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Mathematics is the language and tool of the sciences,
a cultural phenomenon with a rich historical traditions, and a
model of abstract reasoning. Historically, mathematical methods
and thinking have proved extraordinarily successful in physics,
and engineering. Nowadays, it is used successfully in many new
areas, from computer science to biology and finance. A Mathematics
concentration provides a broad education in various areas of mathematics
in a program flexible enough to accommodate many ranges of interest.
The study of mathematics is an excellent preparation
for many careers; the patterns of careful logical reasoning and
analytical problem solving essential to mathematics are also applicable
in contexts where quantity and measurement play only minor roles.
Thus students of mathematics may go on to excel in medicine, law,
politics, or business as well as any of a vast range of scientific
careers. Special programs are offered for those interested in
teaching mathematics at the elementary or high school level or
in actuarial mathematics, the mathematics of insurance. The other
programs split between those which emphasize mathematics as an
independent discipline and those which favor the application of
mathematical tools to problems in other fields. There is considerable
overlap here, and any of these programs may serve as preparation
for either further study in a variety of academic disciplines,
including mathematics itself, or intellectually challenging careers
in a wide variety of corporate and governmental settings.
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 mathematics course.
Each of MATH 115, 185, and 295 is a first course
in calculus and generally credit can be received for only one
course from this list. The Sequence 115116215 is appropriate
for most students who want a complete introduction to calculus.
One of MATH 215, 285, or 395 is prerequisite to most more advanced
courses in Mathematics.
The sequences 156255256, 175176285286, 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 175176 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 and
176 are taught by the discovery method: students are presented
with a great variety of problem 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. MATH 295396 is excellent preparation
for mathematics at the advanced undergraduate and 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 or 295 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, 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 Equation  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.
Special Departmental Policies. All prerequisite
courses must be satisfied with a grade of C or above. Students
with lower grades in prerequisite courses must receive special
permission of the instructor to enroll in subsequent courses.
MATH 105. Data, Functions, and Graphs.
THERE WILL BE JOINT EVENING EXAMINATIONS FOR ALL SECTIONS OF MATH 105, 6:00  8:00 P.M. ALSO A JOINT FINAL.
Instructor(s):
Prerequisites & Distribution: 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 103, 105, and 110. (4). (MSA). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/105/
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 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.
TEXT: Functions Modeling Change, Connally, Wiley Publishing.
MATH 110. PreCalculus (SelfStudy).
Section 001.
Instructor(s):
Prerequisites & Distribution: See Elementary Courses above. Enrollment in MATH 110 is by recommendation of MATH 115 instructor and override only. No credit granted to those who already have 4 credits for precalculus mathematics courses. A maximum of four credits may be earned in MATH 103, 105, and 110. (2). (Excl).
Credits: (2).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html
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 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 in the Fall and must visit the Math Lab to complete paperwork and receive course materials.
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.
THERE WILL BE JOINT EVENING EXAMINATIONS FOR ALL SECTIONS OF MATH 115, 6:008:00 PM. ALSO A JOINT FINAL.
Instructor(s):
Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among MATH 112, 115, 185, and 295. No credit granted to those who have completed MATH 175. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/115/
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.
THERE WILL BE JOINT EVENING EXAMINATIONS FOR ALL SECTIONS OF MATH 116, 6:008:00 PM. ALSO A JOINT FINAL.
Instructor(s):
Prerequisites & Distribution: MATH 115. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/116/
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 128. Explorations in Number Theory.
Section 001.
Instructor(s):
Hugh L Montgomery
Prerequisites & Distribution: High school mathematics through at least Analytic Geometry. Only firstyear students, including those with sophomore standing, may preregister for FirstYear Seminars. All others need permission of instructor. No credit granted to those who have completed a 200 (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).
FirstYear Seminar
Credits: (4).
Course Homepage: No homepage submitted.
This course is intended for nonscience concentrators and students in the preconcentration years with no intended concentration, who want to engage in mathematical reasoning without having to take calculus first. Students will be introduced to elementary ideas of number theory, an area of mathematics that deals with properties of the integers. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying factorizations and greatest common divisors, emphasis will shift to the patterns that emerge when the integers are classified according to the remainder produced upon division by some fixed number ('congruences'). Once some basic tools have been established, applications will be made in several directions. For example, students may derive a precise parameterization of Pythagorean triples a^{2} + b^{2} = c^{2}. Students who like math but don't especially like calculus will want to enroll in this firstyear MSA seminar. Students will do handson experimentation with numerical patterns and will tackle numerical riddles and brainteasers as they focus on empirical discovery and proof of theorems. Students will write their own text in number theory and enjoy their growing ability to think like mathematicians.
