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26:645:611Real 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, Lebesque measure and measurable functions, Lebesque integral, the dominated convergence theorem, comparison of the Riemann and the Lebesque integrals. |
26:645:612Real Analysis II (3) Lebesque Measure Theory: Lebesque measure, Lebesque 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 Lebesque 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. |
26:645:613Functional 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. |
26:645:621Complex 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. |
26:645:622Complex 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. |
26:645:623Selected Topics in Complex Analysis (3) Prerequisite: 26:645:621. |
26:645:631Algebra 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. |
26:645:632Algebra 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. |
26:645:633Selected Topics in Algebra (3) Prerequisite: 26:645:632. |
26:645:634Number 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. |
26:645:635Algebraic Geometry (3) Geometry of projectives spaces, cohomology of coherent sheaves, and schemes. Prerequisites: 26:645:611 and 631. |
26:645:636Theory 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. |
26:645:641Topology I (3) Metric spaces, connectedness, compactness, Tychonoff`s theorem, Baire category theorem, simplicial complexes, CW-complexes, manifolds, fundamental group, covering spaces, VanKampen`s theorem, computations of the fundamental groups of CW-complexes, including graphs, surfaces, knot complements, Sn projective and spaces, Brouwer`s fixed point theorem, simplicial approximation, and general position. |
26:645:642Topology 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. |
26:645:643Differentiable 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. |
26:645:644Geometric and Differential Topology (3) * Cohomology theories, transversality, Poincare duality, topics of instructor`s choice. Prerequisites: 26:645:642 and 643. |
26:645:645Differential Geometry (3) * Riemannian metrics, parallel translation and connections, curvature, exponential map, integrability theorems, and topics of instructor`s choice. Prerequisite: 26:645:643. |
26:645:647Cryptography (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. |
26:645:721Advanced Topics in Complex Analysis (3) Prerequisite: Permission of instructor. |
26:645:731Advanced Topics in Algebra (3) Prerequisite: Permission of instructor. |
26:645:734Advanced Topics in Number Theory (3) Prerequisite: Permission of instructor. |
26:645:736Advanced Topics in Representation Theory (3) Prerequisite: Permission of instructor. |
26:645:741Advanced Topics in Topology (3) Prerequisite: Permission of instructor. |
26:645:742Dynamical 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. |
26:645:744Advanced Topics in Geometry (3) Prerequisite: Permission of the instructor. |
26:645:750Independent Study (BA) Study under supervision and guidance of a faculty member. |
26:645:791Doctoral 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. |
26:645:799Doctoral 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 term; registration for additional credits is permitted with the approval of the adviser, up to a maximum of 12 credits per term. |
26:645:800Matriculation Continued (E1) |
26:645:811Graduate Fellowship (E,BA) |
26:645:866Graduate Assistantship (E,BA) |
26:645:877Teaching Assistantship (E,BA) |
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