College of LS&A

Fall Academic Term '02 Graduate Course Guide

Note: You must establish a session for Fall Academic Term 2002 on wolverineaccess.umich.edu in order to use the link "Check Times, Location, and Availability". Once your session is established, the links will function.

Courses in Mathematics


This page was created at 7:29 PM on Thu, Oct 3, 2002.

Fall Academic Term, 2002 (September 3 - December 20)


MATH 404. Intermediate Differential Equations and Dynamics.

Section 001.

Instructor(s): David A Schneider

Prerequisites: MATH 216, 256 or 286, or 316. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is a course oriented to the solutions and applications of differential equations. Numerical methods and computer graphics are incorporated to varying degrees depending on the instructor. There are relatively few proofs. Some background in linear algebra is strongly recommended. First-order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods. Math 454 is a natural sequel.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 412. Introduction to Modern Algebra.

Instructor(s):

Prerequisites: MATH 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions or the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc. ) and their proofs. Math 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, and complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.

Math 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an honors version of Math 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 416. Theory of Algorithms.

Section 001.

Instructor(s): Hendrikus Gerardus Derksen

Prerequisites: MATH 312 or 412 or EECS 203, and EECS 281. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Many common problems from mathematics and computer science may be solved by applying one or more algorithms well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness. This course has substantial overlap with EECS 586 more or less depending on the instructors. In general, Math 416 will put more emphasis on the analysis aspect in contrast to design of algorithms. Math 516 (given infrequently) and EECS 574 and 575 (Theoretical Computer Science I and II) include some topics which follow those of this course.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 417. Matrix Algebra I.

Instructor(s):

Prerequisites: Three courses beyond MATH 110. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled MATH 513. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/courses/417/

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. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (honors). 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.

Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 419. Linear Spaces and Matrix Theory.

Instructor(s):

Prerequisites: Four terms of college mathematics beyond MATH 110. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Math 419 covers much of the same ground as Math 417 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, and applications to differential and difference equations.

Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more proof-oriented. Math 513 is much more abstract and sophisticated. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 423. Mathematics of Finance.

Instructor(s):

Prerequisites: MATH 217 and 425; EECS 183. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory.

Specific Topics:

  1. Review of basic probability.
  2. The one-period binomial model of stock prices used to price futures.
  3. Arbitrage, equivalent portfolios, and risk-neutral valuation.
  4. Multiperiod binomial model.
  5. Options and options markets; pricing options with the binomial model.
  6. Early exercise feature (American options).
  7. Trading strategies; hedging risk.
  8. Introduction to stochastic processes in discrete time. Random walks.
  9. Markov property, martingales, binomial trees.
  10. Continuous-time stochastic processes. Brownian motion.
  11. Black-Scholes analysis, partial differential equation, and formula.
  12. Numerical methods and calibration of models.
  13. Interest-rate derivatives and the yield curve.
  14. Limitations of existing models. Extensions of Black-Scholes.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 424. Compound Interest and Life Insurance.

Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the Society of Actuaries exam; the material covered will depend somewhat on the instructor. Math 424 is required for students concentrating in actuarial mathematics; others may take Math 147, which deals with the same techniques but with less emphasis on continuous growth situations. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 425 / STATS 425. Introduction to Probability.

Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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. The course will make essential use of the material of Math 116 and 215. 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. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 425 / STATS 425. Introduction to Probability.

Section 001.

Instructor(s): Sergey Fomin (fomin@umich.edu)

Prerequisites: MATH 215, 255, or 285. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/425.html

This course introduces students to the theory of probability and to a number of applications. 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. The material corresponds to most of Chapters 1-7 and part of 8 of Ross.

Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. Your lowest homework set score will be dropped.

This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out. Every student with the total score of 90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).

Homework: There will be approximately 10 problem sets. Text (required): Sheldon Ross, A First Course in Probability, 6th edition, Prentice-Hall, 2002.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 425 / STATS 425. Introduction to Probability.

Section 005.

