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# Courses in Mathematics (Division 428)

This page was created at 4:02 PM on Wed, Dec 13, 2000.

## Fall Term, 2000 (September 6 – December 22)

Open courses in Mathematics

Wolverine Access Subject listing for MATH

Take me to the Fall Term '00 Time Schedule for Mathematics.

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Elementary Mathematics Courses. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. All courses require three years of high school mathematics; four years are strongly recommended and more information is given for some individual courses below. Students with College Board Advanced Placement credit and anyone planning to enroll in an upper-level class should consider one of the Honors sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and thinking through a single course. They are neither prerequisite nor preparation for any further course. No credit will be received for the election of Math 127 or 128 if a student already has received credit for a 200- (or higher) level mathematics course.

Each of Math 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics.

The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. Most students electing Math 295 have completed a thorough high school calculus course. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one or two semesters of AP credit, Math 119 and Math 156. Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 156 is an Honors course intended primarily for science and engineering concentrators and will emphasize both applications and theory. Interested students should consult a mathematics advisor for more details.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 295 after Math 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 an advisor before switching from one sequence to another. In all cases a maximum total of 16 credits may be earned for calculus courses Math 115 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

Students completing Math 116 who are principally interested in the application of mathematics to other fields may continue either to Math 215 (Analytic Geometry and Calculus III) or to Math 216 (Introduction to Differential Equations) – these two courses may be taken in either order. Students who have greater interest in theory or who intend to take more advanced courses in mathematics should continue with Math 215 followed by the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217 (or the Honors version, Math 513) is required for a concentration in Mathematics; it both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations in Math 316. Math 216 is not intended for mathematics concentrators.

Attention Potential Elementary School Teachers: Math 385 is Offered this Spring Term

All elementary teaching certificate candidates are required to take two math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489 in the Winter Term. Due to heavy enrollment pressure, Math 385 will be offered this Spring Term (IIIA 2000) as well. Last Fall Term a number of students were closed out of Math 385. Next Fall Term, class-size limits will be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course. The next Spring Term offering of Math 385 will be in 2002. For further information, contact Prof. Krause at his e-mail address, krause@math.lsa.umich.edu.

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.

### Math. 105. Data, Functions, and Graphs.

#### There will be Joint Evening Examinations for All Sections of Math 105, 6:00 – 8:00 p.m. on Mon., Oct 11 and Thurs., Nov 18. Also a Joint Final.

Prerequisites & Distribution: Students with credit for Math. 103 can elect Math. 105 for only 2 credits. No credit granted to those who have completed any Mathematics course numbered 110 or higher. (4). (MSA). (QR/1).

Credits: (4).

Course Homepage: No Homepage Submitted.

Math 105 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. 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.

 Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office

### Math. 110. Pre-Calculus (Self-Study).

#### Section 001 – Course Meets the Second Half of the Term. Students Work Independently with Guidance from Math Lab Staff. There will be no Formal Lecture.

Prerequisites & Distribution: See Elementary Courses above. Enrollment in Math 110 is by recommendation of Math 115 instructor and override only. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).

Credits: (2).

Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html

The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of Math 105 (Math 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students who complete 110 are fully prepared for Math 115. Students may enroll in Math 110 only on the recommendation of a mathematics instructor after the third week of classes in the Winter and must visit the Math Lab to complete paperwork and receive course materials.

 Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office

### Math. 115. Calculus I.

#### There will be Joint Evening Examinations for All sections of Math 115, 6-8 p.m., Weds, Oct 6 and Nov 10. Also a Joint Final.

Prerequisites & Distribution: 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. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/

