3217 Angell 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*

Frederick W. Gehring, (T.H. Hildebrandt Distinguished University Professor of Mathematics) *Geometric Function Theory, Quasiconformal Mappings, Mobius Groups*

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 LH. Browne Professor of Science) *Commutative Algebra, Algebraic Geometry*

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*

J. Tobias Stafford, *Noetherian Rings, Lie Algebras, Algebraic K-theory, Rings of Differential Operators*

Thomas F. Storer, *Combinatorics*

B. Alan Taylor, *Complex 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*

Manoussos Grillakis, *Partial Differential Equations, Mathematical Physics*

Thomas Hales, *Lie Theory*

Eduard Harabetian, *Partial Differential Equations, Numerical Analysis*

Robert Krasny, *Partial Differential Equations, Fluid Dynamics*

John W. Lott, *Differential Geometry, Mathematical Physics*

Robert Megginson, *Geometry of Banach Spaces*

Allen Moy, *Representation Theory*

Art J. Schwartz, *Analysis, Computer Algebra*

Chung-Tuo Shih, *Probability Theory*

Ralf J. Spatzier, *Differential Geometry*

John R. Stembridge, *Algebraic Combinatorics*

Berit Stensones, *Several Complex Variables*

Alejandro Uribe-Ahumada, *Global Analysis*

**Assistant Professors**

Alexander Barvonik, *Optimization*

David C. Butler, *Algebraic Geometry*

Richard Canary, *Topology*

Carolyn A. Dean, *Noncommutative Algebra*

Estela A. Gavosto, *Several Complex Variables*

Anthony Giaquinto, *Algebra, Deformation Theory, Polynomial Algebras*

Juha Heinonen, *Geometric Function Theory*

Pekka J. Koskela, *Analysis, Potential Theory, Nonlinear PDE*

Ruth Lawrence, *Topology*

Shenglin Lu, *Mathematical Physics, Applied Mathematics*

Eyal Markman, *Algebraic Geometry*

Nigel Pitt, *Number Theory*

Rodolfo Torres, *Harmonic Analysis*

Zhenghan Wang, *Topology, Geometry, Dynamical Systems*

Trevor Wooley, *Number Theory*

**T.H. Hildebrandt Research Assistant Professors**

Neil Dummigan, *Number Theory*

Willian Jockusch, *Combinatorics*

Richard Laugesen, *Complex Analysis*

Irena Swanson*, Commutative Algebra*

**Adjunct Professors**

Curtis Huntington, *Actuarial Science*

Howard Young, *Employee Benefits, Actuarial Science*

**Lecturer**

Patricia Shure, *Mathematics Education*

**Professors Emeriti** Robert C. F. Bartels, Douglas G. Dickson, Charles L. Dolph, 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, Joseph L. Ullman, 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 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.

Each of Math 112, 115, 185, and 195 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 295 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 175-176-285-286, 185-186-285-286, and 195-196-295-296 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 195-296 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 296 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

In rare circumstances and *with permission of a Mathematics advisor* reduced credit may be granted for Math 185 or 195 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 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 either the regular or Honors sequences. A table explaining the possibilities is available from advisors and the Department. The Department encourages strong students to enter beginning Honors courses in preference to 116 or 215.

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 295-296. A working knowledge of a high-level computer language (e.g., FORTRAN, Pascal, or C) at a level equivalent to completion of EECS 183 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), 3011 Angell Hall, (313) 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.

*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 or 531

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

Mathematical Economics

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 195-296, 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

3011 Angell Hall) no later than the second term of their sophomore year.

*Actuarial Mathematics*

*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.

*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, 3011 Angell 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.

*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 296, and no credit can be earned for a prerequisite to a course taken after the course itself.

**101. Elementary Algebra.**
*Only open to designated summer half-term Bridge students. (2). (Excl).*

**103. Intermediate Algebra.**
*Only open to designated summer half-term Bridge students. (2). (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 195. (4). (N.Excl). (BS).*

**115. Calculus I.**
*See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 195. (4). (N.Excl). (BS). (QR/1).*

**116. Calculus II.**
*Math. 115. Credit is granted for only one course from among Math. 116, 186, and 196. (4). (N.Excl). (BS). (QR/2).*

**118. Analytic Geometry and Calculus II for Social Sciences.**
*Math. 115. (4). (N.Excl). (BS).*

**127. Geometry and the Imagination.**
*Three years of high school mathematics including a geometry course. (4). (NS). (BS). (QR/1).*

**128. Explorations in Number Theory.**
*High school mathematics through at least Analytic Geometry. (4). (NS). (BS). (QR/1).*

**147. Mathematics of Finance.**
*Math. 112 or 115. (3). (Excl). (BS).*

**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, and 185. (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, and 186. (4). (N.Excl). (BS). (QR/1).*

**195. Honors Mathematics I.**
*Permission of the Honors advisor. (4). (N.Excl). (BS). (QR/1).*

**196. Honors Mathematics II.**
*Permission of the Honors advisor. (4). (N.Excl). (BS). (QR/1).*

**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).*

**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. Honors Analysis I.**
*Math. 196 or permission of the Honors advisor. (4). (Excl). (BS).*

**296. Honors Analysis II.**
*Math. 295. (4). (Excl). (BS).*

**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. Eng. 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).*

**398. Topics in Modern Mathematics.**
*Junior standing with an interest and some background in mathematics. (3). (Excl). (BS).*

**399. Independent Reading.**
*(1-6). (Excl). (INDEPENDENT). May be repeated for credit.*

**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).*

**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).*

**432. Projective Geometry.**
*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).*

**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).*

**480. Topics in Mathematics.**
*Math. 217 or 417, 412, or 451, or permission of instructor. (3). (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 in I; 2 in IIIb). (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.*

**498. Topics in Modern Mathematics.**
*Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). (Excl). (BS).*

**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).*

**516. Topics in Algorithms.**
*Two mathematics courses at the 300-level or above, or equivalent computer science courses. (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).*

**522. Actuarial Theory of Pensions and Social Security.**
*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).*

**537. Introduction to Differentiable Manifolds.**
*Math. 513 and 590. (3). (Excl). (BS).*

**551. Advanced Multivariable Calculus.**
*Math. 451 and 513; or permission of Honors advisor. (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).*

**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).*

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