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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.
UNIFORM EVENING EXAMS FOR MATH 105: WED, FEB 6 & MAR 20 68 PM. ALSO A UNIFORM FINAL EXAM.
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). May not be repeated for credit.
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 107. Mathematics for the Information Age.
Section 001.
Instructor(s):
Prerequisites & Distribution: Three to four years high school mathematics. (3). (MSA). (QR/1). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
The course investigates topics relevant to the information age in which we live. Topics covered include cryptography, errorcorrecting codes, data compression, fairness in politics, voting systems, population growth, biological modeling.
MATH 110. PreCalculus (SelfStudy).
STUDENTS IN MATH 110 RECEIVE INDIVIDUALIZED SELFPACED INSTRUCTION IN THE MATHEMATICS LABORATORY.
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). May not be repeated for credit.
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.
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 MATH 115: WED, FEB 6 & MAR 20 68 PM. ALSO A UNIFORM 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). May not be repeated for credit.
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.
UNIFORM EVENING EXAMS FOR MATH 116: THURS, FEB 7 & TUES, MAR 19 68 PM. ALSO A UNIFORM 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). May not be repeated for credit.
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 127. Geometry and the Imagination.
Section 001.
Instructor(s):
Prerequisites & Distribution: Three years of high school mathematics including a geometry course. 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). May not be repeated for credit.
FirstYear Seminar
Credits: (4).
Course Homepage: No homepage submitted.
This course introduces students to the ideas and some of the basic results in Euclidean and nonEuclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The next topic is nonEuclidean geometry. This section begins with the independence of Euclid's Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry differs from Euclidean geometry. The last topic is geometry of higher dimensions: coordinatization – the mathematician's tool for studying higher dimensions; construction of higherdimensional analogues of some familiar objects like spheres and cubes; discussion of the proper higherdimensional analogues of some geometric notions (length, angle, orthogonality, etc. ) This course is intended for students who want an introduction to mathematical ideas and culture. Emphasis on conceptual thinking – students will do handson experimentation with geometric shapes, patterns, and ideas.
MATH 147. Introduction to Interest Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 115. No credit granted to those who have completed a 200 (or higher) level mathematics course. (3). (MSA). (BS). May not be repeated for credit.
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 186. Honors Calculus II.
Instructor(s):
Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1). May not be repeated for credit.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence Math 185186285286 is the honors introduction to the calculus. It is taken by students intending to major 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 transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor. Math 116 is a somewhat less theoretical course which covers much of the same material. Math 285 is the natural sequel.
Text: Calculus, 4th edition, James Stewart, Brooks/Cole.
MATH 214. Linear Algebra and Differential Equations.
Section 001.
Instructor(s):
Ion
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). May not be repeated for credit.
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 214. Linear Algebra and Differential Equations.
Section 002.
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). May not be repeated for credit.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~foth/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 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.
Text: Otto Bretscher, Linear algebra with applications, 2nd ed., Prentice Hall, 2001
Grading:The final grade will be computed from the following: Homework 25% Quizzes 10% First midterm exam 20%
Second midterm exam 20% Final exam 25% (Fri, April 18)
Homework problem sets will be assigned in class once a week, to be turned in the following week. No late
homework will be accepted. Please use only paper and pen/pencil. Show all work.
MATH 215. Calculus III.
UNIFORM EVENING EXAMS FOR MATH 215: MON, FEB 11 & THURS, MAR 21 68 PM. 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). May not be repeated for credit.
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: TUES, FEB 12 & THURS, MAR 21, 810 PM. 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). May not be repeated for credit.
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.
MATH 217. Linear Algebra.
Section 001, 002.
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). May not be repeated for credit.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math217/
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 217. Linear Algebra.
Section 003.
Instructor(s):
Wasserman
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). May not be repeated for credit.
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 255. Applied Honors Calculus III.
EVENING EXAMS THURS FEB 7 & WED MAR 20, 68 PM.
Instructor(s):
Prerequisites & Distribution: MATH 156. Credit can be earned for only one of MATH 215, 255, or 285. (4). (MSA). (BS). May not be repeated for credit.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~karni/m255/m255.html
Multivariable calculus, line, surface, and volume integrals; vector fields, Green's theorem, Stokes theorem; divergence theorem, applications. Maple will be used throughout.
