Rutgers, The State University of New Jersey
Graduate School–Newark
 
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American Studies 050
Behavioral and Neural Sciences 112
Biology 120
Business and Science 137
Chemistry 160
Creative Writing 200
Criminal Justice 202
Economics 220
English 350 (Includes American Literature 352)
Environmental Science 375
Environmental Geology 380
Global Affairs 478
History 510
Jazz History and Research 561
Liberal Studies 606
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Mathematical Sciences 645
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Nursing 705
Peace and Conflict Studies 735
Physics, Applied 755
Political Science 790
Psychology 830
Public Administration 834
Sustainability: Urban Eco-sustainability Track
Urban Environmental Analysis and Management
Urban Systems 977 (Joint Ph.D. Program with NJIT and UMDNJ)
Women's and Gender Studies 988
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Camden Newark New Brunswick/Piscataway
Catalogs
  Graduate School–Newark 2012–2014 Programs, Faculty, and Courses Mathematical Sciences 645 Graduate Courses  

Graduate Courses

26:645:611 Real Analysis I (3) Continuity and differentiability of functions of many variables, the chain rule, higher derivatives, Taylor's theorem, maxima and minima, metric spaces, completeness, contraction mapping principle, inverse functions and the implicit function theorem, the Riemann integral and its properties, Lebesgue measure and measurable functions, Lebesgue integral, the dominated convergence theorem, comparison of the Riemann and the Lebesgue integrals.
26:645:612 Real Analysis II (3) Lebesgue Measure Theory: Lebesgue measure, Lebesgue integral, functions of bounded variation, differentiation of integrals, absolute continuity and convex functions, Lp spaces. Minkowski and Hoelder inequalities, convergence, completeness. General Measure Theory: measure spaces and functions, integration, convergence theorems, signed measures, Radon-Nikodym theorem, the Lebesgue Stieltjes integral, product measures and the Fubini theorem, the Hausdorff measure, Baire sets and Borel sets, regularities of Baire and Borel measures, construction of Borel measures, homogeneous spaces. Harmonic Analysis: Fourier analysis on R and R/Z, harmonic analysis on locally compact groups, existence of Haar measure, example: SL(n), Pontryagin duality, Fourier inversion, representation of compact groups, decomposition of L2. Introduction into ODEs and PDEs: existence and uniqueness theorems. Prerequisite: 26:645:611.
26:645:613 Functional Analysis (3) Fundamental principles of linear analysis: Hahn-Banach, uniform boundedness and closed graph theorems, Riesz representation theorem, weak topologies, Riesz theory of compact operators, spectral theory of operators on Hilbert space, and applications to differential and integral equations. Prerequisite: 26:645:611.
26:645:621 Complex Variables I (3) Complex differentiability, Cauchy-Riemann equations, power series and elementary functions. Cauchy's theorem, the Cauchy integral formula, Cauchy's estimates, Morera's theorem. Entire functions, Liouville's theorem. Convergence, differentiation, and integration of sequences and series of holomorphic functions. Local mapping properties of holomorphic functions: isolation of zeros, conformality, inverse function theorem, critical points. Elementary Riemann surfaces. Classification of singularities. Laurent series. The residue theorem and applications: meromorphic functions, the Mittag-Leffler theorem. Holomorphic functions on the Riemann sphere, Möbius transformations. Maximum modulus principle, Schwarz's lemma, conformal maps of the unit disc. The Riemann mapping theorem, the Schwarz reflection principle. Harmonic functions, harmonic conjugates. The Dirichlet problem and the Poisson kernel for the unit disc. Prerequisite: 26:645:611.
26:645:622 Complex Variables II (3) Theory of Riemann surfaces: uniformization theorem, Abel-Jacobi theorem, theorem of Riemann-Roch and related topics including theta functions, the Riemann theta function, Jacobian functions, Jacobian variety, Abelian variety, etc. Prerequisite: 26:645:621.
26:645:623 Selected Topics in Complex Analysis (3) Prerequisite: 26:645:621.
26:645:631 Algebra I (3) Groups: subgroups; homomorphisms; cyclic groups; Lagrange's theorem; quotient groups; symmetric, alternating, and dihedral groups; direct products and sums; free groups; free abelian groups; finitely generated abelian groups; and Sylow theorems. Rings: homomorphisms, integral domains, fields, ideals, prime and maximal ideals, Chinese remainder theorem, factorization in commutative rings, UFD, PID, Euclidean rings, rings of quotients, localization, local rings, polynomial rings, Gauss's lemma, and Eisenstein criterion.
