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Course Criteria
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4.00 Credits
SPRING Prerequisite: MATH 532 or permission of the instructor. The theory of fields and character theory. Topics: Galois theory, finite fields, cyclotomic extensions, transcendental extensions, group rings, Wedderburn's Theorem, Schur orthogonality relations.
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1.00 - 5.00 Credits
FALL Prerequisite: MATH 360 or 460 or a course in Topology. Examines properties of abstract topological spaces and mappings including compactness and connectedness, conditions for metrizability.
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4.00 Credits
Prerequisite: MATH 551. Discusses uniformities and proximities, nets and filters, compactification, completeness, function spaces, quotient spaces and related concepts.
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4.00 Credits
Prerequisite: MATH 552. Selected topics, depending on the interest of the class and instructor, chosen from such areas as point-set topology, linear topological spaces, homotopy theory, homology theory, topological groups and topological dynamics.
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4.00 Credits
WINTER-ODD YEARS Prerequisites: MATH 551. This course presents the concepts of general measure and integration theory including the Lebesque integral and its properties.
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4.00 Credits
SPRING-ODD YEARS Prerequisite: MATH 561. Examines the concept of derivative in a measure theoretic setting, as well as product measures and Fubini's theorem.
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4.00 Credits
FALL Prerequisite: Graduate standing or permission of the instructor. The course focuses on the mathematics of applications, depending on the interests of the class and the instructor. This course may be repeated for credit; topics will be specified in the section subtitle.
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4.00 Credits
WINTER-EVEN YEARS Prerequisite: MATH 551. This course establishes the basic properties of holomorphic functions, including complex derivatives, power series, singularities, residues and the general integral formula of Cauchy. In particular, the course proves such classical results as the Fundamental Theorem of Algebra, the Open Mapping Theorem, the Maximum Principle and the theorems of Weierstrass, Montel or Looman-Menchoff. This course also presents examples of elementary conformal mappings, with optional applications to cartography or physics, from geometric or analytic points of view.
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4.00 Credits
SPRING-EVEN YEARS Prerequisite: MATH 581. Continues MATH 581 through the proofs of advanced results, such as the general Riemann Mapping Theorem, or properties of the special functions of Riemann and Weierstrass. If time permits, may include application to Algebraic Geometry, Number Theory and Coding or extensions to several complex variables, for example.
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5.00 Credits
Prerequisite: Bachelor's degree or permission of instructor. Designed to expose participants to a variety of instructional techniques for teaching mathematics concepts and skills at the K-8 level. Strengths and weaknesses of different techniques, such as lecture demonstration, small-group activities and problem solving are modeled and discussed.
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