Instructor(s): Kristen S Moore (ksmoore@umich.edu)

Prerequisites: MATH 215, 255, or 285. (3).

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2002/fall/math/425/005.nsf

This course introduces students to the theory of probability and to a number of applications. 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. The material corresponds to most of Chapters 1-7 and part of 8 of Ross.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 431. Topics in Geometry for Teachers.

Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math431/

This course is a study of the axiomatic foundations of Euclidean and non-Euclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups; and finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative geometry courses at this level are 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 433. Introduction to Differential Geometry.

Section 001.

Instructor(s): Mario Bonk

Prerequisites: MATH 215 (or 255 or 285), and 217. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; and (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, and classification of minimal surfaces of revolution. 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513), and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 450. Advanced Mathematics for Engineers I.

Instructor(s):

Prerequisites: MATH 215, 255, or 285. No credit granted to those who have completed or are enrolled in MATH 454. (4).

Credits: (4).

Course Homepage: No homepage submitted.

Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given, and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g. Math 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 451. Advanced Calculus I.

Instructor(s):

Prerequisites: MATH 215 and one course beyond MATH 215; or MATH 255 or 285. Intended for concentrators; other students should elect MATH 450. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.) and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits, and continuity; differentiation; integration, and the Fundamental Theorem of Calculus; infinite series; and sequences and series of functions.

There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, 590, 596, and 597.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 454. Boundary Value Problems for Partial Differential Equations.

Instructor(s):

Prerequisites: MATH 216, 256, 286, or 316. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of 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; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both Math 455 and 554 cover many of the same topics but are very seldom offered. Math 454 is prerequisite to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 463. Mathematical Modeling in Biology.

Section 001.

Instructor(s): Carl P Simon (cpsimon@umich.edu)

Prerequisites: MATH 217, 417, or 419; MATH 286, 256, or 316. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course will concentrate on the applications of ordinary differential equations to physiological systems. Partial differential equations will not be covered in detail. Thus, a course in ODEs such as 216 or 316 will be sufficient preparation for this course.

Who could take the course? Basically anybody who is interested in applying mathematical methods to the biological sciences. For instance, students from Biology, Chemistry, Physics, Complex Systems, Biophysics, Biomedical Engineering, Mathematics, Chemical Engineering, Physiology, Microbiology, and Epidemiology.

What kind of background will you need? Basically a course in differential equations, such as 216 or 316. If you have never seen a differential equation before, you may have trouble with the course. You will also need to be familiar and comfortable with computers, as a lot of the work in the course will have to be done on a computer. You will not need to be an expert in biology, as we will learn most of what we need to know as we go.

Topics

  • Review of phase-plane methods for ordinary differential equations.
  • Predator-Prey models.
  • Models of disease transmission. SIR models, and epidemics.
  • Enzyme kinetics.
    • Pseudo-steady-state hypothesis.
    • Cooperativity.
    • Monod-Wyman-Changeux models.
  • Cellular homeostasis.
    • The membrane potential. The Nernst equation. Electrodiffusion. The Goldman-Hodgkin-Katz equation. The constant field approximation.
    • Osmosis.
    • Control of cell volume.
    • Sodium transport and cell volume control.
  • Review of limit cycles and oscillations. The Hopf bifurcation.
  • Excitability.
    • The Hodgkin-Huxley model and action potentials.
    • Two-variable models. FitzHugh-Nagumo model.
    • Phase-plane analysis of the models.
  • The circulatory system.
    • Blood flow and compliance.
    • Guyton's model.
    • Cardiac regulation.
    • Fetal circulation.
    • The Windkessel model.
  • Blood.
    • Myoglobin and hemoglobin.
    • Cooperativity and oxygen transport.
    • Carbon dioxide transport. Bohr and Haldane effects.
  • Muscle.
    • The Hill model.
    • Crossbridge theory and the Huxley model.
    • Discrete binding site models.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 471. Introduction to Numerical Methods.

