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Course Criteria
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1.00 - 3.00 Credits
Topics vary. May be repeated for a total of 12 credits in 267 and 297 combined if there is no duplication in topic. Students may enroll in more than one section of this course each semester.
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3.00 Credits
A written presentation of research results, original for the student but not usually original in the larger sense. The regulations governing the writing of a master of arts thesis in mathematics will apply to the writing of the senior thesis.
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3.00 Credits
Manifolds; submanifolds; tangent and vector bundles. Vector fields and flows, Lie brackets, distributions, and the Frobenius theorem. Sard’s theorem; transversality and intersection theory; degree theory and applications. Tensors and differential forms; the exterior derivative; Stokes’ theorem and integration; de Rham cohomology. Prerequisite: linear algebra and either 242 or 272a.
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3.00 Credits
Connectedness, compactness, countability, and separation axioms. Complete metric spaces. Function spaces.
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3.00 Credits
The fundamental group and covering spaces. Topology of surfaces. Simplicial complexes and homology theory. Homotopy theory. Prerequisite: 242.
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3.00 Credits
Elements of enumerative analysis including permutations, combinations, generating functions, recurrence relations, the principle of inclusion and exclusion, and Polya’s theorem. Some special topics will be treated as class interest and background indicate (e.g., Galois fields, theory of codes, and block designs). Students unfamiliar with permutations, combinations, and basic counting techniques should take 215 prior to 274.
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3.00 Credits
The mathematical theory of networks. Traversing graphs using paths, cycles, and trails. Matchings and other graph factors. Coloring of vertices and edges. Connectivity and its relation to paths and flows. Embeddings of graphs in surfaces, especially the plane. Prerequisite: linear algebra. Students unfamiliar with basic ideas of graph theory, including paths, cycles, and trees, should take 215 prior to 275.
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3.00 Credits
The basic operations on sets. Cardinal and ordinal numbers. The axiom of choice. Zorn’s lemma, and the well-ordering principle. Introduction to the topology of metric spaces, including the concepts of continuity, compactness, connectivity, completeness, and separability. Product spaces. Applications to Euclidean spaces. Strongly recommended for beginning graduate students and for undergraduates who plan to do graduate work in mathematics. Prerequisite: multivariable calculus and linear algebra.
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3.00 Credits
Group theory through Sylow theorems and fundamental theorem of finitely generated abelian groups. Prerequisite: linear algebra. An elementary course in modern algebra (e.g., 223) is strongly recommended.
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3.00 Credits
Introductory theory of commutative rings and fields, and additional topics such as Galois theory, modules over a principal ideal domain and finite dimensional algebras. Prerequisite: linear algebra. An elementary course in modern algebra (e.g., 223) is strongly recommended.
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