Note: You must establish a session on wolverineacccess.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 (Division 428)
This page was created at 8:00 AM on Fri, Oct 20, 2000.
Fall Term, 2000 (September 6 – December 22)
Open courses in Mathematics
Wolverine Access Subject listing for MATH
Take me to the Fall Term '00 Time Schedule for Mathematics.
To see what graduate courses have been added to or changed in Mathematics this week go to What's New This Week.
Math. 404. Intermediate Differential Equations and Dynamics.
Prerequisites & Distribution: Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 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. Firstorder equations, second and higherorder 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 competingspecies and predatorprey models, numerical methods. Math 454 is a natural sequel. WL:2
Math. 412. Introduction to Modern Algebra.
Prerequisites & Distribution: Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in 512. Students with credit for 312 should take 512 rather than 412. One credit granted to those who have completed 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 of 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, 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.
Math. 413. Calculus for Social Scientists.
Prerequisites & Distribution: Not open to freshmen, sophomores or mathematics concentrators. (3).Not open to Mathematics graduate students.
Credits: (3).
Course Homepage: No Homepage Submitted.
A oneterm course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.
Math. 417. Matrix Algebra I.
Prerequisites & Distribution: 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 problemsolving, 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.
Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.
Section 001, 003.
Prerequisites & Distribution: 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).CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m419.html
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 prooforiented 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, innerproduct spaces; unitary, selfadjoint, and orthogonal operators and matrices, 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 prooforiented. 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.
Text:
Otto Bretscher, Linear Algebra with Applications, Prentice Hall, 1997.
Grading:
The final grade will be computed from the following:
Homework: 25 %
Quizzes: 10 %
Two midterm exams: 20 % each
Final exam: 25 %
Homework problem sets will be given once a week, to be turned in the following week. No late homeworks will be accepted. Similarly, quizzes can not be made up, but
your worst score will be dropped.
Exams:
First midterm exam: Thursday, October 12
Second midterm exam: Tuesday, November 14
Final exam: Thursday, December 21
Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.
Section 002, 004.
Instructor(s): J Tobias Stafford (jts@umich.edu)
Prerequisites & Distribution: 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).CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
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 prooforiented 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, innerproduct spaces; unitary, selfadjoint, and orthogonal operators and matrices, 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 prooforiented. 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.
Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.
Section 005, 006.
Prerequisites & Distribution: 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).CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://www.math.lsa.umich.edu/~bkleiner/syllabus.html
Linear equations, GaussJordan elimination, linear transformations and their inverses, matrix algebra, subspaces, linear
independence, bases, orthogonality, GramSchmidt, orthogonal transformations and matrices, least squares, determinants, eigenvalues and eigenvectors, coordinate systems, diagonalization, and quadratic forms.
Text: Linear Algebra with Applications, by Otto Bretscher
The course work includes regular homework assignments, quizzes, 2 midterms, and a final exam.
Grading policy. Coursework will be weighted as follows: Homework 25%, quizzes 10%, two midterms 20% each, and the final exam 25%.
First midterm: October 12, in class.
Second midterm: November 14, in class.
Final exam: Section 005: Thursday December 21, 1:303:30; Section 006: Thursday December 21, 10:3012:30.
Math. 423. Mathematics of Finance.
Prerequisites & Distribution: Math. 217 and 425; CS 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
 Review of basic probability.
 The oneperiod binomial model of stock prices used to price futures.
 Arbitrage, equivalent portfolios and riskneutral valuation.
 Multiperiod binomial model.
 Options and options markets; pricing options with the binomial model.
 Early exercise feature (American options).
 Trading strategies; hedging risk.
 Introduction to stochastic processes in discrete time. Random walks.
 Markov property, martingales, binomial trees.
 Continuoustime stochastic processes. Brownian motion.
 BlackScholes analysis, partial differential equation and formula.
 Numerical methods and calibration of models.
 Interestrate derivatives and the yield curve.
 Limitations of existing models. Extensions of BlackScholes.
Math. 425/Stat. 425. Introduction to Probability.
Prerequisites & Distribution: 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.
Math. 425/Stat. 425. Introduction to Probability.
Prerequisites & Distribution: Math. 215, 255, or 285. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
See Statistics 425..
Math. 427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans.
Section 001.
Prerequisites & Distribution: Junior standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course. Math 521 and/or 522 (which can be taken independently of each other) provide more indepth examination of the actuarial techniques used in employee benefit plans.
Math. 431. Topics in Geometry for Teachers.
Section 001 – Axiomatic Foundations of Euclidean and nonEuclidean Geometry.
Prerequisites & Distribution: Math. 215, 255, or 285. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~pscott/Math431F00.html
This course is a study of the axiomatic foundations of Euclidean and nonEuclidean 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; finite, hyperbolic, and taxicab nonEuclidean 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.
Math. 433. Introduction to Differential Geometry.
Prerequisites & Distribution: Math. 215, or 255 or 285, and Math. 217(3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course is about the analysis of curves and surfaces in 2 and 3space 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, 4vertex theorem; (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, 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.
Math. 450. Advanced Mathematics for Engineers I.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (4).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~uribe/450F00.html
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 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, 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.
Math. 450. Advanced Mathematics for Engineers I.
Section 002.
Prerequisites & Distribution: Math. 215, 255, or 285. (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 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, 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.
Math. 451. Advanced Calculus I.
Section 001.
Instructor(s): Peter L Duren (duren@umich.edu)
Prerequisites & Distribution: 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, the Fundamental Theorem of Calculus; infinite series; 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 115116, 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.
Math. 451. Advanced Calculus I.
Section 002.
Prerequisites & Distribution: 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: http://www.math.lsa.umich.edu/~spatzier/451/451.html
Course Outline:This course will develop calculus rigorously and introduce concepts and ideas important in more advanced mathematics. More specifically, we will discuss the following material: basic logic, number systems, sequences and series, continuity, metric spaces, derivatives and integrals. This a theoretical course with emphasis on precise definitions and proofs both in the lectures and the homework problems.
