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Elementary
Mathematics Courses. In order to accommodate diverse
backgrounds and interests, several course options are available
to beginning mathematics students. All courses require three years
of high school mathematics; four years are strongly recommended
and more information is given for some individual courses below.
Students with College Board Advanced Placement credit and anyone
planning to enroll in an upperlevel class should consider one
of the Honors
sequences and discuss the options with a mathematics advisor.
Students who need additional preparation for calculus are tentatively
identified by a combination of the math placement test (given
during orientation), college admission test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss
this placement information with each student and refer students
to a special mathematics advisor when necessary.
Two courses preparatory to the calculus, MATH 105 and 110, are offered. MATH 105 is a course on data analysis, functions
and graphs with an emphasis on problem solving. MATH 110 is a
condensed halfterm version of the same material offered as a
selfstudy course taught through the Math Lab and is only open
to students in MATH 115 who find that they need additional preparation
to successfully complete the course. A maximum total of 4 credits
may be earned in courses numbered 103, 105, and 110. MATH 103
is offered exclusively in the Summer halfterm for students in
the Summer Bridge Program.
MATH 127 and 128 are courses containing selected topics from
geometry and number theory, respectively. They are intended for
students who want exposure to mathematical culture and thinking
through a single course. They are neither prerequisite nor preparation
for any further course. No credit will be received for the election
of MATH 127 or 128 if a student already has credit for a 200(or
higher) level MATH course.
Each of MATH 115, 185, and 295 is a first course in calculus.
Generally credit can be received for only one of 115 or 185. The
sequence MATH 115116215 is appropriate for must students who
want a complete introduction to calculus. One of 215, 285, or
395 is prerequisite to most more advanced courses in Mathematics.
The sequences MATH 156255256, 175186285286, 185186285286, and 295296395396 are Honors sequences. Students need not be
enrolled in the LS&A Honors Program to enroll in any of these
courses but must have the permission of an Honors advisor. Students
with strong preparation and interest in mathematics are encouraged
to consider these courses.
MATH 185285 covers much of the material of MATH 115215 with
more attention to the theory in addition to applications. Most
students who take MATH 185 have taken a high school calculus course, but it is not required. MATH 175186 assumes a knowledge of calculus
roughly equivalent to MATH 115 and covers a substantial amount
of socalled combinatorial mathematics as well as calculusrelated
topics not usually part of the calculus sequence. MATH 175 is
taught by the discovery method: students are presented with a
great variety of problems and encouraged to experiment in groups
using computers. The sequence MATH 295396 provides a rigorous
introduction to theoretical mathematics. Proofs are stressed over
applications and these courses require a high level of interest
and commitment. Most students electing MATH 295 have completed
a thorough high school calculus course. MATH 295396 is excellent
preparation for mathematics at the advanced undergraduate and
beginning graduate level.
Students with strong scores on either the AB or BC version
of the College Board Advanced Placement exam may be granted credit
and advanced placement in one of the sequences described above;
a table explaining the possibilities is available from advisors
and the Department. In addition, there is one course expressly
designed and recommended for students with one or two semesters
of AP credit, MATH 156. MATH 156 is an Honors course intended
primarily for science and engineering concentrators and will emphasize
both applications and theory. Interested students should consult
a mathematics advisor for more details.
In rare circumstances and with permission of a Mathematics
advisor, reduced credit may be granted for MATH 185 after MATH
115. A list of these and other cases of reduced credit for courses
with overlapping material is available from the Department. To
avoid unexpected reduction in credit, a student should always consult
an advisor before switching from one sequence to another. In all
cases a maximum total of 16 credits may be earned for calculus
courses MATH 115 through 396, and no credit can be earned for
a prerequisite to a course taken after the course itself.
Students completing MATH 116 who are principally interested
in the application of mathematics to other fields may continue
either to MATH 215 (Analytic Geometry and Calculus III) or to
MATH 216 (Introduction to Differential Equations); these two courses
may be taken in either order. Students who have greater interest
in theory or who intend to take more advanced courses in mathematics
should continue with MATH 215 followed by the sequence MATH 217316
(Linear AlgebraDifferential Equations). MATH 217 (or the Honors
version, MATH 513) is required for a concentration in Mathematics;
it both serves as a transition to the more theoretical material
of advanced courses and provides the background required to optimal
treatment of differential equations in MATH 316. MATH 216 is not
intended for mathematics concentrators.
A maximum total of 4 credits may be earned in MATH 103, 105, and 110. A maximum total of 16 credits may be earned for calculus courses MATH 112 through MATH 396, and no credit can be earned for a prerequisite to a course taken after the course itself.
