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Open courses in Mathematics (*Not realtime Information. Review the "Data current as of: " statement at the bottom of hyperlinked page)
Wolverine Access Subject listing for MATH
Fall Term '01 Time Schedule for Mathematics.
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
admissions 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 Math 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 through the Math Lab and directed towards students who are unable to complete a first calculus
course successfully. A maximum total of 4 credits may be earned
in courses numbered 110 and below. 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 received credit for a 200 (or higher) level mathematics
course (except 385, 489 or 497).
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. All
students must have the permission of an Honors advisor to enroll
in any of these courses, but they need not be enrolled in the
LS&amp;A Honors Program. All students with strong preparation
and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard
sequences.
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 (see course description) 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 problems and encouraged to experiment
in groups using computers. The sequence Math 295396 provides
a rigorous introduction to theoretical mathematics. Proofs are
stressed over applications and these courses require a high level
of interest and commitment. Most students electing Math 295 have
completed a thorough high school calculus course. The student
who completes Math 396 is prepared to explore the world of 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
are two courses expressly designed and recommended for students
with one or two semesters of AP credit, Math 119 and Math 156.
Both will review the basic concepts of calculus, cover integration
and an introduction to differential equations, and introduce the
student to the computer algebra system MAPLE. Math 119 will stress
experimentation and computation, while 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, students 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
Math 396, and no credit can be earned for a prerequisite to a
course taken after the course itself.
Students completing Math 116
who are principally interested in the application of mathematics
to other fields may continue either to Math 215 (Analytic Geometry
and Calculus III) or to Math 216 (Introduction to Differential
Equations) – these two courses may be taken in either order. Students
who have greater interest in theory or who intend to take more
advanced courses in mathematics should continue with Math 215
followed by the sequence Math 217316 (Linear AlgebraDifferential
Equations). Math 217 (or the Honors version, Math 513) is required
for a concentration in Mathematics; it both serves as a transition
to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential
equations in Math 316. Math 216 is not intended for mathematics
concentrators.
Attention Potential
Elementary School Teachers: Math 489 is Offered this Spring Term
All elementary teaching certificate
candidates are required to take two mathematics courses, Math 385
and Math 489, either before or after admission to the School of Education.
Math 385 is offered in the Fall, Math 489 in the Winter. Due
to increasing enrollments, Math 489 will be offered this Spring Term (IIIA, 2001) as well. Since class size limits in Winter 2002 will be strictly
enforced, anyone who can elect Math 489 in the Spring Term is urged to
do so. It is the surest way to guarantee oneself a place in the course.
The next Spring Term offering of Math 489 will be in 2003. For further
information, contact Prof. Krause at 7631186 or at his office, 3086
East Hall.
A maximum total of 4 credits
may be earned in Mathematics courses numbered 110 and below. A maximum total of 16 credits may be earned for calculus
courses Math 112 through Math 396, and no credit can be earned
for a prerequisite to a course taken after the course itself.
MATH 105. Data, Functions, and Graphs.
There Will Be Joint Evening Examinations For All Sections Of Math 105, 6:00 – 8:00 P.M. On Mon. Oct. 8 And Thurs Nov. 15.
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. (4). (MSA). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/105/
Math 105 serves both as a preparatory class 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. Math 110 is a condensed halfterm version of the same material offered as a selfstudy course through the Math Lab.
TEXT: Functions Modeling Change, Connally, Wiley Publishing.
MATH 110. PreCalculus (SelfStudy).
Section 001 – Enrollment In Math 110 Is By Permission Of Math115 Instructor And Override Only. Course Meets The Second Half Of The Term. Students Work Independently With Guidance From Math Lab Staff.
Instructor(s):
Prerequisites & Distribution: See Elementary Courses above. Enrollment in Math. 110 is by recommendation of Math. 115 instructor and override only. No credit granted to those who already have 4 credits for precalculus mathematics courses. (2). (Excl).
Credits: (2).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html
The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, halfterm version of Math 105 (Math 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students who complete 110 are fully prepared for Math 115. Students may enroll in Math 110 only on the recommendation of a mathematics instructor after the third week of classes in the Fall and must visit the Math Lab to complete paperwork and receive course materials.
MATH 115. Calculus I.
There Will Be Joint Evening Examinations For All sections Of Math 115, 68 P.M., Weds Oct. 3 And Nov. 7.
Instructor(s):
Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/115/
The sequence Math 115116215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.
Topics include functions and graphs, derivatives and their applications to reallife problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions, and Graphs). Math 116 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 186. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI83 is recommended).
TEXT: Calculus, 2nd edition, HughesHallet, Wiley Publishing
TI83 Graphing Calculator, Texas Instruments.
MATH 116. Calculus II.
There will be joint evening examinations for all sections of Math 116 on Tues. Oct. 9 and Nov. 13.
Instructor(s):
Prerequisites & Distribution: Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/116/
See Math 115 for a general description of the sequence Math 115116215.
Topics include the indefinite integral, techniques of integration, introduction to differential equations, and infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. Math 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 285.
Text: Calculus, 2nd Edition, HughesHallet/Gleason, Wiley Publishing.
TI83 Graphing Calculator, Texas Instruments.
MATH 128. Explorations in Number Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: High school mathematics through at least Analytic Geometry. Only firstyear students, including those with sophomore standing, may preregister for FirstYear Seminars. All others need permission of instructor. No credit granted to those who have completed a 200 (or higher) level mathematics course. (4). (MSA). (BS). (QR/1).
FirstYear Seminar,
Credits: (4).