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, the Fundamental Theorem of Calculus; infinite series; 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 115116, 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.
Text: "Elementary Analysis: The Theory of Calculus" by by Kenneth A. Ross, Springer Verlag
Course Outline:This course will develop calculus rigorously and introduce concepts and ideas important in more advanced mathematics. More specifically, we will discuss the following material: basic logic, number systems, sequences and series, continuity, metric spaces, derivatives and integrals. This a theoretical course with emphasis on precise definitions and proofs both in the lectures and the homework problems.
Grading Policy: homework 40%; midterm 20% each; final exam 20%;
Homework Policy: Homework will be assigned weekly and collected on Monday. You may discuss the homework problems with other students, but you should write up the solutions on your own.
Math. 454. Boundary Value Problems for Partial Differential Equations.
Prerequisites & Distribution: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3).
Credits: (3).
Course Homepage: https://coursetools.ummu.umich.edu/2000/fall/lsa/math/454/001.nsf
This course covers methods of solving partial differential equations (e.g., the heat, wave, Helmholtz and Laplace equations) with specified boundary conditions in
various geometries. We will cover separation of variables, Fourier series, Bessel
functions, spherical harmonics, orthogonal polynomials, Sturm – Liouville theory, eigenfunctions of the Laplacian in several different coordinate systems, conformal
mapping, etc. These methods have applications in fields as diverse as mechanics, quantum mechanics, thermodynamics, aerodynamics, finance, electromagnetism, and many others, and we will take our examples from such disciplines.
Math. 471. Introduction to Numerical Methods.
Section 001.
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/
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 nonlinear 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, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 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.
Math. 471. Introduction to Numerical Methods.
Section 002.
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (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 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 nonlinear 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, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 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.
Math. 481. Introduction to Mathematical Logic.
Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~pgh/courses/481/
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 firstorder 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 (and, or, not, implies), tautologies, and tautological consequence are studied. The heart of the course is the study of firstorder 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 nonstandard 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.
Text: An Introduction to Mathematical Logic by Richard E. Hodel, PWS Publishing Co. 1995
Grading: 25% homework, 30% midterm exam (Thursday 26 October 78:30), 45% final exam.
Math. 485. Mathematics for Elementary School Teachers and Supervisors.
Prerequisites & Distribution: One year of high school algebra. No credit granted to those who have completed or are enrolled in 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 halfterm).
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 in or preparing to teach in the elementary school.
Math. 497. Topics in Elementary Mathematics.
Prerequisites & Distribution: 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 twocourse sequence Math 385489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problemsolving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two onehour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and inservice elementary, middle, or juniorhigh school teachers. Content will vary from term to term.
Math. 499. Independent Reading.
Section 001.
Prerequisites & Distribution: Graduate standing in a field other than mathematics. (14).
Credits: (14).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: 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: http://www.math.lsa.umich.edu/seminars/applied/index.html
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 UM 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.
Math. 513. Introduction to Linear Algebra.
Prerequisites & Distribution: Math. 412. Two credits granted to those who have completed Math. 214, 217, 417, or 419. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Math 513 is the Math Department's most complete and rigorous course in linear algebra. The formal prerequisite is Math 412; however, it is recommended that students have some experience with other, more demanding, proof oriented courses. Examples would include Math 451, Math 525, Math 531, or any higher level course. COMPLETION OF THE 90'S SEQUENCE IS ITSELF AN
EXCELLENT QUALIFICATION FOR MATH 513. The student body is usually a fairly even mix of Honors Math and CS undergraduates and graduate students from mathrelated fields. Math 513 is also good for Master's students in Math.
The text will be, as usual, Linear Algebra, an Introductory Approach, by Curtis. We will study in depth vector spaces and linear transformations over arbitrary fields. We will also cover bilinear and (elementary) quadratic forms and applications to differential
equations. Significant applications will be an important feature of the course.
Weekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%.
Math. 520. Life Contingencies I.
Prerequisites & Distribution: Math. 424 and Math. 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 multidecrement and multiplelife applications directly related to life insurance and pensions. The sequence 520521 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. Recommended text: Actuarial Mathematics (Second Editions) by Bowles et al.
Math. 523. Risk Theory.
Prerequisites & Distribution: Math. 425. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html
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 riskreward tradeoff. 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 BlackScholes option pricing model.
Math. 525/Stat. 525. Probability Theory.
Prerequisites & Distribution: Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 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 510511 are natural sequels.
Math. 532. Topics in Discrete and Applied Geometry.
Section 001 – Crystals and Quasicrystals
Prerequisites & Distribution: One of Math 217, 417, 419 or 513. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Crystalline patterns have always been observed in nature. This kind of regularity has been expressed mathematically in terms of symmetry groups operating on space preserving (periodic) lattice configurations, e.g., of molecule sites. In the mid1980's the adequacy of this point of view was challenged by the discovery of materials (dubbed "quasicrystals". which appeared crystalline in nature, but which violated certain basic restrictions derived from the groupsymmetry paradigm (existence of "forbidden" fivefold point symmetry).
In this course we propose to study first the basics of symmetry groups in geometry and especially crystallographic groups and their relationship to physical crystals, in particular, the symmetry restrictions on lattices in Euclidean threespace, and at least an introduction to the classification of crystallographic symmetries in two and three space dimensions. The middle third of the course, roughly, will treat the rudiments of Fourier analysis and the theory of xray crystallography. We will use these in order to examine and evaluate computer simulations of diffraction patterns and experiments. Time permitting, we will also discuss some relatively current issues in xray crystallography related to protein crystallography. Finally we will discuss aperiodic phenomena in tilings, examine what their regularities are and discuss some very open issues: Are these really a candidate for the modeling of quasicrystals in nature? Are there finite sets of local rules by which one may know how to build up a Penrose pattern to cover the whole plane? How does one classify the set of Penrose tilings geometrically and what information does such a tiling carry?
