3217 Angell Hall

764-0337

Professor Donald J. Lewis, Chair

*May be elected as a departmental concentration program*

**Professors**

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*

Douglas G. Dickson, *Complex Analysis*

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*

John L. Harer, *Topology*

Donald G. Higman, *Group Theory, Algebraic Combinatorics*

Peter G. Hinman, *Mathematical Logic, Recursion Theory, Foundations of Mathematics, Computational Complexity*

Melvin Hochster, (R.L. Wilder Professor of Mathematics) *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*

David W. Masser, *Transcendental Numbers, Commutative Algebra, Complex Variables*

Jack E. McLaughlin, *Group Theory, Linear Algebra*

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*

Joseph L. Ullman, *Orthogonal Polynomials, Classical Analysis*

Arthur G. Wasserman, *Differential Topology, Transformation Groups, Foliations, Applied Mathematics*

David J. Winter, *Algebra, Lie Algebras, Algebraic Groups*

Michael B. Woodroofe, *Probability Theory, Mathematical Statistics*

**Associate Professors**

David E. Barrett, *Several Complex Variables*

Christoph Borgers, *Numerical Solution of Partial Differential Equations*

Jack L. Goldberg, *Special Functions, Linear Algebra*

Eduard Harabetian, *Partial Differential Equations, Numerical Analysis*

Robert Krasny, *Partial Differential Equations, Fluid Dynamics*

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

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, *Global Analysis*

Michael I. Weinstein, *Nonlinear Partial Differential Equations*

**Assistant Professors**

Hans U. Boden, *Topology*

Geza Bohus, *Algebra, Combinatorics*

Stephen M. Buckley, *Harmonic Analysis*

David C. Butler, *Algebraic Geometry*

Richard Canary, *Topology*

Zhenhua Chen, *Several Complex Variables*

Karen E. Clark, *Partial Differential Equations, Applied Mathematics*

Peter Cholak, *Logic, Recursion Theory*

Charles R. Collins, *Numerical Analysis*

Carolyn A. Dean, *Noncommutative Algebra*

Cristina Draghicescu, *Numerical Analysis*

Estela A. Gavosta, *Several Complex Variables*

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

Rita Gitik, *Topology*

Mark W. Gross. *Algebraic Geometry*

Martin Hildebrand, *Probability Analysis*

Yi Hu, *Algebraic Geometry*

Rosa Qi Huang, *Algebraic Combinatorics*

Naihuan Jing, *Lie Theory*

Smadar Karni, *Numerical Analysis*

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

Shirong Lu, *Kac-Moody Algebra*

Eyal Markman, *Algebraic Geometry*

Yair Minsky, *Differential Geometry, Kleinian Groups, Teichmuller Theory*

James W. Shearer, *Partial Differential Equations*

Judith H. Silverman, *Algebraic Topology*

A. Tahvildar-Zadeh, * Partial Differential Equations, Dynamical Systems*

Jeffrey L. Thunder, *Number Theory*

Rodolfo Torres, *Harmonic Analysis*

Rolland J. Trapp, *Topology*

Trevor Wooley, *Number Theory*

Zaifeh Ye, *Several Complex Variables*

J. James Zhang, *Algebra, Combinatorics*

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

Zhiren Jin, *Partial Differential Equations, Differential Geometry*

Willian Jockusch, *Combinatorics*

Stephane Laederich, *Dynamical Systems, Partial Differential Equations*

Zhongmin Shen, *Differential Geometry*

Irena Swanson*, Commutative Algebra*

**Adjunct Professor**

Howard Young, *Employee Benefits, Actuarial Science*

**Lecturer**

Patricia Shure

**Professors Emeriti** Robert C. F. Bartels, Charles L. Dolph, Frank Harary, George E. Hay, Albert E. Heins, Donald A. Jones, Phillip S. Jones, Wilfred Kaplan, Wilfred M. Kincaid, Chung-Nim Lee, 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.** All courses require *three* years of high school mathematics; *four *years are strongly recommended. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. Two courses preparatory to the calculus, Math 105/106 and Math 109/110, are offered in pairs: a lecture-recitation format and a self-study version of the same material through the Math Lab. Math 105/106 is a course in college algebra and trigonometry with an emphasis on functions and graphs. Math 109/110 is a half-term course for students with all the necessary prerequisites for calculus 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 101 and 103 are 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 mathematical thinking through a single course.

Each of Math 112, 113, 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 Math 113-114 is intended for students of the life sciences who require only one year of calculus. The sequence Math 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. Math 215 is prerequisite to most more advanced courses in Mathematics. Math 112 and Math 113-114 *do not provide preparation for any subsequent course.* Math 113 *does not provide preparation for Math 116 or 118.*

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 counselor 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 185-286 covers much of the same material as Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have had 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, 113, 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 counselor 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 counselors and the Department. Other students who have studied calculus in high school may take a Departmental placement exam during the first week of the Fall term to receive advanced placement *without credit* in the 115 sequence.