Course Homepage: No homepage submitted.
No Description Provided.
Check Times, Location, and Availability
MATH 147. Introduction to Interest Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 112 or 115. No credit granted to those who have completed a 200 (or higher) level mathematics course. (3). (MSA). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators should elect Math 424, which covers the same topics but on a more rigorous basis requiring considerable use of calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds, and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. This course is not part of a sequence. Students should possess financial calculators.
Text: Mathematics of Finance, Zima and Brown, McGraw Hill Publishing.
MATH 156. Applied Honors Calculus II.
There Will Be Joint Evening Examinations For All Sections Of Math 156, Thurs, Oct 11 And Wed, Nov 14, 6:00 – 8:00 P.M.
Prerequisites & Distribution: Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html
The sequence 156255256 is an Honors calculus sequence for engineering and science concentrators who scored 4 or 5 on the AB or BC Advanced Placement calculus exam. The course emphasizes computational skills, conceptual understanding, and applications of calculus. Math 156 provides students with the background
needed for a variety of subsequent courses in math, science, and engineering. It also introduces students to MAPLE, a highlevel software tool for doing mathematics on a computer.
 Applications of the Integral:
sigma notation, Riemann sums, definite integral, fundamental theorem of calculus, work, improper integrals, arclength, surface area, moments and center of mass, hydrostatic force, probability
density functions
 Differential Equations:
exponential growth and decay, Newton's law of cooling/heating, logistic equation, stability of equilibrium points
 Series:
sequences, series, geometric series, integral and comparison tests, alternating series, ratio test, power series, radius of convergence, Taylor series, binomial series, applications of Taylor
approximation
 Review (as needed):
integration by parts, trigonometric substitution, partial fractions, trigonometric integrals, l'Hopital's rule
 Special Topics:
asymptotic expansions, Bessel function, complex numbers, Euler's formula, Gamma function, hyperbolic functions, Laplace transform, parametric curves, polar coordinates
Homework: Math 156 has weekly homework assignments. Students may work together and discuss the homework problems with each other, but each student should write up and submit their own set of
solutions. After the assignment is collected, solutions will be available in a looseleaf book at the Undergraduate Library Reserve Desk on the 2nd floor of the Shapiro Library.
Exams: 2 midterm exams and a final exam
TEXT: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 175. Combinatorics and Calculus.
Section 001 – Introduction to Cryptology and Discrete Mathematics
Prerequisites & Distribution: Permission of Honors advisor. No credit granted to those who have completed a 200level or higher Mathematics course. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/175.html
Text (required):
An Introduction to Cryptology and Discrete Math – The Math 175 Coursepack, by C.Greene, P.Hanlon, T.Hsu, and J.Hutchinson. *Available from Dollar Bill, 611 Church Street, $16.09+tax.
Supplementary texts (not required):
Cryptological mathematics, by R.E.Lewand, Mathematical Association of America, 2000.
Introduction to cryptography, by J.A.Buchmann, SpringerVerlag, 2001.
Prerequisites: Math 115 or equivalent (singlevariable calculus) recommended.
Description: This course gives a historical introduction to Cryptology, the science of secret codes. It begins with the oldest recorded codes, taken from hieroglyphic engravings, and ends with the encryption schemes used to maintain privacy during Internet credit card transactions. Since secret codes are based on mathematical ideas, each new kind of encryption method leads in this course to the study of new mathematical ideas and results.
The first part of the course deals with permutationbased codes: substitutional ciphers, transpositional codes, Vigenere ciphers and more complex polyalphabetic substitutions including those created by rotor machines such as the WWII Enigma. The mathematical subjects treated in this section include permutations, modular arithmetic and some elementary statistics.
In the second part of the course, the subject moves to bit stream encryption methods. These include block cipher schemes such as the Data Encryption Standard (DES). The mathematical concepts introduced here are recurrence relations and some more advanced statistical results.
Public key encryption is the subject of the final part of the course. We learn the mathematical underpinnings of DiffieHellman key exchange, RSA, and Knapsack codes. A substantial number of results from elementary number theory are needed and proved in this section of the course.
There is considerable development of problemsolving skills in Math 175. So, students taking the course should have significant mathematical experience and sophistication.
Grading: There are no quizzes and no exams in the course. The grade will be based on homework together with weekly computer labs. This course will not be graded on a curve.
MATH 185. Honors Calculus I.
Section 001.
Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~nevins/185hmwk.html
The sequence Math 185186285286 is the Honors introduction to calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.
Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.
TEXT: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 185. Honors Calculus I.
Section 002, 003.
Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~nmacura/teach.html
The sequence Math 185186285286 is the Honors introduction to calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.
Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.
TEXT: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 185. Honors Calculus I.
Section 004.
Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: No homepage submitted.
The sequence Math 185186285286 is the Honors introduction to calculus. It is taken by students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program.
Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.
TEXT: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 214. Linear Algebra and Differential Equations.
Instructor(s): Morton Brown
Prerequisites & Distribution: Math. 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
This course is an introduction to matrices and linear algebra. This course covers the basic needs to understand a wide variety of applications that use the ideas of linear algebra, from linear programming to mathematical economics. The emphasis is on concepts and problem solving. The course is designed as an alternative to Math 216 for students who need more linear algebra and less differential equations background than provided in 216. The course includes an introduction to the main concepts of linear algebra – matrix operations, echelon form, solution of systems of linear equations, Euclidean vector spaces, linear combinations, independence and spans of sets of vectors in Euclidean space, eigenvectors and eigenvalues, and similarity theory. There are applications to discrete Markov processes, linear programming and solution of linear differential equations with constant coefficients.
