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Course Criteria
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3.00 Credits
Theory of Lebesgue measure and integration, monotone convergence, teh dominated convergence theorem, Fubini's Theorem, Radon-Nikodym theorem,Riesz representation theorem, Lp and lp spaces, functions of finite variation, Stieltjes integral, absolute continuity.
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3.00 Credits
The theory of linear and nonlinear partial differential equations. Topics include: classical theory of elliptic, parabolic and hyperbolic partial differential equations and their solutions, potential theory, maximum principle, existence of weak solutions, regularity of solutinos, Duhamel's principle and Cauchy's problem.
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3.00 Credits
An in-depth study of functions of one complex variable. Topics include: properties of holomorphic, harmonic, meromorphic and entire functions (open mapping, maximum modulus, mean value, Poisson's, Rouche's, Liouville's, Picard's and Mittag-Leffler's theorems), residue theory (residue theorem, argument principle and applications), conformal mappings (Mobius and Christoffel- Schwarz canonical transformations, Riemann mapping theorem), analytic continuation (monodromy theorem, Schwarz reflection principle, Riemann surfaces and multi-valued functions).
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3.00 Credits
The study of the approximation of functions in normed linear spaces. The course emphasizes the theory of interpolation and approximation by polynomials, rational functions and spline functions. Main topics include: best approximation, order of approximation, interpolation, existence and uniqueness of best approximants, theorems by Weierstrass, Haar, Chebyshev, Bernstein, Markov, Korovkin, Schoenberg, and applications.
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3.00 Credits
This course provides a comprehensive study of group theory. The course begins with basic concepts of group theory (binary structures, subgroups, homomorphisms) and continues with the study of normal subgroups, quotient groups and the isomorphism theorems. Further topics to be studied include group actions, Sylow's theorem and the structure of finitely generated abelian groups.
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3.00 Credits
The course provides a comprehensive study of rings and fields. The course begins with the basic concepts (rings, subrings, ideals, quotient rings, homomorphisms), continues with the arithmetic of rings, applications to rings of polynomials and field theory, and concludes with a chapter on Galois theory that links field theory and group theory.
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3.00 Credits
The study and applications of calculus on manifolds. Topics include: atlases, tangent spaces, differentiable maps; immersions and submanifolds, submersions and quotient manifolds; matrix groups and their Lie algebras; vector fields and flows; differential forms, exterior derivative, and Lie derivative.
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3.00 Credits
The study of the topology of geometric objects from the algebraic viewpoint, in particular using homotopy and homology groups. Main topics: Topological manifolds, homotopy, fundamental group, free groups, covering spaces, and homology.
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1.00 Credits
Under supervision of one or more faculty members, each student will choose topics related to his or her concentration, or topics of interest to the class, read and research on them, then make presentations in front of the class or a larger audience. Students will also attend presentations of internal and external speakers on mathematical sciences.
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3.00 Credits
No course description available.
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