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Course Criteria
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3.00 Credits
PQ: MATH 16300 or 19900. This three-course sequence is intended for students who plan to major in mathematics or who require a rigorous treatment of analysis in several dimensions. Both theoretical and problem-solving aspects of multivariable calculus are treated carefully. Topics in MATH 20300 include metric spaces, the topology of Rn, compact sets, the geometry of Euclidean space, and limits and continuous mappings. MATH 20400 deals with partial differentiation, vector-valued functions, extrema, and the inverse and implicit function theorems. MATH 20500 is concerned with and multiple integrals, line and surface integrals, and the theorems of Green, Gauss, and Stokes. This sequence is the basis for all advanced courses in analysis and topology. Autumn, Winter, Spring; Winter, Spring, Autumn; Spring, Autumn, Winter.
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3.00 Credits
PQ: Invitation only. This highly theoretical sequence in analysis is intended for the most able students. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral. Autumn, Winter, Spring.
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3.00 Credits
PQ: MATH 20000 or 20300. This course covers direct and iterative methods of solution of linear algebraic equations and eigenvalue problems. Topics include numerical differentiation and quadrature for functions of a single variable, approximation by polynomials and piece-wise polynomial functions, approximate solution of ordinary differential equations, and solution of nonlinear equations. Spring.
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3.00 Credits
PQ: MATH 15200 or 16200, and PHYS 13200. This course, with concurrent enrollment in PHYS 13300, is required of students who plan to major in physics. Topics include infinite series and power series, complex numbers, linear equations and matrices, partial differentiation, multiple integrals, vector analysis, and Fourier series. Applications of these methods include Maxwell's equations, wave packets, and coupled oscillators. Spring.
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3.00 Credits
PQ: MATH 25500 or 25800. This course focuses on the interplay between abstract algebra (group theory, linear algebra, and the like) and geometry. Several of the following topics are covered: affine geometry, projective geometry, bilinear forms, orthogonal geometry, and symplectic geometry. This course is offered in alternate years. Spring.
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3.00 Credits
PQ: MATH 25500 or 25800. Topics include factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree. Spring.
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3.00 Credits
PQ: MATH 25500 or 25800, or consent of instructor. MATH 25600 or 25900 is strongly recommended. This course covers the projective line and plane curves, both affine and projective. We also study conics and cubics, as well as the group law on the cubic. Abstract curves associated to function fields of one variable are discussed, along with the genus of a curve and the Riemann-Roch theorem. Curves of low genus are emphasized. This course is offered in alternate years. Not offered 2009 C10; will be offered 201 0 -11.
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3.00 Credits
PQ: MATH 16300 or 19900. This sequence covers groups, subgroups, and permutation groups; rings and ideals; some work on fields; vector spaces, linear transformations and matrices, and modules; and canonical forms of matrices, quadratic forms, and multilinear algebra. MATH 25400-25500-25600: Autumn, Winter, Spring; MATH 25400-25500: Winter, Spring.
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3.00 Credits
PQ: MATH 16300 or 19900. This sequence is an accelerated version of MATH 25400-25500-25600. Topics include the theory of finite groups, commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms. We also cover basic field theory, the structure of p-adic fields, and Galois theory. Autumn, Winter, Spring.
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3.00 Credits
PQ: MATH 20300 or 20700, and 25400 or 25700. This course examines topology on the real line, topological spaces, connected spaces and compact spaces, identification spaces and cell complexes, and projective and other spaces. With MATH 27400, it forms a foundation for all advanced courses in analysis, geometry, and topology. Winter.
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