TEXT: Linear Algebra and Its Applications, David Lay, Addison Wesley Publishing.
MATH 215. Calculus III.
There Will Be Joint Evening Examinations For All Sections Of Math 215, 68 P.M., Thur Oct 11, And Thur Nov 15.
Instructor(s):
Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/215/
The sequence Math 115116215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly computer lab using Maple software. Math 285 is a somewhat more theoretical course which covers the same material. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.
TEXT: STUDENTS HAVE CHOICE OF EITHER: Calculus, 4th edition, James Stewart, Brooks/Cole Publishing
or
Multivariable Calculus, 4th edition, James Stewart, Brooks/Cole Publishing.
MATH 216. Introduction to Differential Equations.
There will be joint examinations for all sections of Math 216 on Mon. Oct. 8 and Mon. Nov. 12
Instructor(s):
Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Not intended for Mathematics concentrators. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/216/
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216417 (or 419) and Math 217316. The sequence Math 216417 emphasizes problemsolving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence 217316. Math 286 covers much of the same material in the Honors sequence. The sequence Math 217316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.
Text: Differential Equations, Computing and Modeling, 2nd edition, Edwards and Penney, Prentice Hall Publishing.
MATH 217. Linear Algebra.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~asgari/ma217_fall01.html
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216417 (or 419) and Math 217316. The sequence Math 216417 emphasizes problemsolving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R^{n}; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.
Text: Linear Algebra and Its Applications, 2nd edition, David Lay, Addison Wesley Publishing.
MATH 217. Linear Algebra.
Section 002, 003.
Instructor(s): Evangelos Mouroukos
Prerequisites & Distribution: Math. 215, 255, or 285. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: No homepage submitted.
For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216417 (or 419) and Math 217316. The sequence Math 216417 emphasizes problemsolving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R^{n}; linear dependence, bases, and dimension; linear transformations; eigenvalues and eigenvectors; diagonalization; and inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.
Text: Linear Algebra and Its Applications, 2nd edition, David Lay, Addison Wesley Publishing.
MATH 256. Applied Honors Calculus IV.
Section 001.
Prerequisites & Distribution: Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~millerpd/Courses/256.html
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, and MAPLE.
MATH 256. Applied Honors Calculus IV.
Section 002.
Instructor(s):
Prerequisites & Distribution: Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
Linear algebra, matrices, systems of differential equations, initial and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, and MAPLE.
MATH 285. Honors Calculus III.
Instructor(s):
Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
See Math 185 for a general description of the sequence Math 185186285286.
Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; maximumminimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.
MATH 285. Honors Calculus III.
Section 002.
Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math285/
See Math 185 for a general description of the sequence Math 185186285286.
Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; maximumminimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.
MATH 288. Math Modeling Workshop.
Section 001 – (Drop/Add deadline=September 25).
Instructor(s):
Prerequisites & Distribution: Math. 216 or 316, and Math. 217 or 417. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of three credits.
Mini/Short course
Credits: (1).
Course Homepage: No homepage submitted.
This course is designed to help students understand more clearly how techniques from other undergraduate mathematics courses can be used in concert to solve realworld problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used. During the weekly workshop students will be presented with realworld problems on which techniques of undergraduate mathematics offer insights. They will see examples of (1) how to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can't be solved, and (4) how to present the solution to others. Students will have a chance to use the skills developed by participating in the UM Undergraduate Math Modeling Meet.
MATH 289. Problem Seminar.
Section 001 – (Drop/Add deadline=September 25).
Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit with permission.
Mini/Short course
Credits: (1).
Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math289.html
One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to Honors students. This course is excellent preparation for the Putnam exam. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.
MATH 295. Honors Mathematics I.
Section 001.
Instructor(s):
Prerequisites & Distribution: Prior knowledge of first year calculus and permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: No homepage submitted.
Math 295296395396 is the main Honors calculus sequence. It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts and problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should seriously consider starting with this sequence. The expected background is high school trigonometry and algebra (previous calculus not required). This sequence is not restricted to students enrolled in the LS&A Honors Program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, and some linear algebra. Math 175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel.
MATH 316. Differential Equations.
Instructor(s):
Prerequisites & Distribution: Math. 215 and 217. Credit can be earned for only one of Math. 216, 256, 286, or 316. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to differential equations for students who have studied linear algebra (Math 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. Firstorder equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvectoreigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higherorder equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. Math 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. Math 286 is the Honors version of Math 316. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451.
MATH 333. Directed Tutoring.
Instructor(s): Eugene F Krause
Prerequisites & Distribution: Math. 385 and enrollment in the Elementary Program in the School of Education. (13). (Excl). (EXPERIENTIAL). May be repeated for a total of three credits.
Credits: (13).
Course Homepage: No homepage submitted.
An experiential mathematics course for exceptional upperlevel students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (Math 385 and Math 489) required of all elementary teachers.
MATH 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).
Credits: (3).
Course Homepage: No homepage submitted.
This 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.
Section 001.
Instructor(s):
Prerequisites & Distribution: Engineering 101; one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No homepage submitted.
This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to nonlinear equations, ordinary differential equations, and polynomial approximations. Other topics may include discrete Fourier transforms, twopoint boundaryvalue problems, and MonteCarlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for 571572.
MATH 385. Mathematics for Elementary School Teachers.
Section 001.
Instructor(s): Eugene F Krause
Prerequisites & Distribution: One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in Math. 485. (3). (Excl).
Credits: (3).