MATH 103. Intermediate Algebra.
Instructor(s):
Prerequisites & Distribution: Only open to designated summer halfterm Bridge students. (Excl). May not be repeated for credit. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
Credits: (2 in the halfterm).
Course Homepage: No homepage submitted.
No Description Provided. Contact the Department.
MATH 105. Data, Functions, and Graphs.
THERE ARE JOINT EVENING EXAMS FOR ALL SECTIONS of MATH 105: WED, OCT 1 & MON, NOV 3, 68PM. ALSO A JOINT FINAL. CAUTION! AVOID SCHEDULING ANOTHER CLASS THAT CONFLICTS WITH THESE EVENING EXAMS.
Instructor(s):
Prerequisites & Distribution: (4). (MSA). (QR/1). May not be repeated for credit. Students with credit for MATH 103 can elect MATH 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
Credits: (4).
Course Homepage: No homepage submitted.
MATH 105 serves both as a preparatory course to the calculus sequences and as a terminal course for students who need only this level of mathematics. Students who complete MATH 105 are fully prepared for MATH 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of realworld applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing.
MATH 110. PreCalculus (SelfStudy).
ENROLLMENT IN MATH 110 IS BY PERMISSION OF MATH 115 INSTRUCTOR ONLY. COURSE MEETS SECOND HALF OF THE TERM. STUDENTS WORK INDEPENDENTLY WITH GUIDANCE FROM MATH LAB STAFF.
Instructor(s):
Prerequisites & Distribution: See Elementary Courses above. Enrollment in MATH 110 is by recommendation of MATH 115 instructor and override only. (2). (Excl). May not be repeated for credit. No credit granted to those who already have 4 credits for precalculus mathematics courses. A maximum of four credits may be earned in MATH 101, 103, 105, and 110.
Credits: (2).
Course Homepage: No homepage submitted.
The course covers data analysis by means of functions and graphs. MATH 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, halfterm version of MATH 105 (MATH 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete MATH 115. Students who complete MATH 110 are fully prepared for MATH 115. Students may enroll in MATH 110 only on the recommendation of a mathematics instructor after the third week of classes.
ENROLLMENT IN MATH 110 IS BY PERMISSION OF MATH 115 INSTRUCTOR ONLY. COURSE MEETS SECOND HALF OF THE TERM. STUDENTS WORK INDEPENDENTLY WITH GUIDANCE FROM MATH LAB STAFF.
MATH 115. Calculus I.
UNIFORM EVENING EXAMS FOR ALL SECTIONS OF MATH 115: WED, OCT 8 & WED, NOV 12, 6:008:00 PM. ALSO A UNIFORM FINAL. CAUTION! AVOID SCHEDULING ANOTHER CLASS THAT CONFLICTS WITH THESE EVENING EXAMS.
Instructor(s):
Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit usually is granted for only one course from among 115, 185, and 295. No credit granted to those who have completed MATH 175.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence MATH 115116215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.
Topics include functions and graphs, derivatives and their applications to reallife problems in various fields, and definite integrals. MATH 185 is a somewhat more theoretical course which covers some of the same material. MATH 175 includes some of the material of MATH 115 together with some combinatorial mathematics. A student whose preparation is insufficient for MATH 115 should take MATH 105 (Data, Functions, and Graphs). MATH 116 is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking MATH 186. The cost for this course is over $100 since the student will need a text (to be used for MATH 115 and 116) and a graphing calculator (the Texas Instruments TI83 is recommended).
TEXT: Calculus, 3rd edition, HughesHallet, Wiley Publishing.
TI83 Graphing Calculator, Texas Instruments.
MATH 116. Calculus II.
UNIFORM EVENING EXAMS FOR ALL SECTIONS OF MATH 116: TUES, OCT 7 & TUES, NOV 11, 68PM. ALSO A UNIFORM FINAL. CAUTION! AVOID SCHEDULING ANOTHER CLASS THAT CONFLICTS WITH THESE EVENING EXAMS.
Instructor(s):
Prerequisites & Distribution: MATH 115. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course from among MATH 116, 156, 176, and 186.
Credits: (4).
Course Homepage: No homepage submitted.
See MATH 115 for a general description of the sequence MATH 115116215.
Topics include the indefinite integral, techniques of integration, introduction to differential equations, and infinite series. MATH 186 is a somewhat more theoretical course which covers much of the same material. MATH 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking MATH 285.
Text: Calculus, 3rd Edition, HughesHallet/Gleason, Wiley Publishing.
TI83 Graphing Calculator, Texas Instruments.
MATH 147. Introduction to Interest Theory.
