Mathematics

Graduate courses: descriptions

Undergraduates must have departmental approval to take 5000-6000 level mathematics courses; 7000-8000 level courses are open only to graduate students.

MATH 5000-3. Foundations of Mathematics. Focuses on foundations used in other graduate courses and for specialization in foundations. Includes equivalence relations, orderings, ordinal and cardinal numbers and arithmetic, axiom of choice; first-order logic, models, truth, compactness and completeness theorems, nonstandard analysis, and infinitesimals; and formulation of Goedel's incompleteness theorem. Prereqs., MATH 3130, 3140, and 4310. Undergraduates must have approval of the instructor.

MATH 5030-3, 5040-3. Intermediate Mathematical Physics 1 and 2. Surveys classical mathematical physics, starting with complex variable theory and finite dimensional vector spaces. Topics in ordinary and partial differential equations, the special functions, boundary value problems, potential theory, and Fourier analysis are discussed. Prereqs., MATH 4310 and 4320. Same as PHYS 5030 and 5040.

MATH 5150-3. Linear Algebra. Vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms. Prereq., MATH 3130.

MATH 5430-3. Ordinary Differential Equations. Introduces theory and applications of ordinary differential equations, including existence and uniqueness theorems, qualitative behavior, series solutions, and numerical methods, for scalar equations and systems. Prereqs., MATH 3130 and 4310

MATH 5460-3. Applied Topics in Complex Variables. Same as MATH 4460.

MATH 5470-3. Partial Differential Equations 1. Introduces theory and applications of partial differential equations, including existence, uniqueness, stability, regularity, and solution construction and approximation procedures. Prereq., MATH 4430, or APPM 4350 and APPM 4360, or equivalent.

MATH 5480-3. Partial Differential Equations 2. Continuation of MATH 5470. Connections between partial differential equations and Fourier theory, Green's functions, variational and numerical methods. Applications to fluid dynamics, quantum mechanics, and elsewhere as time permits. Prereq., MATH 5470 or APPM 5470.

MATH 5520-3. Introduction to Mathematical Statistics. Same as MATH 4520 and APPM 4520/5520.

MATH 5540-3. Time Series Analysis. Basic properties, linear extrapolation, and filtering of stationary random functions. Spectral and cross-spectral analysis; estimation of the power spectrum using computers; nonstationary time series; comparison of various computer programs. Prereq., MATH 4510/4520 or instructor consent. Same as MATH 4540 and APPM 4540/5540.

MATH 5600-3. Numerical Analysis 1. Solution of linear systems, least squares approximations, nonlinear algebraic equations, interpolation, and quadrature. Prereqs., calculus, MATH 3130, and CSCI 1200. Same as APPM 5600.

MATH 5610-3. Numerical Analysis 2. Solution of ordinary and partial differential equations; matrix eigenvalue eigenvector problems. Prereq., MATH 5600 or APPM 5600. Same as APPM 5610.

MATH 5800-3. History of Mathematics. Same as MATH 4800.

MATH 6110-3. Introduction to Number Theory. Divisibility properties of integers, congruences, diophantine equations, arithmetic functions, quadratic residues, distribution of primes, and algebraic number fields. Prereq., MATH 3140.

MATH 6130-3, 6140-3. Modern Algebra 1 and 2. Groups, rings and ideals, fields, polynomials, Galois theory. Prereq., MATH 3140.

MATH 6150-3. Commutative Algebra. Serves as an introduction to topics that are used in number theory and algebraic geometry, including radicals of ideals, exact sequences of modules, tensor products, Ext., Tor, localization, primary decomposition of ideals, and Noetherian rings. Prereq., MATH 6140.

MATH 6170-3. Algebraic Geometry. Serves as an introduction to algebraic geometry, including affine and projective varieties, rational maps and morphisms, and differentials and divisors. Additional topics might include Bezout's Theorem, the Riemann-Roch Theorem, elliptic curves, and sheaves and schemes. Prereq., MATH 6140.

MATH 6180-3. Algebraic Number Theory. Serves as an introduction to topics that include number fields and completions, norms, discriminants and differents, finiteness of the ideal class group, Dirichlet's unit theorem, decomposition of prime ideals in extension fields, decomposition, and ramification groups. Prereq., MATH 6110 and 6140.

MATH 6190-3. Analytic Number Theory. Serves as an introduction to topics that include the Riemann Zeta-function and its meromorphic continuation, characters and Dirichlet series, Dirichlet's theorem on primes in arithmetic progression, zero-free regions of the zeta function, and the prime number theory. Prereq., MATH 6110 and 6350.

MATH 6210-3, 6220-3. Introduction to Topology 1 and 2. Elements of general topology, algebraic topology, differentiable manifolds. Prereqs., MATH 3130, 3140, 4310, and 4320.