Course Homepage: No homepage submitted.
All elementary teaching certificate candidates are required to take two
math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489
in the Winter Term. Due to heavy enrollment pressure, Math 385 will be
offered this Spring Term (IIIA 2000) as well. Enrollment is limited to 30 students per section; classsize limits will be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course.
This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Although only two years of high school mathematics are required, a more complete background including precalculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course. Math 489 is the required sequel.
For further information, contact Prof. Krause at his email
address, krause@math.lsa.umich.edu.
MATH 385. Mathematics for Elementary School Teachers.
Section 002.
Instructor(s): Eugene F Krause
Prerequisites & Distribution: One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in Math. 485. (3). (Excl).
Credits: (3).
Course Homepage: No homepage submitted.
All elementary teaching certificate candidates are required to take two
math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489
in the Winter Term. Due to heavy enrollment pressure, Math 385 will be
offered this Spring Term (IIIA 2000) as well. Enrollment is limited to 30 students per section; classsize limits will be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course.
This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Although only two years of high school mathematics are required, a more complete background including precalculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory), and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course. Math 489 is the required sequel.
For further information, contact Prof. Krause at his email
address, krause@math.lsa.umich.edu.
MATH 395. Honors Analysis I.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
This course is a continuation of the sequence Math 295296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. This course studies functions of several real variables. Topics are chosen from elementary linear algebra (vector spaces, subspaces, bases, dimension, and solutions of linear systems by Gaussian elimination); elementary topology (open, closed, compact, and connected sets, and continuous and uniformly continuous functions); differential and integral calculus of vectorvalued functions of a scalar; differential and integral calculus of scalarvalued functions on Euclidean spaces; linear transformations (null space, range, matrices, calculations, linear systems, and norms); and differential calculus of vectorvalued mappings on Euclidean spaces (derivative, chain rule, and implicit and inverse function theorems).
MATH 399. Independent Reading.
Instructor(s):
Prerequisites & Distribution: (16). (Excl). (INDEPENDENT). May be repeated for credit.
Credits: (16).
Course Homepage: No homepage submitted.
Designed especially for Honors students.
MATH 404. Intermediate Differential Equations and Dynamics.
Section 001.
Prerequisites & Distribution: Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 286, or 316. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math404.html
There are three main objectives to this class. First, we will present and explain mathematical methods for obtaining approximate analytical solutions to differential equations that cannot be solved exactly. The material in this section will mostly come from Bender and Orszag and we will present it in a very introductory manner. Second, we will introduce and explain the theories of dynamical systems, specifically bifurcation and chaos, phase portraits, linear stability analysis, and local and global behavior of both linear and nonlinear differential equations. Third, we
will use the computer via matlab and maple to visualize what we are learning and to gain a better understanding of the dynamics. Learning the material will be done through homework that will require both pen and paper analysis and computation support. I would expect all students to have had an introductory course in differential equations and some familiarity with matlab. There will be a special session or two on matlab for students who do not have familiarity with this computation method.
Textbooks
 Advanced Mathematical Methods for Scientists and Engineers (required), Carl M. Bender and Steven A. Orszag, McGrawHill, 1978,.
This book presents the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to 'real' differential equations that arise in physics and engineering. These equations are usually not solvable in closed form and numerical methods may not converge to useful solutions. The aim is to teach the insights that are most useful in approaching new problems and it avoids the special methods and tricks that work only for particular problems. We will use this book in an introductory manner where some of the more advanced topics will not be covered.
 Nonlinear Systems (recommended), P.G. Drazin, Cambridge texts in Applied Mathematics,1994.
 Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields (recommended), J. Guckenheimer and P. Holmes, Springer
Mathematical concepts to be covered
 Ordinary differential equations
 Approximate solutions of linear differential equations
 Approximate solutions of nonlinear differential equations
 Stability analysis, bifurcations, and limit cycles
 Phase portraits
 Perturbation theory
 Boundary layer theory
 Bessel, parabolic cylinder, and Airy functions
 Numerical analysis with Matlab
Learning Objectives and Instructor Expectations
The objective of this course (as worded in Bender and Orszag) is to help young and established scientists and engineers to build the skill necessary to analyze equations that they encounter in the real world. Asymptotic and perturbation analysis are some of the most useful and powerful, as well as beautiful, methods for finding approximate solutions to equations. Combining these techniques with those of dynamical systems and computation provide the student with a powerful tool for analyzing most ordinary differential equations.
Grading
Homework assignments will count as 40% of grade evaluation. There will also be two midterms worth 30% of the grade and a final that counts for 25%. The remaining 5% is student participation.
MATH 412. Introduction to Modern Algebra.
Instructor(s):
Prerequisites & Distribution: Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in Math. 512. Students with credit for Math. 312 should take Math. 512 rather than 412. One credit granted to those who have completed Math. 312. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. Math 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, 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 413. Calculus for Social Scientists.
Section 001.
Instructor(s):
Prerequisites & Distribution: Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
A oneterm course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.
MATH 417. Matrix Algebra I.
Instructor(s):
Prerequisites & Distribution: Three courses beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled Math. 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/courses/417/
Many problems in science, engineering, and mathematics are best formulated in terms of matrices – rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problemsolving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.
Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.
Text: Linear Algebra with Applications, Otto Bretscher, Prentice Hall Publishing.
MATH 419. Linear Spaces and Matrix Theory.
Instructor(s):
Prerequisites & Distribution: Four terms of college mathematics beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). (Excl). (BS).
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No homepage submitted.
Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous prooforiented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, innerproduct spaces; unitary, selfadjoint, and orthogonal operators and matrices, 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.
Text: Linear Algebra with Applications, 3rd edition, Otto Bretscher, Prentice Hall Publishing.
MATH 423. Mathematics of Finance.
Prerequisites & Distribution: Math. 217 and 425; CS 183. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~mikegs/423/index.html
Required Text:
Options, Futures and Other Derivatives by Hull, fourth edition, Prentice Hall 1999.
Background and Goals: 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 derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary.
Contents:
 Forwards and Futures, Hedging using Futures, Bills and Bonds, Swaps, Perfect Hedges.
 OptionsEuropean and American, Trading Strategies, PutCall Parity, BlackScholes formula.
 Volatility, methods for estimating volatilityexponential, GARCH, maximum likelihood.
 Dynamic Hedging, stoploss, BlackScholes, the Greek letters.
 Other Options.
Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam. There will be 8 homework assignments. Each quiz will consist of a slightly modified homework problem.
MATH 424. Compound Interest and Life Insurance.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a prerequisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to reallife situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the Society of Actuaries exam; the material covered will depend somewhat on the instructor. Math 424 is required for students concentrating in actuarial mathematics; others may take Math 147, which deals with the same techniques but with less emphasis on continuous growth situations. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).
MATH 425 / STATS 425. Introduction to Probability.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/425.html
This course introduces students to the theory of probability and to a number of applications. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. The material corresponds to most of Chapters 17 and part of 8 of Ross.
Grade will be based on two 1hour midterm exams, 20% each; 20% homework; 40% final exam. Your lowest homework set score will be dropped.
This course will not be graded on a curve.
Homework will normally be due in class on Fridays. There will be approximately 10 problem sets. The midterm exams are held in class. No makeups will be given.
MATH 425 / STATS 425. Introduction to Probability.
Section 002, 004.
Instructor(s): Jeganathan
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
See Statistics 425.002.
MATH 425 / STATS 425. Introduction to Probability.
Section 003.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~carswell/math425/
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 005.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~stephnsb/cur425/math425005.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 006.
Instructor(s): Amirdjanova
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
See Statistics 425.006.
MATH 431. Topics in Geometry for Teachers.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math431/
This course is a study of the axiomatic foundations of Euclidean and nonEuclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups; and finite, hyperbolic, and taxicab nonEuclidean geometries. Alternative geometry courses at this level are 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.
MATH 433. Introduction to Differential Geometry.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 215, or 255 or 285, and Math. 217. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is about the analysis of curves and surfaces in 2 and 3space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4vertex theorem; and (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, and classification of minimal surfaces of revolution. 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513), and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.
MATH 450. Advanced Mathematics for Engineers I.
Section 001.
Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math450.html
Course Description
There are three main objectives to this class. First, we will introduce the concepts of partial differential equations and complex variables and some basic techniques for analyzing these problems. Second, by studying the application of PDE's to physics, engineering, and biology, the
student will begin to acquire intuition and expertise about how to use these equations to model scientific processes. Finally, by utilizing
numerous numerical techniques, the student will begin to visualize, hence better understand, what a PDE is and how it can be used to study the
Natural Sciences.
Textbooks
 Advanced Engineering Mathematics (required), Erwin Kreyszig, Wiley, 1998.
 Elementary Applied PDE's with Fourier Series and Boundary Value Problems (good reference), R. Haberman, PrenticeHall, 1997.
 Partial Differential Equations (good reference), S.J. Farlow, Wiley ,1982.
Mathematical concepts to be covered:
 Review of sequences and series
 Fourier Series
 Partial Differential Equations
 Applications of PDE's to physics, engineering and biology
 Complex variables
 Conformal mapping
 Numerical analysis with Matlab
Learning Objectives and Instructor Expectations
The objective of this course is to help young and established scientists and engineers to build the skill necessary to analyze equations that they
encounter in the real world. This learning will be done using homework and computer assignements as well as in class assignments that will be
done in groups and presented in class. Every week on Tuesday there will be a quiz or group project to be done in class. Also, time during each
class will be devoted to the discussion of homework problems. Class attendance and participation is expected and is factored into your final
grade.
Grading
Homework assignments will count as 30% of grade evaluation. Quizes and group projects will count for 20%. There will also be one midterm
worth 20% of the grade and a final (Fri Dec 21st, 10:30 – 12:30) that counts for 25%. The remaining 5% is student participation.
There will be no makeup quizzes; instead I will drop the lowest quiz score.
MATH 450. Advanced Mathematics for Engineers I.
Section 002.
Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS).
Credits: (4).
Course Homepage: No homepage submitted.
Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given, and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g. Math 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems; and derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.
Text: Advanced Engineering Mathematics, 8th edition Edward Kreyszig Wiley.
MATH 451. Advanced Calculus I.
Instructor(s):
Prerequisites & Distribution: Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.) and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, 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.
Text: Elementary Analysis, The Theory of Calculus, Kenneth Ross, SpringerVerlag.
MATH 454. Boundary Value Problems for Partial Differential Equations.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundaryvalue problems for secondorder linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the onedimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; and applications to linear inputoutput systems, analysis of data smoothing and filtering, signal processing, timeseries analysis, and spectral analysis. Both Math 455 and 554 cover many of the same topics but are very seldom offered. Math 454 is prerequisite to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556.
no textbook
MATH 463. Mathematical Modeling in Biology.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419; 286, 256, or 316. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~tjacks/Math463.html
It is widely anticipated that Biology and Biomedical science will be the premier sciences of the 21st century. The complexity of the biological
sciences makes interdisciplinary involvement essential and the increasing use of mathematics in biology is inevitable as biology becomes more
quantitative. Mathematical biology is a fast growing and exciting modern application of mathematics which has gained worldwide recognition. In this course, mathematical models that suggest possible mechanisms which may underlie specific biological processes are developed and analyzed. Another major emphasis of the course is illustrating how these models can be used to predict what may follow under currently
untested conditions. The course moves from classical to contemporary models at the population, organ, cellular, and molecular levels.
