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MATH 26300: Introduction to Algebraic Topology
3.00 Credits
University of Chicago
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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MATH 26700: Introduction to Representation Theory of Finite Groups
3.00 Credits
University of Chicago
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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MATH 26700 - Introduction to Representation Theory of Finite Groups
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MATH 26800: Introduction to Commutative Algebra
3.00 Credits
University of Chicago
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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MATH 26800 - Introduction to Commutative Algebra
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MATH 27000: Basic Complex Variables
3.00 Credits
University of Chicago
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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MATH 27200: Basic Functional Analysis
3.00 Credits
University of Chicago
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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MATH 27200 - Basic Functional Analysis
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MATH 27300: Basic Theory of Ordinary Differential Equations
3.00 Credits
University of Chicago
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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MATH 27400: Introduction to Differentiable Manifolds and Integration on Manifolds
3.00 Credits
University of Chicago
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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MATH 27400 - Introduction to Differentiable Manifolds and Integration on Manifolds
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MATH 27500: Basic Theory of Partial Differential Equations
3.00 Credits
University of Chicago
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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MATH 27500 - Basic Theory of Partial Differential Equations
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MATH 27700: Mathematical Logic I
3.00 Credits
University of Chicago
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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MATH 27800: Mathematical Logic II
3.00 Credits
University of Chicago
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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MATH 27800 - Mathematical Logic II
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