MATH 6230-3, 6240-3. Introduction to Differential Geometry 1 and 2. Differential forms in Euclidean 3-space, frame fields, Frenet formulas, calculus of differential forms on surfaces, extrinsic and intrinsic geometry of surfaces, Riemannian geometry of differentiable manifolds, geodesics, curvature, the Gauss-Bonnet theorem. Prereqs., MATH 3130 and 4320.

MATH 6310-3, 6320-3. Introduction to Real Analysis 1 and 2. Zorn's dilemma, metric and normed linear spaces, completions, continuous functions, Riemann-Stieltjes and Lebesque integration, measure theory, Lebesque function spaces, and Fourier analysis. Prereq., MATH 4310.

MATH 6350-3, 6360-3. Functions of a Complex Variable 1 and 2. Complex numbers and complex plane. Cauchy-Riemann equations, complex integration, Cauchy integral theory, infinite series and products, residue theory, conformal mapping, analytic continuation, singularities, elementary special functions. Prereq., MATH 4310.

MATH 6410-3, 6420-3. Calculus of Variations and Control Theory 1 and 2. Classical necessary and sufficient conditions emphasizing the simplest problems; the problem of Lagrange; Hamiltonian and Lagraphical and other quick and approximate procedures emphasizing applications in the behavioral, biological, and physical sciences. Prereq., instructor consent.

MATH 6550-3. Introduction to Stochastic Processes. Systematic study of Markov chains and some of the simpler Markov processes including renewal theory, limit theorems for Markov chains, branching processes, queueing theory, and birth and death processes. Applications to physical and biological sciences. Prereqs., MATH 4510 and 4310, or instructor consent. Same as APPM 6550.

MATH 6620-3. Numerical Solution of Initial Value Problems. Includes multi-step and single-step methods for ODE; stability; stiff equations; difference schemes for heat and wave equations; applications. Prereqs., CSCI 3656 or 5606; MATH 3130, 4310, and 4430.

MATH 6630-3. Numerical Solution of Boundary Value Problems. Includes finite difference solution of two-point boundary problems and elliptic problems; methods of SOR, ADI, conjugate gradients; finite element method; nonlinear problems; applications. Prereq., MATH 3130, 4310, 4430, or 4650.

MATH 6710-3, 6720-3. Mathematical Logic 1 and 2. Alternate years. First-order logic, completeness theorem, introduction to model theory, ultraproducts, Goedel's incompleteness theorems, theory of recursive functions. Prereqs., MATH 4710 and 4730, or instructor consent.

MATH 6730-3, 6740-3. Advanced Set Theory 1 and 2. Cardinal and ordinal arithmetic, generalizations of Ramsey's theorem, independence of the axiom of choice and of the generalized continuum hypothesis. Prereqs., MATH 4710 and 4730, or instructor consent.

MATH 6900 (1-3). Independent Study.

MATH 6950 (1-6). Master's Thesis.

MATH 7030-3, 7040-3. Advanced Mathematical Physics 1 and 2. Hilbert space, theory of distributions, L2-spaces, Sobolev spaces, methods of functional analysis, spectral theory of operators, applications to quantum theory, and group theory. Prereqs., MATH 4310 and 4320, and MATH 4450 or 6350. Same as PHYS 7030 and 7040.

MATH 8230-3, 8240-3. Algebraic Topology 1 and 2. Homology and cohomology theories, homotopy theory, obstruction theory, and applications. Prereqs., MATH 6130 and 6140, MATH 6210 and 6220, or instructor consent.

MATH 8250-3, 8260-3. Mathematical Theory of Relativity 1 and 2. Maxwell equations; Lorentz force; Minkowski space-time; Lorentz, Poincar, and conformal groups; metric manifolds; covariant differentiation; Einstein space-time; cosmologies; unified field theories. Prereq., instructor consent.

MATH 8270-3, 8280-3. Differential Topology 1 and 2. Differentiable manifolds, tangent bundles, vector fields, differential forms. Frobenius theorem, Riemannian metrics, selected topics. Prereqs., MATH 5150, 6210 and 6220, 6310, and 6320.

MATH 8330-3, 8340-3. Functional Analysis 1 and 2. Introduces such topics as Banach spaces (Hahn-Banach theorem, open mapping theorem, etc.), operator theory (compact operators and integral equations, spectral theorem for bounded self-adjoint operators), and Banach algebras (the Gelfand theory). Prereqs., MATH 6310 and 6320.

MATH 8370-3, 8380-3. Harmonic Analysis 1 and 2. Trigonometric series, periodic functions, diophantine approximation, Fourier series. Bohr and Stepanoff almost periodic functions, positive definite functions, the L1 and L2 theory of the Fourier integral. Applications to group theory and differential equations. Prereqs., MATH 5150 and 6320.