Textbook:
Mathematical Models in Biology, First Edition; L. EdelsteinKeshet;
McGrawHill Publishing; 1987.
Other references:
 Mathematical Physiology, J. Keener & J. Sneyd, Springer, 1998.
 Mathematical Biology, J.D. Murray ; Springer Verlag, 1989.
Biological Topics Include
 Single Species and Interacting Population Dynamics
 Modeling Infectious and dynamic diseases
 The Heart, Circulation, and Blood Cell Production
 Regulation of Cell function
 Molecular Interactions and ReceptorLigand Binding
 Biological oscillators: HodgkinHuxley theory of Nerve Membranes
 Intro to Reactiondiffusion and Biological Pattern Formation
Mathematical and Modeling Concepts
Include
 Derivation of Biological Models
 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
Learning Objectives and Instructor Expectations:
Although an interdisciplinary subject such as Mathematical Biology can be made rather difficult, I will attempt to present the course material in
as simple a manner as possible. A basic knowledge of differential equations is recommended; however, I will review in detail the mathematical
tool necessary to analyze the models we study. More theoretical aspects, such as proofs, will not be presented. Biological applications will be
emphasized although no previous knowledge of biology is assumed. With each topic discussed I give a brief description of the biological
background sufficient to understand, develop, and study the models of interest. Upon completion of this course, students will have a working
knowledge of how mathematics and biology can be combined to enhance both fields.
Grading: Homework assignments will count as 1/4 of the grade evaluation. There will also be two exams which will count as 1/4 each. The final 1/4 of the
grade rests upon the completion of a substantial research paper describing a modeling project chosen with my assistance. An inclass
presentation of the project is also required.
Computer Lab:
There will be a MATLAB based computer component of the course. No prior knowledge of MATLAB is required. The computer lab will be
used to visualize solutions, and dynamic behavior of complex biological models.
MATH 471. Introduction to Numerical Methods.
Section 001.
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~hansjohn/m471.html
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for nonlinear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.
Text: An Introduction to Numerical Analysis, Kendall Atkinson Wiley.
MATH 471. Introduction to Numerical Methods.
Section 002.
Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one highlevel computer language. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/471f01/main.html
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for nonlinear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students; the sequence Math 571572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.
Text: An Introduction to Numerical Analysis, Kendall Atkinson Wiley.
MATH 481. Introduction to Mathematical Logic.
Section 001.
Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and firstorder logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite. In the first third of the course the notion of a formal language is introduced and propositional connectives (and, or, not, implies), tautologies, and tautological consequence are studied. The heart of the course is the study of firstorder predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including nonstandard analysis. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.
Text: An Introduction to Mathematical Logic by Richard E. Hodel, PWS Publishing Co. 1995.
Grading: 25% homework, 30% midterm exam (Thursday 26 October 78:30), 45% final exam.
MATH 485. Mathematics for Elementary School Teachers and Supervisors.
Section 001.
Instructor(s): Eugene F Krause
Prerequisites & Distribution: One year of high school algebra. No credit granted to those who have completed or are enrolled in Math. 385. (3). (Excl). (BS). May not be included in a concentration plan in mathematics.
Credits: (3; 2 in the halfterm).
Course Homepage: No homepage submitted.
The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching or preparing to teach in the elementary school.
MATH 497. Topics in Elementary Mathematics.
Section 001 – Topic?
Instructor(s): Eugene F Krause
Prerequisites & Distribution: Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.
Credits: (3).
Course Homepage: No homepage submitted.
No Description Provided.
Check Times, Location, and Availability
MATH 497. Topics in Elementary Mathematics.
Section 531.
Instructor(s):
Prerequisites & Distribution: Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits.
Credits: (3).
Course Homepage: No homepage submitted.
This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required twocourse sequence Math 385489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problemsolving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two onehour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and inservice elementary, middle, or juniorhigh school teachers. Content will vary from term to term.
MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.
Section 001.
Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.
Credits: (1).
Course Homepage: http://www.math.lsa.umich.edu/seminars/applied/index.html
The Applied and Interdisciplinary Mathematics(AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from UM and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. Math 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.
Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.
MATH 513. Introduction to Linear Algebra.
Section 001.
Prerequisites & Distribution: Math. 412. Two credits granted to those who have completed Math. 214, 217, 417, or 419. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math513.html
Math 513 is the Math Department's most complete and rigorous course in linear algebra. We will study in depth vector spaces and linear transformations over arbitrary fields. We will also cover bilinear and (elementary) quadratic forms and applications to differential equations. Significant applications will be an important feature of the course.
Grading: Weekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%.
Text: Linear Algebra, an Introductory Approach by Curtis, Springer Verlag.
MATH 520. Life Contingencies I.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 424 and Math. 425. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multidecrement and multiplelife applications directly related to life insurance and pensions. The sequence 520521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc. Recommended text: Actuarial Mathematics (Second Editions) by Bowles et al.
MATH 523. Risk Theory.
Section 001.
Prerequisites & Distribution: Math. 425. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html
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.
