
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3.00 Credits
Hilbert's finiteness theorem for reductive groups. Properties of the orbits and the orbit space. Classical invariant theory. Hilbert-Mumford-Richardson theorem. Rosenlicht's theorem on the existence of invariants. Matsushima criterion. Richardson's theorem on the principal stabilizer. Chevalley-Luna-Richardson theorem. Linear actions with a non-trivial stabilizer. Polar representations. Methods of the orbit classification. Applications to classical problems of linear algebra. Other topics.
Prerequisite:
Prereq: 18.705
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