Instructor(s):
Prerequisites & Distribution: MATH 115. (3). (MSA). (BS). May not be repeated for credit. No credit granted to those who have completed a 200 (or higher) level mathematics course.
Credits: (3).
Course Homepage: No homepage submitted.
This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators should elect MATH 424, which covers the same topics but on a more rigorous basis requiring considerable use of calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds, and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. This course is not part of a sequence. Students should possess financial calculators.
MATH 156. Applied Honors Calculus II.
UNIFORM EVENING EXAMS FOR ALL SECTIONS OF MATH 156: WED, OCT 8 AND WED, NOV 12, 6:008:00 PM. ALSO A UNIFORM FINAL.
Prerequisites & Distribution: Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit is granted for only one course among MATH 116, 156, 176, 186, and 296.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html
Math 156 is a 2nd term Honors calculus course for engineering and science students. The course emphasizes applications of calculus, computational skills, and conceptual understanding. The
prerequisite is a score of 4 or 5 on the Advanced Placement calculus AB or BC exam. Math 156 provides students with the calculus background they need for subsequent courses in engineering, math, and
science.
The aim of Math 156 is to provide students with the math background they need for
subsequent courses. Math 156 strikes a balance between theory and application. Theorems
are stated carefully and several are proven, but technical details are omitted (though
reference is made to higherlevel math courses where such issues are discussed). The
proofs are presented in easily understood steps, based on the course coordinator's prior
experience. Examples are given to illustrate the theory.
The course starts by reviewing the definition of the integral as a limit of Riemann sums.
The students presumably have seen this topic in their AP class, but many comment that
the Math 156 treatment is different. The class moves quickly to topics that most students
haven't seen before, including improper integrals, and applications such as work, center of
mass, arclength, surface area, hydrostatic force, and probability density functions. Math
156 avoids the traditional segment on "methods of integration for their own sake"; instead
the methods are discussed as they arise in concrete problems. Taylor approximation is
discussed in some depth with emphasis on applications the students will likely encounter
in later courses (e.g., farfield expansion for the electrostatic potential of a pair of charged particles).
 Part I. Integration
 sigma notation
 area
 definite integral
 fundamental theorem of calculus
 indefinite integrals
 work
 improper integrals
 arclength
 surface area
 hydrostatic force, center of mass
 probability density functions
 Part II. Differential Equations
 modeling with differential equations
 exponential growth and decay
 logistic equation
 Part III. Series
 sequences
 series
 integral test
 comparison test
 alternating series
 absolute convergence, ratio test
 power series
 Taylor series
 binomial series
 applications of Taylor polynomials
 Additional Topics (time permitting)
 parametric curves
 area defined by parametric curves
 polar coordinates
 complex numbers
 Review (as needed)
 substitution
 inverse trigonometric functions
 hyperbolic functions
 L'Hopital's rule
 integration by parts
 trigonometric integrals
 trigonometric substitution
 partial fractions
The enrollment is roughly 1/2 engineering majors and 1/2 science majors. The class meets 4 times per week and each
class is 50 minutes long. There are uniform weekly homework assignments and uniform
exams (two 90 minute midterms and one 2 hour final exam). The homework assignments
include problems from the assigned text and customized problems. The students are introduced to MAPLE in a computer lab. There is brief exposure
to special topics such as Bessel function, Gamma function, error function, fractal sets, Laplace transform, polar coordinates, complex numbers.
Text: "Calculus" by James Stewart, 5th edition, Brooks/Cole Publishing Company
MATH 185. Honors Calculus I.
Section 001, 002.
Prerequisites & Distribution: Permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 175 or 295.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~asgari/ma185_fall03.html
The sequence MATH 185186285286 is the Honors introduction to calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.
Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. MATH 115 is a somewhat less theoretical course which covers much of the same material. MATH 186 is the natural sequel.
MATH 185. Honors Calculus I.
Section 003, 004.
Prerequisites & Distribution: Permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 175 or 295.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence MATH 185186285286 is the Honors introduction to calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.
Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. MATH 115 is a somewhat less theoretical course which covers much of the same material. MATH 186 is the natural sequel.
MATH 214. Linear Algebra and Differential Equations.
Prerequisites & Distribution: MATH 115 and 116. (4). (MSA). (BS). (QR/1). May not be repeated for credit. 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.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m214.html
This course is intended for secondyear students who might otherwise take MATH 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect MATH 217.
While MATH 216 includes 34 weeks of Linear Algebra as a tool in the study of Differential Equations, MATH 214 will include roughly three weeks of Differential Equations as an application of Linear Algebra.