The course will involve lectures, regular problem sets and a term project, but no exams. In addition, there will be computer simulations to do (packages provided).
The texts will be: "Groups and Symmetry", M.A. Armstrong (Springer), "Quasicrystals and Geometry", by M. Senechal (Cambridge), "Miles of Tiles", by C. Radin (American Mathematical Society), as well as some notes on Fourier analysis, and the book "Principles of Protein XRay Crystallography", by J. Drenth (Springer) [not required]. Webbased materials will also be used.
Background prerequisites will be flexible. The course is suitable for undergraduates with a background of calculus, linear algebra and, perhaps, the rudiments of groups, and graduate students in mathematics or areas of possible application: chemistry, physics, engineering and biology. If in doubt, contact the instructor.
Math. 537. Introduction to Differentiable Manifolds.
Prerequisites & Distribution: 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 twosemester 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 manifoldswithboundary. 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, LeviCivita connections, geodesics and curvature. Homework assignments will be given periodically, with the frequency depending on whether or not we get a grader. There will also be a final exam.
The textbooks will be "Differential Topology" by Victor Guillemin and Alan Pollack, PrenticeHall, and "Morse Theory" by John
Milnor, Princeton University Press. Math 591 or the equivalent is a prerequisite. I will assume a knowledge of differentiable
manifold theory as covered in Sections 1.11.4 of the book by Guillemin and Pollack. The titles of these sections are "Definitions", "Derivatives and tangents", "The inverse function theorem and immersions" and "Submersions". If a prospective student has not seen this material before, it might be helpful to look at Chapter 1 of Guillemin and Pollack. I will review this material at the
beginning of the academic term.
Math. 555. Introduction to Functions of a Complex Variable with Applications.
Prerequisites & Distribution: Math. 450 or 451. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Text: Complex Variables and Applications, 5th ed. (Churchill and Brown);
Student Body: largely engineering and physics graduate students with some math and engineering undergrads
Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable. Concepts and calculations are emphasized over proofs.
Content: Differentiation and integration of complex valued functions of a complex variable, series mappings, residues, applications. Evaluation of improper real integrals. This corresponds to Chapters 19 of Churchill. Alternatives: Math 596 (Analysis
I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs
rather than calculations.
Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.
There will be homework, midterm and a final.
Math. 556. Methods of Applied Mathematics I.
Prerequisites & Distribution: 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, SturmLiouville problems, and eigenfunction expansions. We will then move on to the Fourier Transform, RiemannLebesgue 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, Schrodinger's equation, Laplace's equation.
Text: Fourier Analysis and its Applications by G.B. Folland
Grading: homework 60%, midterm 15%, final exam 25%. Homework is key in this class. You are expected to hand in carefully completed homework.
Math. 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419. (3).CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering 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.
Math. 562/IOE 511/Aero. 577. Continuous Optimization Methods.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419. (3).CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No Homepage Submitted.
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasiNewtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.
Math. 565. Combinatorics and Graph Theory.
Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/565.html
Student work expected: several problem sets.
Synopsis: The ultimate fun course, with a focus on problem solving, showcasing the gems of enumerative and algebraic combinatorics. The course will cover a
dozen of virtually independent topics, chosen solely on the basis of their
beauty. Topics will include generating functions, algebraic graph theory, partially ordered sets, combinatorics of polytopes, matching theory, enumeration of tilings, partitions, and Young tableaux.
Reference texts (none required):
 [BS]
 A.Bjorner and R.P.Stanley, A combinatorial miscellany,
Cambridge University Press, to appear.
 [GS]
 I.Gessel and R.P.Stanley, Algebraic enumeration, in Handbook of Combinatorics, MIT Press, 1995.
 [vW]
 J.H. van Lint and R.M.Wilson, A course in combinatorics , Cambridge University Press, 1996.
 [EC]
 R.P.Stanley, Enumerative combinatorics, vol.12, Cambridge University Press, 19971999.
Potential topics to be covered:
 Hooklength formula.
 De Bruijn sequences.
 Enumeration of trees.
 Stirling numbers.
 Spectra of graphs.
 Walks on a cube.
 Sperner theory.
 Inversions and major index.
 qbinomial coefficients.
 Distributive lattices.
 Gaussian coefficients.

 Tableaux and involutions.
 Schensted's correspondence.
 Catalan numbers.
 Matrixtree theorem.
 Eulerian tours.
 Domino tilings.
 Polya theory.
 The marriage theorem.
 Assignment polytope.
 Cyclic polytopes.
 Permutohedra.

Math. 571. Numerical Methods for Scientific Computing I.
Prerequisites & Distribution: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Prerequisites: linear algebra on the level of Math 419 or Math 513, working knowledge of computer programming (any language).
Background: This course deals with numerical methods for solving linear systems of equations ($Ax{=}b$) and the eigenvalue problem ($Ax{=}\lambda x$). A main motivation will be systems that arise from discretizing elliptic boundary value problems by
finitedifference and variational methods, in one and two space dimensions. Material will include chapters 16 of Ciarlet's book and Briggs' notes. Additional topics (e.g., least squares problem, conjugate gradient method, GMRES) may be presented if there is
enough time.
Alternatives: There is no real alternative. Math 471 (Introduction to numerical methods) covers a small part of the same material at a lower level. Math 571 and 572 may be taken in either order.
Subsequent Courses: Math 671 (Analysis of numerical methods I) is an advanced topics course in numerical analysis. Topics vary.
Texts:
 Introduction to Numerical Linear Algebra and Optimisation, by P. G. Ciarlet, Cambridge University Press.
 A Multigrid Tutorial, by W. L. Briggs, SIAM.
Math. 575. Introduction to Theory of Numbers I.
