Mathematics
2072 East Hall
764-0337
Professor B. A. Taylor, Chair
May be elected as a departmental concentration program
Professors
David E. Barrett, Several Complex Variables
Andreas R. Blass, Logic, Set Theory, Category Theory, Computational Complexity,
Combinatorics
Morton Brown, Topology
Daniel M. Burns, Jr., Complex Analysis, Algebraic and Differential Geometry
Joseph G. Conlon, Mathematical Physics, Applied Mathematics
Igor V. Dolgachev, Algebraic Geometry
Peter L. Duren, Real and Complex Analysis, Univalent Functions, Harmonic
Analysis, Probability
Paul G. Federbush, Rigorous Quantum Field Theory and Statistical Mechanics
John Erik Fornaess, Several Complex Variables, Analysis
Robert L. Griess, Jr., Finite Group Theory, Group Extension Theory, Simple
Groups
Philip J. Hanlon, Combinatorics
Donald G. Higman, Group Theory, Algebraic Combinatorics
Peter G. Hinman, Mathematical Logic, Recursion Theory, Foundations of
Mathematics, Computational Complexity
Melvin Hochster (R.W. and L.H. Browne Professor of Science), Commutative
Algebra, Algebraic Geometry
Curtis Huntington, Actuarial Science
James M. Kister, Geometric Topology, Transformation Groups
Eugene F. Krause, Mathematics Education
Donald J. Lewis, Diophantine Equations, Algebraic Numbers and Function
Fields
James S. Milne, Algebraic Geometry and Number Theory
Hugh L. Montgomery, Number Theory, Distribution of Prime Numbers, Fourier
Analysis, Analytic Inequalities, Probability
Gopal Prasad, Representation Theory
M. S. Ramanujan, Functional Analysis, Nuclear Spaces
Jeffrey B. Rauch, Partial Differential Equations
Frank A. Raymond, Topology, Transformation Groups
G. Peter Scott, Geometric Topology, Combinatorial Group Theory
Carl P. Simon, Dynamical Systems, Singularity Theory, Mathematical Economics,
Mathematical Epidemiology, Applied Mathematics
Joel A. Smoller, Nonlinear Partial Differential Equations
Ralf J. Spatzier, Differential Geometry
J. Tobias Stafford, Noetherian Rings, Lie Algebras, Algebraic K-theory,
Rings of Differential Operators
John R. Stembridge, Algebraic Combinatorics
Thomas F. Storer, Combinatorics
B. Alan Taylor, Complex Analysis
Alejandro Uribe-Ahumada, Global Analysis
Arthur G. Wasserman, Differential Topology, Transformation Groups, Foliations,
Applied Mathematics
Michael I. Weinstein, Nonlinear Partial Differential Equations
David J. Winter, Algebra, Lie Algebras, Algebraic Groups
Michael B. Woodroofe, Probability Theory, Mathematical Statistics
Associate Professors
Anthony M. Bloch, Geometric Mechanics, Nonlinear Control Theory
Christoph Borgers, Numerical Solution of Partial Differential Equations
Jack L. Goldberg, Special Functions, Linear Algebra
Thomas Hales, Lie Theory
Eduard Harabetian, Partial Differential Equations, Numerical Analysis
Juha Heinonen, Geometric Function Theory
Robert Krasny, Partial Differential Equations, Fluid Dynamics
Igor Kriz, Homotopy Theory
John W. Lott, Differential Geometry, Mathematical Physics
Robert Megginson, Geometry of Banach Spaces
Allen Moy, Representation Theoy
Art J. Schwartz, Analysis, Computer Algebra
Chung-Tuo Shih, Probability Theory
Berit Stensones, Several Complex Variables
Trevor Wooley, Number Theory
Assistant Professors
John N. Aarsvold, Applied Analysis
Alexandre Barvinok, Combinatorics, Optimization
Michael Bean, Number Theory
Michael Bennett, Number Theory, Diophantine Approximation
Richard Canary, Topology
Timothy Chow, Geometry, Analysis
Donatella Delfino, Commutative Algebra
Vesselin Gasharov, Combinatorics
Rita Gitik, Topology
Mark Gockenbach, Applied Mathematics
Lizhen Ji, Geometry, Analysis
Richard Jordan, Applied Mathematics, Fluid Mechanics
Ruth Lawrence, Topology
Liviu Nicolaescu, Global Analysis
Mingqing Ouyang, Geometry, Topology
Guillaume Sanje-Mpacko, Group Theory, Hecke Algebras
Nikolai Savaliev, Geometry, Topology
Peter Smereka, Bubbly Fluids
Renming Song, Mathematical Physics
Joanna Staniszkis, Noncommutative Algebra
Edward Taylor, Geometry, Kleinian Groups
Siu-Kei Tin, Dynamical Systems
Lan Wang, Number Theory
Caryn Werner, Algebraic Geometry
Glen Whitney, Logic
T.H. Hildebrandt Research Assistant Professors
Jeremy D. Avigad, Proof Theory, Logic
Ioannis Emmanouil, Algebraic Geometry
Timothy Hsu, Group Theory
Alexandru Nica, Operator Algebra
Tonghai Yang, Number Theory, Algebraic Geometry
Adjunct Professors
Howard Young, Employee Benefits, Actuarial Science
Lecturer
Patricia Shure, Mathematics Education
Professors Emeriti Robert C. F. Bartels, Douglas G. Dickson,
Frederick W. Gehring (T.H.Hildebrandt Distinguished University Professor
of Mathematics), Frank Harary, George E. Hay, Donald A. Jones, Phillip S.
