Courses in MATHEMATICS (DIVISION 428)

NOTE: WL:3 for all courses.

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.

105. Data, Functions, and Graphs. Students with credit for Math. 103 can elect Math. 105 for only 2 credits. (4). (Excl). (QR/1).

Math 105 is a preparatory class to the calculus sequences. Students who complete 105 are fully prepared for Math 115. This is a course on analyzing data by means of functions and graphs. The emphasis is on mathematical modeling of real-world applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are assessed during the term by periodic testing. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab. The course prepares students for Math 115

106. Algebra and Analytic Trigonometry (Self-Paced). Students with credit for Math. 103 can elect Math. 106 for only 2 credits. No credit granted to those who have completed or are enrolled in Math 105. (4). (Excl).

Self-study version of Math 105. There are no lectures or sections. Students enrolling in Math 106 must visit the Math Lab during the first full week of the term to complete paperwork and to receive course materials. Students study on their own and consult with tutors in the Math Lab whenever needed. Progress is measured by tests following each chapter and by scheduled midterm and final exams. Math 106 students take the same midterm and final exams as Math 105 students. More detailed information is available from the Math Lab.

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).

Math 110 is a preparatory course for the calculus sequence. Students who complete Math 110 are fully prepared for Math 115. The course is a condensed, half-term version of Math 105 designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students enrolling in Math 110 must visit the Math Lab to complete paperwork and receive course materials. The course covers data analysis by means of functions and graphs. The course prepares students for Math 115.

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).

This is a one-term survey course that provides the basics of elementary calculus. Emphasis is placed on intuitive understanding of concepts and not on rigor. Topics include differentiation with application to curve sketching and maximum-minimum problems, antiderivatives and definite integrals. Trigonometry is not used. This course does not mesh with any of the courses in the other calculus sequences.

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).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Algebra and Trigonometry) or its self-paced equivalent Math 106. Math 116 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 186.

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).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing and questioning skills. Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. Math 215 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 285.

118. Analytic Geometry and Calculus II for Social Sciences. Math. 115. No credit to those having completed 116 or 186. (4). (N.Excl). (BS).

Math 118, a sequel to Math 115, is a combination of the techniques and concepts from Math 116, 215, and 216 that are most useful in the social and decision sciences (especially economics and business). Topics covered include: logarithms, exponentials, elementary integration techniques (substitution, by parts and partial fractions), infinite sequences and series, systems of linear equations, matrices, determinants, vectors, level sets, partial derivatives, Lagrange multipliers for constrained optimization, and elementary differential equations. (Students planning to take Math 215 and 216 should still take 116, although one can pass from 118 to 215 with a bit of work and redundancy.)

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).

The sequence Math 119-219 is intended for students who have earned a score of 4 or better on either the AB or BC version of the Advanced Placement Exam in Mathematics. No familiarity with computers is necessary. The material covered will be approximately that of Math 116 and 215. In addition, students are taught to use the computer algebra system MAPLE (on the Macintosh) a symbolic algebra program which aids the student in visualization, computation and organization as a tool to do routine calculations, to visualize and to explore. MAPLE is thoroughly integrated into the course and the use of MAPLE is permitted (encouraged) on homework and tests. Students are presented with challenging unstructured problems done in groups. Learning to work well with others is an important (and satisfying) part of the course. The emphasis is on concepts and problem-solving rather than theory and proof. Topics include applications of the definite integral, separable differential equations, inverse functions, infinite sequences and series, conics and parametric curves. Math 186 (Fall) is a quite similar course in the Honors sequence with greater emphasis on applications to the physical sciences and engineering. Math 219 is the natural sequel. Students who complete Math 119 and continue to Calculus III should elect Math 219 which is a special MAPLE-oriented version of Math 215. A student who has done very well in this course could enter the Honors sequence at this point by taking 285.

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

This course introduces students to the ideas and some of the basic results in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient Greece, the course includes the construction of new geometric objects from old ones by projecting and by taking slices. The next topic is non-Euclidean geometry. This section begins with the independence of Euclid's Fifth Postulate and with the construction of spherical and hyperbolic geometries in which the Fifth Postulate fails; how spherical and hyperbolic geometry differs from Euclidean geometry. The last topic is geometry of higher dimensions: coordinatization the mathematician's tool for studying higher dimensions; construction of higher-dimensional analogues of some familiar objects like spheres and cubes; discussion of the proper higher-dimensional analogues of some geometric notions (length, angle, orthogonality, etc.) This course is intended for students who want an introduction to mathematical ideas and culture. Emphasis on conceptual thinking students will do hands-on experimentation with geometric shapes, patterns and ideas. Grades based on homework and a final project. No exams. Text: Beyond the Third Dimension (Thomas Banchoff, 1990).