Text: Multivariable Calculus, 4th edition, James Stewart, Brooks/Cole.
MATH 286. Honors Differential Equations.
Instructor(s):
Prerequisites & Distribution: MATH 285. Credit can be earned for only one of MATH 216, 256, 286, or 316. (3). (MSA). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
See Math. 186 for a general description of the sequence Math 185186285286.
Topics include firstorder differential equations, higherorder linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, RungeKutta). If time permits, Picard's Theorem will be proved. Math 216 and 316 cover much of the same material. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 286 is also preparation for more theoretical courses such as Math 451.
MATH 289. Problem Seminar.
Section 001.
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 296. Honors Mathematics II.
Instructor(s):
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among MATH 116, 156, 176, 186, and 296. (4). (Excl). (BS). (QR/1). May not be repeated for credit.
Credits: (4).
Course Homepage: No homepage submitted.
The sequence Math 295296395396 is a more intensive honors sequence than 185186285286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. The precise content depends on material covered in 295 but will generally include topics such as infinite series, power series, Taylor expansion, metric spaces. Other topics may include applications of analysis, Weierstrass Approximation theorem, elements of topology, introduction to linear algebra, complex numbers.
MATH 310. Elementary Topics in Mathematics.
Section 001 – Math Games & Theory of Games.
Instructor(s):
Prerequisites & Distribution: Two years of high school mathematics. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
The current offering of the course focuses on game theory. Students study the strategy of several games where mathematical ideas and concepts can play a role. Most of the course will be occupied with the strructure of a variety of two person games of strategy: tictactoe, tictactoe misere, the French military game, hex, nim, the penny dime game, and many others. If there is sufficient interest students can study: dots and boxes, go moku, and some aspects of checkers and chess. There will also be a brief introduction to the classical Von Neuman/Morgenstern theory of mixed strategy games.
MATH 312. Applied Modern Algebra.
Section 001.
Instructor(s):
Thompson
Prerequisites & Distribution: MATH 217. Only one credit granted to those who have completed MATH 412. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~hmthomps/312.html
Prerequisites: Math 215, 255, or 285 and Math 217
Text: L.N. Childs. A Concrete Introduction to Higher Algebra.
Second Edition. Springer, 1995. This text will be supplemented with
handouts.
Background and Goals: One of the main goals of the course (along
with every course in the algebra sequence) is to expose students to rigorous,
prooforiented mathematics. Students are required to have taken Math 217,
which should provide a first exposure to this style of mathematics. A distinguishing
feature of this course is that the abstract concepts are not studied in
isolation. Instead, each topic is studied with the ultimate goal being
a realworld application.
Content: groups, rings, and fields, including modular arithmetic,
polynomial rings, linear algebra over finite fields, and permutation groups.
Applications from areas such as errorcorrecting codes, cryptography, computational
algebra, and the Polya method of enumeration.
Alternatives: Math 412 (Introduction to Modern Algebra) is a
more abstract and prooforiented course with less emphasis on applications
and is better preparation for most pure mathematics courses. Math 567 is
a more advanced course on coding theory.
Subsequent Courses: Math 312 is one of the alternative prerequisites
for Math 416 (Theory of Algorithms), and several advanced EECS courses
make substantial use of the material of Math 312. Another good followup
course is Math 475 (Elementary Number Theory).
Course Work: There will be weekly homework assignments, two inclass
midterms, and a final.
Grading: Each midterm will be worth 20% of the grade, the fianl
35%, and the homework 25%.
Examination Dates: The first midterm is scheduled for Friday,
February 7, the second is scheduled for Friday, March 28,and the final
is scheduled for Friday, April 18 from 4:00 PM to 6:00 PM.
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). May not be repeated for credit.
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.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 385 and enrollment in the Elementary Program in the School of Education. Permission of instructor required. (13). (Excl). (EXPERIENTIAL). May be repeated for credit for a maximum of 3 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 351. Principles of Analysis.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 215 and 217. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
The content of this course is similar to that of Math 451 but Math 351 assumes less background. This course covers topics that might be of greater use to students considering a Mathematical Sciences concentration or a minor in Math. Course content includes: analysis of the real line, rational and irrational numbers, infinity – large and small, limits, convergence, infinite sequences and series, continuous functions, power series and differentiation.