Jones , Wilfred Kaplan, Wilfred M. Kincaid, Chung-Nim Lee, Jack E. McLaughlin,
Cecil J. Nesbitt, Carl M. Pearcy, George Piranian, Maxwell O. Reade, Ronald
H. Rosen, Charles J. Titus, and James G. Wendel.
Mathematics is sometimes called the Queen of the Sciences; because of its
unforgiving insistence on accuracy and rigor it is a model for all of science.
It is a field which serves science but also stands on its own as one of
the greatest edifices of human thought. Much more than a collection of calculations,
it is finally a system for the analysis of form. Alone among the sciences,
it is a discipline where almost every fact can and must be proved.
The study of mathematics is an excellent preparation for many careers; the
patterns of careful logical reasoning and analytical problem solving essential
to mathematics are also applicable in contexts where quantity and measurement
play only minor roles. Thus students of mathematics may go on to excel in
medicine, law, politics, or business as well as any of a vast range of scientific
careers. Special programs are offered for those interested in teaching mathematics
at the elementary or high school level or in actuarial mathematics, the
mathematics of insurance. The other programs split between those which emphasize
mathematics as an independent discipline and those which favor the application
of mathematical tools to problems in other fields. There is considerable
overlap here, and any of these programs may serve as preparation for either
further study in a variety of academic disciplines, including mathematics
itself, or intellectually challenging careers in a wide variety of corporate
and governmental settings.
Elementary 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 upper-level 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 half-term version of the same
material offered as a self-study 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 half-term for students in
the Summer Bridge Program.
Math 127, 128, and 147 are courses containing selected topics from geometry
number theory, and financial mathematics. 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, 128, or 147 if
a student has already received credit for a 200(or higher)-level mathematics
course.
Each of Math 112, 115, 185, and 295 is a first course in calculus and generally
credit can be received for only one course from this list. Math 112 is designed
for students of business and the social sciences who require only one term
of calculus. It neither presupposes nor covers any trigonometry. The sequence
115-116-215 is appropriate for most students who want a complete introduction
to calculus. Math 118 is an alternative to Math 116 intended for students
of the social sciences who do not intend to continue to Math 215. One of
Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics.
Math 112 does not provide preparation for any subsequent course.
Students planning a career in medicine should note that some medical schools
require a course in calculus. Generally either Math 112 or 115 will satisfy
this requirement, although most science concentrations require at least
a year of calculus. Math 112 is accepted by the School of Business Administration,
but Math 115 is prerequisite to concentration in Economics and further math
courses are strongly recommended.
The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396
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&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 185-285 covers much of the material of Math 115-215 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 175-176
assumes a knowledge of calculus roughly equivalent to Math 115 and covers
a substantial amount of so-called combinatorial mathematics (see course
description) as well as calculus-related 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 295-396 provides
a rigorous introduction to theoretical mathematics. Proofs are stressed
over applications and these courses require a high level of interest and
commitment. 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 semester
of AP credit, Math 119 and Math 156 (Fall). Both will review the basic concepts
of calculus, cover integration and an introduction to differential equations,
and introduce the student to the use of the computer algebra system MAPLE.