MATH 147. Introduction to Interest Theory.
Section 001.
Instructor(s):
Demosthenis S Demosthenous
Prerequisites & Distribution: MATH 115. No credit granted to those who have completed a 200 (or higher) level mathematics course. (3). (MSA). (BS).
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.
THERE WILL BE JOINT EVENING EXAMINATIONS FOR ALL SECTIONS OF MATH 156, 6:00  8:00 P.M. ALSO A JOINT FINAL.
Instructor(s):
Prerequisites & Distribution: Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among MATH 116, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html
The sequence Math 156255256 is an honors calculus sequence for engineering and science concentrators who scored 4 or 5 on the AB or BC Advanced Placement calculus exam. Topics include Riemann sums, the definite integral, fundamental theorem of calculus, applications of integral calculus (e.g. arclength, surface area, work, hydrostatic pressure, center of mass), improper integrals, infinite sequences and series, differential equations, complex numbers. MAPLE will be used throughout.
MATH 175. Combinatorics and Calculus.
Section 001.
Prerequisites & Distribution: Permission of Honors advisor. No credit granted to those who have completed a 200level or higher Mathematics course. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/175.html
This course is an alternative to Math 185 as an entry to the honors sequence. The sequence Math 175176 is a twoterm introduction to Combinatorics, Dynamical Systems, and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. Students are expected to have some previous experience with the basic concepts and techniques of calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each term in a group project. Grades will be based on homework and projects with a strong emphasis on homework. Personal computers will be a valuable experimental tool in this course, and students will be asked to learn to program in either BASIC, PASCAL, or FORTRAN. There are two major topic areas: enumeration theory and graph theory. The section on enumeration theory will emphasize classical methods for counting including:
 binomial theorem and its generalizations;
 solving recursions;
 generating functions; and
 the inclusionexclusion principle.
In the process, we will discuss infinite series. The section on graph theory will include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological results and applications to computer science and will include:
 connectivity;
 trees, Prufer codes, and data structures;
 planar graphs, Euler's formula and Kuratowski's Theorem; and
 coloring graphs, chromatic polynomials, and orientation.
This material has many applications in the field of Computer Science. Math 176 is the standard sequel.
MATH 185. Honors Calculus I.
Instructor(s):
Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among MATH 115, 185, and 295. No credit granted to those who have completed MATH 175. (4). (MSA). (BS). (QR/1).
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.
Instructor(s):
Prerequisites & Distribution: MATH 115 and 116. 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. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
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 3 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 MATH 215; ALSO A UNIFORM FINAL.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of MATH 215, 255, or 285. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/215/
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 MATH 216; ALSO A UNIFORM FINAL.
Instructor(s):
Prerequisites & Distribution: MATH 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. Credit can be earned for only one of MATH 216, 256, 286, or 316. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/216/
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.
Text: Differential Equations, Computing and Modeling, 2nd edition, Edwards and Penney, Prentice Hall Publishing.
MATH 217. Linear Algebra.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. 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. (4). (MSA). (BS). (QR/1).
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.
Text: Linear Algebra and Its Applications, 2nd edition, David Lay, Addison Wesley Publishing.
MATH 256. Applied Honors Calculus IV.
Instructor(s):
Prerequisites & Distribution: MATH 255. Credit can be earned for only one of MATH 216, 256, 286, or 316. (4). (MSA). (BS).
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. Credit can be earned for only one of MATH 215, 255, or 285. (4). (MSA). (BS).
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.
Section 001.
Instructor(s):
Hendrikus Gerardus Derksen
Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit with permission.
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.
Section 001.
Instructor(s):
Mario Bonk
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among MATH 115, 185, and 295. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: No homepage submitted.
Math 295296395396 is the main Honors calculus sequence. It is aimed at talented students who intend to concentrate 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. Credit can be earned for only one of MATH 216, 256, 286, or 316. (3). (Excl). (BS).
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 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):
Eugene F Krause (krause@umich.edu)
Prerequisites & Distribution: MATH 385 and enrollment in the Elementary Program in the School of Education. (13). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.