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 concentrate 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 (Data, Functions, and Graphs). 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. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office ### Math. 115. Calculus I. #### Section 100 – Students in Math 115 Section 100 Receive Individualized Self-Paced Instruction in the Mathematics Laboratory in Room B860 E H. Students Must Go to the Math Lab During the First Full Week of Classes. Prerequisites & Distribution: 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. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/ 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 concentrate 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 5, P/I ### Math. 116. Calculus II. #### There will be Joint Evening Examinations for All Sections of Math 116, 6-8 p.m., Tues., Oct. 12 and Tues., Nov 16. Also a Joint Final. Prerequisites & Distribution: Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/courses/116/index.shtml See Math 115 for a general description of the sequence Math 115-116-215. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office ### Math. 147. Introduction to Interest Theory. #### Instructor(s): Prerequisites & Distribution: Math. 112 or 115. No credit granted to those who have completed a 200- (or higher) level mathematics course. (3). (MSA). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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 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. This course is not part of a sequence. Students should possess financial calculators.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office ### Math. 156. Applied Honors Calculus II. #### There will be Joint Evening Examinations for All Sections of Math 156, Thurs, Oct 14 and Weds, Nov 17, 6:00 – 8:00 p.m. Also a Joint Final. #### Instructor(s): Robert Krasny (krasny@umich.edu) Prerequisites & Distribution: Score of 4 or 5 on the AB or BC Advanced Placement calculus exam. Credit is granted for only one course among Math 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html Math 156 is a 2nd semester Honors calculus course for engineering and science students. The course is offered in the Fall semester and is designed for entering freshmen who have Advanced Placement credit for 1st semester calculus. The course emphasizes computational skills, conceptual understanding, and applications of calculus. Math 156 provides students with the background needed for a variety of subsequent courses in math, science and engineering. It also introduces students to MAPLE, a high-level software tool for doing math on a computer. Prerequisite: score of 4 or 5 on the Advanced Placement AB calculus exam Syllabus: 1. Applications of the Integral: sigma notation, Riemann sums, definite integral, fundamental theorem of calculus, work, improper integrals, arclength, surface area, moments and center of mass, hydrostatic force, probability density functions 2. Differential Equations: exponential growth and decay, Newton's law of cooling/heating, logistic equation, stability of equilibrium points 3. Series: sequences, series, geometric series, integral and comparison tests, alternating series, ratio test, power series, radius of convergence, Taylor series, binomial series, applications of Taylor approximation Review: (as needed) integration by parts, trigonometric substitution, partial fractions, trigonometric integrals, l'Hopital's rule Special Topics: asymptotic expansions, Bessel function, complex numbers, Euler's formula, Gamma function, hyperbolic functions, Laplace transform, parametric curves, polar coordinates Exams: 2 midterm exams and a final exam Homework: Math 156 has weekly homework assignments. Students may work together and discuss the homework problems with each other, but each student should write up and submit their own set of solutions. After the assignment is collected, solutions will be available in a loose-leaf book at the Undergraduate Library Reserve Desk on the 2nd floor of the Shapiro Library. Sequel: • Math 255 – Multivariable Calculus • Math 256 – Differential Equations and Linear Algebra  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 175. Combinatorics and Calculus. #### Section 001 – Introduction to Cryptology and Discrete Mathematics #### Instructor(s): Sergey Fomin (fomin@umich.edu) Prerequisites & Distribution: Permission of Honors advisor. No credit granted to those who have completed a 200-level or higher Mathematics course. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~fomin/175.html Text (required):An Introduction to Cryptology and Discrete Math – The Math 175 Coursepack, by C.Greene, P.Hanlon, T.Hsu, and J.Hutchinson. Prerequisites: Math 115 or equivalent (Single-variable calculus) recommended. This course gives a historical introduction to Cryptology, the science of secrete codes. It begins with the oldest recorded codes, taken from hieroglyphic engravings, and ends with the encryption schemes used to maintain privacy during Internet credit card transactions. Since secret codes are based on mathematical ideas, each new kind of encryption method leads in this course to the study of new mathematical ideas and results. The first part of the course deals with permutation-based codes: substitutional ciphers, transpositional codes, Vigenere ciphers and more complex polyalphabetic substitutions including the those created by rotor machines such as the Enigma. The mathematical subjects treated in this section include permutations, modular arithmetic and some elementary statistics. In the second part of the course, the subject moves to bit stream encryption methods. These include block cipher schemes such as the Data Encryption Standard (DES). The mathematical concepts