Math 119 will stress experimentation and computation, while Math 156 is
intended primarily for engineering and science majors, and will emphasize
both applications and theory. Interested students are advised to 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 one of Math 112
or 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 a advisor before switching
from one sequence to another. In all cases a maximum total of 16 credits
may be earned for calculus courses Math 112 through Math 296, and no credit
can be earned for a prerequisite to a course taken after the course itself.
Students completing Math 215 may continue either to Math 216 (Introduction
to Differential Equations) or to the sequence Math 217-316 (Linear Algebra-Differential
Equations). Math 217-316 is required for all students who intend
to take more advanced courses in mathematics, particularly for those who
may concentrate in mathematics. Math 217 both serves as a transition to
the more theoretical material of advanced courses and provides the background
required for optimal treatment of differential equations.
Prerequisites to Concentration. Most programs require completion
of one of the sequences ending with Math 215-217, 285-217, or 395-396. A
working knowledge of a high-level computer language such as FORTRAN or C
or a computer algebra system (such as Math 403), at a level equivalent to
completion of a course of three or more credits; and eight credits of Physics,
preferably Physics 140/141 and 240/241, are recommended for all programs
and required for some. For detailed requirements consult the brochure
Undergraduate Programs available from the Undergraduate Mathematics
Program Office (UMPO), 2082 East Hall, (734) 763-4223.
Concentration Programs. A student considering concentration in mathematics
should consult a mathematics concentration advisor in the UMPO as early
as possible and certainly by the first term of the sophomore year. The Department
offers many different concentration programs with varying requirements;
failure to meet some of these at the intended time may delay completion
of the program and graduation. A concentration plan must be designed with
and approved by a concentration advisor. The departmental brochure Undergraduate
Programs should be regarded as the most comprehensive and up-to-date
guide to the options and requirements for concentration programs in mathematics.
All the information in that brochure and much more is available online via
the World Wide Web. From the department's home page at: http://www.math.lsa.umich.edu
select the item "Degree Programs and Faculty."
Pure Mathematics
(Students should consult the pamphlet Undergraduate Programs of the Department
of Mathematics for its program requirements which take precedence over
the descriptions in this Bulletin.)
a. Four basic courses (one course from each of the following four groups):
Modern Algebra: Math 412 or 512
Differential Equations: Math 286 or 316
Analysis: Math 451
Geometry/Topology: Math 432, 433, 490, 531 or 590
b. Four elective courses (mathematics) chosen from a list of approved electives
and approved by a concentration advisor.
c. One cognate course outside the Mathematics Department, but having advanced
mathematical content.
Mathematical Sciences Program
(Students should consult the pamphlet Undergraduate Programs of the Department
of Mathematics for its program requirements which take precedence over
the descriptions in this Bulletin.)
Additional prerequisite: one term of computer programming (EECS 183 or the
equivalent), and for the Numerical and Applied Analysis option, 8 credits
of physics.
a. Four basic courses (one course from each of the following four groups):
Differential Equations: Math 286 or 316
Discrete Math/Modern Algebra: Math 312, 412 or 512
Analysis: Math 450 or 451
Probability: Math 425 or 525
b. At least three courses from ONE of the Program Options listed below (the
list of possible electives for each option is given in the Undergraduate
Programs pamphlet described above):
Discrete and Algorithmic Methods
Numerical and Applied Analysis
Operations Research and Modelling
Probabilistic Methods
Mathematics of Finance and Risk Management
Mathematical Economics
Mathematical Physics
Control Systems
c. Two additional advanced mathematics (or related) courses, approved by
a concentration advisor.
Honors Concentration
Outstanding students may elect an Honors concentration in Mathematics. The
Honors Program is designed not only for students who expect to become mathematicians
but also for students whose ultimate professional goal lies in the humanities,
law, medicine or the sciences.
Students intending an Honors concentration are strongly advised to take
one of the Honors introductory calculus sequences (175 or 185)-286 or 295-396,
or some combination of the two. Eight credits of physics and familiarity
with a high-level computer language are strongly recommended.
(Students should consult the pamphlet Undergraduate Programs of the Department
of Mathematics for its program requirements which take precedence over
the descriptions in this Bulletin.)
a. Four basic courses (one course from each of the following four groups):
Linear Algebra: Math 513
Modern Algebra: Math 512
Analysis: Math 451
Geometry/Topology: Math 433, 490, 590 or 531
b. Four elective (mathematics) courses, chosen with the approval of the
Honors advisor.
c. One cognate course from outside the Mathematics department, but containing
significant mathematical content, chosen with the approval of the Honors
advisor.