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

This course is intended for non-science concentrators and students in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to take calculus first. Students will be introduced to elementary ideas of number theory, an area of mathematics that deals with properties of the integers. Students will make use of software provided for IBM PCs to conduct numerical experiments and to make empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying factorizations and greatest common divisors, emphasis will shift to the patterns that emerge when the integers are classified according to the remainder produced upon division by some fixed number ('congruences'). Once some basic tools have been established, applications will be made in several directions. For example, students may derive a precise parameterization of Pythagorean triples a2 + b2 = c2.

147. Introduction to Interest Theory. Math. 112 or 115. (3). (Excl). (BS).

This course is designed for students who seek an introduction to the mathematical concepts and techniques employed by financial institutions such as banks, insurance companies, and pension funds. Actuarial students, and other mathematics concentrators, should elect Math 424 which covers the same topics but on a more rigorous basis requiring considerable use of the calculus. Topics covered include: various rates of simple and compound interest, present and accumulated values based on these; annuity functions and their application to amortization, sinking funds and bond values; depreciation methods; introduction to life tables, life annuity, and life insurance values. The course is not part of a sequence. Students should possess financial calculators.

175. Combinatorics and Calculus. Permission of Honors advisor. (4). (N.Excl). (BS). (QR/1).

This course is an alternative to Math 185 as an entry to the Honors sequence. The sequence Math 175-176 is a two-term introduction to Combinatorics, Dynamical Systems, and Calculus. The topics are integrated over the two terms although the first term will stress combinatorics and the second term will stress the development of calculus in the context of dynamical systems. Students are expected to have some previous experience with the basic concepts and techniques of calculus. The course stresses discovery as a vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate each semester in a group project. Grades will be based on homework and projects with a strong emphasis on homework. Personal computers will be a valuable experimental tool in this course and students will be asked to learn to program in either BASIC, PASCAL or FORTRAN. There are two major topic areas: enumeration theory and graph theory. The section on enumeration theory will emphasize classical methods for counting including (1) binomial theorem and its generalizations; (2) solving recursions; (3) generating functions; and (4) the inclusion- exclusion principle. In the process, we will discuss infinite series. The section on graph theory will include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological results and applications to computer science and will include (1) connectivity; (2) trees, Prufer codes, and data structures; (3) planar graphs, Euler's formula and Kuratowski's Theorem; and (4) coloring graphs, chromatic polynomials, and orientation. This material has many applications in the field of Computer Science. Math 176 is the standard sequel.

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).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics will be included at the discretion of the instructor. Math 115 is a somewhat less theoretical course which covers much of the same material. Math 186 is the natural sequel.

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).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. The version of Math 186 given in the Fall semester is intended for students students who have earned a score of 4 or better on either the AB or BC version of the Advanced Placement Exam in Mathematics, will include applications from the physical sciences and engineering, and will make use of the computer algebra system MAPLE. Topics covered include transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor.

203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent enrollment in one term of calculus. No programming experience is assumed. (1). (Excl).

Recent years have seen the development of several powerful software packages, known as Computer Algebra Systems, for doing mathematics on the computer. These programs have the capacity to solve problems numerically, graphically, and symbolically in calculus, linear algebra, differential equations, statistics, and many areas of science and engineering. This one-credit mini-course is a brief introduction to the two most popular of these systems, Maple and Mathematica. It will be of interest to all students whose career interests require mathematical skills. No programming experience is assumed. Students should have taken or be concurrently enrolled in a first course in calculus. The elementary features of Maple and Mathematica will be introduced and applied to various types of problems in algebra and calculus. Alternatives 403 is a more thorough introduction to either Maple and Mathematica. Subsequent Courses This course introduces the student to a tool which can be useful in almost any course which uses mathematics.

215. Calculus III. Math. 116 or 186. (4). (Excl). (BS). (QR/1).

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to major in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a midterm and final exam. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. There is a weekly computer lab using MAPLE software. Math 285 is a somewhat more theoretical course which covers the same material. For students intending to major in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is Math 217. Students who intend to take only one further mathematics course and need differential equations should take Math 216.

216. Introduction to Differential Equations. Math. 215. (4). (Excl). (BS).

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. This course is not intended for mathematics concentrators, who should elect the sequence 217-316. Math 286 covers much of the same material in the Honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.