The following is a tentative outline of the course:
 Systems of linear equations, matrices, row operations, reduced row echelon form, free variables, basic variables, basic solution, parametric description of the solution space. Rank of a matrix.
 Vectors, vector equations, vector algebra, linear combinations of vectors, the linear span of vectors.
 The matrix equation Ax = b. Algebraic rules for multiplication of matrices and vectors.
 Homogeneous systems, principle of superposition.
 Linear independence.
 Applications, Linear models.
 Matrix algebra, dot product, matrix multiplication.
 Inverse of a matrix.
 Invertible matrix theorem.
 Partitioned matrices.
 2dimensional discrete dynamical systems.
 Markov process, steady state.
 Transition matrix, eigenvector, steady state lines (affine hulls).
 Geometry of two and three dimensions: affine hulls, linear hulls, convex hulls, half planes, distance from point to a plane, optimization.
 Introduction to linear programming.
 The geometry of transition matrices in 2 dimensions (rotations, shears, ellipses, eigenvectors).
 Transition matrices for 3D (rotations, orthogonal matrices, symmetric matrices)
 Determinants.
 2 and 3dimensional determinant as area and volume.
 Eigenvectors and Eigenvalues.
 Eigenvectors.
 Complex numbers including Euler's formula.
 Complex eigenvalues and their geometric meaning.
 Review of ordinary differential equations.
 Systems of ordinary differential equations in 2 dimensions.
Regular problem sets and exams.
MATH 215. Calculus III.
UNIFORM EVENING EXAMS FOR ALL SECTIONS OF MATH 215: THURS, OCT 9 & THURS, NOV 13, 68PM. ALSO A UNIFORM FINAL.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 215, 255, or 285.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence MATH 115116215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly computer lab using Maple software. MATH 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is MATH 217. Students who intend to take only one further mathematics course and need differential equations should take MATH 216.
TEXT: STUDENTS HAVE CHOICE OF EITHER:
Calculus, 4th edition, James Stewart, Brooks/Cole Publishing, or
Multivariable Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 216. Introduction to Differential Equations.
UNIFORM EVENING EXAMS FOR ALL SECTIONS OF MATH 216: WED, OCT 8 & MON, NOV 10, 68PM. ALSO A UNIFORM FINAL.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.
Credits: (4).
Course Homepage: No homepage submitted.
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, MATH 216417 (or 419) and MATH 217316. The sequence MATH 216417 emphasizes problemsolving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence MATH 217316. MATH 286 covers much of the same material in the honors sequence. The sequence MATH 217316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. MATH 404 covers further material on differential equations. MATH 217 and 417 cover further material on linear algebra. MATH 371 and 471 cover additional material on numerical methods.
MATH 217. Linear Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (4). (MSA). (BS). (QR/1). May not be repeated for credit. 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.
Credits: (4).
Course Homepage: No homepage submitted.
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, MATH 216417 (or 419) and MATH 217316. The sequence MATH 216417 emphasizes problemsolving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence MATH 217316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering MATH 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R^{n}; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. MATH 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. MATH 513 covers more in a much more sophisticated way. The intended course to follow MATH 217 is 316. MATH 217 is also prerequisite for MATH 412 and all more advanced courses in mathematics.
MATH 256. Applied Honors Calculus IV.
Instructor(s):
Prerequisites & Distribution: MATH 255. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.
Credits: (4).
Course Homepage: No homepage submitted.
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, and MAPLE.
MATH 285. Honors Calculus III.
Instructor(s):
Prerequisites & Distribution: MATH 176 or 186, or permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. Credit can be earned for only one of MATH 215, 255, or 285.
Credits: (4).
Course Homepage: No homepage submitted.
See MATH 185 for a general description of the sequence MATH 185186285286.
Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; maximumminimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. MATH 215 is a less theoretical course which covers the same material.
MATH 289. Problem Seminar.
Instructor(s):
Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit. Repetition requires permission of the department.
Credits: (1).
Course Homepage: No homepage submitted.
One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to honors students. This course is excellent preparation for the Putnam exam. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.
MATH 295. Honors Mathematics I.
Instructor(s):
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. (4). (MSA). (BS). (QR/1). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 185.
Credits: (4).
Course Homepage: No homepage submitted.
MATH 295296395396 is the main Honors calculus sequence. It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts and problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should seriously consider starting with this sequence. The expected background is high school trigonometry and algebra (previous calculus not required). This sequence is not restricted to students enrolled in the LS&A Honors Program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, and some linear algebra. MATH 175 and MATH 185 are less intensive Honors courses. MATH 296 is the intended sequel.