MATH 354. Fourier Analysis and its Applications.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in MATH 454. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. A substantial portion of the time is spent on both scientific/technological applications (e.g., signal processing, Fourier optics), and applications in other branches of mathematics (e.g., partial differential equations, probability theory, number theory). Students will do some computer work, using MATLAB, an interactive programming tool that is easy to use, but no previous experience with computers is necessary.
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. May not be repeated for credit.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://www.math.lsa.umich.edu/courses/371/
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.
Textbook:
A. Ralston and P. Rabinowitz. A First Course in Numerical Analysis. Dover Publishing, 2001.
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.
Grading
25%  6 Homework assignments 25%  4 Projects 30%  2 Exams 20%  1 Final exam
MATH 396. Honors Analysis II.
Section 001.
Prerequisites & Distribution: MATH 395. (4). (Excl). (BS). May not be repeated for credit.
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~spatzier/396.html
Course Outline: Math 396 is the fourth course in the math honors sequence Math 295296395396. This
sequence offers a thorough preparation for any upper level and even beginning graduate course in
mathematics and other areas that use mathematics (in particular the sciences and engineering). In this fourth
course in this sequence, we will continue developping calculus with rigor and clarity, emphasize the concepts
and at the same time enrich the basic material by drawing on examples and applications from other areas of
mathematics. While the basic material will be a fairly rigorous development of calculus, our point of view will
push the ideas and concepts that haven proved crucial in other important mathematical theories, e.g. complex
analysis and topology. More specifically, we will discuss the calculus of several variables, the idea of manifolds
and how to do calculus on them.
Grading Policy: homework 50%; midterm 25%; final exam 25%;
Homework Policy: Homework will be assigned weekly and collected on Wednesday. You may discuss the
homework problems with other students, but you should write up the solutions on your own.
Text: "Calculus on Manifolds" by Michael Spivak W.A. Benjamin
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 412. Introduction to Modern Algebra.
Section 001 – Abstract Algebra.
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). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~brichert/class/412.html
This course is an introduction to Abstract Algebra, with an emphasis on the logic and
mathematical techniques underlying this beautiful subject. The initial topics include ones common to every branch
of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational
numbers, real numbers, complex numbers). These are then applied to the study of particular types of
mathematical structures: 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.
Text: Abstract Algebra, Thomas W. Hungerford, Second Edition, Harcourt College Publishers, 1997.
Coursework: There will be weekly homework assignments. We will have two inclass midterms, and a final exam.
Grades: Grades will be based on the exams and homework, although performance in class may be taken into
account. Each midterm counts 20%, the final 40%, and the homework 20%.
MATH 412. Introduction to Modern Algebra.
Section 002.
Instructor(s):
De Fernex
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). May not be repeated for credit.
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 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). May not be repeated for credit.
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). May not be repeated for credit.
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 422 / BE 440. Risk Management and Insurance.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 115, junior standing, and permission of instructor. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
We will explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the customer's needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches.
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/winter/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.
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 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). 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 001.
Prerequisites & Distribution: MATH 215, 255, or 285. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://wwwpersonal.umich.edu/~jrs/math425.html
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 002.
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/~sadovska/425/425.html
Course description: Basic concepts of probability are introduced, and applications to other sciences are noted.
Most of the reasoning is rooted either in combinatorics or in calculus. The emphasis is on concepts, calculations
and problemsolving, rather than on formal proofs. Serious use is made of material from Math 116 and Math 215,
but no prior knowledge of combinatorics is assumed. Specific topics include methods of both discrete and
continuous probability, conditional probability, independent events, random variables, jointly distributed random
variables, expectations, variances, and limit laws.
Exams: There will be two midterm exams and a final exam.
Homework: Daily homework will be assigned for each section we cover. These assignments will not be collected.
In addition, you will be given weekly problem sets which will be collected and graded.
Grading: The course grade will be determined as follows.
Homework: Exam 1: 20%
Exam 2: 20%
Final exam: 30%.
Text: A First Course in Probability, Sixth Edition, by Sheldon Ross, PrenticeHall, 2002. The course covers most of Chapters 17, and a part of Chapter 8.
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: http://www.math.lsa.umich.edu/~kalinin/Teach/425/425index.html
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.
Text: Sheldon Ross, A First Course in Probability (6th ed.), PrenticeHall, 2002.