Students who, in the judgment of the departmental Honors committee, have
completed an Honors concentration with distinction are granted a citation
upon graduating. Interested students should discuss their program and the
specific requirements for obtaining the citation with a Mathematics Honors
advisor (appointments scheduled in 2082 East Hall) no later than the second
term of their sophomore year.
Actuarial Mathematics
(Students should consult the pamphlet Undergraduate Programs of the Department
of Mathematics for its program requirements which take precedence over
the descriptions in this Bulletin.)
Additional prerequisite: At least one course in each of the following fields:
Accounting (271, 272, 471), Computer Science (183, 280), and Economics (201,
202, 400).
a. Five basic courses (one from each of the following five groups):
1. Differential Equations: Math 286 or 316
2. Probability: Math 425 or 525
3. Analysis: Math 450 or 451
4. Statistics: Stat 426
5. Numerical Analysis: Math 371 or 471 (preferred)
b. Three special actuarial courses, including Math 424 and 520,
and one of Math 521 or 522.
c. Two additional courses in areas relating to Actuarial Science, approved
by an advisor.
Teaching Certificate
It is essential that students planning to obtain a teaching certificate
consult a teaching certificate advisor, either Professor Krause (LS&A) or
Professor Coxford (Education), prior to beginning their concentration
program.
Additional prerequisite: One term of computer programming, EECS 183 or the
equivalent.
(Students should consult the pamphlet Undergraduate Programs of the Department
of Mathematics for its program requirements which take precedence over
the descriptions in this Bulletin.)
a. Four basic courses, one from each of the following four groups (chosen
with the approval of a teaching certificate advisor):
1. Discrete Math/Modern Algebra: Math 312, 412, or 512
2. Geometry: Math 431, 432 or 531
3. Probability: Math 425 or 525
4. Secondary Mathematics: Math 486
b. Seven specific education courses, totaling 28 credits. Consult the Undergraduate
Programs pamphlet for the list of courses.
c. A major or minor in a second academic area (normally requires 20-24 credits
in a structured program other than Mathematics. Consult the Bulletin
of the School of Education for acceptable programs).
d. Two additional courses, which must include a course in the Psychology
Department, and a minimum of one additional mathematics course.
Students should consult with Professor Coxford in their sophomore year to
be admitted to the certification program and to schedule practice teaching.
Advising. Appointments are scheduled at the Undergraduate Mathematics
Program Office, 2082 East Hall. Students are strongly urged to consult with
a concentration advisor each term before selecting courses for the following
term.
Foreign Languages. The language requirement of the A.B. or B.S. degrees
with concentration in mathematics may be satisfied in any of the languages
acceptable to the College. However, students planning to do graduate work
in mathematics should be aware that at most universities one of the requirements
for a Ph.D. degree is a demonstration of the ability to read mathematical
texts in two of the three languages French, German, and Russian.
Special Departmental Policies. All prerequisite courses must be satisfied
with a grade of C- or above. Students with lower grades in prerequisite
courses must receive special permission of the instructor to enroll in subsequent
courses.
William Lowell Putnam Competition. A departmental team participates
in the annual William Lowell Putnam Competition sponsored by the Mathematical
Association of America. Interested students with exceptional mathematical
aptitude are asked to contact the department office for detailed information.
The department also sponsors other competitions and activities.
Summer Research. The department has opportunities for a limited number
of undergraduate students to pursue on-site summer research under the auspices
of the Research Experience for Undergraduates (REU) program. Students pursue
a 7-8 week summer research project under the mentorship of regular departmental
faculty, and are paid a stipend for this work. Contact the UMPO for further
details.
Courses in Mathematics (Division 428)
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.
103. Intermediate Algebra. Only open to designated summer half-term
Bridge students. IIIb. (2 in the half-term). (Excl).
105. Data, Functions, and Graphs. Students with credit for
Math. 103 can elect Math. 105 for only 2 credits. (4). (Excl). (QR/1).
110. Pre-Calculus (Self-Study). See Elementary Courses
above. No credit granted to those who already have 4 credits for pre-calculus
mathematics courses. (2). (Excl).