MATH 316. Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit. Credit can be earned for only one of MATH 216, 256, 286, or 316.
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to differential equations for students who have studied linear algebra (MATH 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. Firstorder equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvectoreigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higherorder equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. MATH 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. MATH 286 is the Honors version of MATH 316. MATH 471 and/or MATH 572 are natural sequels in the area of differential equations, but MATH 316 is also preparation for more theoretical courses such as MATH 451.
MATH 333. Directed Tutoring.
Instructor(s):
Prerequisites & Distribution: Enrollment in the secondary teaching certificate program with concentration in mathematics. Permission of instructor required. (13). (Excl). (EXPERIENTIAL). May be repeated for credit for a maximum of 3 credits. Offered mandatory credit/no credit.
Credits: (13).
Course Homepage: No homepage submitted.
An experiential mathematics course for elementary teachers. Students tutor precalculus (Math. 105) or calculus (Math. 115) in the Math. Lab. They also participate in a biweekly seminar to discuss mathematical and methodological questions. Mandatory Credit/No Credit grading.
MATH 371 / ENGR 371. Numerical Methods for Engineers and Scientists.
Instructor(s):
Prerequisites & Distribution: ENGR 101; one of MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in Math 471. 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.
This is a survey course of the basic numerical methods which are used to solve practical scientific problems.
Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction
to MATLAB, an interactive program for numerical linear algebra. Convergence theorems are discussed and
applied, but the proofs are not emphasized.
Objectives of the course
 Develop numerical methods for approximately solving problems from continuous mathematics on the
computer
 Implement these methods in a computer language (MATLAB)
 Apply these methods to application problems
Computer language:
In this course, we will make extensive use of Matlab, a technical computing environment for numerical
computation and visualization produced by The MathWorks, Inc. A Matlab manual is available in the MSCC Lab.
Also available is a MATLAB tutorial written by Peter Blossey.
MATH 385. Mathematics for Elementary School Teachers.
Instructor(s):
Carolyn A Dean
Prerequisites & Distribution: One year each of high school algebra and geometry. (3). (Excl). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 485.
Credits: (3).
Course Homepage: No homepage submitted.
All elementary teaching certificate candidates are required to take two
math courses, MATH 385 and MATH 489, either before or after admission to
the School of Education. MATH 385 is offered in the Fall Term, MATH 489
in the Winter Term. Enrollment is limited to 30 students per section; classsize limits will be STRICTLY enforced. Anyone who can elect MATH 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course.
This course, together with its sequel MATH 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Although only two years of high school mathematics are required, a more complete background including precalculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course. MATH 489 is the required sequel.
For further information, contact Prof. Dean, cdean@umich.edu.
MATH 395. Honors Analysis I.
Section 001.
Instructor(s):
Mario Bonk
Prerequisites & Distribution: MATH 296 or permission of the Honors advisor. (4). (Excl). (BS). May not be repeated for credit.
Credits: (4).
Course Homepage: No homepage submitted.
This course is a continuation of the sequence MATH 295296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. This course studies functions of several real variables. Topics are chosen from elementary linear algebra (vector spaces, subspaces, bases, dimension, and solutions of linear systems by Gaussian elimination); elementary topology (open, closed, compact, and connected sets, and continuous and uniformly continuous functions); differential and integral calculus of vectorvalued functions of a scalar; differential and integral calculus of scalarvalued functions on Euclidean spaces; linear transformations (null space, range, matrices, calculations, linear systems, and norms); and differential calculus of vectorvalued mappings on Euclidean spaces (derivative, chain rule, and implicit and inverse function theorems).
MATH 399. Independent Reading.
Instructor(s):
Prerequisites & Distribution: Permission of instructor required. (16). (Excl). (INDEPENDENT). May be repeated for credit.
Credits: (16).
Course Homepage: No homepage submitted.
Designed especially for Honors students.
MATH 404. Intermediate Differential Equations and Dynamics.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256 or 286, or 316. (3). (Excl). (BS). May not be repeated for credit.
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.
MATH 412. Introduction to Modern Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285; and 217. (3). (Excl). (BS). May not be repeated for credit. 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.
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.
MATH 416. Theory of Algorithms.
Instructor(s):
Prerequisites & Distribution: MATH 312 or 412 or EECS 203, and EECS 281. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Many common problems from mathematics and computer science may be solved by applying one or more algorithms — welldefined 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 pseudoprimality testing (in connection with coding questions). Algorithm types such as divideandconquer, 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.
MATH 417. Matrix Algebra I.
Instructor(s):
Prerequisites & Distribution: Three courses beyond MATH 110. (3). (Excl). (BS). May not be repeated for credit. 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.