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).

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of Engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. These courses are explicitly designed to introduce the student to both the concepts and applications of their subjects and to the methods by which the results are proved. Therefore the student entering Math 217 should come with a sincere interest in learning about proofs. The topics covered include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of Rn; linear dependence, bases, and dimension; linear transformations; Eigenvalues and Eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material with more emphasis on computation and applications and less emphasis on proofs. Math 513 covers more in a much more sophisticated way. The intended course to follow Math 217 is 316. Math 217 is also prerequisite for Math 412 and all more advanced courses in mathematics.

219. Calculus III Using MAPLE. Math. 119. (4). (Excl).

Math 219 is calculus of several variables limited to students who have taken Math 119. Students are presented with challenging unstructured problems done in groups. Topics include vector algebra and vector functions, introduction to Fourier series, analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation; line, surface, and volume integrals and applications; vector fields and integration; Green's Theorem and Stokes' Theorem. Math 215 covers much of the same material with less use of MAPLE. For students intending to concentrate in mathematics or who have some interest in the theory of mathematics as well as its applications, the appropriate sequel is 217. Students who intend to take only one further mathematics course and need differential equations should take 216.

285. Honors Analytic Geometry and Calculus III. Math. 186 or permission of the Honors advisor. (4). (Excl). (BS).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Topics include vector algebra and vector functions; analytic geometry of planes, surfaces, and solids; functions of several variables and partial differentiation, maximum-minimum problems; line, surface, and volume integrals and applications; vector fields and integration; curl, divergence, and gradient; Green's Theorem and Stokes' Theorem. Additional topics may be added at the discretion of the instructor. Math 215 is a less theoretical course which covers the same material.

286. Honors Differential Equations. Math. 285. (3). (Excl). (BS).

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to major in mathematics, science, or engineering as well as students heading for many other fields who want a somewhat more theoretical approach. Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also included. This sequence is not restricted to students enrolled in the LS&A Honors Program. Topics include first-order differential equations, higher-order linear differential equations with constant coefficients, an introduction to linear algebra, linear systems, the Laplace Transform, series solutions and other numerical methods (Euler, Runge-Kutta). If time permits, Picard's Theorem will be proved. Math 216 and 316 cover much of the same material. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 286 is also preparation for more theoretical courses such as Math 451.

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.

This course is designed to help students understand more clearly how techniques from other undergraduate mathematics courses can be used in concert to solve real-world problems. After the first two lectures the class will discuss methods of attacking problems. For credit a student will have to describe and solve an individual problem and write a report on the solution. Computing methods will be used. During the weekly workshop students will be presented with real-world problems on which techniques of undergraduate mathematics offer insights. They will see examples of (1) how to approach and set up a given modeling problem systematically, (2) how to use mathematical techniques to begin a solution of the problem, (3) what to do about the loose ends that can't be solved, and (4) how to present the solution to others. Students will have a chance to use the skills developed by participating in the UM Undergraduate Math Modelling Meet.

289. Problem Seminar. (1). (Excl). (BS). May be repeated for credit with permission.

One of the best ways to develop mathematical abilities is by solving problems using a variety of methods. Familiarity with numerous methods is a great asset to the developing student of mathematics. Methods learned in attacking a specific problem frequently find application in many other areas of mathematics. In many instances an interest in and appreciation of mathematics is better developed by solving problems than by hearing formal lectures on specific topics. The student has an opportunity to participate more actively in his/her education and development. This course is intended for superior students who have exhibited both ability and interest in doing mathematics, but it is not restricted to Honors students. This course is excellent preparation for the Putnam exam. Students and one or more faculty and graduate student assistants will meet in small groups to explore problems in many different areas of mathematics. Problems will be selected according to the interests and background of the students.

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).

The sequence Math 295-296-395-396 is a more intensive Honors sequence than 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This is not restricted to students enrolled in the LS&A Honors Program. This course presents an introduction to mathematical analysis with emphasis on proofs and theory. The precise content may vary with the instructor, but generally will cover such topics as Functions of one variable and their representation by graphs, set theory, construction of the real number field, limits of sequences and functions, continuity, elementary functions, derivatives and integrals with applications, parametric representation, polar coordinates, applications of mathematical induction. Additional topics may include countability, topology of the real numbers, infinite series, and uniform continuity.

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).