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 *strongly recommended* 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.** Mathematics 316, 286, or 296. A working knowledge of some high-level computer language (e.g., Fortran, Pascal, or C) at a level equivalent to the completion of EECS 183 (283) is recommended for all programs and required for the Mathematical Sciences Program. Eight credits of physics, preferably Physics 140/141 and 240/241, are recommended for the Pure Mathematics and Mathematical Sciences Programs and required for the Numerical and Applied Analysis Option of the Mathematical Sciences Program.

**Concentration Programs.** A concentration plan in mathematics should be developed in consultation with, and *must* be approved by, a concentration advisor. Students who are interested in concentrating in mathematics should consult the brochure *Undergraduate Programs of the Department of Mathematics* in the Undergraduate Mathematics Program Office, 3011 Angell Hall. That publication is to be regarded as the most comprehensive and up-to-date guide to concentration programs in mathematics. Each election must receive prior approval of a Mathematics Department junior/senior counselor. The specific courses in a plan depend on each student's particular interests and goals. The following programs are offered:

1. Pure Mathematics Program.

(Students should consult the brochure 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)

1. Modern Algebra: Mathematics 412 or 512;

2. Linear Algebra: Mathematics 217, 419, or 513;

3. Analysis: Mathematics 451;

4. Geometry/Topology: Mathematics 432, 433, 490, or 531.

b. Four elective courses chosen from a list of approved electives and approved by a counselor.

c. One cognate course outside the Mathematics Department, but having advanced mathematical content.

2. *Mathematical Sciences Program*

(Students should consult the brochure *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)

1. Discrete Mathematics/Modern Algebra: Mathematics 312, 412, or 512;

2. Linear Algebra: Mathematics 217, 417, 419, or 513;

3. Probability: Mathematics 425 or 525;

4. Analysis: Mathematics 450 or 451.

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* brochure described above.)

1. Discrete and Algorithmic Methods

2. Numerical and Applied Analysis

3. Operations Research and Modelling

4. Probabilistic Methods

5. Mathematical Economics

6. Control Systems

c. Two additional advanced mathematics (or related) courses.

3. *Actuarial Mathematics Program*.

(Students should consult the brochure *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, Computer Science, and Economics.

a. Five basic courses (one course from each of the following five groups)

1. Linear Algebra: Mathematics 217, 417, 419, or 513;

2. Probability: Mathematics 425 or 525;

3. Analysis: Mathematics 450 or 451;

4. Statistics: Statistics 426;

5. Numerical Analysis: Mathematics 371 or 471

b. Three special actuarial courses: These must include Mathematics 424 and 520 and at least one of Mathematics 521 and 522.

c. Two courses in related areas from a list of approved courses.

**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.

**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 should consult the brochure *Undergraduate Programs of the Department of Mathematics* for its program requirements which take precedence over the descriptions in this *Bulletin*.)

Additional Prerequisite: Completion of one of the sequences ending in 286 or 296. Familiarity with a high-level computer language is strongly urged. (Exceptional students who have not elected one of the basic Honors sequences can also become candidates for an Honors concentration with the approval of the Chairman of the Mathematics Honors Committee.)

a. Four basic courses (one from each of the following four groups) :

1. Linear Algebra: Mathematics 513;

2. Modern Algebra: Mathematics 512;

3. Analysis: Mathematics 451;

4. Geometry/Topology: Mathematics 433, 490, 590, or 531.

b. Four elective courses chosen with the approval of the Honors counselor.

c. One cognate course outside the Mathematics Department, but having advanced mathematical content.

Students who, in the judgment of the departmental Honors committee, have completed an Honors concentration with distinction are granted a citation upon graduation. Interested students should discuss their program and the specific requirements for obtaining the citation with a Mathematics Honors counselor in 1210 Angell Hall no later than the second term of the sophomore year.

**Advising and Counseling.** 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.

**Teaching Certificate.** Prerequisites: One of the three-term calculus sequences terminating in one of the courses Mathematics 215, 285, or 295; and one term of computer programming, EECS 183 (283) or equivalent.

Because the requirements for a teaching certificate depend on whether the student is enrolled in LS&A or the School of Education, it is essential that the student consult either Professor Coxford in the School of Education or Professor Krause in the Mathematics Department *before* electing the courses listed below.

a. Five basic mathematics courses (one course from each group listed below)

1. Linear Algebra: Mathematics 217 or 417;

2. Discrete Mathematics/Modern Algebra: Mathematics 312, or 412, or 512;

3. Geometry: Mathematics 431, 432, or 531;

4. Probability: Mathematics 425 or 525;

5. Secondary Mathematics: Mathematics 486.

b. Minor or major in a second academic area.

This normally requires 20-24 credits in a structured program in an academic area other than mathematics. The *Bulletin* of the School of Education contains a complete list of acceptable majors and minors and their requirements.

c. Seven required courses totalling twenty-eight credits in Education.

Students should consult with Professor Arthur Coxford at the School of Education in their sophomore year to be admitted to the certification program and to schedule practice teaching.

d. Two additional courses: one introductory psychology course and one mathematics course.

**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.

Copyright © 1992-3

The Regents of the University of Michigan, Ann Arbor, MI 48109 USA

1.734.764.1817 (University Operator)