MATH 425 / STATS 425. Introduction to Probability.
Section 007 – Section 007 ONLY satisfies the upperlevel writing requirement.
Instructor(s):
Burns Jr
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 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). May not be repeated for credit.
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). 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 452. Advanced Calculus II.
Section 001.
Instructor(s):
Prerequisites & Distribution: MATH 217, 417, or 419; and MATH 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include:
 partial derivatives and differentiability;
 gradients, directional derivatives, and the chain rule;
 implicit function theorem;
 surfaces, tangent plane;
 maxmin theory;
 multiple integration, change of variable, etc.; and
 Green's and Stokes' theorems, differential forms, exterior derivatives.
Math 551 is a higherlevel course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. Math 452 is prerequisite to Math 572 and is good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 537, 635).
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). May not be repeated for credit.
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 462. Mathematical Models.
Section 001.
Prerequisites & Distribution: MATH 216, 256, 286, or 316; and MATH 217, 417, or 419. Students with credit for MATH 362 must have department permission to elect MATH 462. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~dmbortz/Math462.html
Course Description
Introductory survey of applied mathematics with emphasis on modeling of physical and biological problems in terms of differential equations. Formulation, solution, and interpretation of the results.
Mathematical and Modeling Concepts to be covered
 Concepts of Modelling
 Dimensions, Units, Dimensional Analysis
 Differential equations
 Concepts of equilibria and stability
 Nonlinearity, limit cycles, bifurcations
 Asymptotics and Perturbation theory
 Examples with partial differential equations
 Parameter estimating techniques
Textbook:
There are no required texts for this class.
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 or 472. (3). (Excl). (BS). May not be repeated for credit.
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 475. Elementary Number Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: At least three terms of college mathematics are recommended. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This is an elementary introduction to number theory, especially congruence arithmetic. Number theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and cyphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly numbertheoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially selfcontained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity, and quadratic fields. Math 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with Math 412 which stresses the algebraic content. Math 475 may be followed by Math 575 and is good preparation for Math 412. All of the advanced number theory courses, Math 675, 676, 677, 678, and 679, presuppose the material of Math 575, although a good student may get by with Math 475. Each of these is devoted to a special subarea of number theory.
MATH 476. Computational Laboratory in Number Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: Prior or concurrent enrollment in MATH 475 or 575. (1). (Excl). (BS). May not be repeated for credit.
Credits: (1).
Course Homepage: No homepage submitted.
Students will be provided software with which to conduct numerical explorations. Students will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the laboratory should boost the student's performance in Math 475 or Math 575. Students in the lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for numbertheoretic purposes, e.g., for factoring. No exams.
MATH 485. Mathematics for Elementary School Teachers and Supervisors.
Section 001.
Instructor(s):
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. May not be repeated for credit.
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 486. Concepts Basic to Secondary Mathematics.
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 is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade. Topics covered have included problem solving; sets, relations and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. This material is covered in the course pack and scattered points in the text book. There is no real alternative, but the requirement of Math 486 may be waived for strong students who intend to do graduate work in mathematics. Prior completion of Math 486 may be of use for some students planning to take Math 312, 412, or 425.
MATH 489. Mathematics for Elementary and Middle School Teachers.
Instructor(s):
Prerequisites & Distribution: MATH 385 or 485. May not be used in any graduate program in mathematics. (3). (Excl). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course, together with its predecessor Math 385, 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. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including precalculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problemsolving strategies are used throughout the course.
MATH 490. Introduction to Topology.
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/~kalinin/Teach/490/490index.html
This course in an introduction to both pointset and algebraic topology. Although much of the presentation is theoretical and prooforiented, the material is wellsuited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, courses containing some group theory (Math 412 or 512) and advanced calculus (Math 451) are desirable although not absolutely necessary. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. These include pointset topology, examples of topological spaces, orientable and nonorientable surfaces, fundamental groups, homotopy, and covering spaces. Metric and Euclidean spaces are emphasized. Math 590 is a deeper and more difficult presentation of much of the same material which is taken mainly by mathematics graduate students. Math 433 is a related course at about the same level. Math 490 is not prerequisite for any later course but provides good background for Math 590 or any of the other courses in geometry or topology.
MATH 498. Topics in Modern Mathematics.
Section 001 – Polynomial Equations.
