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Math Home Math Graduate Home Graduate School Home Admission Requirments Financial Support Graduate Bulletin MA and MS Programs MM and MAT Programs Doctoral Program Faculty Research Interests Master Schedule Math Calendar International Information Search USC  THIS SITE		Course Descriptions for (MATH) 7xx and 8xx 700 — Linear Algebra. (3) Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms. 701 — Algebra I. (3) Algebraic structures, sub-structures, products, homomorphisms, and quotient structures of groups, rings, and modules. 702 — Algebra II. (3) (Prereq: MATH 701) Fields and field extensions. Galois theory, topics from, transcendent field extensions, algebraically closed fields, finite fields. 703 — Analysis I. (3 each) Compactness, completeness, continuous functions. Outer measures, measurable sets, extension theorem and Lebesgue-Stieltjes measure. Integration and convergence theorems. Product measures and Fubini's theorem. Differentiation theory. Theorems of Egorov and Lusin. Lp-spaces. Analytic functions: Cauchy-Riemann equations, elementary special functions. Conformal mappings. Cauchy's integral theorem and formula. Classification of singularities, Laurent series, the Argument Principle. Residue theorem, evaluation of integrals and series. 704 — Analysis II. (3 each) Compactness, completeness, continuous functions. Outer measures, measurable sets, extension theorem and Lebesgue-Stieltjes measure. Integration and convergence theorems. Product measures and Fubini's theorem. Differentiation theory. Theorems of Egorov and Lusin. Lp-spaces. Analytic functions: Cauchy-Riemann equations, elementary special functions. Conformal mappings. Cauchy's integral theorem and formula. Classification of singularities, Laurent series, the Argument Principle. Residue theorem, evaluation of integrals and series. 705 — Analysis III. (3) (Prereq: MATH 703, 704) Signed and complex measures, Radon-Nikodym theorem, decomposition theorems. Metric spaces and topology, Baire category, Stone-Weierstrass theorem, Arzela-Ascoli theorem. Introduction to Banach and Hilbert spaces, Riesz representation theorems. 708 — Foundations of Computational Mathematics I. (3) (Prereq: MATH 554 or equivalent upper level undergraduate course in Real Analysis) Approximation of functions by algebraic polynomials, splines, and trigonometric polynomials; numerical differentiation; numerical integration; orthogonal polynomials and Gaussian quadrature; numerical solution of nonlinear systems, unconstrained optimization. 709 — Foundations of Computational Mathematics II. (3) (Prereq: MATH 544 or 526, or equivalent upper level undergraduate course in Real Analysis) Vectors and matrices; QR factorization; conditioning and stability; solving systems of equations; eigenvalue/eigenvector problems; Krylov subspace iterative methods; singular value decomposition. Includes theoretical development of concepts and practical algorithm implementation. 710 — Probability Theory I. {=STAT 710} (3) (Prereq: STAT 511, 512, or MATH 703) Probability spaces, random variables and distributions, expectations, characteristic functions, laws of large numbers, and the central limit theorem. 711 — Probability Theory II. {=STAT 711} (3) (Prereq: MATH 710) More about distributions, limit theorems, Poisson approximations, conditioning, martingales, and random walks. 720 — Applied Mathematics I. (3) (Prereq: MATH 555 and MATH 520 or equivalent) Methods for solving equations from applied mathematics and the natural sciences, including a study of boundary value problems, integral equations, and eigenvalue problems using transform techniques, Green's functions, and variational principles. 721 — Applied Mathematics II. (3) (Prereq: MATH 720) Topics in partial differential equations with emphasis on the equations of the natural sciences; includes classifications of higher order equations, existence and uniqueness of solutions, theory of characteristics, basic properties of elliptic and parabolic equations, Dirichlet and Neumann problems, and the Cauchy problem for hyperbolic equations. 722 — Numerical Optimization. (3) (Prereq: graduate standing or consent of the department) Topics in optimization; includes linear programming, integer programming, gradient methods, least squares techniques, and discussion of existing mathematical software. 723 — Differential Equations. (3) (Prereq: MATH 703, 704 or permission of instructor) Elliptic equations: fundamental solutions, maximum principles, Green’s function, energy method and Dirichlet principle; Sobolev spaces: weak derivatives, extension and trace theorems; weak solutions and Fredholm alternative, regularity, eigenvalues and eigenfunctions. 724 — Differential Equations II. (3) (Prereq: MATH 723) Detailed study of the following topics: method of characteristics; Hamilton-Jacobi equations; conservation laws; heat equation; wave equation; linear parabolic equations; linear hyperbolic equations. 725 — Approximation Theory. (3) (Prereq or coreq: MATH 703) Approximation of functions; existence, uniqueness and characterization of best approximants; Chebyshev's theorem; Chebyshev polynomials; degree of approximation; Jackson and Bernstein theorems; B-splines; approximation by splines; quasi-interpolants; spline interpolation. 