MATH 8410-3, 8420-3. Mathematical/Computational Fluid Dynamics 1 and 2. Mathematical treatment of basic Navier-Stokes partial differential equations describing fluid dynamics, including the Euler and Stokes equations as approximations for high and low speed flows. Emphasizes both analytical considerations and computational methods. Prereq., instructor consent.

MATH 8430-3, 8440-3. Theory of Ordinary Differential Equations 1 and 2. Prereqs., MATH 5150, 6320, and instructor consent.

MATH 8750-3, 8760-3. Lattices and General Algebra 1 and 2. Modular, distributive, Brouwerian, and Boolean lattices. Applications to algebra and topology. Homomorphism, congruence relations, direct factorization, free algebras, varieties. Prereqs., MATH 4730, 6130, and 6140.

MATH 8900 (1-3). Independent Study.

MATH 8990-10. Doctoral Dissertation. All doctoral students must register for not fewer than 30 hours of dissertation credit as part of the requirements for the degree. For a detailed discussion of doctoral dissertation credit, refer to the Graduate School portion of the catalog.

Topics

MATH 6174-3. Topics in Combinatorial Analysis. Topics such as finite combinatorial analysis, combinatorial questions entering in topology, infinite permutations and transformations, graph theory. Prereq., instructor consent.

MATH 6404-3. Topics in Applied Mathematics. Selected topics in mathematical problems arising from various applied fields such as mechanics, electromagnetic theory, and economics. Prereq., instructor consent.

MATH 6534-3. Topics in Mathematical Probability. Prereqs., advanced calculus and MATH 4510.

MATH 8104-3. Modular Forms. Serves as an introduction to topics that include the upper-half plane and its geometry, modular forms, congruence subgroups, cusps, Fourier expansions, Theta series, Poincar series, Hecke operators, and relations to Dirichlet series. Prereq., MATH 6130 and 6350.

MATH 8114-3, 8124-3. Topics in Number Theory 1 and 2. May include theory of algebraic numbers, L-series and zeta functions, the zeta functions of an algebraic variety, character sums, multiplicative and additive number theory, diophantine equations and approximations, or other topics chosen by instructor. Prereq., MATH 6120 or instructor consent.

MATH 8134-3. Diophantine Approximation. Serves as an introduction to topics that include heights, Thue-Siegel-Roth Theorem; S-unit equations, and applications to Diophantine equations.

MATH 8144-3. Transcendental Number Theory. Serves as an introduction to topics that include Louiville's Theorem, methods of Gelfond-Schneider and Schneider-Lang, linear forms in logarithms, and transcendence measures. Prereq., MATH 6115 and 6350.

MATH 8174-3, 8184-3. Topics in Algebra 1 and 2. Detailed study of advanced topics not covered in modern algebra or other courses, to be chosen by instructor. Prereq., modern algebra. MATH 8174 is not required for MATH 8184.

MATH 8304-3, 8314-3. Topics in Analysis 1 and 2. Advanced topics in analysis include Lie groups, Banach algebras, operator theory, ergodic theory, representation theory, etc.

Prereqs., MATH 8330 and 8340, or instructor consent.

MATH 8324-3, 8334-3. Topics in Real Variables 1 and 2. Abstract measure theory, function spaces, and other topics. Prereqs., MATH 6310 and 6320, or instructor consent.

MATH 8364-3, 8374-3. Topics in Complex Variables 1 and 2. Advanced topics in complex analysis: Riemann surfaces, several complex variables, special functions, rational approximation, potential theory, etc. Prereq., instructor consent.

MATH 8714-3, 8724-3. Topics in Logic 1 and 2. Selected advanced topics in logic or foundations to be chosen by the instructor. Prereq., instructor consent.

Seminars

Normally, about half of the following seminars are given each year. The same seminar number may be repeated for credit.

MATH 5905-1. Mathematics Teacher Training. Designed to train students to become effective teachers. Students teach a mathematics course, meeting weekly with faculty to discuss problems particular to the teaching of mathematics. Prereqs., graduate standing and experience as a teaching assistant.

MATH 8115-3. Seminar: Number Theory.

MATH 8135-3. Seminar: Algebra.

MATH 8205-3. Seminar: Topology.

MATH 8315-3. Seminar: Analysis.

MATH 8325-3. Seminar: Functional Analysis.

MATH 8405-3. Seminar: Applied Mathematics.

MATH 8435-3. Seminar: Differential Equations.

MATH 8505-3. Seminar: Probability Theory and Statistics.

MATH 8605-3. Seminar: Numerical Analysis.

MATH 8705-3. Seminar: Logic and Foundations of Mathematics.

MATH 8805-3. Seminar.

MATH 8815-3. ULAM Seminar.