Instructor(s):
Derksen
Prerequisites & Distribution: Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
In many areas one encounters systems of polynomial equations in several variables. There is a systematic way of solving polynomial equations which will be discussed in the course. We will introduce ideals and affine varieties. This will give the students a gentle introduction into algebraic geometry. The main tool is the socalled Buchberger algorithm for computing Groebner bases of ideals. Besides solving polynomial equations, these methods can be applied to a lot of other computational problems as well. We will discuss various applications in algebraic geometry. We will also discuss other applications, such as robotics, automated theorem proving in geometry, integer programming, algebraic geometry and satisfiability in propositional logic.
There are many computer algebra systems where the algorithms are implemented. There will be a weekly lab where students can familiarize themselves with computer algebra systems such as MAPLE. We will practice to translate problems such that they can be solved using Groebner bases packages, such as the one implemented in MAPLE.
The course is aimed at undergraduate mathematics students and graduate computer science students. Students should be familiar with linear algebra, but no knowledge of algebraic geometry is needed.
The book(s) which will be used will be announced later.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Instructor(s):
Karni
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 credit for a maximum of 6 credits.
Credits: (1).
Course Homepage: No homepage submitted.
Audience: Math 501 is a required course for the AIM program, and is primarily intended for these graduate students. Other graduate students from mathematics, physics, engineering, or other applied sciences may also find this course of interest and should contact the instructor prior to registration if interested.
Background and Goals: During their first three years of study, students in the AIM graduate program are required to enroll in Math 501 in both the Fall and Winter terms. In part, this seminar course is coordinated with the Applied and Interdisciplinary Mathematics Research Seminar. The AIM Student Seminar will (i) present the background to the research to be discussed at a more advanced level in the week's AIM Research Seminar, (ii) put the work in context and enable discussion of the importance of the results, and (iii) generally provide an introduction to the topic of the research seminar. Thus students gain meaningful exposure to a broad range of problems. Through direct speaking opportunities in class, the AIM Student Seminar also teaches students to give presentations to an interdisciplinary audience. Both aspects of Math 501 listening and speaking, are vital to general interdisciplinary training, and hence Math 501 is an important part of the AIM graduate program. In Math 501, students will learn both what other students are doing and also what the current of modern research is, and in this way the course will foster interactions and camaraderie among AIM students and faculty.
Course Requirements: Vigorous, active participation is expected during class time, and in addition to participating in the student seminar, students are required to attend all AIM Research Seminars held on Friday afternoons from 3 to 4 PM, and need to allocate time for this in their schedules. This onecredit course is graded on a credit/nocredit basis.
There is no textbook for the course.
MATH 512. Algebraic Structures.
Section 001.
Instructor(s):
Griess
Prerequisites & Distribution: MATH 451 or 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: mainly undergrad math concentrators with a few grad students from other fields
Background and Goals: This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous prooforiented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended
Content: The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitelygenerated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.
Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course covering about half of the material of Math 512. The sequence Math 593594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412.
Subsequent Courses: Together with Math 513, this course is excellent preparation for the sequence Math 593594.
MATH 513. Introduction to Linear Algebra.
Section 001.
Instructor(s):
Nevins
Prerequisites & Distribution: MATH 412. Two credits granted to those who have completed MATH 214, 217, 417, or 419. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to the theory of abstract vector spaces and linear transformations, which are fundamental structures in mathematics and form part of the basic toolkit for many areas of mathematics, science and engineering. The course will emphasize concepts and proofs and will include sufficient calculation to enable students to cement their understanding and apply the ideas of the course in a variety of fields.
Topics to be covered: Vector spaces (over arbitrary fields), linear transformations, bases, matrices, eigenvectors, bilinear and quadratic forms, and Jordan canonical form. As time permits we will cover some of the additional topics in areas to which our tools may be applied, such as differential questions and coding theory.
Text: Sheldon Axler, Linear Algebra Done Right, SpringerVerlag
MATH 521. Life Contingencies II.
Section 001.
Instructor(s):
Huntington
Prerequisites & Distribution: MATH 520. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course is a continuation of Mathematics 520 (a yearlong sequence). It covers the topics of reserving models for life insurance; multiplelife models including joint life and last survivor contingent insurances; multipledecrement models including disability, retirement and withdrawal; insurance models including expenses; and business and regulatory considerations.