Prerequisites & Distribution: Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (1, 3).
Credits: (1, 3).
Course Homepage: No Homepage Submitted.
Prerequisites: Basic analysis and modern algebra (equivalent to the level of Math 451 and 412)
Text: Class notes and problem sheets will be selfcontained and comprehensive. The standard source should be: an Introduction to the Theory of Numbers (Niven, Zuckerman and Montgomery, 5th edition, Wiley, 1991), or failing that: Introduction to Number
Theory (L.K. Hua, SpringerVerlag, 1982).
Interested in graduate study in Number Theory? Heard about Public Key Cryptosystems and want to find out what makes them tick? Intrigued by number theory after hearing about Fermat's Last Theorem? Then Math 575 is the course for you!
Math 575 has recently been revised with the intention of providing graduate students with a solid introduction to Number Theory suitable for continuing through the subsequent courses in the graduate program in Number Theory here at Michigan. As such, it will also provide a means for undergraduates interested in Number Theory to prepare for graduate study elsewhere. Graduate students not directly interested in Number Theory will be able to complete their renaissance education by hearing one of the epic tales of mathematical conquest on a Homeric scale.
Students wishing to take the course should already have significant experience in writing proofs, and should have a basic understanding of analysis and abstract algebra (groups, rings, fields).
Content: We begin with a reasonably brisk discussion of the basic notions: Euclidean algorithm, primes and unique factorisation, congruences, Chinese Remainder Theorem (Public Key Cryptosystems), primitive roots, quadratic reciprocity and binary quadratic
forms. The second half of the course is devoted to topics (as time permits) which lead to substantial lines of research covered in subsequent graduate courses: diophantine equations, quadratic fields, padic numbers, elliptic curves, diophantine approximation and transcendence, arithmetic functions, continued fractions, distribution of prime numbers.
Coursework: Approximately one assignment every two weeks, containing both easy and challenging questions, together with 2 straightforward inclass midterms and a takehome final exam (4 days available to complete this). Anyone wishing to discuss the course, or Number Theory in general is welcome to talk with me.
Math. 590. Introduction to Topology.
Prerequisites & Distribution: Math. 451. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math590/
Text: Sieradski, An Introduction to Topology and Homotopy, PWSKent.
Topology provides a foundation for many areas of mathematics and is itself an active area of research. This course is an
introduction to the subject and will emphasize the construction of proofs.
Topics include metric spaces, abstract topological spaces, continuous functions, connectedness, compactness, the fundamental
group and surfaces.
The student will engage in brief presentations, problem sets, a midterm and a final exam.
Math. 591. General and Differential Topology.
Prerequisites & Distribution: Math. 451. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~pscott/Math591F00.html
This course will cover the prerequisites for the portions of the topology qualifying examination, which involve pointset and differential topology. We will start by introducing abstract topological spaces and their basic properties. We will examine in detail the properties of connectedness and compactness. Then we will focus on the quotient topology, group actions and orbit spaces.
The course will end by studying manifolds and differential topology, where topics covered will include tangent spaces, the regular
value theorem. Whitney's embedding theorem and transversality. Students with a strong background in pointset and differential topology may want to consider taking Math 537 instead.
Math. 593. Algebra I.
Prerequisites & Distribution: Math. 513. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~idolga/teach.html
Prerequisites: First courses in abstract algebra and linear algebra (Math 513 or 419 and Math 512).
This is the first part of a twosemester course in basic algebra. The main topic is linear algebra of modules over any ring. We shall classify modules over principal ideal domains and deduce from this the theory of Jordan forms of matrices over a field. Other topics include multilinear algebra (tensors and exterior algebra), structure of symmetric bilinear forms over arbitrary fields, orthogonal groups, Clifford algebras, elements of homological algebra.
The work will be evaluated on the basis of homework problem solutions and one final exam.
Textbook: S. Lang Algebra, 3d edition. AddisonWesley 1993. We plan to cover Chapters 3,4,13,14,15,16,19,20 from this book.
Math. 596. Analysis I.
Prerequisites & Distribution: Math. 451. Students with credit for Math. 555 may elect Math 596 for two credits only. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Complex numbers, geometric properties, stereographic projection, basic spherical geometry. Complex functions, differentiability and the CauchyRiemann equations, the Laplace operator, elementary analytic functions and linear fractional transformations, construction of conformal mappings.
Contour integrals, Cauchy's theorem and the Cauchy integral formula, Taylor series and Laurent expansions, Liouville's theorem, unique continuation, Morera's theorem. Residue theorem and applications. Analytic continuation and Schwarz reflection principle.
Argument principle, Rouche's theorem, Hurwitz's theorem, local univalence. Maximum modulus theorem, the Schwarz lemma and some generalizations. Harmonic functions, Poisson formula and Jensen's thoerem.
Meromorphic functions, MittagLeffler's theorem, infinite products, Weierstrass' theorem, removeable singularities, CaseratiWeierstrass theorem.
Normal families, Montel's theorem, Riemann mapping theorem. There will be approximately weekly problem sets, two midterm
exams and a final. A written term project can be done in lieu of the second midterm exam.
Math. 602. Real Analysis II.
Section 001.
Prerequisites & Distribution: Math 590 and 597. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course is a basic course for anyone who is interested in analysis, topology or applied mathematics. It deals with fundamental tools (Hilbert Spaces, Banach Spaces, operators and distribution theory), needed in these disciplines. I plan to consider the
following topics: category, contraction mappings, Hahn Banach Theorem, Banach spaces, dual spaces, Holder and Minkowski
inequalities, Riesz representation theorem, closed graph and open mapping theorems, uniform boundedness principle, condensation of singularities, Hilbert space, orthonormal sets and Fourier expansions, bounded operators on Hilbert Space, compact operators, Schauder fixed point theorem, theory of distributions.
