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Course Criteria
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3.00 Credits
PQ: MATH 26200. Topics include the fundamental group of a space; Van Kampen's theorem; covering spaces and groups of covering transformation; existence of universal covering spaces built up out of cells; and theorems of Gauss, Brouwer, and Borsuk-Ulam. Spring.
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3.00 Credits
PQ: MATH 25900 or 25600. Topics include group algebras and modules, semisimple algebras and the theorem of Maschke; characters, character tables, orthogonality relations and calculation; and induced representations and characters. Applications to permutation groups and solvability of groups are also included. Autumn.
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3.00 Credits
PQ: MATH 25900 or 25600. Topics include basic definitions and properties of commutative rings and modules, Noetherian and Artinian modules, exact sequences, Hilbert basis theorem, tensor products, localizations of rings and modules, associated primes and primary decomposition, Artin-Rees Lemma, Krull intersection theorem, completions, dimension theory of Noetherian rings, integral extensions, normal domains, Dedekind domains, going up and going down theorems, dimension of finitely generated algebras over a field, Affine varieties, Hilbert Nullstellensatz, dimension of affine varieties, product of affine varieties, and the dimension of intersection of subvarieties. Winter.
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3.00 Credits
PQ: MATH 20500 or 20900. Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping. Autumn, Spring.
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3.00 Credits
PQ: MATH 20900 or 27000. Topics include Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, Lp- spaces, Fourier transforms, Plancherel's theorem for Rn, and spectral properties of bounded linear operators. Winter.
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3.00 Credits
PQ: MATH 27000 or PHYS 22100. This course covers first-order equations and inequalities, Lipschitz condition and uniqueness, properties of linear equations, linear independence, Wronskians, variation-of-constants formula, equations with constant coefficients and Laplace transforms, analytic coefficients, solutions in series, regular singular points, existence theorems, theory of two-point value problem, and Green's functions. Winter.
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3.00 Credits
PQ: MATH 26200. Topics include exterior algebra; differentiable manifolds and their basic properties; differential forms; integration on manifolds; and the theorems of Stokes, DeRham, and Sard. With MATH 26200, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Spring.
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3.00 Credits
PQ: MATH 27300. This course covers classification of second-order equations in two variables, wave motion and Fourier series, heat flow and Fourier integral, Laplace's equation and complex variables, second-order equations in more than two variables, Laplace operators, spherical harmonics, and associated special functions of mathematical physics. Spring.
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3.00 Credits
PQ: MATH 25400 or 25700. This course introduces mathematical logic. Topics include propositional and predicate logic and the syntactic notion of proof versus the semantic notion of truth (e.g., soundness, completeness). We also discuss the G del completeness theorem, the compactness theorem, and applications of compactness to algebraic problems. Autumn.
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3.00 Credits
PQ: MATH 27700 or equivalent. Topics include number theory, Peano arithmetic, Turing compatibility, unsolvable problems, G del's incompleteness theorem, undecidable theories (e.g., the theory of groups), quantifier elimination, and decidable theories (e.g., the theory of algebraically closed fields). Winter.
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