introduced here are recurrence relations and some more advanced statistical results. Public key encryption is the subject of the final part of the course. We learn the mathematical underpinnings of Diffie-Hellman key exchange, RSA and Knapsack codes. A substantial number of results from elementary number theory are needed and proved in this section of the course. There is considerable development of problem-solving skills in Math 175. So, students taking the course should have significant mathematical experience and sophistication. We recommend that students who sign up for this course have credit for Math 115 or have taken a course at least to the level of Math 115. There are no quizzes and no exams in the course. The course is based in large part on a series of homework sets in which students are asked to solve problems according to Steps 1-6 above. There will also be weekly computer labs and a final project which is an elaborate cryptanalysis problem which serves as a capstone experience the course.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 185. Honors Calculus I. #### Instructor(s): Prerequisites & Distribution: Permission of the Honors advisor. Credit is granted for only one course from among Math. 112, 113, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: No Homepage Submitted. The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to concentrate 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 214. Linear Algebra and Differential Equations. #### Instructor(s): Prerequisites & Distribution: Math 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. Two credits granted to those who have completed or are enrolled in Math. 216. No credit granted to those who have completed or are enrolled in Math 513. (4). (MSA). (BS). Credits: (4). Course Homepage: No Homepage Submitted. This course is intended for second-year students who might otherwise take Math 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect Math 217. While Math 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, Math 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra. The textbook is Linear Algebra and its Applications by David Lay. The following is a tentative outline of the course: • Systems of linear equations, matrices, row operations, reduced row echelon form, free variables, basic variables, basic solution, parametric description of the solution space. Rank of a matrix. • Vectors, vector equations, vector algebra, linear combinations of vectors, the linear span of vectors. • The matrix equation Ax = b. Algebraic rules for multiplication of matrices and vectors. • Homogeneous systems, principle of superposition. • Linear independence. • Applications, Linear models • Matrix algebra, dot product, matrix multiplication. • Inverse of a matrix. • Invertible matrix theorem. • Partitioned matrices. • 2-dimensional discrete dynamical systems. • Markov process, steady state. • Transition matrix, eigenvector, steady state lines (affine hulls). • Geometry of two and three dimensions: affine hulls, linear hulls, convex hulls, half planes, distance from point to a plane, optimization. • Introduction to linear programming. • The geometry of transition matrices in 2 dimensions (rotations, shears, ellipses, eigenvectors). • Transition matrices for 3-D (rotations, orthogonal matrices, symmetric matrices) • Determinants. • 2- and 3-dimensional determinant as area and volume. • Eigenvectors and Eigenvalues • Eigenvectors • Complex numbers including Euler's formula. • Complex eigenvalues and their geometric meaning • Review of ordinary differential equations. • Systems of ordinary differential equations in 2 dimensions. Regular problem sets and exams.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 215. Calculus III. #### There will be Joint Evening Examinations for All Sections of Math 215, 6-8 p.m., Thurs Oct 14, And Weds Nov 18. Also a Joint Final. Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/courses/215/ 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 concentrate 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 concentrate 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office ### Math. 216. Introduction to Differential Equations. #### There will be Joint Evening Examinations for All Sections of Math 216, 6-8 p.m., Mon, Oct 11 And Nov 15. Also a Joint Final. Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/courses/216/ 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. There is a weekly computer lab using MATLAB software. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No waitlist – go to department office ### Math. 217. Linear Algebra. #### Instructor(s): Prerequisites & Distribution: Math. 215, 255, or 285. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in 513. (4). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### MATH 256. Applied Honors Calculus IV. #### Section. Prerequisites & Distribution: Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS). Credits: (4). Course Homepage: No Homepage Submitted. No Description Provided Check Times, Location, and Availability ### Math. 256. Applied Honors Calculus IV. #### Section 001, 002. #### Instructor(s): Ralf W Wittenberg (ralf@umich.edu) Prerequisites & Distribution: Math. 255. Credit can be earned for only one of Math. 216, 256, 286, or 316. (4). (MSA). (BS). Credits: (4). Course Homepage: No Homepage Submitted. This course is an introduction to the study of differential equations, and to some linear algebra. More specifically, topics we will cover will include first-order equations (especially linear and separable equations), numerical methods, linear homogeneous and nonhomogeneous second-order equations, systems of differential equations, and the qualitative theory of linear and nonlinear dynamical systems. If time permits, we will also discuss Laplace transforms and/or series solutions of differential equations. Modeling and applications will be stressed throughout. While the emphasis will be on analytical investigations and solutions, we will also be using computer software, especially Maple.