Instructor(s):

Prerequisites: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. No credit granted to those who have completed or are enrolled in MATH 371. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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 proven. 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, and the Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571-572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 481. Introduction to Mathematical Logic.

Section 001.

Instructor(s): Andreas R Blass

Prerequisites: MATH 412 or 451 or equivalent experience with abstract mathematics. (3).

Credits: (3).

Course Homepage: No homepage submitted.

All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite. In the first third of the course the notion of a formal language is introduced and propositional connectives ({\i and, or, not, implies}), tautologies, and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including non-standard analysis. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 485. Mathematics for Elementary School Teachers and Supervisors.

Section 001.

Instructor(s): Eugene F Krause (krause@umich.edu)

Prerequisites: One year of high school algebra. No credit granted to those who have completed or are enrolled in MATH 385. (3). May not be included in a concentration plan in mathematics. Does not apply to any math degree programs.

Credits: (3; 2 in the half-term).

Course Homepage: No homepage submitted.

The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching or preparing to teach in the elementary school.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 497. Topics in Elementary Mathematics.

Section 001.

Instructor(s): Eugene F Krause (krause@umich.edu)

Prerequisites: MATH 489. (3). May be repeated for a total of six credits.

Credits: (3).

Course Homepage: No homepage submitted.

This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from term to term.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 499. Independent Reading.

Instructor(s):

Prerequisites: Graduate standing in a field other than mathematics. (1-4). (INDEPENDENT).

Credits: (1-4).

Course Homepage: No homepage submitted.

This course is intended for graduate students in fields other than mathematics who require mathematical skills not otherwise available though existing courses.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of Instructor

MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.

Section 001.

Instructor(s): Peter D Miller (millerpd@umich.edu)

Prerequisites: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.

Credits: (1).

Course Homepage: No homepage submitted.

The Applied and Interdisciplinary Mathematics(AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. Math 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.

Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 513. Introduction to Linear Algebra.

Section 001.

Instructor(s): Richard Paul Horja

Prerequisites: MATH 412. Two credits granted to those who have completed MATH 214, 217, 417, or 419. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course (e.g., Math 451 or 512) is recommended. 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; and Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned with applications. The natural sequel to Math 513 is 593. Math 513 is also prerequisite to several other courses (Math 537, 551, 571, and 575) and may always be substituted for Math 417 or 419.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 520. Life Contingencies I.

Section 001.

Instructor(s): Curtis E Huntington (chunt@umich.edu)

Prerequisites: MATH 424 and 425. (3).

Credits: (3).

Course Homepage: No homepage submitted.

The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 523. Risk Theory.

Section 001.

Instructor(s): Mattias Jonsson (mattiasj@umich.edu)

Prerequisites: MATH 425. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the risk-reward tradeoff. Topics include:

  • Review of probability.
  • Bachelier and Lundberg models of investment and loss aggregation.
  • Fallacy of time diversification and its generalizations.
  • Geometric Brownian motion and the compound Poisson process.
  • Modeling of individual losses which arise in a loss aggregation process.
  • Distributions for modeling size loss, statistical techniques for fitting data, and credibility.
  • Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory.
  • The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the Black-Scholes option pricing model.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 525 / STATS 525. Probability Theory.

Section 001.

Instructor(s): Charles R Doering

Prerequisites: MATH 450 or 451. Students with credit for MATH 425/STATS 425 can elect MATH 525/STATS 525 for only one credit. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, 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. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510-511 are natural sequels.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 537. Introduction to Differentiable Manifolds.

Section 001.