Credits: (3).
Course Homepage: No homepage submitted.
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 MATH 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. Linear Spaces and Matrix Theory.
Instructor(s):
Prerequisites & Distribution: Four terms of college mathematics beyond MATH 110. (3). (Excl). (BS). May not be repeated for credit. 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.
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 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, 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 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 423. Mathematics of Finance.
Instructor(s):
Prerequisites & Distribution: MATH 217 and 425; EECS 183. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/423/001.nsf
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 424. Compound Interest and Life Insurance.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/424/002.nsf
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 reallife 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).
MATH 425 / STATS 425. Introduction to Probability.
Instructor(s):
Mathematics faculty
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
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. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.
MATH 425 / STATS 425. Introduction to Probability.
Instructor(s):
Statistic faculty
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
See Statistics 425.
MATH 425 / STATS 425. Introduction to Probability.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
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 / STATS 425. Introduction to Probability.
Section 003.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
See Statistics 425.003.
MATH 425 / STATS 425. Introduction to Probability.
Section 007.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~daschnei/math425/Math425HomePage.htm
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. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.
MATH 425 / STATS 425. Introduction to Probability.
Section 008.
Instructor(s):
Kausch
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/425/008.nsf
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. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.
MATH 427 / HB 603. Retirement Plans and Other Employee Benefit Plans.
Section 001.
Prerequisites & Distribution: Junior standing. (3). (Excl). May not be repeated for credit.
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 MATH 522 (which can be taken independently of each other) provide more indepth examination of the actuarial techniques used in employee benefit plans. No textbook
MATH 429. Internship.
Instructor(s):
Prerequisites & Distribution: Concentration in Mathematics. (1). (Excl). (EXPERIENTIAL). May be elected up to three times for credit. Internship credit is not retroactive and must be prearranged. May not apply toward a Mathematics concentration. May be used to satisfy the Curriculum Practical Training (CPT) required of foreign students. Offered mandatory credit/no credit.
Credits: (1).
Course Homepage: No homepage submitted.
Credits is granted for a fulltime internship of at least eight weeks that is used to enrich a student's academic experience and/or allows the student to explore careers related to his/her academic studies.
MATH 431. Topics in Geometry for Teachers.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math431/
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; and finite, hyperbolic, and taxicab nonEuclidean geometries. Alternative geometry courses at this level are MATH 432 and 433. Although it is not strictly a prerequisite, MATH 431 is good preparation for MATH 531.
MATH 433. Introduction to Differential Geometry.
Section 001.
Prerequisites & Distribution: MATH 215 (or 255 or 285), and 217. (3). (Excl). (BS). May not be repeated for credit.
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 (MATH 215) and linear algebra (preferably MATH 217). Some exposure to differential equations (MATH 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; 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. MATH 537 is a substantially more advanced course which requires a strong background in topology (MATH 590), linear algebra (MATH 513), and advanced multivariable calculus (MATH 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). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.
Credits: (4).
Course Homepage: No homepage submitted.
Background and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.
Content: Topics covered include; Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed.
MATH 450. Advanced Mathematics for Engineers I.
Section 002.
Prerequisites & Distribution: MATH 215, 255, or 285. (4). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.
Credits: (4).
Course Homepage: Http://www.math.lsa.umich.edu/~glyng/Teaching/Math450.html
Background and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.
Content: Topics covered include; Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed.
MATH 451. Advanced Calculus I.
Instructor(s):
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). (Excl). (BS). May not be repeated for credit.
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 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 001.
Instructor(s):
Peter L Duren
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). (Excl). (BS). May not be repeated for credit.
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 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 454. Boundary Value Problems for Partial Differential Equations.
Section 001.
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. 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.
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 boundaryvalue problems for secondorder 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 onedimensional 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 inputoutput systems, analysis of data smoothing and filtering, signal processing, timeseries 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.
MATH 454. Boundary Value Problems for Partial Differential Equations.
Section 002.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit. 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.
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 boundaryvalue problems for secondorder 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 onedimensional 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 inputoutput systems, analysis of data smoothing and filtering, signal processing, timeseries 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.
MATH 463. Mathematical Modeling in Biology.
Section 001.
Prerequisites & Distribution: MATH 217, 417, or 419; MATH 286, 256, or 316. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/463/001.nsf
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.
MATH 471. Introduction to Numerical Methods.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.
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 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, 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 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). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~hansjohn/m471.html
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 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, 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 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.
Instructor(s):
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). May not be repeated for credit.
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 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. PHIL 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.
MATH 485. Mathematics for Elementary School Teachers and Supervisors.