112. Brief Calculus. See Elementary Courses above.
Credit is granted for only one course from among Math. 112, 113, 115, 185
and 295. (4). (N.Excl). (BS).
115. Calculus I. 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. (4). (N.Excl). (BS). (QR/1).
116. Calculus II. Math. 115. Credit is granted for only one
course from among Math. 116, 119, 186, and 296. (4). (N.Excl). (BS). (QR/2).
119. Calculus II Using MAPLE. Math. 115 or equivalent. Credit
is granted for only one course from among Math. 114, 116, 119, 186, and
296. (4). (Excl).
127. Geometry and the Imagination. Three years of high school
mathematics including a geometry course. No credit granted to those who
have completed a 200- (or higher) level mathematics course. (4). (NS). (BS).
(QR/1).
128. Explorations in Number Theory. High school mathematics
through at least Analytic Geometry. No credit granted to those who have
completed a 200- (or higher) level mathematics course. (4). (NS). (BS).
(QR/1).
147. Introduction to Interest Theory. Math. 112 or 115. No
credit granted to those who have completed a 200- (or higher) level mathematics
course. (3). (Excl). (BS).
156. Applied Honors Calculus II. Score of 4 or 5 on the AB
or BC Advanced Placement calculus exam. (4). (Excl).
175. Combinatorics and Calculus. Permission of Honors advisor.
(4). (N.Excl). (BS). (QR/1).
176. Dynamical Systems and Calculus. Math. 175 or permission
of instructor. (4). (N.Excl). (BS).
185. Honors Analytic Geometry and Calculus I. Permission of
the Honors advisor. Credit is granted for only one course from among Math.
112, 113, 115, 185, and 295. (4). (N.Excl). (BS). (QR/1).
186. Honors Analytic Geometry and Calculus II. Permission
of the Honors advisor. Credit is granted for only one course from among
Math. 114, 116, 119, 186, and 296. (4). (N.Excl). (BS). (QR/1).
203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent
enrollment in one term of calculus. No programming experience is assumed.
(1). (Excl).
215. Calculus III. Math. 116 or 186. (4). (Excl). (BS). (QR/1).
216. Introduction to Differential Equations. Math. 215. (4).
(Excl). (BS).
217. Linear Algebra. Math. 215. No credit granted to those
who have completed or are enrolled in Math. 417, 419, or 513. (4). (Excl).
(BS). (QR/1).
219. Calculus III Using MAPLE. Math. 119. (4). (Excl).
255. Applied Honors Calculus III. Math 156. (4). (Excl).
256. Applied Honors Calculus IV. Math 255. (3). (Excl).
285. Honors Analytic Geometry and Calculus III. Math. 186
or permission of the Honors advisor. (4). (Excl). (BS).
286. Honors Differential Equations. Math. 285. (3). (Excl).
(BS).
288. Math Modeling Workshop. Math. 216 or 316, and Math. 217
or 417. (1). (Excl). (BS). Offered mandatory credit/no credit. May be elected
for a total of 3 credits.
289. Problem Seminar. (1). (Excl). (BS). May be repeated for
credit with permission.
295(195). Honors Mathematics I. 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). (N.Excl).
(BS). (QR/1).
296(196). Honors Mathematics II. Prior knowledge of first
year calculus and permission of the Honors advisor. Credit is granted for
only one course from among Math. 116, 119, 186, and 296. (4). (N.Excl).
(BS). (QR/1).
312. Applied Modern Algebra. Math. 217. (3). (Excl). (BS).
316. Differential Equations. Math. 215 and 217, or equivalent.
Credit can be received for only one of Math. 216 or Math. 316, and credit
can be received for only one of Math. 316 or Math. 404. (3). (Excl). (BS).
333. Directed Tutoring. Math. 385 and enrollment in the Elementary
Program in the School of Education. (1-3). (Excl). (EXPERIENTIAL). May be
repeated for a total of three credits.
350/Aero. 350. Aerospace Engineering Analysis. Math. 216 or
316 or the equivalent. (3). (Excl). (BS).
354. Fourier Analysis and its Applications. Math. 216, 316,
or 286. No credit granted to those who have completed or are enrolled in
Math. 454. (3). (Excl). (BS).
362. Applications of Calculus and Linear Algebra. Math. 216
or 217. (3). (Excl). (BS).
371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering
103 or 104, or equivalent; and Math. 216. I and II. (3). (Excl). (BS).