Instructor(s): John W Lott (lott@umich.edu)

Prerequisites: MATH 513 and 590. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course will be a painless introduction to differential topology and differential geometry, meaning the study of spaces and their curvatures. It is the first part of a two-term sequence. The material in this course is crucial for students who wish to study differential geometry, topology, algebraic geometry, several complex variables, Lie groups, and dynamical systems. It is also relevant for other branches of mathematics, such as partial differential equations. We'll start out by doing calculus on manifolds, introducing and using differential forms. We'll prove Stokes' theorem for compact oriented manifolds-with-boundary. We'll also define the de Rham cohomology groups of a manifold and prove their basic properties. Then I'll spend some time on Morse theory. This theory shows how, given a generic function on a manifold, one obtains a decomposition of the manifold into simple building blocks called handles. Morse theory is a basic tool in topology and was used in Smale's famous proof of the Poincare conjecture in more than four dimensions, although we will not go into this. Finally, we'll cover some basic Riemannian geometry, including Riemannian metrics, Levi-Civita connections, geodesics, and curvature. Homework assignments will be given periodically. There will also be a final exam.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 555. Introduction to Functions of a Complex Variable with Applications.

Section 001.

Instructor(s): Jeffrey B Rauch

Prerequisites: MATH 450 or 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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 principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, and applications. Evaluation of improper real integrals and fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 556. Methods of Applied Mathematics I.

Section 001 Initial value Problems and Boundary Value Problems.

Instructor(s): Peter D Miller (millerpd@umich.edu)

Prerequisites: MATH 217, 419, or 513; 451 and 555. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is an introduction to analytical methods for initial value problems and boundary value problems. This course should be useful to students in mathematics, physics, and engineering. We will begin with systems of ordinary differential equations. Next, we will study Fourier Series, Sturm-Liouville problems, and eigenfunction expansions. We will then move on to the Fourier Transform, Riemann-Lebesgue Lemma, inversion formula, the uncertainty principle, and the sampling theorem. Next we will cover distributions, weak convergence, Fourier transforms of tempered distributions, weak solution of differential equations, and Green's functions. We will study these topics within the context of the heat equation, wave equation, Schrödinger's equation, and Laplace's equation.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 561 / IOE 510 / SMS 518. Linear Programming I.

Section 001.

Instructor(s): Marina A Epelman

Prerequisites: MATH 217, 417, or 419. (3). CAEN lab access fee required for non-Engineering students.

Credits: (3).

Lab Fee: CAEN lab access fee required for non-Engineering students.

Course Homepage: No homepage submitted.

Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 562 / IOE 511 / AEROSP 577. Continuous Optimization Methods.

Section 001.

Instructor(s): Romesh Saigal (rsaigal@umich.edu)

Prerequisites: MATH 217, 417, or 419. (3). CAEN lab access fee required for non-Engineering students.

Credits: (3).

Lab Fee: CAEN lab access fee required for non-Engineering students.

Course Homepage: No homepage submitted.

Survey of continuous optimization problems. Unconstrained optimization prob-lems: unidirectional search techniques; gradient, conjugate direction, quasi-Newton methods. Introduction to constrained optimization using techniques of unconstrained optimization through penalty transformations, augmented Lagrangians, and others. Discussion of computer programs for various algorithms.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 565. Combinatorics and Graph Theory.

Section 001.

Instructor(s): Mark A Skandera

Prerequisites: MATH 412 or 451 or equivalent experience with abstract mathematics. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one proof-oriented course. Graph Theory topics include Trees; k -connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5-Color Theorem; Kuratowski's Theorem; and the Matrix-Tree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q -series, linear recurrences and rational generating functions, and Pólya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 571. Numerical Methods for Scientific Computing I.

Section 001.

Instructor(s): Divakar Viswanath

Prerequisites: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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. 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, and the conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; and numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 1

MATH 575. Introduction to Theory of Numbers I.

Section 001.

Instructor(s): Mark E Dickinson (dickinsm@umich.edu)

Prerequisites: MATH 451 and 513. Students with credit for MATH 475 can elect MATH 575 for 1 credit. (1, 3).

Credits: (1, 3).

Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/575f02/

Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and preferably Math 512. Proofs are emphasized, but they are often pleasantly short. A computational laboratory (Math 476, 1 credit) will usually be offered as a supplement to this course. Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, and application of these ideas to the solution of classical problems such as Fermat's last 'theorem'. Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. Math 475 is a non-honors version of Math 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. All of the advanced number theory courses (Math 675, 676, 677, 678, and 679) presuppose the material of Math 575. Each of these is devoted to a special subarea of number theory.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 590. Introduction to Topology.