Contents: 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 homework assignments. Each quiz will consist of a slightly modified homework problem.
8 quizzes= 8x10=80 points midterm= 60 points
final= 80 points
Total= 220 points
Required Text:
Loss Modelsfrom Data to Decisions by Klugman, Panjer and Willmot, Wiley 1998.
MATH 524. Topics in Actuarial Science II.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 424, 425, and 520; and Stats. 426. (3). (Excl). (BS). May be repeated for a total of 9 credits.
Credits: (3).
Course Homepage: No homepage submitted.
No Description Provided.
Check Times, Location, and Availability
MATH 525 / STATS 525. Probability Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510511 are natural sequels.
An Introduction to Probability Theory and Its Applications, 3rd edition, William Feller Wiley.
recommended – Introduction to Probability Theory Hoel, Port, Stone HoughtonMifflin.
MATH 532. Topics in Discrete and Applied Geometry.
Section 001 – Topic?
Instructor(s):
Prerequisites & Distribution: One of Math. 217, 417, 419 or 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
No Description Provided.
Check Times, Location, and Availability
MATH 537. Introduction to Differentiable Manifolds.
Section 001.
Prerequisites & Distribution: Math. 513 and 590. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course will be a painless introduction to differential topology and differential geometry, meaning the study of spaces and their curvatures. It is the first part of a twosemester sequence. The material in this course is crucial for students who wish to study differential geometry, topology, algebraic geometry, several complex variables, Lie groups and dynamical systems. It is also relevant for other branches of mathematics, such as partial differential equations. We'll start out by doing calculus on manifolds, introducing and using differential forms. We'll prove Stokes' theorem for compact oriented manifoldswithboundary. We'll also define the de Rham cohomology groups of a manifold and prove their basic properties. Then I'll spend some time on Morse theory. This theory shows how, given a generic function on a manifold, one obtains a decomposition of the manifold into simple building blocks called handles. Morse theory is a basic tool in topology and was used in Smale's famous proof of the Poincare conjecture in more than four dimensions, although we will not go into this. Finally, we'll cover some basic Riemannian geometry, including Riemannian metrics, LeviCivita connections, geodesics, and curvature. Homework assignments will be given periodically, with the frequency depending on whether or not we get a grader. There will also be a final exam.
The textbooks will be Differential Topology by Victor Guillemin and Alan Pollack, PrenticeHall, and Morse Theory by John Milnor, Princeton University Press. Math 591 or the equivalent is a prerequisite. I will assume a knowledge of differentiable manifold theory as covered in Sections 1.11.4 of the book by Guillemin and Pollack. The titles of these sections are "Definitions", "Derivatives and tangents", "The inverse function theorem and immersions" and "Submersions". If a prospective student has not seen this material before, it might be helpful to look at Chapter 1 of Guillemin and Pollack. I will review this material at the beginning of the academic term.
MATH 555. Introduction to Functions of a Complex Variable with Applications.
Section 001.
Prerequisites & Distribution: Math. 450 or 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~rauch/courses.html
This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, and applications. Evaluation of improper real integrals and fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.
Text: Complex Variables with Applications, 6th edition, Brown/Churchill, McGraw Hill.
MATH 556. Methods of Applied Mathematics I.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 217, 419, or 513; 451 and 555. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is an introduction to analytical methods for initial value problems and boundary value problems. This course should be useful to students in mathematics, physics, and engineering. We will begin with systems of ordinary differential equations. Next, we will study Fourier Series, SturmLiouville problems, and eigenfunction expansions. We will then move on to the Fourier Transform, RiemannLebesgue Lemma, inversion formula, the uncertainty principle and the sampling theorem. Next we will cover distributions, weak convergence, Fourier transforms of tempered distributions, weak solution of differential equations and Green's functions. We will study these topics within the context of the heat equation, wave equation, Schrodinger's equation, and Laplace's equation.
Text: Fourier Analysis and its Applications by G.B. Folland.
Grading: homework 60%, midterm 15%, final exam 25%. Homework is key in this class. You are expected to hand in carefully completed homework.
MATH 561 / IOE 510 / SMS 518. Linear Programming I.
Section 001.
Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: http://wwwpersonal.engin.umich.edu/~murty/510/index.html
Prerequisites: A course in linear or matrix algebra.
Background Required: Elementary matrix algebra(concept of linear
independence, bases, matrix inversion, pivotal methods for solving linear
equations), geometry of R^{n} including convex sets and affine
spaces.
Reference Books:
 K. G. Murty, Operations Research: Deterministic Optimization Models,
Prentice Hall, 1995.
 K. G. Murty, Linear Programming, Wiley, 1983.
 M.S. Bazaraa, J. J. Jarvis, and H. D. Shirali, Linear Programming
and Network Flows, Wiley, 1990.
 R. Saigal, Linear Programming: A Modern Integrated Analysis,
Kluwer, 1995.
 D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Athena, 1997.
 R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language
for Mathematical Programming Scientific Press, 1993.
Course Content:
 Linear Programming models and their various applications. Separable
piecewise linear convex function minimization problems, uses in
curve fitting and linear parameter estimation. Approaches for solving
multiobjective linear programming models, the Goal programming technique.
 What useful planning information can be derived from an LP model (marginal
values and their planning uses).
 Pivot operations on systems of linear equations, basic vectors, basic solutions, and bases. Brief review of the geometry of convex polyhedra.
 Duality and optimality conditions for LP.
 Revised primal and dual simplex methods for LP.
 Infeasibility analysis, marginal analysis, cost coefficient
and right hand side constant ranging, and other sensitivity analyses.