385. Mathematics for Elementary School Teachers. One year
each of high school algebra and geometry. No credit granted to those who
have completed or are enrolled in 485. (3). (Excl).
395(295). Honors Analysis I. Math. 296 or permission of the
Honors advisor. (4). (Excl). (BS).
396(296). Honors Analysis II. Math. 395. (4). (Excl). (BS).
399. Independent Reading. (1-6). (Excl). (INDEPENDENT). May
be repeated for credit.
403. Mathematical Modeling Using Computer Algebra Systems. Math.
116 and junior standing. (3). (Excl). (QR/1).
404. Intermediate Differential Equations. Math. 216. No credit
granted to those who have completed Math. 286 or 316. (3). (Excl). (BS).
412. Introduction to Modern Algebra. Math. 215 or 285; and
217. No credit granted to those who have completed or are enrolled in 512.
Students with credit for 312 should take 512 rather than 412. One credit
granted to those who have completed 312. (3). (Excl). (BS).
413. Calculus for Social Scientists. Not open to freshmen,
sophomores or mathematics concentrators. (3). (Excl). (BS).
416. Theory of Algorithms. Math. 312 or 412 or CS 303, and
CS 380. (3). (Excl). (BS).
417. Matrix Algebra I. Three courses beyond Math. 110. No
credit granted to those who have completed or are enrolled in 217, 419,
or 513. (3). (Excl). (BS).
419/EECS 400/CS 400. Linear Spaces and Matrix Theory. Four
terms of college mathematics beyond Math 110. No credit granted to those
who have completed or are enrolled in 217 or 513. One credit granted to
those who have completed Math. 417. I and II. (3). (Excl). (BS).
420. Matrix Algebra II. Math. 217, 417 or 419. (3). (Excl).
(BS).
422. Topics in Actuarial Mathematics I. Math. 216 or permission
of instructor. (3). (Excl). (BS).
423. Mathematics of Finance. Math. 217 and 425; CS 183. (3).
(Excl). (BS).
424. Compound Interest and Life Insurance. Math. 215 or permission
of instructor. (3). (Excl). (BS).
425/Stat. 425. Introduction to Probability. Math. 215. (3).
(N.Excl). (BS).
427/Social Work 603. Retirement Plans and Other Employee Benefit
Plans. Junior standing. (3). (Excl).
431. Topics in Geometry for Teachers. Math. 215. (3). (Excl).
(BS).
433. Introduction to Differential Geometry. Math. 215. (3).
(Excl). (BS).
450. Advanced Mathematics for Engineers I. Math. 216, 286,
or 316. (4). (Excl). (BS).
451. Advanced Calculus I. Math. 215 and one course beyond
Math. 215; or Math. 285. Intended for concentrators; other students should
elect Math. 450. (3). (Excl). (BS).
452. Advanced Calculus II. Math. 217, 417, or 419; and Math.
451. (3). (Excl). (BS).
454. Boundary Value Problems for Partial Differential Equations.
Math. 216, 286 or 316. Students with credit for Math. 354, 455 or
554 can elect Math. 454 for 1 credit. (3). (Excl). (BS).
455. Boundary-Value Problems and Complex Variables. Math.
450. Intended primarily for undergraduates; graduate students by permission
of advisor. No credit granted to those who have completed 454 or 555. (4).
(Excl). (BS).
462. Mathematical Models. Math. 216, 286 or 316; and 217,
417, or 419. Students with credit for 362 must have department permission
to elect 462. (3). (Excl). (BS).
471. Introduction to Numerical Methods. Math. 216, 286, or
316; and 217, 417, or 419; and a working knowledge of one high-level computer
language. (3). (Excl). (BS).
475. Elementary Number Theory. (3). (Excl). (BS).
476. Computational Laboratory in Number Theory. Prior or concurrent
enrollment in Math. 475 or 575. (1). (Excl). (BS).
481. Introduction to Mathematical Logic. Math. 412 or 451
or equivalent experience with abstract mathematics. (3). (Excl). (BS).
485. Mathematics for Elementary School Teachers and Supervisors.
One year of high school algebra or permission of instructor. No credit
granted to those who have completed or are enrolled in 385. May not be included
in a concentration plan in mathematics. I and IIIb. (3; 2 in the half-term).
(Excl). (BS).
486. Concepts Basic to Secondary Mathematics. Math. 215. (3).
(Excl). (BS).