726 — Numerical Differential Equations I. (3) (Prereq: MATH 708, 709 or permission of instructor) Elliptic equations: fundamental solutions, maximum principles, Green’s function, energy method and Dirichlet principle; Sobolev spaces: weak derivatives, extension and trace theorems; weak solutions and Fredholm alternative, regularity, eigenvalues and eigenfunctions. 727 — Numerical Differential Equations II. (3) (Prereq: MATH 726) Ritz and Galerkin weak formulation. Finite element, mixed finite element, collocation methods for elliptic, parabolic, and hyperbolic PDEs, including development, implementation, stability, consistency, convergence analysis, and error estimates. 728 — Selected Topics in Applied Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 729 — Nonlinear Approximation. (3) (Prereq: MATH 703) Nonlinear approximation from piecewise polynomial (spline) functions in the univariate and multivariate case, characterization of the approximation spaces via Besov spaces and interpolation, Newman's and Popov's theorems for rational approximation, characterization of the approximation spaces of rational approximation, nonlinear n-term approximation from bases in Hilbert spaces and from unconditional bases in Lp(p>1), greedy algorithms, application of nonlinear approximation to image compression. 730 — General Topology I. (3 each) Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces. 731 — General Topology II. (3 each) Topological spaces, filters, compact spaces, connected spaces, uniform spaces, complete spaces, topological groups, function spaces. 732 — Algebraic Topology I. (3 each) (Prereq: MATH 730 or 705, and 701) The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cup-product, triangulable spaces. 733 — Algebraic Topology II. (3 each) (Prereq: MATH 730 or 705, and 701) The fundamental group, homological algebra, simplicial complexes, homology and cohomology groups, cup-product, triangulable spaces. 734 — Differential Geometry. (3) (Prereq: MATH 550) Differentiable manifolds; classical theory of surfaces and hypersurfaces in Euclidean space; tensors, forms and integration of forms; connections and covariant differentiation; Riemannian manifolds; geodesics and the exponential map; curvature; Jacobi fields and comparison theorems, generalized Gauss-Bonnet theorem. 735 — Lie Groups. (3) (Prereq: MATH 705 or 730) Manifolds; topological groups, coverings and covering groups; Lie groups and their Lie algebras; closed subgroups of Lie groups; automorphism groups and representations; elementary theory of Lie algebras; simply connected Lie groups; semisimple Lie groups and their Lie algebras. 738 — Selected Topics in Geometry and Topology. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 741 — Algebra III. (3 each) (Prereq: MATH 702) Theory of rings, modules, fields, bilinear forms, and advanced topics in matrix theory. 742 — Algebra IV. (3 each) (Prereq: MATH 702) Theory of rings, modules, fields, bilinear forms, and advanced topics in matrix theory. 743 — Lattice Theory. (3) (Prereq: MATH 702) Sublattices, homomorphisms and direct products of lattices; freely generated lattices; modular lattices and projective geometries; the Priestley and Stone dualities for distributive and Boolean lattices; congruence relations on lattices. 744 — Matrix Theory. (3) (Prereq: MATH 700) Extremal properties of positive definite and hermitian matrices, doubly stochastic matrices, totally non-negative matrices, eigenvalue monotonicity, Hadamard-Fisher determinantal inequalities. 746 — Commutative Algebra. (3) (Prereq: MATH 701) Prime spectrum and Zariski topology; finite, integral, and flat extensions; dimension; depth; homological techniques, normal and regular rings. 747 — Algebraic Geometry. (3) (Prereq: MATH 701) Properties of affine and projective varieties defined over algebraically closed fields, rational mappings, birational geometry and divisors especially on curves and surfaces, Bezout's theorem, Riemann-Roch theorem for curves. 748 — Selected Topics in Algebra. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 750 — Fourier Analysis. (3) (Prereq: MATH 703 and 704) The Fourier transform on the circle and line, convergence of Fejer means; Parseval's relation and the square summable theory, convergence and divergence at a point; conjugate Fourier series, the conjugate function and the Hilbert transform, the Hardy-Littlewood maximal operator and Hardy spaces. 751 — The Mathematical Theory of Wavelets. (3) (Prereq: MATH 703) The L1 and L2 theory of the Fourier transform on the line, bandlimited functions and the Paley-Weiner theorem, Shannon-Whittacker Sampling Theorem, Riesz systems, Mallat-Meyer multiresolution analysis in Lebesgue spaces, scaling functions, wavelet constructions, wavelet representation and unconditional bases, nonlinear approximation, Riesz's factorization lemma, and Daubechies' compactly supported wavelets. 752 — Complex Analysis. (3) (Prereq: MATH 703, 704) Normal families, meromorphic functions, Weierstrass product theorem, conformal maps and the Riemann mapping theorem, analytic continuation and Riemann surfaces, harmonic and subharmonic functions. 754 — Several Complex Variables. (3) (Prereq: MATH 703 and 704) Properties of holomorphic functions of several variables, holomorphic mappings, plurisubharmonic functions, domains of convergence of power series, domains of holomorphy and pseudoconvex domains, harmonic analysis in several variables. 