Section 001.

Instructor(s): Arthur G Wasserman

Prerequisites: MATH 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 591. General and Differential Topology.

Section 001.

Instructor(s): Andreas R Blass

Prerequisites: MATH 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, and classification of surfaces. Math 592 is the natural sequel.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 593. Algebra I.

Section 001.

Instructor(s): Robert K Lazarsfeld

Prerequisites: MATH 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to 512. Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, and exterior algebras.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 596. Analysis I.

Section 001.

Instructor(s): Peter L Duren

Prerequisites: MATH 451. Students with credit for MATH 555 may elect MATH 596 for two credits only. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topics include: review of analysis in R2 including metric spaces, differentiable maps, and Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, and linear fractional transformations; Cauchy's theorem and Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, and argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families; and Riemann mapping theorem. Math 595 covers some of the same material with greater emphasis on applications and less attention to proofs.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 602. Real Analysis II.

Section 001.

Instructor(s): Joel Smoller (smoller@umich.edu)

Prerequisites: MATH 590 and 597. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 605. Several Complex Variables.

Section 001.

Instructor(s): Berit Stensønes (berit@umich.edu)

Prerequisites: MATH 596 and 597. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 612. Lie Algebra and their Representatives.

Section 001.

Instructor(s): Gopal Prasad

Prerequisites: MATH 593 and 594; Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Topics include: Representation Theory of semisimple Lie algebras over the complex numbers; Weyl's Theorem, root systems, Harish Chandra's Theorem, Weyl's formulae, and Kostant's Multiplicity Theorem; and Lie groups, their Lie algebras, and further examples of representations.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 614. Commutative Algebra.

Section 001.

Instructor(s): J Tobias Stafford

Prerequisites: MATH 593. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Topics include: a review of commutative rings and modules; local rings and localization; Noetherian and Artinian rings; Integral independence; valuation rings, Dedekind domains, completions, and graded rings; and Dimension theory.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 619. Topics in Algebra.

Section 001 Groups & Lattices.

Instructor(s): Robert Griess (rlg@umich.edu)

Prerequisites: MATH 593. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 626 / STATS 626. Probability and Random Processes II.

Section 001 Deterministic and Stochastic Optimal Control.

Instructor(s): Joseph Conlon (conlon@umich.edu)

Prerequisites: MATH 625. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math626.html

This course will be an introduction to optimal control theory. The goal is to develop the tools that are important to understanding control theory problems. Three examples illustrate the range of problems:

  1. A rocket has engines which can be adjusted in flight. Find the settings of the engines so that the rocket gets to its target in minimum time.
  2. A portfolio consists of a variable combination of cash and stocks. Amounts are withdrawn from the portfolio at a variable rate for consumption. Find the optimal distribution of the portfolio between cash and stocks to maximise the expected utility of consumption.
  3. A company must transport a given product from m origins to n destinations. Given the amount of product at each origin and the amount required at each destination, find the minimum cost of doing the transportation.

Control problems always involve minimising a "cost function" as a function of the adjustable parameters= "controls" in the problem. In problem (a) the cost function is the time to target. Problem (a) is an example of deterministic control theory. Problem (b) is the Merton portfolio optimisation problem, an example of stochastic control theory. Problem (c) is the Monge-Kantorovich mass transportation problem, an example in linear programming. The mathematics involved with the three problems are closely related.

The first part of the course explores the relationship between deterministic control problems and Hamiltonian mechanics. Central to this is the result that the cost function satisfies a first order partial differential equation, the Hamilton-Jacobi equation, known as the Bellman equation in control theory.

The second part of the course considers problems of stochastic control theory. The cost function is now an expectation value of a functional of the dynamical variables. It satisfies a second order parabolic or elliptic partial differential equation which can be fully nonlinear. We study two examples of these, the Burgers' equation and the Monge-Ampere equation. We also explore the connection between control theory and prediction theory. In prediction theory one tries to predict the value of a random variable from observations of other random variables correlated to the variable of interest. We show that an exactly solvable problem in prediction theory, the Kalman filter, is equivalent to a control problem with linear dynamics and quadratic cost function.