Text: Actuarial Mathematics (2nd Edition) by Bowers, Gerber, Hickman, Jones and Nesbitt (Society of Actuaries).
MATH 523. Risk Theory.
Section 001.
Instructor(s):
David Schneider
Prerequisites & Distribution: MATH 425. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~daschnei/math523/math523.html
Prerequisites: A solid background in probability theory at the 400 level, Math 425 or equivalent.
Background and Goals: 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. It provides background for the professional exams in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society.
Content: 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. It provides background
for the professional exams in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society.
We intend to cover the following topics: Standard distributions used for claim frequency models and for loss
variables, theory of aggregate claims, compound Poisson claims model, discrete time and continuous time
models for the aggregate claims variable, the ChapmanKolmogorov equation for expectations of aggregate
claims variables, the Brownian motion process, estimating the probability of ruin, reinsurance schemes and their
implications for profit and risk. Credibility theory, classical theory for independent events, least squares theory
for correlated events, examples of random variables where the least squares theory is exact.
Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam.
There will be 8 Problem sets. Each quiz will consist of a slightly modified homework problem.
8 quizzes: 37% of grade.
midterm= 27% of grade.
final= 36% of grade.
Textbook:
Loss Modelsfrom Data to Decisions by Klugman, Panjer and Willmot, Wiley 1998. The book is on reserve at the
Shapiro Science Library.
MATH 525 / STATS 525. Probability Theory.
Section 001.
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). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~sadovska/525/525.html
This course presents a rigorous study of the mathematical theory of probability. The
emphasis will be on fundamental concepts and proofs but examples and applications will also be discussed. The
course covers basic results and methods of both discrete and continuous probability theory, conditional probability
and conditional expectation, discrete and continuous random variables, convergence of random variables and
other topics.
Exams: There will be a midterm and a cumulative final exam.
Homework, etc.: Weekly homework assignments will be collected and graded. Attendance and participation in
lectures is expected.
Grading: The course grade will be determined as follows.
Homework: 45% Midterm Exam: 20% Final exam: 35%.
Text: Probability & Random Processes, Third Edition, by G. Grimmett and D. Stirzaker, Oxford University Press
(2001).
MATH 526 / STATS 526. Discrete State Stochastic Processes.
Section 001.
Instructor(s):
Doering
Prerequisites & Distribution: MATH 525 or EECS 501. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Recommended: Good understanding of advanced calculus covering limits, series, the notion of continuity, differentiation and the Riemann integral; Linear algebra including eigenvalues and eigenfunctions.
This is a core course for the graduate program in Applied & Interdisciplinary Mathematics (AIM).
Background and Goals: The theory of stochastic processes is concerned with systems which change in accordance with probability laws. Many applications can be found in physics, engineering, computer sciences, economics, financial mathematics and biological sciences, as well as in other branches of mathematical analysis such as partial differential equations. The purpose of this course is to provide an introduction to the theory of stochastic processes. It is a second course in mathematical probability which should be of interest to students of mathematics and statistics as well as students from other disciplines in which stochastic processes have found significant applications. Special efforts will be made to both explore the rich diversity of applications as well as to make students aware of mathematical subtleties underlying stochastic processes.
Content: The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queueing theory; an introduction to Brownian motion; stationary processes and martingales.
Coursework: weekly (or biweekly) problem sets will count for 50% of the grade, a midterm exam will count for 20% of the grade. The final exam will count for 30%.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001 – Intro to Complex Variables.
Instructor(s):
Stensones
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program
Background and Goals: 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. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.
Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 19 of Churchill.
Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications.
Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.
MATH 558. Ordinary Differential Equations.
Section 001 – Applied Nonlinear Dynamics.
Instructor(s):
Smereka
Prerequisites & Distribution: MATH 450 or 451. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Existence and uniqueness of theorems for flows, linear systems, Floquet theory, PoincaréBendixson theory, Poincaré maps, periodic solutions, stability theory, Hopf bifurcations, chaotic dynamics.
MATH 561 / IOE 510 / SMS 518. Linear Programming I.
Section 001.
Instructor(s):
Amy Ellen Mainville Cohn
Prerequisites & Distribution: MATH 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students. May not be repeated for credit.
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 566. Combinatorial Theory.
Section 001 – Introduction to enumerative and algebraic combinatorics.