The sequence Math 295-296-395-396 is a more intensive Honors sequence than 185-186-285-286. The material includes all of that of the lower sequence and substantially more. The approach is theoretical, abstract, and rigorous. Students are expected to learn to understand and construct proofs as well as do calculations and solve problems. The expected background is a thorough understanding of high school algebra and trigonometry. No previous calculus is required, although many students in this course have had some calculus. Students completing this sequence will be ready to take advanced undergraduate and beginning graduate courses. This sequence is not restricted to students enrolled in the LS&A Honors Program. Topics will be chosen from: logarithms and exponentials, sups and infs, sequences and series, techniques of integration, Bolzano-Weierstrass Theorem, uniform continuity and convergence, C* and analytical functions, Weierstrass Approximation Theorem, metric spaces: Rn and C0[a,b], completeness and compactness, topics in linear algebra: vector spaces, linear dependence and bases, matrix operations.

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).

This is an introduction to differential equations for students who have studied linear algebra (Math 217). It treats techniques of solution (exact and approximate), existence and uniqueness theorems, some qualitative theory, and many applications. Proofs are given in class; homework problems include both computational and more conceptually oriented problems. First-order equations: solutions, existence and uniqueness, and numerical techniques; linear systems: eigenvector-eigenvalue solutions of constant coefficient systems, fundamental matrix solutions, nonhomogeneous systems; higher-order equations, reduction of order, variation of parameters, series solutions; qualitative behavior of systems, equilibrium points, stability. Applications to physical problems are considered throughout. Math 216 covers somewhat less material without the use of linear algebra and with less emphasis on theory. Math 286 is the Honors version of Math 316. Math 471 and/or 572 are natural sequels in the area of differential equations, but Math 316 is also preparation for more theoretical courses such as Math 451.

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.

An experiential mathematics course for elementary teachers. Students would tutor elementary (Math. 102) or intermediate (Math. 104) algebra in the Math. Lab. They would also participate in a weekly seminar to discuss mathematical and methodological questions.

350/Aero. 350. Aerospace Engineering Analysis. Math. 216 or 316 or the equivalent. (3). (Excl). (BS).

This is a three-hour lecture course in engineering mathematics which continues the development and application of ideas introduced in Math 215 and 216. The course is required in the Aerospace Engineering curriculum, and covers subjects needed for subsequent departmental courses. The major topics discussed include vector analysis, Fourier series, and an introduction to partial differential equations, with emphasis on separation of variables. Some review and extension of ideas relating to convergence, partial differentiation, and integration are also given. The methods developed are used in the formulation and solution of elementary initial- and boundary-value problems involving, e.g., forced oscillations, wave motion, diffusion, elasticity, and perfect-fluid theory. There are two or three one-hour exams and a two-hour final, plus about ten homework assignments, or approximately one per week, consisting largely of problems from the text. The text is Mathematical Methods in the Physical Sciences by M.L. Boas.

371/Engin. 371. Numerical Methods for Engineers and Scientists. Engineering 103 or 104, or equivalent; and Math. 216. (3). (Excl). (BS).

This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571-572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for 571-572.

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).

This course, together with its sequel Math 489, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include problem solving, sets and functions, numeration systems, whole numbers (including some number theory) and integers. Each number system is examined in terms of its algorithms, its applications, and its mathematical structure. There is no alternative course. Math 489 is the required sequel.

395(295). Honors Analysis I. Math. 296 or permission of the Honors advisor. (4). (Excl). (BS).

This course is a continuation of the sequence Math 295-296 and has the same theoretical emphasis. Students are expected to understand and construct proofs. This course studies functions of several real variables. Topics are chosen from elementary linear algebra: vector spaces, subspaces, bases, dimension, solutions of linear systems by Gaussian elimination; elementary topology: open, closed, compact, and connected sets, continuous and uniformly continuous functions; differential and integral calculus of vector-valued functions of a scalar; differential and integral calculus of scalar-valued functions on Euclidean spaces; linear transformations: null space, range, matrices, calculations, linear systems, norms; differential calculus of vector-valued mappings on Euclidean spaces: derivative, chain rule, implicit and inverse function theorems.

396(296). Honors Analysis II. Math. 395. (4). (Excl). (BS).

This course is a continuation of Math 395 and has the same theoretical emphasis. Students are expected to understand and construct proofs. Differential and integral calculus of functions on Euclidean spaces. Students who have successfully completed the sequence Math 295-396 are generally prepared to take a range of advanced undergraduate and graduate courses such as Math 512, 513, 525, 590, and many others.

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

Designed especially for Honors students.

