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Course Criteria
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3.00 Credits
MODERN ALGEBRA II (3) Continuation of MATH 5220/420. Prerequisite: MATH 5220/420. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
COMPLEX ANALYSIS II (3) Harmonic functions and the Dirichlet problem. Introduction to Riemann surfaces. Negative curvature and Picard's Big Theorem. Further topics may include: Hardy spaces, Corona theorem, a deeper study of Riemann surfaces, the uniformization theorem. Prerequisites: Math 5440/422 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
COMMUTATIVE ALGEBRA (3) Introduction to commutative rings and modules over commutative rings. Chain conditions, Noetherian and Artinian rings. Localization. Finitely generated algebras over a field, varieties. Further topics may include: discrete valuation rings and Dedekind domains, completions, Nullstellansatz. Prerequisites: Math 5220/420, Math 5230/421. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
MODERN APPLIED ALGEBRA (3) Algebraic theory with applications to theoretical computing. Topics include finite state automata, Turing machines, computability, the theoretical limits of computers, and coding theory. Prerequisite: MATH 4250/361. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
INTRODUCTION TO COMPUTATIONAL ALGEBRAIC GEOMETRY (3) Geometry of curves/surfaces defined by polynomial equations. Emphasis on concrete computations with polynomials using computer packages, interplay between algebra and geometry as well as algebra and biology. Algebra and topology presented as needed. Prerequisites: MATH 5220/420. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
THEORY AND APPLICATION OF INFINITE SERIES (3) Infinite series of real and complex terms including the summability methods of Abel and Cesaro; tests for convergence of series of positive constants including those of Cauchy, Maclaurin, Gauss; alternating series; conditional convergence and Riemann?s theorem; absolute and uniform convergence; theorems of Weierstrass, Cauchy, Abel, Levi and Sternitz; introduction to the theory of divergent series. Prerequisite: MATH 4410/358. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
ANALYSIS I (3) Functions of bounded variation, Riemann-Stieljes integral, topology of the real line, measure theory, measurable functions, Lebesgue integral, other selected topics. Prerequisites: MATH 4410/358. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
ANALYSIS II (3) Continuation of Math 5420/415: Radon-Nikodym theorem, Lp spaces, Riesz Representation theorem, functions of several variables, inverse and implicit function theorems, smooth manifolds, tangent and cotangent bundles, vector bundles, differential forms, pullback, wedge product, integration, Poincare lemma. Prerequisites: MATH 5420/415. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
COMPLEX VARIABLES I (3) Complex numbers and polygenic and monogenic functions, theory of residues, Taylor and Laurent series, Cauchy-Riemann and Laplace equations. Prerequisite: MATH 4450/356 or 4410/358. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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3.00 Credits
FUNCTIONAL ANALYSIS (3) Hahn-Banach theorem, weak topologies, operators on Hilbert and Banach spaces, normal, self-adjoint and compact operators, geometric and spectral analysis of linear operators, generalized functions. At instructor's discretion: applications to Fourier series, numerical analysis, probability and dynamics, or differential equations will be discussed. Prerequisites: MATH 5420/415. 3.000 Credit Hours 3.000 Lecture hours Levels: Graduate Schedule Types: Lecture College of Arts and Sciences College Mathematics & Computer Science Department
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