Prerequisites & Distribution: MATH 216, 256, 286, or 316. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/566.html
 INTRODUCTION
 Cayley's Theorem
 De Bruijn Sequences
 Hooklength formula
 ALGEBRAIC GRAPH THEORY
 Spectra of graphs
 Walks on a cube
 Sperner's theorem
 Matrixtree theorem
 Eulerian tours
 Domino tilings
 PARTITIONS AND TABLEAUX
 Partitions. Pentagonal Number Theorem
 Young's lattice
 The Schensted correspondence
 Tableaux and involutions
 CLASSICAL ENUMERATION
 Catalan numbers
 Stirling numbers
 Inversions and major index
 q binomial coefficients
 Rook polynomials
 Polya theory
 DISCRETE GEOMETRY
 Theorems of P.Hall and G.König
 Birkhoff's theorem. The assignment polytope
 Cyclic polytopes
 Permutohedra
 The weak order of the symmetric group
MATH 567. Introduction to Coding Theory.
Section 001.
Instructor(s):
Yu
Prerequisites & Distribution: One of MATH 217, 419, 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This course is an introduction to coding theory, focusing on the mathematical background for linear errorcorrecting codes. It will begin with a discussion of Shannon's theorem and channel capacity. The definition of linear codes will be given along with a review of necessary tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes will be studied including the Hamming, BCH, cyclic, Melas, ReedMuller, and ReeSolomon codes. An introduction to the problem of decoding will be included, starting with syndrome decoding and covering weight enumerator polynomials and the MacWilliams Sloane identity. Further topics to be included range from consideration of asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form. Student Body: Undergraduate math majors and EECS graduate students
Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of errorcorrecting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology.
Content: Introduction to coding theory focusing on the mathematical background for errorcorrecting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, ReedMuller, and ReedSolomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the MacWilliams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form.
Alternatives: none
Subsequent Courses: Math 565 (Combinatorics and Graph Theory) and Math 556 (Methods of Applied Math I) are natural sequels or predecessors. This course also complements Math 312 (Applied Modern Algebra) in presenting direct applications of finite fields and linear algebra.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001 – numerical linear algebra.
Instructor(s):
Epperson
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: No homepage submitted.
Course Description: This will be an introduction to numerical linear algebra, in which we will study solution techniques for the linear systems problem (both direct and iterative), the least squares problem, and the algebraic eigenvalue problem. The text will be: Fundamentals of Matrix Computations, by David Watkins. The course will emphasize both theory and implementation of the methods, so proficiency in a computing language is necessary. A good background in linear algebra is also necessary. The course grade will be determined by weekly homework assignments, a midterm, and a final exam.
Subsequent Courses: Math 572 (Numer Meth for Sci Comput II) 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 572. Numerical Methods for Scientific Computing II.
Section 001.
Instructor(s):
Karni
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: No homepage submitted.
Prerequisites: Solid background in advanced calculus and linear algebra is needed. 454 is not required but is good to have. Knowledge of a computing programming language such as Fortran, C or the Matlab computing environment is mandatory.
This course is an introduction to numerical methods for boundaryvalue and initialvalue problems. The course will cover numerical methods for ordinary differential equations and for linear elliptic, parabolic and hyperbolic partial differential equations. Nonlinear hyperbolic partial differential equations may also be discussed, if time permits.
The course will focus on the derivation of methods, on their accuracy, stability and convergence properties, as well as on practical aspects of their efficient implementation. The course should be useful to students in mathematics, physics and engineering.
Text: Finite Difference Methods for Differential Equations. Notes by Randall J. LeVeque, available as a coursepack from Ulrichs.
MATH 582. Introduction to Set 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://wwwpersonal.umich.edu/~pgh/Math582/
Set Theory is at the same time (1) a branch of mathematics, (2) a tool used in practically every other branch of
mathematics, and (3) the best medium for understanding the foundations of mathematics. This course is mainly a
study of (1), but much of the motivation comes from (2) and (3) and these aspects will be covered from time to
time. Everyone who has taken a course in any sort of abstract mathematics, be it algebra, analysis, or topology,
has used notions such as "set", "function" "equivalence relation", "linear ordering", etc.; a lowlevel goal of the
course is to improve familiarity and comfort with these common mathematical notions. Deeper topics are
wellorderings, ordinal numbers, cardinal numbers, and their properties. Set theory as a separate discipline really
began with Cantor's discovery (in the late 19th century) that infinite sets can have different sizes, and the
consequences and refinements of this fact will be a centerpiece of the course. We will also discuss historically
troublesome assertions such as the Axiom of Choice and the Continuum Hypothesis.