403. Mathematical Modeling Using Computer Algebra Systems. Math. 116 and junior standing. (3). (Excl). (QR/1).

Many fields of study including the Natural Sciences, Engineering, Economics and Statistics use mathematics regularly and extensively both as a tool and as a means for modeling phenomena. Since the realistic models usually lead to problems not solvable by simple analytic techniques either because they involve too many parameters or are highly nonlinear new methods are needed to give the students insight into the problem. One rather new powerful technique for doing this is the so-called Computer Algebra (CA) system. These systems manipulate symbols as easily as hand held calculators manipulate numbers. So, for example, MATHEMATICA (the CA system used in this course) can compute the indefinite integral of tan x, expand ex sin x in power series, find the general solution of y" + y = cos t, and so on. In essence, MATHEMATICA is an "expert" mathematical assistant. Using MATHEMATICA easily and productively is the primary goal of Math 403. There are no final exams but rather students work in teams to produce a term project using MATHEMATICA. There are two hours of lecture and 1 hour of actual computer work per week. Weekly demonstrations of computer competency in using MATHEMATICA amounts to 50% of the term grade. The term project comprises the remaining 50%. No previous computer programming is required or needed. (Goldberg)

404. Intermediate Differential Equations. Math. 216. No credit granted to those who have completed Math. 286 or 316. (3). (Excl). (BS).

This is a course oriented to the solutions and applications of differential equations. Numerical methods and computer graphics are incorporated to varying degrees depending on the instructor. There are relatively few proofs. Some background in linear algebra is strongly recommended. First-order equations, second and higher-order linear equations, Wronskians, variation of parameters, mechanical vibrations, power series solutions, regular singular points, Laplace transform methods, eigenvalues and eigenvectors, nonlinear autonomous systems, critical points, stability, qualitative behavior, application to competing-species and predator-prey models, numerical methods. Math 454 is a natural sequel.

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).

This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. Math 217 are equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of two particular types of mathematical structures: rings and groups. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied. Math 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an Honors version of Math 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 513, 581, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.

413. Calculus for Social Scientists. Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS).

A one-term course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.

416. Theory of Algorithms. Math. 312 or 412 or CS 303, and CS 380. (3). (Excl). (BS).

Many common problems from mathematics and computer science may be solved by applying one or more algorithms - well-defined procedures that accept input data specifying a particular instance of the problem and produce a solution. Students entering Math 416 typically have encountered some of these problems and their algorithmic solutions in a programming course. The goal here is to develop the mathematical tools necessary to analyze such algorithms with respect to their efficiency (running time) and correctness. Different instructors will put varying degrees of emphasis on mathematical proofs and computer implementation of these ideas. Typical problems considered are: sorting, searching, matrix multiplication, graph problems (flows, travelling salesman), and primality and pseudo-primality testing (in connection with coding questions). Algorithm types such as divide-and-conquer, backtracking, greedy, and dynamic programming are analyzed using mathematical tools such as generating functions, recurrence relations, induction and recursion, graphs and trees, and permutations. The course often includes a section on abstract complexity theory including NP completeness. This course has substantial overlap with EECS 586 more or less depending on the instructors. In general, Math 416 will put more emphasis on the analysis aspect in contrast to design of algorithms. Math 516 (given infrequently) and EECS 574 and 575 (Theoretical Computer Science I and II) include some topics which follow those of this course.

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).

Many problems in science, engineering, and mathematics are best formulated in terms of matrices rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, Eigenvalues and Eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations. Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

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. (3). (Excl). (BS).

Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations. Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more proof-oriented. Math 513 is much more abstract and sophisticated. EECS 400 is the same course. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571

423. Mathematics of Finance. CS 183 and Math. 217, 316, and 425. (3). (Excl). (BS).

This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary. Topics include risk and return theory, portfolio theory, capital asset pricing model, random walk model, stochastic processes, Black-Scholes Analysis, numerical methods and interest rate models.

424. Compound Interest and Life Insurance. Math. 215 or permission of instructor. (3). (Excl). (BS).

This course explores the concepts underlying the theory of interest and then applies them to concrete problems. The course also includes applications of spreadsheet software. The course is a pre-requisite to advanced actuarial courses. It also helps students prepare for the Part 4A examination of the Casualty Actuarial Society and the Course 140 examination of the Society of Actuaries. The course covers compound interest (growth) theory and its application to valuation of monetary deposits, annuities, and bonds. Problems are approached both analytically (using algebra) and geometrically (using pictorial representations). Techniques are applied to real-life situations: bank accounts, bond prices, etc. The text is used as a guide because it is prescribed for the Society of Actuaries exam; the material covered will depend somewhat on the instructor. Math 424 is required for students concentrating in actuarial mathematics; other may take Math 147, which deals with the same techniques but with less emphasis on continuous growth situations. Math 520 applies the concepts of Math 424 together with probability theory to the valuation of life contingencies (death benefits and pensions).