Instructor(s):
Prerequisites & Distribution: One year of high school algebra. (3). (Excl). (BS). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 385. May not be included in a concentration plan in mathematics.
Credits: (3).
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.
Section 001.
Prerequisites & Distribution: MATH 489. (3). (Excl). (BS). May be repeated for credit for a maximum of 6 credits.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/497/001.nsf
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 and
algebra.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Instructor(s):
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). (BS). May be repeated for credit for a maximum of 6 credits. Offered mandatory credit/no credit.
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 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.
Section 001.
Prerequisites & Distribution: MATH 412. (3). (Excl). (BS). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.
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 prooforiented 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; 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 prooforiented 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 MATH 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.
MATH 513. Introduction to Linear Algebra.
Section 002.
Prerequisites & Distribution: MATH 412. (3). (Excl). (BS). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~hmthomps/513.html
Student Body: a mix of math and computer science undergrads and nonmath majors
Background and Goals: 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.
Content: Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear
transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential
equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form.
Alternatives: MATH 419 (Lin. Spaces and Matrix Thy) covers much of the same material using the same text, but there is more stress on computation and applications. MATH 217 (Linear Algebra) is similarly
prooforiented but significantly less demanding than MATH 513. MATH 417 (Matrix Algebra I) is much less
abstract and more concerned with applications.
Subsequent Courses: The natural sequel to MATH 513 is MATH 593 (Algebra I). 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.
Text: Friedberg, Insel & Spence. Linear Algebra. Fourth Edition. Springer Prentice Hall, 2003.
Course Work: There will be weekly homework assignments, an inclass midterms, and a final.
Grading: The midterm will be worth 25% of the grade, the final 45%, and the homework 30%.
MATH 520. Life Contingencies I.
Section 001.
Prerequisites & Distribution: MATH 424 and 425. (3). (Excl). (BS). May not be repeated for credit.
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. MATH 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 MATH 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 MATH 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.
MATH 523. Risk Theory.
Instructor(s):
Prerequisites & Distribution: MATH 425. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/523/001.nsf
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 / STATS 525. Probability Theory.
Section 001.
Instructor(s):
Charles R Doering
Prerequisites & Distribution: MATH 451 (strongly recommended) or 450. MATH 425 would be helpful. (3). (Excl). (BS). May not be repeated for credit.
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 MATH 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, STATS 426, and the sequence STATS 510511 are natural sequels.
An Introduction to Probability Theory and Its Applications, 3rd edition, William Feller Wiley.
recommended — Introduction to Probability Theory Hoel, Port, Stone HoughtonMifflin.
MATH 537. Introduction to Differentiable Manifolds.
Prerequisites & Distribution: MATH 513 and 590. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~bkleiner/math537.html
Background and Goals: This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses MATH 513 and MATH 590. Its goal is to introduce the basic concepts and results of differential topology and differential geometry.
Content: Manifolds, vector fields and flows, differential forms, Stokes' theorem, Lie group basics, Riemannian metrics, LeviCivita connection, geodesics
Alternatives: MATH 433 (Intro to Differential Geometry) is an undergraduate version which covers less material in a less sophisticated way.
Subsequent Courses: MATH 635 (Differential Geometry)
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~baik/Teaching/
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.
MATH 556. Methods of Applied Mathematics I.
Section 001.
Prerequisites & Distribution: MATH 217, 419, or 513; 451 and 555. (3). (Excl). (BS). May not be repeated for credit.
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, Schrödinger's equation, and Laplace's equation.
MATH 561 / IOE 510 / SMS 518. Linear Programming I.
Section 001.
Instructor(s):
Marina A Epelman
Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. 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.
No Description Provided. Contact the Department.
MATH 562 / IOE 511 / AEROSP 577. Continuous Optimization Methods.
Section 001.
Instructor(s):
Murty
Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit. 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.
No Description Provided. Contact the Department.
MATH 565. Combinatorics and Graph Theory.
Section 001.
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~plhersh/565a03.html
The syllabus will describe this more thoroughly, but this course is designed primarily for students in math, computer science, and related fields.
The first half of the course is on graph theory and some complexity theory, while the second half deals with some major topics from algebraic and geometric
combinatorics: partially ordered sets, simplicial complexes as they arise in combinatorics, and matroids. In the course of examining these topics, we also will briefly discuss and use a few of the major techniques from enumerative combinatorics, namely bijective proofs, generating functions and inclusionexclusion via
Möbius functions. The second half of the course will not follow the textbook quite as closely as the first half.
Textbook: "Combinatorics and Graph Theory", by Mark Skandera.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001.