For each of the above topics, I will usually give applications, for example: local existence theorem for ordinary differential
equations, existence of a continuous nowhere differentiable function, closable operators, existence of a periodic continuous
function with a divergent Fourier series at a point, the mean ergodic theorem, existence of fundamental solutions, etc.
There will not be any exams; instead I will give out homework problem sets which I shall grade.
Math. 605. Several Complex Variables.
Section 001.
Prerequisites & Distribution: Math. 596 and 597. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This will be an introductory course in multivariate complex analysis, covering such topics as pseudoconvexity, envelopes of holomorphy, polynomial hulls, holomorphic extension problems, geometry of boundaries, and the inhomogeneous
CauchyRiemann equations. Additional topics (such as structure of analytic sets, intrinsic metrics, extremal problems) will be covered as time permits.
Homework will be assigned approximately every other week.
Math. 612. Lie Algebra and their Representatives.
Section 001.
Prerequisites & Distribution: Math. 593 and 594; Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Text: J. E. Humphreys, Introduction to Lie Algebras and Representation Theory.
Semisimple Lie algebras and their representations are at the crossroads of many important branches of mathematics – witness the ubiquity of Dynkin diagrams, for example. This course should be valuable for those interested in representation theory, as well as those with interests in allied areas, such as (algebraic) combinatorics, group theory, and noncommutative algebra.
We will cover the basic theory of Lie algebras, with emphasis on the complex semisimple case. We plan to cover most of the topics
in Humphreys' book, with the highest priority given to the (finitedimensional) representation theory.
A solid grounding in linear algebra, and familiarity with the basic concepts of abstract algebra will be necessary. If you have any
questions about the course, feel free to contact me.
Math. 614. Commutative Algebra.
Section 001.
Prerequisites & Distribution: Math. 593. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Review of commutative rings and modules. Local rings and localization. Noetherian and Artinian rings. Integral independence. Valuation rings, Dedekind domains, completions, graded rings. Dimension theory.
Math. 619. Topics in Algebra.
Section 001.
Prerequisites & Distribution: Math. 593. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 626/Stat. 626. Probability and Random Processes II.
Section 001.
Prerequisites & Distribution: 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. Control theory is concerned with the evolution of a dynamical system which contains parameters. The parameters are to be adjusted so as to minimize a certain functional called the performance criterion or cost function. A typical performance criterion would be the distance of the dynamic particle from a fixed point, the target point. The first part of the course explores the relationship between control problems and Hamiltonian mechanics. Central to this is the result that the performance criterion satisfies a first order partial differential equation, the HamiltonJacobi equation, known as the Bellman equation in control theory. The second part of the course considers randomly perturbed dynamical systems. The performance criterion 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 MongeAmpere equation. The third part of the course explores 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 performance criterion. In the final part of the course we will be concerned with the MongeKantorivich mass transfer problem. The original transport problem was proposed by Monge in the
1780's. Monge proposed the problem as how best to move a pile of soil to an excavation with the least amount of work.
Kantorovich and Koopmans studied the problem further in the 1940's in the context of the optimum utilisation of the transportation system. They obtained the 1975 Nobel prize in economics for their work. One can obtain the solution to the mass transfer problem with quadratic cost function by solving the MongeAmpere equation. Hence this problem is related to problems which arise in stochastic control theory.
Prequisite: Some knowledge of differential equations and probability theory (at the 400 level). Grades: The grade on the course will be determined by a student's performance on homework sets.
Recommended book: "Deterministic and Stochastic Optimal Control" by W. Fleming and R. Rishel, Springer 1975 (reprinted 1999).
Math. 631. Introduction to Algebraic Geometry.
Section 001.
Prerequisites & Distribution: Math. 594 and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This is an introduction to algebraic geometry. We will discuss affine and projective varieties, including such notions as dimension, singularity, degree, divisors, intersections, and differentials. We will study algebraic curves, including the RiemannRoch theorem.
Text: Shafarevich, Basic Algebraic Geometry, I (and II)
Recommended Text: Harris, Algebraic Geometry, A First Course
Math. 636. Topics in Differential Geometry.
Section 001 – Rigidity Phenomena in Geometry, Group Theory and Dynamics
Prerequisites & Distribution: Math. 635. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
I will discuss various rigidity phenomena in geometry, group theory and dynamics starting with the by now classical rigidity theorems of Mostow and Margulis. The proofs of these theorems draw from several sources: differential and conformal geometry, group theory, ergodic theory and representation theory. In fact, at this point there are at least four completely different proofs of
Mostow's theorem. The results and the methods used have instigated a tremendous amount of activity and a slew of important
results that transcend from differential geometry into group theory and number theory as well dynamics and ergodic theory. I will
discuss some of these ideas and results which may include the theorem of Besson, Courtois and Gallot on minimal volume growth and applications, the paucity of compact manifolds modelled on homogeneous spaces and the dynamics of "large" group actions.
The course will assume familiarity with basic differential geometry and Lie group theory, but will otherwise be essentially selfcontained.
Math. 637. Lie Groups.
Section 001.
Prerequisites & Distribution: Math. 635. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 655. Topics in Fluid Dynamics.
Section 001.
Prerequisites & Distribution: Math. 555, 556, 557, 558; Graduate standing. (3).May be repeated for credit.
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 656. Partial and Differential Equations I.
Section 001.
Prerequisites & Distribution: Math. 558, 596 and 597, and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 658. Ordinary Differential Equations.
Section 001.
Prerequisites & Distribution: A course in differential equations (e.g.404 or 558). Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Prerequisite: a course in differential equations.
This course will discuss certain aspects of the modern theory of ordinary differential equations and dynamical systems, with
emphasis on applications to mechanics and nonlinear control theory. Topics will include the qualitative theory of ODE's on
manifolds, nonlinear stability theory, Lagrangian and Hamiltonian mechanics, mechanical systems with constraints including
nonholonomic systems and the Dirac theory of constraints, reduction and symmetries, integrable systems, mechanical systems with forces, and some key ideas in the control of nonlinear mechanical systems and optimal control. The geometric underpinning of many of these concepts will be discussed.
