600 Topics in Numerical Analysis (3, max 12)

601 Optimization Theory and Techniques (3, SpSm) Necessary and sufficient conditions for existence of extrema with equality constraints; gradient methods; Ritz methods; eigenvalue problems; optimum control problems; inequality constraints; mathematical programming. Prerequisite: MATH 502ab.

602 Galerkin Approximation Methods in Partial Differential Equations (3) Galerkin methods of approximating solutions of elliptic boundary value problems in one and several dimensions; includes the use of spline functions and triangularizations. Prerequisite: departmental approval.

605 Topics in Probability (3, max 12)

610 Topics in Algebra (3, max 12)

611 Homological Algebra (3) Functors, modules, gradings, resolutions, homology, derived functors, spectral sequences, Kunneth theorem, homology and cohomology of groups, bar resolutions, homological dimension, relative homological algebra. Prerequisite: MATH 510ab.

612 Topics in Commutative Ring Theory (3, max 12) Localization, structure of Noetherian rings, integral extensions, valuation theory, graded rings, characteristic functions, local algebra, dimension theory. Prerequisite: MATH 510ab.

613 Topics in Noncommutative Ring Theory (3, max 12) Jacobson radical, nil radical, nil rings and nil-potence, chain conditions, polynomial identity and group rings. Goldie theorems, current research. Prerequisite: MATH 510ab.

620 Topics in Complex Analysis (3, max 12)

621 Riemann Surfaces (3) Topology of Riemann surfaces, Dirichlet problem, existence of analytic functions, function field of a Riemann surface, algebraic curves, uniformization theory, the Riemann-Roch theorem. Prerequisite: MATH 520, MATH 440.

625 Topics in Real Analysis (3, max 12)

626 Linear Integral Equations (3) Equation of Volterra and Fredholm type, adjoint theory and Fredholm alternative, spectral theory and Sturm-Liouville problems, singular equations and Wiener-Hopf theory. Prerequisite: MATH 525ab.

627 Nonlinear Integral Equations (3) Volterra and Fredholm equations of Hammerstein type. Fixed point and implicit function theorems, equations of Urysohn type, dependence on parameters and bifurcation theory. Prerequisite: MATH 525ab.

630 Topics in Number Theory (3, max 12)

635 Topics in Differential Geometry (3, max 12) Topics to be chosen from the following: geometry of complex manifolds, relations between topology and curvature, homogeneous spaces, symmetric spaces, geometry of submanifolds. Prerequisite: MATH 535ab.

640 Algebraic Topology (3) Eilenberg-MacLane spaces, spectra, extraordinary homology and cohomology, homology in fiber spaces, manifolds, duality, characteristic classes, sheaves. Prerequisite: MATH 540.

641 Topics in Topology (3, max 12)

650 Seminar in Statistical Consulting (3)

665 Topics in Ordinary Differential Equations (3, max 12)

677 Mathematical Biosciences (3, max 12) A survey of the types of mathematical problems that arise in biology and medicine with emphasis upon the use of digital and analog computers. Prerequisite: MATH 465.

680 Nonlinear Functional Analysis (3) Calculus in Banach spaces, degree theory, fixed point theorems. Study of compact, monotone, accretive and nonexpansive operators. Prerequisite: MATH 580.

681 Selected Topics in Functional Analysis (3, max 12) Course content will vary with professor and academic year offered. It will include topics of current interest in both linear and nonlinear functional analysis and their applications.

685 Topics in Mathematical Control Theory (3, max 12)

689 Topics in Mathematical Physics (3, max 12)

700 Seminar in Numerical Analysis (3)

705 Seminar in Probability (3)

710 Seminar in Algebra (3)

725 Seminar in Analysis (3)

730 Seminar in Number Theory (3)

735 Seminar in Differential Geometry (3)

740 Seminar in Topology (3)

761 Seminar in Programming and Computability (3)

765 Seminar in Ordinary Differential Equations (3)

780 Seminar in Functional Analysis (3)

790 Research (1-12, FaSpSm) Research leading to the doctorate. Maximum units which may be applied to the degree to be determined by the department. Graded CR/NC.

794abcdz Doctoral Dissertation (2-2-2-2-0, FaSpSm) Credit on acceptance of dissertation. Graded IP/CR/NC.

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