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 285. Honors Calculus III. #### Section 002. #### Instructor(s): Robert Megginson (meggin@umich.edu) Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~meggin/math285/index.html Required Text: Calculus by Stewart, fourth edition, Brooks/Cole, 1999. Background and Goals: The sequence Math 185-186-285-286 is the Honors introduction to 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. Content: 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. Alternatives: Math 215 (Calculus III) is a somewhat less theoretical course which covers much of the same material. Subsequent Courses: Math 216 (Introduction To Differential Equations), Math 286 (Honors Differential Equations) or Math 217 (Linear Algebra).  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 285. Honors Calculus III. #### Section 003. #### Instructor(s): Dror Varolin (varolin@umich.edu) Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~varolin/285.html See Math 185 for a general description of the sequence Math 185-186-285-286. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 285. Honors Calculus III. #### Section 004. #### Instructor(s): Robert L Griess Jr (rlg@umich.edu) Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). Credits: (4). Course Homepage: No Homepage Submitted. See Math 185 for a general description of the sequence Math 185-186-285-286. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 289. Problem Seminar. #### Instructor(s): Prerequisites & Distribution: (1). (Excl). (BS). May be repeated for credit with permission. Credits: (1). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 295. Honors Mathematics I. #### Instructor(s): Prerequisites & Distribution: 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). (MSA). (BS). (QR/1). Credits: (4). Course Homepage: No Homepage Submitted. Math 295-296-395-396 is the main Honors calculus sequence. It is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts, problem solving, as well as the underlying theory and proofs of important results. Students interested in taking advanced mathematical courses later should consider seriously starting with this sequence. The expected background is high school trigonometry and algebra (previous calculus not required). This sequence is not restricted to students enrolled in the LS&A Honors Program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, some linear algebra. Math 175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 316. Differential Equations. #### Instructor(s): Prerequisites & Distribution: Math. 215 and 217. Credit can be earned for only one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 333. Directed Tutoring. #### Instructor(s): Prerequisites & Distribution: 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. Credits: (1-3). Course Homepage: No Homepage Submitted. An experiential mathematics course for exceptional upper-level students in the elementary teacher certification program. Students tutor needy beginners enrolled in the introductory courses (Math 385 and Math 489) required of all elementary teachers.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 354. Fourier Analysis and its Applications. #### Instructor(s): Kristen Moore (ksmoore@umich.edu) Prerequisites & Distribution: Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math. 454. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. Fourier Analysis is a powerful tool for solving problems in signal processing, optics, heat conduction, sound propagation, and CAT scanning. This course is an introduction to Fourier analysis at an elementary level, emphasizing applications. The main topics are Fourier series, discrete Fourier transforms, and continuous Fourier transforms. We will spend a substantial portion of the time on some of the scientific and technological applications described above as well as the applications to other branches of mathematics, such as partial differential equations and probability theory. Students will do some computer work using Matlab or Mathematica, interactive programming tools that are easy to use, but no previous experience with computers is necessary.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 371/Engin. 371. Numerical Methods for Engineers and Scientists. #### Instructor(s): Zhong-Hui Duan (zduan@umich.edu) Prerequisites & Distribution: Engineering 101; one of Math. 216, 256, 286, or 316. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/ 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 385. Mathematics for Elementary School Teachers. #### Instructor(s): Eugene Krause (krause@umich.edu) Prerequisites & Distribution: One year each of high school algebra and geometry. No credit granted to those who have completed or are enrolled in 485. (3). (Excl). Credits: (3). Course Homepage: No Homepage Submitted. All elementary teaching certificate candidates are required to take two math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489 in the Winter Term. Due to heavy enrollment pressure, Math 385 will be offered this Spring Term (IIIA 2000) as well. Enrollment is limited to 30 students per section; class-size limitswill be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course. 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. 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. For further information, contact Prof. Krause at his e-mail address, krause@math.lsa.umich.edu.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 395. Honors Analysis I. #### Instructor(s): Prerequisites & Distribution: Math. 296 or permission of the Honors advisor. (4). (Excl). (BS). Credits: (4). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 399. Independent Reading. #### Instructor(s): Prerequisites & Distribution: (1-6). (Excl). (INDEPENDENT). May be repeated for credit. Credits: (1-6). Course Homepage: No Homepage Submitted. Designed especially for Honors students.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 404. Intermediate Differential Equations and Dynamics. #### Instructor(s): Prerequisites & Distribution: Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 286, or 316. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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. WL:2  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 412. Introduction to Modern Algebra. #### Instructor(s): Prerequisites & Distribution: Math. 215, 255, 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). Credits: (3). Course Homepage: No Homepage Submitted. 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 or 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 particular types of mathematical structures such as groups, rings, and fields. 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, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 413. Calculus for Social Scientists. #### Instructor(s): Prerequisites & Distribution: Not open to freshmen, sophomores or mathematics concentrators. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 417. Matrix Algebra I. #### Instructor(s): Prerequisites & Distribution: Three courses beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled Math. 513. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/courses/417/ 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory. #### Section 001, 003. #### Instructor(s): Alexandre I Barvinok (barvinok@umich.edu) Prerequisites & Distribution: Four terms of college mathematics beyond Math 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m419.html 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. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571. Text: Otto Bretscher, Linear Algebra with Applications, Prentice Hall, 1997. Grading: The final grade will be computed from the following: Homework: 25 % Quizzes: 10 % Two midterm exams: 20 % each Final exam: 25 % Homework problem sets will be given once a week, to be turned in the following week. No late homeworks will be accepted. Similarly, quizzes can not be made up, but your worst score will be dropped. Exams: First midterm exam: Thursday, October 12 Second midterm exam: Tuesday, November 14 Final exam: Thursday, December 21  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory. #### Section 002, 004. #### Instructor(s): J Tobias Stafford (jts@umich.edu) Prerequisites & Distribution: Four terms of college mathematics beyond Math 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: No Homepage Submitted. 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. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 419/EECS 400/CS 400. Linear Spaces and Matrix Theory. #### Section 005, 006. #### Instructor(s): Bruce A Kleiner (bkleiner@umich.edu) Prerequisites & Distribution: Four terms of college mathematics beyond Math 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: http://www.math.lsa.umich.edu/~bkleiner/syllabus.html Linear equations, Gauss-Jordan elimination, linear transformations and their inverses, matrix algebra, subspaces, linear independence, bases, orthogonality, Gram-Schmidt, orthogonal transformations and matrices, least squares, determinants, eigenvalues and eigenvectors, coordinate systems, diagonalization, and quadratic forms. Text: Linear Algebra with Applications, by Otto Bretscher The course work includes regular homework assignments, quizzes, 2 midterms, and a final exam. Grading policy. Coursework will be weighted as follows: Homework 25%, quizzes 10%, two midterms 20% each, and the final exam 25%. First midterm: October 12, in class. Second midterm: November 14, in class. Final exam: Section 005: Thursday December 21, 1:30-3:30; Section 006: Thursday December 21, 10:30-12:30.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 423. Mathematics of Finance. #### Instructor(s): Prerequisites & Distribution: Math. 217 and 425; CS 183. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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 reflect observed market features, and to provide the necessary mathematical tools for their analysis and implementation. The course will introduce the stochastic processes used for modeling particular financial instruments. However, the students are expected to have a solid background in basic probability theory. Specific Topics 1. Review of basic probability. 2. The one-period binomial model of stock prices used to price futures. 3. Arbitrage, equivalent portfolios and risk-neutral valuation. 4. Multiperiod binomial model. 5. Options and options markets; pricing options with the binomial model. 6. Early exercise feature (American options). 7. Trading strategies; hedging risk. 8. Introduction to stochastic processes in discrete time. Random walks. 9. Markov property, martingales, binomial trees. 10. Continuous-time stochastic processes. Brownian motion. 11. Black-Scholes analysis, partial differential equation and formula. 12. Numerical methods and calibration of models. 13. Interest-rate derivatives and the yield curve. 14. Limitations of existing models. Extensions of Black-Scholes.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 425/Stat. 425. Introduction to Probability. #### Instructor(s): Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 425/Stat. 425. Introduction to Probability. #### Instructor(s): Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS). Credits: (3). Course Homepage: No Homepage Submitted. See Statistics 425..  