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Graduate Course Offerings Please note: courses are only available to graduate students at the George Washington University
Algebra I is the basic algebra graduate course offering, however it is not a prerequisite for Algebra II and III. Students may elect to take those as desired. Algebra I introduces the ideas of group theory including symmetric groups, free abelian groups, finitely generated abelian groups, Sylow theorems, and solvable groups. The course also touches on factorization in commutative rings, rings of polynomials, chain conditions, semisimple rings, Wedderburn-Artin theorems, Galois theory. The topics of Algebra II pick up where Algebra I left off. It allows graduate students to continue their study of algebra and prepare them for further exploration of the topic. This course offering will lead into subjects discussed in Algebra III. Algebra III is an extension of the material of Math 201-2, including Frobenius' theorem on associative division algebras, the Hurwitz problem on composition of forms, valuation theory, formally real fields, rings without finiteness conditions, elements of homological algebra with applications. Algebraic topics are chosen from Lie groups and Lie algebras, non-associative algebras, abelian groups, classical groups, algebraic number theory, representation theory, algebraic geometry, and ring theory. Topic in Algebra requires previous completion of Math 201-202. Math 206 be repeated for credit with permission of the instructor. Math 211 requires previous completion of Math 139, or an equivalent course. Topics covered in Complex Analysis include topology of the complex plane; complex differentiation and integration; Cauchy's theorem and its consequences; Taylor and Laurent series; classification of singularities; residue theory; conformal mapping; and the Riemann mapping theorem. Graduate course requires completion of Math 139, or an equivalent course. Math 214 allows graduate students to further their understanding of mathematical theories through its exploration of Lebesgue measure and integration in abstract spaces, probability measures, absolute continuity, the Radon-Nikodym theorem, measures on product spaces, and the Fubini theorem. The course also touches on LP spaces and their properties. The prerequisite for Math 215 is either completion of Math 214, or an equivalent course. Topics covered in this course include Topological and metric spaces; Tychonoff theorem; Banach spaces; linear functionals and operators; Hahn-Banach, closed graph, and open-mapping theorems; uniform boundedness; Hilbert spaces; eigenvalues,and projections. Enrollment in Math 216 requires only for the graduate student to have received permission from the instructor. Possible topics include Banach algebras, function algebras, spectral theory for bounded and unbounded operators, harmonic analysis on topological groups and semigroups, topological vector spaces and operator algebras. It is possible to repeat Math 216 for credit with permission of the professor. The prerequisites for this course include knowledge of matrix theory as well as previous completion of Math 140. In this course graduate students will explore the existence and uniqueness of solutions, continuity and differentiability of solutions with respect to initial conditions as well as the properties of linear systems, phase portraits, planar systems and Poincaré endixson theory. Course requires previous completion of Math 140, however enrollment is also granted with the permission of the instructor. This course explores classical techniques for the solution of linear partial differential equations, Poisson's equation, heat equation, and wave equation. Topics covered also include the existence and uniqueness of solutions, maximum principles, separation of variables, Fourier series, eigenfunction expansions, and Green's functions. Enrollment in this course requires completion of Math 140, or the approval of the instructor. This course places emphasis on modern theory and analytical techniques applied to the solution of partial differential equations. Topics include Sobolev spaces, generalized solutions, strong solutions and regularity; Sobolev imbedding theorem; Rellich-Kondrachov theorem; Leray-Schauder fixed-point theorems; nonlinear eigenvalue problems. Math 222 is inherently different than other graduate level courses. This course requires the student to have prior knowledge of a programming language as well as completion of Math 33. In this course the student will explore computer arithmetic and round-off errors, solution of linear and nonlinear systems, interpolation and approximations, numerical differentiation and integration, as well as igenvalues and eigenvectors. Initial and boundary value problems for ordinary differential equations. Error propagation, convergence and stability. Finite difference and finite element methods for partial differential equations. Prerequisite: Math 111 and knowledge of a programming language. Ergodicity, mixing, the K-property and the Bernoulli property. Poincaré recurrence, the Rohlin lemma, the ergodic theorem, and entropy theory. Additional topics from isomorphism theory, spectral theory, the theory of joinings, and coding theory. Prerequisite: Math 214 or permission of instructor. Linear and nonlinear systems, flows, Poincaré maps, structural stability. Examples of chaotic systems in the physical sciences. Local bifurcations, center manifold theory, normal forms, the averaging theorem. Hyperbolic invariant sets, strange attractors, the Smale horseshoe, symbolic dynamics. Prerequisite: Math 124 and 140 or permission of instructor. Possible topics include, but are not limited to, the calculus of variations, control theory, nonlinear partial differential equations, and mathematical programming. May be repeated for credit with permission. Math 32 requires previous completion of Math 111 and 124. In addition to these general requirements, It is also requested that the graduate student has a previous knowledge of a programming language. The course will explore numerical methods in regards to software, as well as provdide introductions to the methods, tools, and ideas of numerical computation. Math 261 requires either the permission of the instructor or completion of undergraduate level algebra and linear algebra. The course will provide an introudction to fundamental methods and current resesearch problems in partially ordered sets, enumeration, tableaux, and partitions. Model theory: the relation between a formal language (syntax) and its interpretations (semantics). Consistency, completeness, and compactness. Tarski's theorem on the inexpressibility of truth. Godel's incompleteness theorem and its impact on mathematics.
Fundamental groups and the Van Kampen theorem; simplicial complexes, simplicial homology, and Euler characteristic; singular homology, Mayer-Vietoris sequences. Topics may include cohomology, cup products, and Poincaré duality; classification of surfaces; knots and their fundamental groups. Prerequisite: Math 281 or permission of instructor. Topics may include hyperbolic structures on surfaces and 3-manifolds; knot theory; topology of 3-manifolds; topology of 4-manifolds. Prerequisite: Math 282 or permission of the instructor. May be repeated for credit with permission. |