Prerequisites & Distribution: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math571.html
Prerequisites: a course in linear algebra (e.g. Math 217, 417, 419, 513 or equivalent)
Text: "Numerical Linear Algebra" by L. N. Trefethen and D. Bau (SIAM)
This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is illconditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties.
The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended.
Topics: 1. warmup: vector and matrix norms, orthogonal matrices, projection matrices, singular value decomposition (SVD); 2. least squares problems: QR factorization, GramSchmidt orthogonalization, Householder triangularization, normal equations; 3. backward error analysis: stability, condition number, IEEE floating point arithmetic; 4. direct methods for Ax=b: Gaussian elimination, LU factorization, pivoting, Cholesky factorization; 5. eigenvalues and eigenvectors: Schur factorization, reduction to Hessenberg and tridiagonal form, power method, inverse iteration, shifts, Rayleigh quotient iteration, QR algorithm; 6. iterative methods for Ax=b: Krylov subspace, Arnoldi iteration, GMRES, conjugate gradient method, preconditioning; 7. applications: image compression using the SVD, least squares data fitting, finitedifference schemes for a twopoint boundary value problem, Dirichlet problem for the Laplace equation
krasny@umich.edu, 7633505, 4830 EH
MATH 575. Introduction to Theory of Numbers I.
Section 001 — [3 credits].
Instructor(s):
Kannan Soundararajan
Prerequisites & Distribution: MATH 451 and 513. (1, 3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 475 can elect MATH 575 for 1 credit.
Credits: (1, 3).
Course Homepage: No homepage submitted.
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, 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, padic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. MATH 475 is a nonHonors 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.
MATH 590. Introduction to Topology.
Section 001.
Instructor(s):
Arthur G Wasserman
Prerequisites & Distribution: MATH 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Background and Goals: Math 590 is an introduction to point set topology. It is quite theoretical and requires extensive construction of proofs.
Content: Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, covering spaces and other topics.
Text: Topology, Second Edition, by Munkres, Prentice Hall
The course will cover (most of) chapters 2 through 6 of the text.
Grades will be based on weekly individual homework assignments, class participation, two tests and the final exam.
MATH 591. General and Differential Topology.
Section 001.
Prerequisites & Distribution: MATH 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Math 451 or the equivalent is a prerequisite. In addition, we will assume a knowledge of Chapter 1 of Munkres' book. Students may wish to go over this chapter before the course.
This course will be a introduction to general topology and differential topology. We will spend roughly half of the semester on each of these topics.
Math 591 is the first part of a twosemester sequence in topology, the sequel being Math 592. The course will be a preparation for part of the topology QR exams. Students who already have a good background in general and differential topology should consider taking Math 537 instead.
We'll cover:
Topological Spaces and Continuous Functions
Quotient Topologies
Connectedness and Local Connectedness
Compactness and Local Compactness
Countability and Separation Axioms
Urysohn's Lemma
Tychonoff's Theorem
Complete Metric Spaces
Manifolds and Smooth Maps
Derivatives and Tangents
Immersions and Submersions
Transversality
Homework assignments will be given periodically. There will also be a midterm exam and a final exam.
Text :
Topology, a First Course by James Munkres, PrenticeHall and Differential Topology by Victor Guillemin and Alan Pollack, PrenticeHall.
We will cover roughly Chapters 24 of Munkres' book, parts of Chapters 5 and 7, and Chapter 1 of Guillemin and Pollack.
MATH 593. Algebra I.
Section 001.
Instructor(s):
William E Fulton
Prerequisites & Distribution: MATH 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisite: Math 513
Background and Goals: 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 Math 512 (Algebraic Structures).
Content: 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, exterior algebras.
Subsequent Courses: Math 594 (Algebra II) and Math 614 (Commutative Algebra I).
Text: J. Rotman, "Advanced Modern Algebra", PrenticeHall, 2002.
MATH 596. Analysis I.
Section 001.
Instructor(s):
Mario Bonk
Prerequisites & Distribution: MATH 451. (3). (Excl). (BS). May not be repeated for credit. Students with credit for MATH 555 may elect MATH 596 for two credits only.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisite: MATH 451 or equivalent (basic principles of analysis).
Background and Goals: 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.
Content: Review of analysis in R^{2} including metric spaces, differentiable maps, Jacobians; analytic functions, CauchyRiemann equations, conformal mappings, linear fractional transformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem.
Alternatives: MATH 555 (Intro to Complex Variables) covers some of the same material with greater emphasis on applications and less attention to proofs.
Subsequent Courses: MATH 597 (Analysis II (Real)), MATH 604 (Complex Analysis II), and MATH 605 (Several Complex Variables).
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