In the final part of the course we will be concerned with problem (c), the Monge-Kantorovich mass transfer problem. The original transport problem was proposed by Monge in the 1780's. The solution by Kantorovich and Koopmans was awarded the 1975 Nobel prize in economics. We shall show how the problem is solved by going to the dual linear program. The continuous version of the dual problem gives a solution of the Monge-Ampere equation, which has already occurred earlier in the course.

Prequisite: Some knowledge of differential equations and probability theory ( at the 500 level).

Grading: Grades will be based on performance in the homework sets.

Text: Deterministic and Stochastic Optimal Contro by W. Fleming and R. Rishel, Springer 1975 ( reprinted 1999).

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 631. Introduction to Algebraic Geometry.

Section 001.

Instructor(s): Igor Dolgachev (idolga@umich.edu)

Prerequisites: MATH 594 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 636. Topics in Differential Geometry.

Section 001 Dynamical Systems & Group Actions.

Instructor(s): Ralf Spatzier (spatzier@umich.edu)

Prerequisites: MATH 635. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 638. Introduction to Representation Theory.

Section 001.

Instructor(s): Mahdi Asgari (asgari@umich.edu)

Prerequisites: MATH 597. Graduate standing. (3). May be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 657. Nonlinear Partial Differential Equations.

Section 001.

Instructor(s): Smoller

Prerequisites: MATH 656. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 658. Ordinary Differential Equations.

Section 001.

Instructor(s): Anthony Bloch (abloch@umich.edu)

Prerequisites: A course in differential equations (e.g., MATH 404 or 558). Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 665. Combinatorial Theory II.

Section 001.

Instructor(s): John Stembridge

Prerequisites: MATH 664 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 671. Analysis of Numerical Methods I.

Section 001 Fast Methods for Particle Simulations.

Instructor(s): Robert Krasny (krasny@umich.edu)

Prerequisites: MATH 571, 572, and Graduate standing. (3). May be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math671b.html

Topics: FFT, potential theory, particle-particle methods (treecodes, FMM), particle-mesh methods (PIC, P3M), Ewald summation, multigrid

Text: A Multigrid Tutorial, W.L. Briggs, V. E. Emson & S. F. McCormick, SIAM

The course will survey some widely used numerical methods and introduce students to an active area of research in scienti c computing. The methods to be discussed are fast, meaning that they require fewer operations to perform a certain task than a straightforward algorithm. A well-known example is the fast Fourier transform (FFT). The FFT reduces the operation count for computing the discrete Fourier transform of a vector of length N from O(N2) to O(N logN). This makes a huge di erence in practice and as a result, the FFT is a basic tool in fields such as signal processing and spectral analysis. As a warm-up, we'll start by deriving the FFT algorithm. The bulk of the course will deal with fast numerical methods for evaluating the potential energy and forces due to long-range particle interactions. This is an important component in molecular dynamics and Monte- Carlo simulations, and applications arise in many areas such as astrophysics (gravitational interaction), chemistry and materials science (electrostatic interaction), and uid dynamics (vortex interaction). In a system with N particles, O(N2) operations are required to evaluate the pairwise interactions by straightforward direct summation. The FFT can be applied when the particles are uniformly spaced, but other techniques are necessary to achieve a speedup for nonuniform particle distributions. First we'll consider particlemesh algorithms including particle-in-cell, P3M (Hockney-Eastwood), Ewald summation, and particle-mesh Ewald (Darden-York-Pedersen). The next topic will be particle-cluster algorithms including hierarchical treecodes (Barnes-Hut) and the fast multipole method (Greengard-Rokhlin). We'll review the spherical harmonics expansion of the Coulomb potential on which these methods are based. There is great interest in optimizing the performance of particle-cluster algorithms and I'll present some recently developed adaptive techniques. The final topic (time permitting) will be the multigrid method and its application to evaluating particle interactions, including an Ewald-multigrid algorithm (Darden-Sagui) and multilevel summation (Brandt). The course grade will be based on homework.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 676. Theory of Algebraic Numbers.