425/Stat. 425. Introduction to Probability. Math. 215. (3). (N.Excl). (BS).
Section 001 and 002.
See Statistics 425.

Sections 003 and 004. This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. Math concentrators should be sure to elect sections of the course that are taught by mathematics (not Statistics) faculty. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

427/Social Work 603. Retirement Plans and Other Employee Benefit Plans. Junior standing. (3). (Excl).

An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course. Math 521 and/or 522 (which can be taken independently of each other) provide more in-depth examination of the actuarial techniques used in employee benefit plans.

431. Topics in Geometry for Teachers. Math. 215. (3). (Excl). (BS).

This course is a study of the axiomatic foundations of Euclidean and non-Euclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tesselations; triangle groups; finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative geometry courses at this level are 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.

433. Introduction to Differential Geometry. Math. 215. (3). (Excl). (BS).

This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, classification of minimal surfaces of revolution. 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513) and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.

450. Advanced Mathematics for Engineers I. Math. 216, 286, or 316. (4). (Excl). (BS).

Although this course is designed principally to develop mathematics for application to problems of science and engineering, it also serves as an important bridge for students between the calculus courses and the more demanding advanced courses. Students are expected to learn to read and write mathematics at a more sophisticated level and to combine several techniques to solve problems. Some proofs are given and students are responsible for a thorough understanding of definitions and theorems. Students should have a good command of the material from Math 215, and 216 or 316, which is used throughout the course. A background in linear algebra, e.g., Math 217, is highly desirable, as is familiarity with Maple software. Topics include a review of curves and surfaces in implicit, parametric, and explicit forms; differentiability and affine approximations; implicit and inverse function theorems; chain rule for 3-space; multiple integrals; scalar and vector fields; line and surface integrals; computations of planetary motion, work, circulation, and flux over surfaces; Gauss' and Stokes' Theorems, derivation of continuity and heat equation. Some instructors include more material on higher dimensional spaces and an introduction to Fourier series. Math 450 is an alternative to Math 451 as a prerequisite for several more advanced courses. Math 454 and 555 are the natural sequels for students with primary interest in engineering applications.

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).

This course has two complementary goals: (1) a rigorous development of the fundamental ideas of Calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc.), and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. The material usually covered is essentially that of Ross' book Elementary Analysis: The Theory of Calculus. Chapter I deals with the properties of the real number system including (optionally) its construction from the natural and rational numbers. Chapter II concentrates on sequences and their limits, Chapters III and IV on the application of these ideas to continuity of functions, and sequences and series of functions. Chapter V covers the basic properties of differentiation and Chapter VI does the same for (Riemann) integration culminating in the proof of the Fundamental Theorem of Calculus. Along the way there are presented generalizations of many of these ideas from the real line to abstract metric spaces. There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, 590, 596, and 597.

452. Advanced Calculus II. Math. 217, 417, or 419; and Math. 451. (3). (Excl). (BS).

This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include (1) partial derivatives and differentiability, (2) gradients, directional derivatives, and the chain rule, (3) implicit function theorem, (4) surfaces, tangent plane, (5) max-min theory, (6) multiple integration, change of variable, etc. (7) Green's and Stokes' theorems, differential forms, exterior derivatives. Math 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. Math 452 is prerequisite to Math 572 and is good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 537, 635)

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).

This course is devoted to the use of Fourier series and other orthogonal expansions in the solution of boundary-value problems for second-order linear partial differential equations. Emphasis is on concepts and calculation. The official prerequisite is ample preparation. Classical representation and convergence theorems for Fourier series; method of separation of variables for the solution of the one-dimensional heat and wave equation; the heat and wave equations in higher dimensions; spherical and cylindrical Bessel functions; Legendre polynomials; methods for evaluating asymptotic integrals (Laplace's method, steepest descent); Fourier and Laplace transforms; applications to linear input-output systems, analysis of data smoothing and filtering, signal processing, time-series analysis, and spectral analysis. Both Math 455 and 554 cover many of the same topics but are very seldom offered. Math 454 is prerequisite to Math 571 and 572, although it is not a formal prerequisite, it is good background for Math 556.

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).