Recommended text: Typed notes by instructor will be handed out. In addition, J. Marsden and T. Ratiu, Mechanics and Symmetry
is recommended. Other books such as V. Arnold, Mathematical Methods of Classical Mechanics will be referenced as well as the primary mathematical literature.
Grading: The course grade will be based on completion of some problem sets, a small essay/project and general class participation.
Math. 669. Topics in Combinatorial Theory.
Section 001 – Combinatorics and Group Representations: A Tour of SL(n) and GL(n).
Prerequisites & Distribution: Math. 565, 566, or 664, and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course will be a survey of (a fraction of) the many interesting applications of Representation Theory to Combinatorics, and viceversa. Some of the most interesting results in Combinatorics have been derived by means of representationtheoretic tools.
Enumeration of plane partitions, unimodality theorems, and the RogersRamanujan identities are all examples of this. In the opposite direction, the symmetry groups that occur most frequently in nature (the symmetric groups, the classical groups) have
representations and characters with extensive combinatorial structure.
The course will be divided into two parts. Part I will begin with a selfcontained development of the representation theory of finite
groups, followed by a detailed study of the case of symmetric groups and closely related groups.
Part II will be concerned with the classical groups; primarily GL(n) and its cousins (SL(n), U(n), etc). In order to have more time to
discuss the combinatorial aspects, we will probably not present detailed proofs of all of the fundamental theorems in this part of the course. (For example, existence of Haar measure will not be proved).
The text "Representation Theory: A First Course" by W. Fulton and J. Harris (SpringerVerlag, 1991) is a good text for the
representation theory side. The combinatorial side will be cobbled together from multiple sources.
Prerequisite for the course is a basic familiarity with algebra (equivalent to Math 594, say). Previous exposure to combinatorics, or a course in Lie algebras, will be advantageous but not necessary.
Math. 676. Theory of Algebraic Numbers.
Section 001.
Instructor(s): Christopher Skinner (cskinner@umich.edu)
Prerequisites & Distribution: Math. 575 and 594. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
This course will be a survey of (a fraction of) the many interesting applications of Representation Theory to Combinatorics, and viceversa. Some of the most interesting results in Combinatorics have been derived by means of representationtheoretic tools. Enumeration of plane partitions, unimodality theorems, and the RogersRamanujan identities are all examples of this. In the opposite direction, the symmetry groups that occur most frequently in nature (the symmetric groups, the classical groups) have representations and characters with extensive combinatorial structure.
The course will be divided into two parts. Part I will begin with a selfcontained development of the representation theory of finite groups, followed by a detailed study of the case of symmetric groups and closely related groups.
Part II will be concerned with the classical groups; primarily GL(n) and its cousins (SL(n), U(n), etc). In order to have more time to discuss the combinatorial aspects, we will probably not present detailed proofs of all of the fundamental theorems in this part of the course. (For example, existence of Haar measure will not be proved).
The text "Representation Theory: A First Course" by W. Fulton and J. Harris (SpringerVerlag, 1991) is a good text for the representation theory side. The combinatorial side will be cobbled together from multiple sources.
Prerequisites: Math 593 and 594 or knowledge of groups, rings, and modules and basic Galois theory.
This course will be an introduction to the fundamental objects of study in algebraic number theory: number fields, local fields, rings of integers, valuations, factorization, class groups, and unit groups. The Fundamental Theorem of Arithmetic states that every
nonzero integer m can be written as a product of powers of prime numbers:
m = up_1^r_1...p_s^r_s, u=+1 or 1, p_i a prime, r_i>0,
and that this expression is essentially unique. The first goal of this course will be to understand the extent to which this holds (or
fails to hold) in number fields. This leads to the notions of the ring of integers of a number field and class group. The latter
measures the extent to which unique factorization fails and is still an object of much study (and mystery).
Since factorization is in general only up to units, to understand a number field we must understand the units in its ring of integers.
As a first step we will prove Dirichlet's theorem about the structure of the group of units in a number field.
We will also cover local fields and their rings of integers. For us, the most important example of a local field will be the completion of a number field with respect to a valuation. Local fields play an indispensable role in understanding global fields.
Many of the general concepts introduced in this course will be studied in more detail for quadratic fields (number fields generated
by the roots of a quadratic polynomial) and for cyclotomic fields (number fields generated by roots of unity). As time permits we
may also cover applications to diophantine equations, such as the early efforts to prove Fermat's Last Theorem.
Knowledge of the material covered in this course will be a prerequisite for next term's course on Class Field Theory, which completely describes the Abelian extensions of a number field or local field. It will also be a prerequisite for subsequent courses on the arithmetic of elliptic curves and modular forms.
Course Work: There will be no exams but maybe some homework.
Texts: There will be no required textbook. I will pass out a list of books that I consider to best cover the material in this course.
These will be on reserve at the library.
Math. 677. Diophantine Problems.
Section 001.
Instructor(s): Wooley
Prerequisites & Distribution: Math. 575. Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Prerequisites: Math 575 (or equivalent introduction to number theory), basic analysis and modern algebra (alpha courses amply
exceed requirements).
The ancient mystic art of diophantine problems, practiced by the ancient Greeks and Egyptians, burnished by the brilliant Gauss, and built into a magnificent edifice of modern mathematical achievement by the master craftsmen of the 19th and 20th Centuries, most recently with work of Wiles on Fermat's Last Theorem.... Math 677 takes you on an epic journey through the fertile valleys of this subject.
Interested?
Content: This course is intended to be an introduction to diophantine equations and inequalities (in which one seeks integral solutions of equations and inequalities), especially those parts of the subject accessible to analytic methods and ideas stemming
from the theory of diophantine approximations.