Credits: (13).
Course Homepage: No homepage submitted.
An experiential mathematics course for exceptional upperlevel students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (Math 385 and Math 489) required of all elementary teachers.
MATH 371 / ENGR 371. Numerical Methods for Engineers and Scientists.
Instructor(s):
Prerequisites & Distribution: ENGR 101; one of MATH 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 471. (3). (Excl). (BS). 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, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to nonlinear equations, ordinary differential equations, and polynomial approximations. Other topics may include discrete Fourier transforms, twopoint boundaryvalue problems, and MonteCarlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for Math 571572.
MATH 385. Mathematics for Elementary School Teachers.
Instructor(s):
Eugene F Krause (krause@umich.edu)
Prerequisites & Distribution: One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in MATH 485. (3). (Excl).
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. Krause at his email
address, krause@math.lsa.umich.edu.
MATH 395. Honors Analysis I.
Section 001.
Instructor(s):
Gopal Prasad
Prerequisites & Distribution: MATH 296 or permission of the Honors advisor. (4). (Excl). (BS).
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: (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.
Section 001.
Instructor(s):
David A Schneider
Prerequisites & Distribution: MATH 216, 256 or 286, or 316. (3). (Excl). (BS).
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. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312. (3). (Excl). (BS).
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.
Section 001.
Instructor(s):
Hendrikus Gerardus Derksen
Prerequisites & Distribution: MATH 312 or 412 or EECS 203, and EECS 281. (3). (Excl). (BS).
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. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled MATH 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/courses/417/
Many problems in science, engineering, and mathematics are best formulated in terms of matrices – rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problemsolving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.
Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.
MATH 419. Linear Spaces and Matrix Theory.
Instructor(s):
Prerequisites & Distribution: Four terms of college mathematics beyond MATH 110. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513. (3). (Excl). (BS).
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).
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory.
Specific Topics
 Review of basic probability.
 The oneperiod binomial model of stock prices used to price futures.
 Arbitrage, equivalent portfolios, and riskneutral valuation.
 Multiperiod binomial model.
 Options and options markets; pricing options with the binomial model.
 Early exercise feature (American options).
 Trading strategies; hedging risk.
 Introduction to stochastic processes in discrete time. Random walks.
 Markov property, martingales, binomial trees.
 Continuoustime stochastic processes. Brownian motion.
 BlackScholes analysis, partial differential equation, and formula.
 Numerical methods and calibration of models.
 Interestrate derivatives and the yield curve.
 Limitations of existing models. Extensions of BlackScholes.
MATH 424. Compound Interest and Life Insurance.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to 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):
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS).
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 001.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/425.html
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of
both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random
variables, expectations, variances, covariances. The material corresponds to most of Chapters 17 and part of 8 of Ross.
Grade will be based on two 1hour midterm exams, 20% each; 20% homework; 40% final exam. Your lowest homework set score will be dropped.
This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out. Every student with the total score of
90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).
Homework: There will be approximately 10 problem sets. Text (required): Sheldon Ross, A First Course in Probability, 6th edition, PrenticeHall, 2002.
MATH 425 / STATS 425. Introduction to Probability.
Section 005.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://coursetools.ummu.umich.edu/2002/fall/math/425/005.nsf
This course introduces students to the theory of probability and to a
number of applications. Topics include the basic results and methods
of both discrete and continuous probability theory: conditional
probability, independent events, random variables, jointly distributed
random variables, expectations, variances, covariances. The material
corresponds to most of Chapters 17 and part of 8 of Ross.
MATH 431. Topics in Geometry for Teachers.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS).
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 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.
MATH 433. Introduction to Differential Geometry.
Section 001.
Instructor(s):
Mario Bonk
Prerequisites & Distribution: MATH 215 (or 255 or 285), and 217. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is about the analysis of curves and surfaces in 2 and 3space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4vertex theorem; and (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, and classification of minimal surfaces of revolution. 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513), and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.
MATH 450. Advanced Mathematics for Engineers I.
Instructor(s):
Prerequisites & Distribution: MATH 215, 255, or 285. No credit granted to those who have completed or are enrolled in MATH 454. (4). (Excl). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given, and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g. Math 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.
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).
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.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450. (3). (Excl). (BS).
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).
Credits: (3).
Course Homepage: No homepage submitted.