26:645:632 Algebra II (3) Modules: left, right, and bimodules; direct sums and products of modules; homomorphisms; exact sequences; free modules; vector spaces; Hom and duality of modules; tensor products; modules over a PID; and elementary divisors. Galois theory: finite extensions, algebraic extensions, minimal polynomials, Galois extensions, fundamental theorem of Galois theory, elementary symmetric functions, splitting fields, algebraic closure, normal and separable extensions, fundamental theorem of algebra, Galois group of a polynomial, finite fields, cyclic extensions, trace and norm, Hilbert's theorem 90, and cyclotomic extensions. Prerequisite: 26:645:631.
26:645:633 Selected Topics in Algebra (3) Prerequisite: 26:645:632.
26:645:634 Number Theory (3) Algebraic number fields, rings of algebraic integers, discriminant, Dedekind domains, unique factorization into prime ideals, ramification theory in Galois extensions, finiteness of ideal class number, Dirichlet's unit theorem, quadratic and cyclotomic fields, the quadratic reciprocity law, the Dedekind zeta function, Dirichlet's class number formula, p-adic fields, and ideles and adeles. Prerequisite: 26:645:631.
26:645:635 Algebraic Geometry (3) Geometry of projectives spaces, cohomology of coherent sheaves, and schemes. Prerequisites: 26:645:611 and 631.
26:645:636 Theory of Lie Groups and Lie Algebra (3) General structure of Lie groups and Lie algebras, semisimple Lie groups, and character theory of compact Lie groups. Prerequisites: 26:645:612 and 632.
26:645:641 Topology I (3) Metric spaces, connectedness, compactness, Tychonoff's theorem, Baire category theorem, simplicial complexes, CW-complexes, manifolds, fundamental group, covering spaces, Van Kampen's theorem, computations of the fundamental groups of CW-complexes, including graphs, surfaces, knot complements, Sn projective and spaces, Brouwer fixed point theorem, simplicial approximation, and general position.
26:645:642 Topology II (3) Singular homology, axioms, Mayer-Vietoris sequence, orientations, homology of CW-complexes including surfaces and projective spaces, higher homotopy groups, homotopy long exact sequences of pairs and fibrations, and Whitehead and Hurewicz theorems. Prerequisite: 26:645:641.
26:645:643 Differentiable Manifolds (3) Inverse and implicit function theorems, differential forms, Sard's theorem, Stokes' theorem, degree of a map, tangent and related bundles, deRham cohomology, Riemannian metrics, connections, and the intrinsic and extrinsic geometry of surfaces in 3-space.
26:645:644 Geometric and Differential Topology (3) Cohomology theories, transversality, Poincare duality, topics of instructor's choice. Prerequisite: 26:645:642. 
26:645:645 Differential Geometry (3) Riemannian metrics, parallel translation and connections, curvature, exponential map, integrability theorems, and topics of instructor's choice. Prerequisite: 26:645:643.
26:645:647 Cryptography (3) Review of basic material from algebra and number theory, primality tests, factorization methods, simple cryptosystems, public key cryptography, the RSA algorithm, discrete logs, the knapsack problem and related cryptosystems, and applications to electronic banking and electronic cash.
26:645:721 Advanced Topics in Complex Analysis (3) Prerequisite: Permission of instructor.
26:645:731 Advanced Topics in Algebra (3) Prerequisite: Permission of instructor.
26:645:734 Advanced Topics in Number Theory (3) Prerequisite: Permission of instructor.
26:645:736 Advanced Topics in Representation Theory (3) Prerequisite: Permission of instructor.
26:645:741 Advanced Topics in Topology (3) Prerequisite: Permission of instructor.
26:645:742 Dynamical Systems (3) Introduction to the mathematical study of chaos and fractals from examples in one-dimensional real and complex dynamical systems. Prerequisites: 26:645:611, 612, 621, 641. Recommended: 26:645:622.
26:645:744 Advanced Topics in Geometry (3) Prerequisite: Permission of the instructor.
26:645:750 Independent Study (BA) Study under supervision and guidance of a faculty member.
26:645:791 Doctoral Seminar (3) A seminar in which faculty, students, and invited speakers present summaries of advanced topics in the mathematical sciences. Students and faculty discuss research procedures and dissertation organization and content. Doctoral students present their own research for discussion and criticism. Corequisite: 26:645:799.
26:645:799 Doctoral Dissertation and Research (BA) Research in the mathematical sciences carried out under the supervision of a faculty member. Culminates in a written dissertation to be published in a leading research journal. Prerequisite: Doctoral candidacy. Corequisite: 26:645:791. A minimum of 24 credits is required. The student must register for at least 6 credits per semester; registration for additional credits is permitted with the approval of the adviser, up to a maximum of 12 credits per semester.
26:645:800 Matriculation Continued (E1)
26:645:811 Graduate Fellowship (E,BA)
26:645:866 Graduate Assistantship (E,BA)
26:645:877 Teaching Assistantship (E,BA)
 
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