755 — Applied Functional Analysis. (3) (Prereq: MATH 703) Banach spaces, Hilbert spaces, spectral theory of bounded linear operators, Fredholm alternatives, integral equations, fixed point theorems with applications, least square approximation. 756 — Functional Analysis I. (3 each) (Prereq: MATH 704) Linear topological spaces; Hahn-Banach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras. 757 — Functional Analysis II. (3 each) (Prereq: MATH 704) Linear topological spaces; Hahn-Banach theorem; closed graph theorem; uniform boundedness principle; operator theory; spectral theory; topics from linear differential operators or Banach algebras. 758 — Selected Topics in Analysis. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 760 — Set Theory. (3) An axiomatic development of set theory: sets and classes; recursive definitions and inductive proofs; the axiom of choice and its consequences; ordinals; infinite cardinal arithmetic; combinatorial set theory. 761 — The Theory of Computable Functions. (3) Models of computation; recursive functions, random access machines, Turing machines, and Markov algorithms; Church's Thesis; universal machines and recursively unsolvable problems; recursively enumerable sets; the recursion theorem; the undecidability of elementary arithmetic. 762 — Model Theory. (3) First order predicate calculus; elementary theories; models, satisfaction, and truth; the completeness, compactness, and omitting types theorems; countable models of complete theories; elementary extensions; interpolation and definability; preservation theorems; ultraproducts. 768 — Selected Topics in Foundations of Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 770 — Discrete Optimization. (3) The application and analysis of algorithms for linear programming problems, including the simplex algorithm, algorithms and complexity, network flows, and shortest path algorithms. No computer programming experience required. 774 — Discrete Mathematics I. (3) An introduction to the theory and applications of discrete mathematics. Topics include enumeration techniques, combinatorial identities, matching theory, basic graph theory, and combinatorial designs. 775 — Discrete Mathematics II. (3) (Prereq: MATH 774 or consent of the instructor) A continuation of MATH 774. Additional topics will be selected from: the structure and extremal properties of partially ordered sets, matroids, combinatorial algorithms, matrices of zeros and ones, and coding theory. 776 — Graph Theory I. (3) The study of the structure and extremal properties of graphs, including Eulerian and Hamiltonian paths, connectivity, trees, Ramsey theory, graph coloring, and graph algorithms. 777 — Graph Theory II. (3) (Prereq: MATH 776 or consent of instructor) Continuation of MATH 776. Additional topics will be selected from: reconstruction problems, independence, genus, hypergraphs, perfect graphs, interval representations, and graph-theoretical models. 778 — Selected Topics in Discrete Mathematics. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 780 — Elementary Number Theory. (3) Diophantine equations, distribution of primes, factoring algorithms, higher power reciprocity, Schnirelmann density, and sieve methods. 782 — Analytic Number Theory I. (3) (Prereq: MATH 580 and 552) The prime number theorem, Dirichlet's theorem, the Riemann zeta function, Dirichlet's L-functions, exponential sums, Dirichlet series, Hardy-Littlewood method partitions, and Waring's problem. 783 — Analytic Number Theory II. (3) (Prereq: MATH 580 and 552) The prime number theorem, Dirichlet's theorem, the Riemann zeta function, Dirichlet's L-functions, exponential sums, Dirichlet series, Hardy-Littlewood method partitions, and Waring's problem. 784 — Algebraic Number Theory. (3) (Prereq: MATH 546 and 580) Algebraic integers, unique factorization of ideals, the ideal class group, Dirichlet's unit theorem, application to Diophantine equations. 785 — Transcendental Number Theory. (3) (Prereq: MATH 580) Thue-Siegel-Roth theorem, Hilbert's seventh problem, diophantine approximation. 788 — Selected Topics in Number Theory. (3) Course content varies and will be announced in the schedule of classes by suffix and title. 790 — Graduate Seminar. (1) (Although this course is required of all candidates for the master's degree it is not included in the total credit hours in the master's program.) 797 — Mathematics into Print. (3) The exposition of advanced mathematics emphasizing the organization of proofs and the formulation of concepts; computer typesetting systems for producing mathematical theses, books, and articles. 798 — Directed Readings and Research. (1-6) (Prereq: full admission to graduate study in mathematics). 799 — Thesis Preparation. (1-9) For master's candidates. 890 — Graduate Seminar. (1-3) A review of current literature in specified subject areas involving student presentations. Content varies and will be announced in the schedule of classes by suffix and title. Minimum of 3 credit hours required of all doctoral students. (Pass-Fail grading) 899 — Dissertation Preparation. (1-12) For doctoral candidates.
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