 Algorithm for transportation models.
 Bounded variable primal simplex method.
 Brief review of Interior point methods for LP.
Work:
 Weekly Homework Assignments.
 Midterm
 Final Exam
 Two Computational Projects to be solved using AMPL.
Approximate weights for determining final grade are: Homeworks (15%), Midterm (20%), Final Exam (50%), Computer Projects (15%).
MATH 562 / IOE 511 / AEROSP 577. Continuous Optimization Methods.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for nonEngineering students.
Credits: (3).
Lab Fee: CAEN lab access fee required for nonEngineering students.
Course Homepage: No homepage submitted.
Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasiNewtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.
MATH 565. Combinatorics and Graph Theory.
Section 001.
Instructor(s):
Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one prooforiented course. Graph Theory topics include Trees; k connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5Color Theorem; Kuratowski's Theorem; and the MatrixTree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q series, linear recurrences and rational generating functions, and Pólya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms.
MATH 571. Numerical Methods for Scientific Computing I.
Section 001.
Prerequisites & Distribution: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math571.html
This course is a rigorous introduction to numerical linear algebra with applications to 2point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, GaussSeidel iteration, the SOR method, an introduction to the multigrid method, and the conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; and numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.
Text:
Introduction to Numerical Linear Algebra & Optimisation Philippe Ciarlet Cambridge.
A Multigrid Funtional William Briggs SIAM.
Numerical Linear Algebra Trefethen & Bav SIAM.
MATH 575. Introduction to Theory of Numbers I.
Section 001.
Instructor(s): Christopher Skinner (cskinner@umich.edu)
Prerequisites & Distribution: Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (1, 3). (Excl). (BS).
Credits: (1, 3).
Course Homepage: No homepage submitted.
Math 575 is intended to provide graduate students with an introduction to number theory sufficient for continuing in the graduate program in number theory here at Michigan. As such, it will also provide a means for undergraduates interested in number theory to prepare for graduate study elsewhere. Graduate students not directly interested in number theory will be able to complete their distribution requirements while learning the mathematics behind such everyday applications as Public Key Cryptography.
The first half of the course will be a brisk (but thorough) discussion of the basic notions: Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, primitive roots, Public Key Cryptography, quadratic reciprocity, binary quadratic forms, and basic arithmetic functions. The second half of the course will be devoted to more advanced topics as time permits: diophantine equations, quadratic fields, padic numbers, diophantine approximation, arithmetic functions, continued fractions, distribution of prime numbers.
Grading: A homework assignment every two weeks, 2 inclass exams, and a takehome final exam.
Text: Introduction to the Theory of Numbers, 5th edition by Niven, Zuckerman and Montgomery.
MATH 590. Introduction to Topology.
Section 001.
Prerequisites & Distribution: Math. 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
Topology provides a foundational framework for geometry, including algebraic and differential topology. This course is an introduction to the subject and will emphasize the construction of proofs.
We will begin with metric spaces, abstract topological spaces, continuous functions, and the properties of connectedness, compactness and separability, then move on to the more geometric notions of homotopy, covering spaces and the fundamental group. Additional topics may include triangulations and the classification of surfaces. There will be problem sets, a midterm and a final exam.
Text: (required) Munkres. Topology: a first course. PrenticeHall, 2nd ed., 2000.
recommended) Armstrong. Basic Topology. UTM, SpringerVerlag, 1983.
Additional readings and handouts as appropriate.
MATH 591. General and Differential Topology.
Section 001.
Prerequisites & Distribution: Math. 451. (3). (Excl). (BS).
Credits: (3).
Course Homepage: http://www.math.lsa.umich.edu/~spatzier/591/591.html
Text: required: Topology by Munkres, 2nd edition, Prentice Hall optional: Differential Topology, Guillemin/Pollack, Prentice Hall
Prerequisites: Familiarity with undergraduate analysis is recommended
Course Outline: This course is an introduction to point set and differential topology. Specifically, we will introduce abstract topological spaces and their basic properties. Then we will discuss the properties of connectedness and compactness. We will construct new topological spaces from old ones, such as subspaces, products and quotients, group actions and orbit spaces. Finally, we will introduce manifolds and differential topology, in particular tangent spaces, the regular value theorem, Whitney's embedding theorem and transversality. Students with a strong background in pointset and differential topology may want to consider taking Math 537 instead. Math 591 will cover the prerequisites for the portions of the topology QR which concern point set topology and differential topology.
Problem Session: TBA
Grading Policy: homework 40%; midterm 30%; final exam 30%.
MATH 593. Algebra I.
Section 001.
Instructor(s): Robert Lazarsfeld (rlaz@umich.edu)
Prerequisites & Distribution: Math. 513. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
Math 593 is the first half of the basic introductory graduate algebra course. It is designed to prepare students for the qualifying review, and to provide the foundation in algebra necessary for graduate work in mathematics.
Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras.
Grading: Homework – possibly done in groups – will be assigned and collected regularly. There will be two inclass hour exams, and a final.
MATH 596. Analysis I.
Section 001.
Prerequisites & Distribution: Math. 451. Students with credit for Math. 555 may elect Math. 596 for two credits only. (3). (Excl). (BS).
Credits: (3).
Course Homepage: No homepage submitted.
This is the standard beginning course in complex analysis with standard content (as described in the Department's web page, for example).
Grading: There will be regular homework assignments, midterm exam (possibly two), and a final.
Text: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable by Lars Valerian Ahlfors, (3rd Edition), McGrawHill Higher Education.
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