Proposed Mathematics Courses for 2006-2007

• MATH 504/5/6 Modern Algebra (Aut/Win/Spr)
  Three-quarter sequence covering group theory; field theory and Galois theory; commutative rings and modules, linear algebra, theory of forms; representation theory, associative rings and modules; commutative algebra and elementary algebraic geometry. Prerequisite: MATH 404 or equivalent.

• MATH 514 Networks and Combinatorial Optimization (Aut)
  Networks and directed graphs. Paths and trees. Feasible and optimal flows and potentials. Transportation problems, matching and assignment problems. Algorithms and applications. Prerequisite: MATH 308 or AMATH 352 and MATH 324. Offered: jointly with AMATH 514.

• MATH 515 Fundamentals of Optimization (Win)
  Maximization and minimization of functions of finitely many variables subject to constraints. Basic problem types and examples of applications; linear, convex, smooth, and nonsmooth programming. Optimality conditions. Saddlepoints and dual problems. Penalties, decomposition. Overview of computational approaches. Prerequisite: linear algebra and advanced calculus. Offered: jointly with IND E 515/AMATH 515.

• MATH 516 Numerical Optimization (Spr)
  Methods of solving optimization problems in finitely many variables, with or without constraints. Steepest descent, quasi-Newton methods. Quadratic programming and complementarity. Exact penalty methods, multiplier methods. Sequential quadratic programming. Cutting planes and nonsmooth optimization. Prerequisite: MATH 515. Offered: jointly with AMATH 516.

• MATH 521/522/523 Advanced Probability (Aut/Win/Spr)
  Measure theory and integration, independence, laws of large numbers, Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales. Prerequisite: either MATH 426 or MATH 576. Offered: jointly with STAT 521/522/523.

• MATH 524/5/6 Real Analysis (Aut/Win/Spr)
  Three-quarter sequence covering the theory of measure and integration, point set topology, Banach spaces, L^p spaces, applications to the theory of functions of one and several real variables. Additional topics to be chosen by instructor. Prerequisite: MATH 426 or equivalent.

• MATH 527/528/529 Functional Analysis (Aut/Win/Spr)
  First quarter of a three-quarter sequence. Review of Banach, Hilbert, and L^p spaces; locally convex spaces (duality and separation theory, distributions, and function spaces); operators on locally convex spaces (adjoints, closed graph/open mapping and Banach-Steinhaus theorems); Banach algebras (spectral theory, elementary applications); spectral theorem for Hilbert space operators. Additional topics chosen by instructor. A working knowledge of real variables, general topology, and complex variables is assumed.

• MATH 534/5/6 Complex Analysis (Aut/Win/Spr)
  Three-quarter sequence covering complex numbers, analytic functions, contour integration, power series, analytic continuation, sequences of analytic functions, conformal mapping of simply connected regions, and related topics. Prerequisite: MATH 426.

• MATH 544/5/6 Topology and Geometry of Manifolds (Aut/Win/Spr)
  Three-quarter sequence covering general topology, the fundamental group, covering spaces, topological and differentiable manifolds, vector fields, flows, the Frobenius theorem, Lie groups, homogeneous spaces, tensor fields, differential forms, Stokes's theorem, deRham cohomology. Prerequisite: MATH 404 and MATH 426 or equivalent.

• MATH 554/5/6 Linear Analysis (Aut/Win/Spr)
  Three-quarter sequence covering advanced linear algebra and matrix analysis, ordinary differential equations (existence and uniqueness theory, linear systems, numerical approximations), Fourier analysis, introductions to functional analysis and partial differential equations, distribution theory. Prerequisite: MATH 426 and familiarity with complex analysis at the level of 427 (the latter may be obtained concurrently).

• MATH 557/558/559 Introduction to Partial Differential Equations (Aut/Win/Spr)
  First quarter of a three-quarter sequence. Review of the theory of distributions and the Fourier transform. Detailed study of main linear equations: wave equation, Laplace's equation, and the heat equation. Sobolev spaces and regularity of solutions of elliptic equations. Theory of pseudodifferential operators. Initial value problem for hyperbolic equations and methods of geometrical optics. Fourier integral operators. The Dirichlet problem and eigenfunction expansions for elliptic equations. Prerequisite: MATH 556.

• MATH 564/565/566 Algebraic Topology (Aut/Win/Spr)
  First quarter of a three-quarter sequence covering classical and modern approaches; complexes and their homology theory; applications; fixed points, products and Poincare duality; axiomatic approach. Prerequisite: MATH 506 and MATH 544, or equivalent.

• MATH 581/582/583 Special Topics in Mathematics(Aut/Win/Spr)
  Advanced topics in various areas of mathematics.

• MATH 600 Independent Study or Research
  
• MATH 700 Master's Thesis
  
• MATH 800 Doctoral Dissertation