96-97 LS&A Bulletin

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).