This course will concentrate on the applications of ordinary differential
equations to physiological systems. Partial differential equations will
not be covered in detail. Thus, a course in ODEs such as 216 or 316 will
be sufficient preparation for this course.
Who could take the course? Basically anybody who is interested
in applying mathematical methods to the biological sciences. For instance,
students from Biology, Chemistry, Physics, Complex Systems, Biophysics,
Biomedical Engineering, Mathematics, Chemical Engineering, Physiology, Microbiology, and Epidemiology.
What kind of background will you need? Basically a course in differential
equations, such as 216 or 316. If you have never seen a differential equation
before, you may have trouble with the course. You will also need to be familiar
and comfortable with computers, as a lot of the work in the course will
have to be done on a computer. You will not need to be an expert in biology,
as we will learn most of what we need to know as we go.
Topics
 Review of phaseplane methods for ordinary differential equations.
 PredatorPrey models.
 Models of disease transmission. SIR models, and epidemics.
 Enzyme kinetics.
 Pseudosteadystate hypothesis.
 Cooperativity.
 MonodWymanChangeux models.
 Cellular homeostasis.
 The membrane potential. The Nernst equation. Electrodiffusion. The
GoldmanHodgkinKatz equation. The constant field approximation.
 Osmosis.
 Control of cell volume.
 Sodium transport and cell volume control.
 Review of limit cycles and oscillations. The Hopf bifurcation.
 Excitability.
 The HodgkinHuxley model and action potentials.
 Twovariable models. FitzHughNagumo model.
 Phaseplane analysis of the models.
 The circulatory system.
 Blood flow and compliance.
 Guyton's model.
 Cardiac regulation.
 Fetal circulation.
 The Windkessel model.
 Blood.
 Myoglobin and hemoglobin.
 Cooperativity and oxygen transport.
 Carbon dioxide transport. Bohr and Haldane effects.
 Muscle.
 The Hill model.
 Crossbridge theory and the Huxley model.
 Discrete binding site models.
MATH 471. Introduction to Numerical Methods.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. No credit granted to those who have completed or are enrolled in MATH 371. (3). (Excl). (BS).
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 481. Introduction to Mathematical Logic.
Section 001.
Instructor(s):
Andreas R Blass
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
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 ({\i and, or, not, implies}), tautologies, and tautological consequence are studied. The heart of the course is the study of firstorder predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including nonstandard analysis. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.
MATH 485. Mathematics for Elementary School Teachers and Supervisors.
Section 001.
Instructor(s):
Eugene F Krause (krause@umich.edu)
Prerequisites & Distribution: One year of high school algebra. No credit granted to those who have completed or are enrolled in MATH 385. (3). (Excl). (BS). May not be included in a concentration plan in mathematics.
Credits: (3; 2 in the halfterm).
Course Homepage: No homepage submitted.
The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching or preparing to teach in the elementary school.
MATH 497. Topics in Elementary Mathematics.
Section 001.
Instructor(s):
Eugene F Krause (krause@umich.edu)
Prerequisites & Distribution: MATH 489. (3). (Excl). (BS). May be repeated for a total of six credits.
Credits: (3).
Course Homepage: No homepage submitted.
This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required twocourse sequence Math 385489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problemsolving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two onehour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and inservice elementary, middle, or juniorhigh school teachers. Content will vary from term to term.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.
Credits: (1).
Course Homepage: No homepage submitted.
The Applied and Interdisciplinary Mathematics(AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from 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.
Instructor(s):
Richard Paul Horja
Prerequisites & Distribution: MATH 412. Two credits granted to those who have completed MATH 214, 217, 417, or 419. (3). (Excl). (BS).
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; and Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly 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 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 520. Life Contingencies I.
Section 001.
Prerequisites & Distribution: MATH 424 and 425. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multidecrement and multiplelife applications directly related to life insurance and pensions. The sequence 520521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.
MATH 523. Risk Theory.
Section 001.
Prerequisites & Distribution: MATH 425. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc.) and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the riskreward tradeoff. Topics include:
 Review of probability.
 Bachelier and Lundberg models of investment and loss aggregation.
 Fallacy of time diversification and its generalizations.
 Geometric Brownian motion and the compound Poisson process.
 Modeling of individual losses which arise in a loss aggregation process.
 Distributions for modeling size loss, statistical techniques for fitting data, and credibility.
 Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory.
 The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the BlackScholes option pricing model.
MATH 525 / STATS 525. Probability Theory.