489. Mathematics for Elementary and Middle School Teachers. Math.
385 or 485, or permission of instructor. May not be used in any graduate
program in mathematics. (3). (Excl).
490. Introduction to Topology. Math. 412 or 451 or equivalent
experience with abstract mathematics. (3). (Excl). (BS).
497. Topics in Elementary Mathematics. Math. 489 or permission
of instructor. (3). (Excl). (BS). May be repeated for a total of six credits.
512. Algebraic Structures. Math. 451 or 513 or permission
of the instructor. No credit granted to those who have completed or are
enrolled in 412. Math. 512 requires more mathematical maturity than Math.
412. (3). (Excl). (BS).
513. Introduction to Linear Algebra. Math. 412 or permission
of instructor. Two credits granted to those who have completed Math. 417;
one credit granted to those who have completed Math 217 or 419. (3). (Excl).
(BS).
520. Life Contingencies I. Math. 424 and Math. 425; or permission
of instructor. (3). (Excl). (BS).
521. Life Contingencies II. Math. 520; or permission of instructor.
(3). (Excl). (BS).
523. Risk Theory. Math. 425. (3). (Excl). (BS).
524. Mortality Studies. Math. 520; or permission of instructor.
(3). (Excl). (BS).
525/Stat. 525. Probability Theory. Math. 450 or 451; or permission
of instructor. Students with credit for Math. 425/Stat. 425 can elect Math.
525/Stat. 525 for only 1 credit. (3). (Excl). (BS).
526/Stat. 526. Discrete State Stochastic Processes. Math.
525 or EECS 501. (3). (Excl). (BS).
531. Transformation Groups in Geometry. Math. 215. (3). (Excl).
(BS).
533. Geometric Algebra. Math. 513. I. (3). (Excl). (BS).
535. Introduction to Algebraic Curves. Math. 513. (3). (Excl).
(BS).
537. Introduction to Differentiable Manifolds. Math. 513 and
590. (3). (Excl). (BS).
555. Introduction to Functions of a Complex Variable with Applications.
Math. 450 or 451. Students with credit for Math. 455 or 554 can elect
Math. 555 for one hour credit. (3). (Excl). (BS).
556. Methods of Applied Mathematics I. Math. 555 or 554. (3).
(Excl). (BS).
557. Methods of Applied Mathematics II. Math. 556. (3). (Excl).
(BS).
558(658). Ordinary Differential Equations. Math. 450 or 451.
(3). (Excl). (BS).
559. Selected Topics in Applied Mathematics. Math. 451 and
419, or equivalent. (3). (Excl). (BS). May be elected for a total of 6 credits.
561/SMS 518 (Business Administration)/IOE 510. Linear Programming
I. Math. 217, 417, or 419. (3). (Excl). (BS).
562/IOE 511/Aero. 577/EECS 505/CS 505. Continuous Optimization Methods.
Math. 217, 417 or 419. I. (3). (Excl). (BS).
565. Combinatorics and Graph Theory. Math. 412 or 451 or equivalent
experience with abstract mathematics. (3). (Excl). (BS).
566. Combinatorial Theory. Math. 216, 286 or 316; or permission
of instructor. (3). (Excl). (BS).
571. Numerical Methods for Scientific Computing I. Math. 217,
419, or 513; and 454 or permission of instructor. (3). (Excl). (BS).
572. Numerical Methods for Scientific Computing II. Math.
217, 419, or 513; and 454 or permission of instructor. (3). (Excl). (BS).
575. Introduction to Theory of Numbers I. Math. 451 and 513;
or permission of instructor. Students with credit for Math. 475 can elect
Math. 575 for 1 credit. (3). (Excl). (BS).
582. Introduction to Set Theory. Math. 412 or 451 or equivalent
experience with abstract mathematics. (3). (Excl). (BS).
590. Introduction to Topology. Math. 451. (3). (Excl). (BS).
591. General and Differential Topology. Math. 451. (3). (Excl).
(BS).
592. Introduction to Algebraic Topology. Math. 591. (3). (Excl).
(BS).
593. Algebra I. Math. 513. (3). (Excl). (BS).
594. Algebra II. Math. 593. (3). (Excl). (BS).
596. Analysis I. Math. 451. (3). (Excl). (BS). Students with
credit for Math. 555 may elect Math 596 for two credits only.
597. Analysis II. Math. 451 and 513. (3). (Excl). (BS).