The focus of this course is the application of a variety of mathematical techniques to solve real-world problems. Students will learn how to model a problem in mathematical terms and use mathematics to gain insight and eventually solve the problem. Concepts and calculations, including use of spreadsheets and programming, are emphasized, but proofs are included. Construction and analysis of mathematical models in the natural or social sciences. Content varies considerably with instructor. Recent versions: Use and theory of dynamical systems (chaotic dynamics, ecological and biological models, classical mechanics), and mathematical models in physiology and population biology. Math 286 and 316 cover some of the dynamics without the applications

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).

This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, Dirichlet problem for the Laplace equation. Math 371 is a less sophisticated version intended principally for Sophomore and Junior engineering students; the sequence Math 571-572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.

481. Introduction to Mathematical Logic. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite. In the first third of the course the notion of a formal language is introduced and propositional connectives ('and', 'or', 'not', 'implies'), tautologies and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including non-standard analysis. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.

497. Topics in Elementary Mathematics. Math. 489 or permission of instructor. (3). (Excl). (BS). May be repeated for a total of six credits.

This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from term to term.

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).

This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 195- 296 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals, and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p. Math 412 is a substantially lower level course over about half of the material of Math 512. The sequence Math 593-594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Together with Math 513, this course is excellent preparation for the sequence Math 593-594.

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).

This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory. For students with only the minimal prerequisite, this is a demanding course; at least one additional proof-oriented course (e.g., Math 451 or 512) is recommended. Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; Eigenvalues and Eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. Math 419 covers much of the same material using the same text, but there is more stress on computation and applications. Math 217 is similarly proof-oriented but significantly less demanding than Math 513. Math 417 is much less abstract and more concerned with applications. The natural sequel to Math 513 is 593. Math 513 is also prerequisite to several other courses: Math 537, 551, 571, and 575, and may always be substituted for Math 417 or 419.

520. Life Contingencies I. Math. 424 and Math. 425; or permission of instructor. (3). (Excl). (BS).

The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.

523. Risk Theory. Math. 425. (3). (Excl). (BS).

This course explores the applications of stochastic models to risk business (uncertain financial payments). The emphasis is on concepts and calculations with some proofs. Students should have a good background in probability (at least a B+ in Math 425). Topics include utility theory, application to buying general insurance to reduce risk, compound distribution models for risk portfolios, application of stochastic processes to the ruin problem and to reinsurance.

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).

This course is a thorough and fairly rigorous study of the mathematical theory of probability. There is substantial overlap with 425, but here more sophisticated mathematical tools are used and there is greater emphasis on proofs of major results. Math 451 is preferable to Math 450 as preparation, but either is acceptable. Topics include the basic results and methods of both discrete and continuous probability theory. Different instructors will vary the emphasis between these two theories. EECS 501 also covers some of the same material at a lower level of mathematical rigor. Math 425 is a course for students with substantially weaker background and ability. Math 526, Stat 426, and the sequence Stat 510-511 are natural sequels.

533. Geometric Algebra. Math. 513. I. (3). (Excl). (BS).

Introduction to coordinates, projective geometry as the geometry of subspaces of a vector space, the projective linear and orthogonal groups, and affine geometry and the affine group.

535. Introduction to Algebraic Curves. Math. 513. (3). (Excl). (BS).

Plane algebraic curves, linear systems of curves, rational curves, analysis of singularities, parametrizations, formal power series, and places of a curve. Bezout's theorem, rational transformations, Luroth's theorem, dual curves, ideal of a curve, and linear series.

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

This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses 590, 513, and 551. Its goal is to introduce the basic concepts and results of Differential Topology and Differential Geometry. Topics may include: Inverse and Implicit function theorem in Rn, differentiable manifolds, tangent and cotangent bundles, exterior differential forms, vector fields, partitions of unity, integration on manifolds, Stokes' Theorem, the divergence theorem. Topics in Riemannian Geometry include Riemannian metrics, covariant differentiation and connections, torsion tensor, Levi-Civita connection, Riemann curvature tensor, Gaussian, sectional, Ricci, scalar and mean curvatures, 2-dimensional case, hypersurface case, Gauss and Codazzi equations, length and energy of curves, geodesics, completeness (Hopf-Rinow Theorem), exponential map, Cartan-Hadamard Theorem. Math 433 is an undergraduate version which covers much less material in a less sophisticated way.

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).

This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. Math 596 covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications. Math 555 is prerequisite to many advanced courses in science and engineering fields.