All of these will be considered from both the nonaxiomatic and axiomatic perspectives. The axiomatic approach is
both more necessary in set theory than is other branches of mathematics and more fruitful. It is necessary partly
because of the discovery that intuitions about sets can easily go astray and lead to paradox and contradiction. It is
fruitful because a relatively simple set of axioms suffices to generate all of the theorems of set theory. Since
essentially all mathematical notions can be expressed in terms of sets, the axiomatization of set theory is in
effect an axiomatization of all of mathematics. Hence the context of axiomatic set theory is wellsuited for dealing
with the philosophical issue of what it means for a mathematical assertion to be true or provable. These
considerations lead to a necessarily brief discussion of consistency and independence results.
The announced prerequisites of Math 412 or 451 have more to do with general level of mathematical
sophistication than specific content. The course is wellsuited to math majors, honors or not, beginning graduate
students, and mathematically minded students of philosophy or computer science. If you have any doubts about
the level of the course, please talk with me. A course in mathematical logic is not presupposed. We will follow the
book of Y.N. Moschovakis, Notes on Set Theory (SpringerVerlag, ISBN 0387941800 and 3540941800). There
will be several problem assignments and perhaps a takehome final exam.
Homework sets will be assigned periodically during the term
MATH 592. Introduction to Algebraic Topology.
Section 001.
Instructor(s):
Lott
Prerequisites & Distribution: MATH 591. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
This is a first course in algebraic topology. It covers the material from this area that appears on the topology qualifying review exam. The topics include the fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, EilenbergSteenrod axioms, and Brouwer's and Lefschetz' fixedpoint theorems,
Text : "Algebraic Topology" by Allen Hatcher, Cambridge University Press
MATH 594. Algebra II.
Section 001 – Group Theory and Galois Theory.
Instructor(s):
Conrad
Prerequisites & Distribution: MATH 593. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Prerequisites: Knowledge of basic linear algebra over an arbitrary field, prior exposure to the concept of a finite group and polynomials in 1 variable over a field (unique factorization, division algorithm, etc.), and a capacity for abstract thought.
Course description: This course will cover the basic elements of group theory and Galois theory, as preparation for the qualifying exam in algebra. We'll begin with a tour of the standard facts from finite group theory with an emphasis on those notions which are important for more general groups (algebraic groups, Lie groups, etc.). This may include a brief discussion of some concepts in the representation theory of finite groups if time permits. Once these basics are handled, we turn out attention to the theory of fields (including characteristic p!) and the historical reason why groups were first introduced by Galois: to do Galois theory! I think that Galois theory is one of the most aweinspiring topics in algebra. By the end of the course, we will have completely solved several classical problems, including how to determine which types of constructions are possible with a straightedge and compass, how to give an `essentially' algebraic proof of the socalled Fundamental Theorem of Algebra, and how to prove that it is impossible (in a very precise sense) to solve the general nth degree polynomial 'in radicals' when n is at least 5 (and how one can derive the classical formulas for n < 5).
Textbook: Abstract Algebra, by Dummit and Foote.
Homework/exams: There will be weekly homework and takehome exams (two midterms and a final). Late homework will not be accepted for any reason, but the two lowest homework grades will be dropped. It is your responsibility to make sure your homework is turned in on time. Your final grade will be based on 50% homeworks, 20% midterm, and 30% final exam.
MATH 597. Analysis II.
Section 001 – Real Analysis.
Instructor(s):
Barrett
Prerequisites & Distribution: MATH 451 and 513. (3). (Excl). (BS). May not be repeated for credit.
Credits: (3).
Course Homepage: No homepage submitted.
Topics will include: Lebesgue measure on the real line and in Rn; general measures; measurable functions; integration; monotone convergence theorem; Fatou's lemma; dominated convergence theorem; Fubini's theorem; function spaces; Holder and Minkowski inequalities; functions of bounded variation; differentiation theory; Fourier analysis. Additional topics such as Sobolev spaces to be covered as time permits.
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