The course begins with an introduction to exponential sums and the HardyLittlewood method, with a discussion of Weyl's
inequality, Hua's Lemma, and the simplest treatment of Waring's problem and diagonal diophantine equations (with a discussion of the associated padic solubility problem). We discuss also distribution modulo 1 of sequences of the shape alpha n^k, and related problems. Next we discuss the DavenportHeilbronn method for solving diagonal diophantine inequalities, and quantitative
estimates will be investigated according to interest. The final part of the course will be devoted to methods from diophantine approximation, starting with Roth's Theorem and Thue equations. Subject to time constraints, we will also provide an introduction to Baker's estimates for linear forms in logarithms.
Coursework: Approximately one assignment every two weeks, containing both easier and more challenging problems. No exams.
Text: Class notes and problem sheets will be selfcontained and comprehensive. Some standard sources: The HardyLittlewood
method (R. C. Vaughan, Cambridge Tract No. 125, C.U.P., 1997), Diophantine Inequalities (R. C. Baker, London Mathematical
Society Monographs, No. 1, O.U.P., 1986), Transcendental Number Theory (A. Baker, 2nd ed., Cambridge University Press, 1990).
Anyone wishing to discuss the course, or Number Theory in general, is welcome to talk with me (wooley@math.lsa.umich.edu).
Math. 681. Mathematical Logic.
Section 001.
Prerequisites & Distribution: Mathematical maturity appropriate for a 600level course. Graduate standing. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~pgh/courses/681/
The goal of the first half of the course is a thorough and rigorous exposition of the completeness and compactness theorems of
firstorder logic. The syntax of firstorder languages, their interpretations in mathematical structures, and the relationship of truth and formal proof are treated in detail. Notions of decidability, definability, and consistency are constant themes. The remainder of the academic term treats more advanced topics in logic and model theory, such as nonstandard models of arithmetic and analysis and an introduction to the Gödel incompleteness and undecidability theorems.
In addition to developing logic as a an area of mathematics, some attention will be devoted to the role of logic as a foundation for the rest of mathematics. The facts that (1) most mathematical objects can be represented as sets, and (2) there are firstorder theories of sets which suffice to prove most of the theorems of mathematics, provide a precise meaning to the assertion that a
mathematical statement is a theorem.
There are no specific prerequisites, but mathematical sophistication appropriate to a 600level course is expected. A previous course in logic is helpful but by no means required. The text will be notes prepared by the lecturer and there will be numerous suggested books for supplementary reading.
Math 681 is a prerequisite for Math 682 (Set Theory), which includes the consistency and independence of the Axiom of Choice and the (Generalized) Continuum Hypothesis, Math 683 (Model Theory), and Math 684 (Recursion Theory) which develops the Gödel incompleteness and undecidability results and the elements of the theory of degrees of unsolvability and recursively enumerable sets.
Math. 694. Differential Topology II.
Section 001.
Prerequisites & Distribution: Math. 337 and 591 and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Transversality, embedding theorems, vector bundles and selected topics from the theories of cobordism, surgery, and characteristic classes.
Math. 695. Algebraic Topology I.
Section 001.
Prerequisites & Distribution: Math. 591 and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Math 695 is a one term course in algebraic topology. The prerequisite for the course is math 592 or equivalently, some
familiarity with singular homology. Topics to be covered include cohomology, homotopy groups and obstruction theory. Math 696
is the natural follow on course.
There will not be any exams but there will be occasional homework assignments.
Math. 697. Topics in Topology.
Section 001 – Deformation Theory.
Prerequisites & Distribution: Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
The first portion of this course will focus on the deformation theory of hyperbolic surfaces. A hyperbolic structure on a surface is a
metric on the surface in which small balls are isometric to small balls in the hyperbolic plane. Every surface of genus bigger than one admits a hyperbolic structure, in fact many different hyperbolic structures. We will study the Teichmuller space of all hyperbolic structures on a fixed surface from both the geometric (FenchelNielsen coordinates) and complex analytic (quasiconformal maps)
viewpoints. Teichmuller spaces, and their quotient Moduli spaces, come up naturally in the fields of geometry, lowdimensional topology, algebraic geometry and complex analysis.
The content of the second portion of the course will be determined by the interests of the students. Possible topics include:
deformation theory of hyperbolic 3manifolds, spectral theory on hyperbolic surfaces, and compactifications of Teichmuller and Moduli spaces.
This course will be geared towards beginning graduate students with an interest in topology, geometry or complex analysis. The only prerequisites will be the material in 591 and 592 on differentiable manifolds and covering spaces. In particular, the course will
begin with an elementary introduction to hyperbolic geometry and the basic properties of hyperbolic surfaces.
Math. 700. Directed Reading and Research.
Prerequisites & Distribution: Graduate standing. (13). (INDEPENDENT).
Credits: (13).
Course Homepage: No Homepage Submitted.
No Description Provided
Check Times, Location, and Availability
Math. 703. Topics in Complex Function Theory I.
Section 001 – Geometric Measure Theory
Prerequisites & Distribution: Math. 604. Graduate standing. (3).May be taken for credit more than once.
Credits: (3).
Course Homepage: No Homepage Submitted.
Traditionally, Geometric Measure Theory consists of a study of the
geometry of (Borel) measures in Euclidean nspace. This study includes, in particular, the analysis of "rectifiable" sets of integral
dimension and various fractal (nonintegral dimensional) phenomena. Today Geometric Measure Theory is a vast subject which touches several branches of mathematics. For example, fractal sets and their measure theoretic analysis are important not only in
mathematical analysis but in geometry and topology as well; even in geometric group theory one studies the metric properties and Hausdorff dimension of the boundary of a group. Similarly, the theory of currents is used in mathematics from the calculus of
variations to algebraic geometry.