Section 001.

Instructor(s): Christopher M Skinner

Prerequisites: MATH 575 and 594. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 678. Modular Forms.

Section 001.

Instructor(s): Kannan Soundararajan

Prerequisites: MATH 575 and 596. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 681. Mathematical Logic.

Section 001.

Instructor(s): Peter G Hinman

Prerequisites: Mathematical maturity appropriate for a 600-level MATH course. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pgh/courses/681/

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 694. Differential Topology II.

Section 001.

Instructor(s): John W Lott (lott@umich.edu)

Prerequisites: MATH 337 and 591 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Topics include transversality, embedding theorems, vector bundles, and selected topics from the theories of cobordism, surgery, and characteristic classes.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 695. Algebraic Topology I.

Section 001.

Instructor(s): G Peter Scott

Prerequisites: MATH 591 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 700. Directed Reading and Research.

Instructor(s):

Prerequisites: Graduate standing. (1-3). (INDEPENDENT).

Credits: (1-3).

Course Homepage: No homepage submitted.

Designed for individual students who have an interest in a specific topic (usually that has stemmed from a previous course). An individual instructor must agree to direct such a reading, and the requirements are specified when approval is granted.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of Instructor

MATH 703. Topics in Complex Function Theory I.

Section 001 Topic?

Instructor(s): John E Fornaess

Prerequisites: MATH 604. Graduate standing. (3). May be taken for credit more than once.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 709. Topics in Modern Analysis I.

Section 001 Topic?

Instructor(s): Lizhen Ji

Prerequisites: MATH 597. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 715. Advanced Topics in Algebra.

Section 001 Topic?

Instructor(s): Melvin Hochster

Prerequisites: Graduate standing. (3). May be taken more than once for credit.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 731. Topics in Algebraic Geometry.

Section 001 Topic?

Instructor(s): William E Fulton

Prerequisites: Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 990. Dissertation/Precandidate.

Instructor(s):

Prerequisites: Election for dissertation work by doctoral student not yet admitted as a Candidate. Graduate standing. (1-8). (INDEPENDENT). May be repeated for credit.

Credits: (1-8; 1-4 in the half-term).

Course Homepage: No homepage submitted.

Election for dissertation work by doctoral student not yet admitted as a Candidate.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of Instructor

MATH 993. Graduate Student Instructor Training Program.

Section 001.

Instructor(s):

Prerequisites: Graduate standing and appointment as GSI in Mathematics Department. (1).

Credits: (1).

Course Homepage: No homepage submitted.

A seminar for all beginning graduate student instructors, consisting of a two day orientation before the term starts and periodic workshops/meetings during the Fall Term. Beginning graduate student instructors are required to register for this course.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

MATH 995. Dissertation/Candidate.

Instructor(s):

Prerequisites: Graduate School authorization for admission as a doctoral Candidate. Graduate standing. (8). (INDEPENDENT). May be repeated for credit.

Credits: (8; 4 in the half-term).

Course Homepage: No homepage submitted.

Graduate School authorization for admission as a doctoral Candidate. N.B. The defense of the dissertation (the final oral examination) must be held under a full term Candidacy enrollment period.

Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, Permission of Instructor

Undergraduate Course Listings for MATH.


Page


This page was created at 7:29 PM on Thu, Oct 3, 2002.


lsa logo

University of Michigan | College of LS&A | Student Academic Affairs | LS&A Bulletin Index | Department Homepage

This page maintained by LS&A Academic Information and Publications, 1228 Angell Hall

Copyright © 2002 The Regents of the University of Michigan,
Ann Arbor, MI 48109 USA +1 734 764-1817

Trademarks of the University of Michigan may not be electronically or otherwise altered or separated from this document or used for any non-University purpose.