Section 001.
Instructor(s):
Charles R Doering
Prerequisites & Distribution: MATH 450 or 451. Students with credit for MATH 425/STATS 425 can elect MATH 525/STATS 525 for only one credit. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510511 are natural sequels.
MATH 537. Introduction to Differentiable Manifolds.
Section 001.
Prerequisites & Distribution: MATH 513 and 590. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course will be a painless introduction to differential topology and differential geometry, meaning the study of spaces and their curvatures. It is the first part of a twoterm sequence. The material in this course is crucial for students who wish to study differential geometry, topology, algebraic geometry, several complex variables, Lie groups, and dynamical systems. It is also
relevant for other branches of mathematics, such as partial differential equations. We'll start out by doing calculus on manifolds, introducing and using differential forms. We'll prove Stokes' theorem for compact oriented manifoldswithboundary. We'll also
define the de Rham cohomology groups of a manifold and prove their basic properties. Then I'll spend some time on Morse
theory. This theory shows how, given a generic function on a manifold, one obtains a decomposition of the manifold into simple
building blocks called handles. Morse theory is a basic tool in topology and was used in Smale's famous proof of the Poincare
conjecture in more than four dimensions, although we will not go into this. Finally, we'll cover some basic Riemannian geometry, including Riemannian metrics, LeviCivita connections, geodesics, and curvature. Homework assignments will be given periodically. There will also be a final exam.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
Instructor(s):
Jeffrey B Rauch
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, and applications. Evaluation of improper real integrals and fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.
MATH 556. Methods of Applied Mathematics I.
Section 001 – Initial value Problems and Boundary Value Problems.
Prerequisites & Distribution: MATH 217, 419, or 513; 451 and 555. (3). (Excl). (BS).
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). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No homepage submitted.
Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.
MATH 562 / IOE 511 / AEROSP 577. Continuous Optimization Methods.
Section 001.
Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No homepage submitted.
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques; gradient, conjugate direction, quasiNewton methods. Introduction to constrained optimization using techniques of unconstrained optimization through penalty transformations, augmented Lagrangians, and others. Discussion of computer programs for various algorithms.
MATH 565. Combinatorics and Graph Theory.
Section 001.
Instructor(s):
Mark A Skandera
Prerequisites & Distribution: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one prooforiented course. Graph Theory topics include Trees; k connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5Color Theorem; Kuratowski's Theorem; and the MatrixTree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q series, linear recurrences and rational generating functions, and Pólya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001.
Instructor(s):
Divakar Viswanath
Prerequisites & Distribution: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is a rigorous introduction to numerical linear algebra with applications to 2point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, GaussSeidel iteration, the SOR method, an introduction to the multigrid method, and the conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; and numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.
MATH 575. Introduction to Theory of Numbers I.
Section 001.
Prerequisites & Distribution: MATH 451 and 513. Students with credit for MATH 475 can elect MATH 575 for 1 credit. (1, 3). (Excl). (BS).
Credits: (1, 3).
Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/575f02/
Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of Math 451 and should have a basic understanding of groups, rings, and fields, at least at the level of Math 412 and preferably Math 512. Proofs are emphasized, but they are often pleasantly short. A computational laboratory (Math 476, 1 credit) will usually be offered as a supplement to this course. Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, and application of these ideas to the solution of classical problems such as Fermat's last 'theorem'. Other topics will depend on the instructor and may include continued fractions, 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).
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to topology with an emphasis on the settheoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590.
MATH 591. General and Differential Topology.
Section 001.
Instructor(s):
Andreas R Blass
Prerequisites & Distribution: MATH 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, and classification of surfaces. Math 592 is the natural sequel.
MATH 593. Algebra I.
Section 001.
Instructor(s):
Robert K Lazarsfeld
Prerequisites & Distribution: MATH 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to 512. Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, and exterior algebras.
MATH 596. Analysis I.
Section 001.
Instructor(s):
Peter L Duren
Prerequisites & Distribution: MATH 451. Students with credit for MATH 555 may elect MATH 596 for two credits only. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topics include: review of analysis in R^{2} including metric spaces, differentiable maps, and Jacobians; analytic functions, CauchyRiemann equations, conformal mappings, and linear fractional transformations; Cauchy's theorem and Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, and argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families; and Riemann mapping theorem. Math 595 covers some of the same material with greater emphasis on applications and less attention to proofs.
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