In the first part of the course, we cover basic classical material (Hausdorff measures, rectifiability, Federer's structure theorem, currents) in Euclidean space. Towards the end, we shall discuss more recent developments (uniform rectifiability of David and Semmes, rectifiability in nonRiemannian sets, etc.). In particular, much of the current research activity in the field takes place in
general metric space settings, where one finds a number of accessible research topics. I shall try to choose topics so as to be useful to students from different areas of concentration.
A course in measure theory (such as Math 597) is the only prerequisite for this class. There will be no assigned text but the book
"Geometry Of Sets And Measures In Euclidean Spaces: Fractals And Rectifiability" by P. Mattila, Cambridge 1995, is
recommended.
Math. 709. Topics in Modern Analysis I.
Section 001 – Complex Dynamics in Higher Dimensions.
Prerequisites & Distribution: Math. 597. Graduate standing. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~mattiasj/teaching/709/
This is a course on complex dynamics in higher dimensions, which for us will mean iterations of holomorphic selfmaps of a complex manifold. Of primary interest is the question "what happens in the long run?'', i.e., how can we understand the asymptotic
behavior as we iterate more and more times. Do the iterates behave "tamely" or "chaotically"?
There are two main sources of inspiration and techniques for higherdimensional complex dynamics. The first one is the field of
differentiable dynamics, i.e. the study of iterations of smooth mappings of Riemannian manifolds. This is an an old field, which
regained interest with the work of Smale in the sixties and subsequently underwent a rapid development, still continuing today.
The second source is the field of onedimensional complex dynamics, which has been an extremely active field during the last twenty years, and accountable for two recent Fields medals.
In higherdimensional complex dynamics, ideas from both these fields are combined, and a rich and beautiful (but far from complete) theory has been developed during the last ten years, due to work of Fornaess and Sibony, Bedford and Smillie, and others. The field is still very active, and the purpose of this course is to lead to research problems. There are no formal prerequisites. Only basic knowledge about one complex variable is essential. Familiarity with onedimensional complex dynamics, (real) hyperbolic dynamics, ergodic theory, several complex variables and pluripotential theory will be helpful, but the necessary concepts will be developed along the way.
There will be no text.
Math. 711. Advanced Algebra.
Section 001.
Instructor(s): Robert Greiss (rlg@umich.edu)
Prerequisites & Distribution: Math. 594 or 612 and Graduate standing. (3).
Credits: (3).
Course Homepage: No Homepage Submitted.
Prerequisites: alpha level algebra courses (593 and 594) or equivalent.
This beta course will present a mixture of topics from general finite group theory, the theory of finite simple groups and basics of
groups acting on error correcting codes and lattices. Below is roughly what I have in mind.
The transfer, local subgroups, control of fusion, normal complements. Extension theory, including the SchurZassenhaus theorem.
The Hall theory which generalizes Sylow theory to sets of primes for solvable groups. Brief description of cohomology of groups. A
brief introduction to classical groups and their geometries. Simplicity of some classical groups (PSL(n,q) and some orthogonal, symplectic and unitary groups). High transitivity of certain actions of classical groups.
Groups acting on lattices including Weyl groups and other groups. Relevant coding theory will be developed. We will present the
recent Pieces of Eight theory which gives a new theory of the sporadic simple groups of Mathieu and Conway (twelve of the twenty six sporadic groups are involved here). This is a setting where we see many important examples of finite simple groups acting on combinatorial structures, e.g., rank 3 graphs.
There will be expository lectures towards the end on more advanced topics and areas for further research.
Math. 731. Topics in Algebraic Geometry.
Section 001.
Prerequisites & Distribution: Graduate standing. (3).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~idolga/teach.html
The goal of this course is to explain, in terms familiar to mathematicians, some fundamental ideas in modern physics related to
quantum field theory, supersymmetry, string theory and dualities. During the last twenty years these ideas led to the discovery of
many new exciting geometrical structures, and were also used to attack some of the central problems in mathematics. An approximate contents of the course is the following:
 Action principle.
 Quantum mechanics.
 Quantization of classical fields.
 Bozonic strings.
 Superstrings.
 Vertex operators.
 Path integrals.
 Quantum field theory.
 Conformal field theory.
 Dualities in string theory.
No background in physics will be needed. We shall review everything we shall need from mathematics (Lie algebras and their
representations, Clifford algebras and spinors, Hilbert spaces, Loop spaces, modular forms, Kahler manifolds).
The basic references are:
 Quantum fields and strings: A course for mathematicians, vols. 1, 2. Amer. Math. Soc. Providence. 1999.
 V. Vafa. A Harvard course on string theory for mathematicians (handwritten notes).
Math. 990. Dissertation/Precandidate.
Prerequisites & Distribution: Election for dissertation work by doctoral student not yet admitted as a Candidate. Graduate standing. (18). (INDEPENDENT). May be repeated for credit.
Credits: (18; 14 in the halfterm).
Course Homepage: No Homepage Submitted.
Election for dissertation work by doctoral student not yet admitted as a Candidate.
Math. 993. Graduate Student Instructor Training Program.
Prerequisites & Distribution: 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 class.
Math. 995. Dissertation/Candidate.
Prerequisites & Distribution: Graduate School authorization for admission as a doctoral Candidate. Graduate standing. (8). (INDEPENDENT). May be repeated for credit.
Credits: (8; 4 in the halfterm).
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.
This page was created at 8:01 AM on Fri, Oct 20, 2000.
University of Michigan  College of LS&A  LS&A Research and Graduate Education  Rackham Bulletin Index  Rackham School of Graduate Studies
This page maintained by LS&A Academic Information and Publications, 1228 Angell Hall
Copyright © 2000 The Regents of the University of Michigan,
Ann Arbor, MI 48109 USA +1 734 7641817
Trademarks of the University of Michigan may not be electronically or otherwise altered or separated from this document or used for any nonUniversity purpose.