556. Methods of Applied Mathematics I. Math. 555 or 554. (3). (Excl). (BS).

Together with its sequel 557, this is a thorough and rigorous introduction to the methods of applied mathematics for graduate students in mathematics, engineering, and physics. Students are expected to master both the proofs and applications of major results. In addition to the complex analysis prerequisite, a strong background in linear algebra ( Math 419 or 513) is recommended. Topics vary with the instructor but often include Green's functions for ordinary differential equations, distributions, integral operators on L2, Hilbert spaces, complete orthonormal sets, compact self-adjoint operators, and scattering theory on R. Other topics may include material on the basic partial differential equations of mathematical physics (Laplace's Equation, the Heat Equation, the Wave Equation, and Schrödinger's Equations) such as Fourier series and boundry-value problems, Sturm-Liouville theory, Fourier transformations, fundamental solutions, energy integrals, and potential theory. Math 557 is the intended sequel. Other possibilities include Math 656 (Partial Differential Equations) and 658 (Ordinary Differential Equations.)

558(658). Ordinary Differential Equations. Math. 450 or 451. (3). (Excl). (BS).

Existence and uniqueness of theorems for flows, linear systems, Floquet theory, Poicare-Bendixson theory, Poincare maps, periodic solutions, stability theory, Hopf bifurcations, chaotic dynamics.

559. Selected Topics in Applied Mathematics. Math. 451 and 419, or equivalent. (3). (Excl). (BS). May be elected for a total of 6 credits.

This course in intended for students with a fairly strong background in pure mathematics, but not necessarily any experience with applied mathematics. Proofs and concepts, as will as intuitions arising from the field of application will be stressed. This course will focus on a particular area of applied mathematics in which the mathematical ideas have been strongly influenced by the application. It is intended for students with a background in pure mathematics, and the course will develop the intuitions of the field of application as well as the mathematical proofs. The applications considered will vary with the instructor and may come from physics, biology, economics, electrical engineering, and other fields. Recent examples have been: Dynamical Systems, Statistical Mechanics, Solitons, and Nonlinear Waves.

561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. Math. 217, 417, or 419. (3). (Excl). (BS).

Formulation of problems from the private and public sectors using the mathematical model of linear programming. Development of the simplex algorithm; duality theory and economic interpretations. Postoptimality (sensitivity) analysis; applications and interpretations. Introduction to transportation and assignment problems; special purpose algorithms and advanced computational techniques. Students have opportunities to formulate and solve models developed from more complex case studies and use various computer programs.

562/IOE 511/Aero. 577/EECS 505/CS 505. Continuous Optimization Methods. Math. 217, 417 or 419. (3). (Excl). (BS).

Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.

565. Combinatorics and Graph Theory. Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one proof-oriented course. Graph Theory topics include Trees; k -connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5-Color Theorem; Kuratowski's Theorem; and the Matrix-Tree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q -series, linear recurrences and rational generating functions, and Polya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms.

571. Numerical Methods for Scientific Computing I. Math. 217, 419, or 513; and 454 or permission of instructor. (3). (Excl). (BS).

This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.

590. Introduction to Topology. Math. 451. (3). (Excl). (BS).

This is an introduction to topology with an emphasis on the set-theoretic aspects of the subject. It is quite theoretical and requires extensive construction of proofs. Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, surfaces and manifolds, fundamental theorem of algebra, and other topics. Math 490 is a more gentle introduction that is more concrete, somewhat less rigorous, and covers parts of both Math 590 and 591. Combinatorial and algebraic aspects of the subject are emphasized over the geometrical. Math 591 is a more rigorous course covering much of this material and more. Both Math 591 and 537 use much of the material from Math 590.

591. General and Differential Topology. Math. 451. (3). (Excl). (BS).

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces, vector fields, submanifolds, inverse function theorem, immersions, submersions, partitions of unity, Sard's theorem, embedding theorems, transversality, classification of surfaces. Math 592 is the natural sequel.

593. Algebra I. Math. 513. (3). (Excl). (BS).

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to 512. Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras.

594. Algebra II. Math. 593. (3). (Excl). (BS).

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Topics include group theory, permutation representations, simplicity of alternating groups for n > 4, Sylow theorems, series in groups, solvable and nilpotent groups, Jordan-Hölder Theorem for groups with operators, free groups and presentations, fields and field extensions, norm and trace, algebraic closure, Galois theory, transcendence degree.

596. Analysis I. Math. 451. (3). (Excl). (BS). Students with credit for Math. 555 may elect Math 596 for two credits only.

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Review of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem. Math 595 covers some of the same material with greater emphasis on applications and less attention to proofs.


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