Besides the descriptions here, see the Department of Mathematics website for more information: http://www.math.lsa.umich.edu/courses/

Analytical

417 Matrix Algebra I

Prerequisite: 3 courses beyond Math 110. I, II, IIIa, and IIIb. (3)

Many problems in science, engineering, and mathematics are best formulated in terms of matrices - rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, Eigenvalues and Eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

419 Linear Spaces and Matrix Theory (EECS 400)

Prerequisite: 4 courses beyond Math 110. I and II. (3)

Math 419 covers much of the same ground as Math 417 (Matrix Algebra I) but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations, determinants, eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations.

425 Introduction to Probability (Stat 425)

Prerequisite: Math 215, 255, or 285. I, II, IIIa, and IIIb. (3)

This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis.

454 Boundary Value Problems for Partial Differential Equations

Prerequisite: Math 216, or Math 256, or Math 286, or Math 316. I and III (3)

This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of initial-value and boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; eigenfunction expansions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Laplace's equation and harmonic functions, including the maximum principle. As time permits, additional topics will be selected from: Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis; dispersive wave equations; the method of stationary phase; the method of characteristics.

464 Inverse Problems

Prerequisite: Math 217, 417, or 419, and 216, 256, 286, or 316. Sporadically (3)

Solution of an inverse problem is a central component of fields such as medical tomography, geophysics, non-destructive testing, and control theory. The solution of any practical inverse problem is an interdisciplinary task. Each such problem requires a blending of mathematical constructs and physical realities. Thus, each problem has its own unique components; on the other hand, there is a common mathematical framework for these problems and their solutions. This framework is the primary content of the course. This course will allow students interested in the above-named fields to have an opportunity to study mathematical tools related to the mathematical foundations. The course content is motivated by a particular inverse problem from a field such as medical tomography (transmission, emission), geophysics (remote sensing, inverse scattering, tomography), or non-destructive testing. Mathematical topics include ill-posedness (existence, uniqueness, stability), regularization (e.g. Tikhonov, least squares, modified least squares, variation, mollification), pseudoinverses, transforms (e.g. k-plane, Radon, X-ray, Hilbert), special functions, and singular-value decomposition. Physical aspects of particular inverse problems will be introduced as needed, but the emphasis of the course is investigation of the mathematical concepts related to analysis and solution of inverse problems.

513 Introduction to Linear Algebra

Prerequisite: Math 412 or 451 or permission of instructor. I and II. (3) Two credits granted to those who have completed MATH 214, 217, 417, or 419

This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications.

525 Probability Theory (Stat 525)

Prerequisite: Math 450 or 451. I. (3)

This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with Math 425 (Intro. to Probability), but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories.

555 Introduction to Functions of a Complex Variable with Applications

Prerequisite: Math 450 or 451. I, II, IIIa (3)

This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply princip les to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics.

556 Methods of Applied Mathematics I

Prerequisite: Math 217, 419, or 513; and 451 and 555. I. (3)

This is an introduction to methods of applied analysis with emphasis on Fourier analysis for differential equations. Initial and boundary value problems are covered. Students are expected to master both the proofs and applications of major results. The prerequisites include linear algebra, advanced calculus and complex variables. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Topics may vary with the instructor but often include Fourier series, separation of variables for partial differential equations, heat conduction, wave motion, electrostatic fields, Sturm-Liouville problems, Fourier transform, Green's functions, distributions, Hilbert space, complete orthonormal sets, integral operators, spectral theory for compact self-adjoint operators.

557 Methods of Applied Mathematics II

Prerequisite: Math 450 or 451. Sporadically. (3)

This course is an introduction to dynamical systems (differential equations and iterated maps). The aim is to survey a broad range of topics in the theory of dynamical systems with emphasis on techniques and results that are useful in applications. Chaotic dynamics will be discussed. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Topics may include bifurcation theory, phase plane analysis for linear systems, Floquet theory, nonlinear stability theory, dissipative and conservative systems, Poincare-Bendixson theorem, Lagrangian and Hamiltonian mechanics, nonlinear oscillations, forced systems, resonance, chaotic dynamics, logistic map, period doubling, Feigenbaum sequence, renormalization, complex dynamics, fractals, Mandelbrot set, Lorenz model, homoclinic orbits, Melnikov's method, Smale horseshoe, symbolic dynamics, KAM theory, homoclinic chaos

558 Applied Nonlinear Dynamics

Prerequisite: Math 217, 419, or 513; 454 and 555. II. (3)

This is an introduction to methods of asymptotic analysis including asymptotic expansions for integrals and solutions of ordinary and partial differential equations. The prerequisites include linear algebra, advanced calculus and complex variables. Math 556 is not a prerequisite. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Topics include stationary phase, steepest descent, characterization of singularities in terms of the Fourier transform, regular and singular perturbation problems, boundary layers, multiple scales, WKB method. Additional topics depend on the instructor but may include non-linear stability theory, bifurcations, applications in fluid dynamics (Rayleigh-Benard convection), combustion (flame speed).

Numerical Analysis

471 Introduction to Numerical Methods

Prerequisite: Math 216, 256, 286, or 316; Math 217, 417, or 419; and a working knowledge of one high-level computer language. I, II, IIIb. (3).

This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation.

571 Numerical Methods for Scientific Computing I

Prerequisite: Math 217, 417, 419, or 513 and Math 450, 451, or 454 or permission. I and II. (3) Math 571 and 572 may be taken in either order.

This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors.

572 Numerical Methods for Scientific Computing II

Prerequisite: Math 217, 417, 419, or 513 and one of Math 450, 451, or 454 or permission. II. (3) Math 571 and 572 may be taken in either order.

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Graduate students from engineering and science departments and strong undergraduates are also welcome. The course is an introduction to numerical methods for solving ordinary differential equations and hyperbolic and parabolic partial differential equations. Fundamental concepts and methods of analysis are emphasized. Students should have a strong background in linear algebra and analysis, and some experience with computer programming. This course is a core course for the Applied and Intersciplinary Mathematics (AIM) graduate program. Content varies somewhat with the instructor. Numerical methods for ordinary differential equations; Lax's equivalence theorem; finite difference and spectral methods for linear time dependent PDEs: diffusion equations, scalar first order hyperbolic equations, symmetric hyberbolic systems.

671 Analysis of Numerical Methods

Prerequisite: Math 571, 572, or permission. I. (3)

This is a course on special topics in numerical analysis and scientific computing. Subjects of current research interest will be included. Recent topics have been: Finite difference methods for hyperbolic problems, Multigrid methods for elliptic bound ary value problems. Students can take this class for credit repeatedly.

Particular Recommendations

Most students should consider Math 471 or 571/572; students in the fission option especially should take Math 571/572. A course in linear algebra (Math 417, 419 or 513) is a recommended prerequisite.

Math 555 is highly recommended for students in the fusion option.

Math 556/557 would be an excellent pair for all doctoral students.

Math 464 may be of particular interest to Measurements and REM students.