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans. #### Section 001. Prerequisites & Distribution: Junior standing. (3). (Excl). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 431. Topics in Geometry for Teachers. #### Section 001 – Axiomatic Foundations of Euclidean and non-Euclidean Geometry. #### Instructor(s): Peter Scott (pscott@umich.edu) Prerequisites & Distribution: Math. 215, 255, or 285. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~pscott/Math431F00.html 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; tessellations; 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 433. Introduction to Differential Geometry. #### Instructor(s): Prerequisites & Distribution: Math. 215, or 255 or 285, and Math. 217(3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 450. Advanced Mathematics for Engineers I. #### Section 001. #### Instructor(s): Alejandro Uribe-Ahumada (uribe@umich.edu) Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS). Credits: (4). Course Homepage: http://www.math.lsa.umich.edu/~uribe/450F00.html 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 450. Advanced Mathematics for Engineers I. #### Section 002. #### Instructor(s): John W Lott (lott@umich.edu) Prerequisites & Distribution: Math. 215, 255, or 285. (4). (Excl). (BS). Credits: (4). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 451. Advanced Calculus I. #### Section 001. #### Instructor(s): Peter L Duren (duren@umich.edu) Prerequisites & Distribution: Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 451. Advanced Calculus I. #### Section 002. #### Instructor(s): Ralf Spatzier (spatzier@umich.edu) Prerequisites & Distribution: Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~spatzier/451/451.html Course Outline:This course will develop calculus rigorously and introduce concepts and ideas important in more advanced mathematics. More specifically, we will discuss the following material: basic logic, number systems, sequences and series, continuity, metric spaces, derivatives and integrals. This a theoretical course with emphasis on precise definitions and proofs both in the lectures and the homework problems. 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. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions. 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. Text: "Elementary Analysis: The Theory of Calculus" by by Kenneth A. Ross, Springer Verlag Course Outline:This course will develop calculus rigorously and introduce concepts and ideas important in more advanced mathematics. More specifically, we will discuss the following material: basic logic, number systems, sequences and series, continuity, metric spaces, derivatives and integrals. This a theoretical course with emphasis on precise definitions and proofs both in the lectures and the homework problems. Grading Policy: homework 40%; midterm 20% each; final exam 20%; Homework Policy: Homework will be assigned weekly and collected on Monday. You may discuss the homework problems with other students, but you should write up the solutions on your own.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 454. Boundary Value Problems for Partial Differential Equations. #### Instructor(s): Timothy Callahan (timcall@umich.edu) Prerequisites & Distribution: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3). (Excl). (BS). Credits: (3). Course Homepage: https://coursetools.ummu.umich.edu/2000/fall/lsa/math/454/001.nsf This course covers methods of solving partial differential equations (e.g., the heat, wave, Helmholtz and Laplace equations) with specified boundary conditions in various geometries. We will cover separation of variables, Fourier series, Bessel functions, spherical harmonics, orthogonal polynomials, Sturm – Liouville theory, eigenfunctions of the Laplacian in several different coordinate systems, conformal mapping, etc. These methods have applications in fields as diverse as mechanics, quantum mechanics, thermodynamics, aerodynamics, finance, electromagnetism, and many others, and we will take our examples from such disciplines.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 471. Introduction to Numerical Methods. #### Section 001. #### Instructor(s): Zhong-Hui Duan (zduan@umich.edu) Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/ 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 471. Introduction to Numerical Methods. #### Section 002. #### Instructor(s): Assen Dontchev L (dontchev@umich.edu) Prerequisites & Distribution: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 481. Introduction to Mathematical Logic. #### Instructor(s): Peter Hinman (pgh@umich.edu) Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~pgh/courses/481/ 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. Text: An Introduction to Mathematical Logic by Richard E. Hodel, PWS Publishing Co. 1995 Grading: 25% homework, 30% midterm exam (Thursday 26 October 7-8:30), 45% final exam.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 485. Mathematics for Elementary School Teachers and Supervisors. #### Instructor(s): Prerequisites & Distribution: One year of high school algebra. No credit granted to those who have completed or are enrolled in 385. (3). (Excl). (BS). May not be included in a concentration plan in mathematics. Credits: (3; 2 in the half-term). Course Homepage: No Homepage Submitted. The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching in or preparing to teach in the elementary school.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 497. Topics in Elementary Mathematics. #### Instructor(s): Prerequisites & Distribution: Math. 489. (3). (Excl). (BS). May be repeated for a total of six credits. Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 501. Applied & Interdisciplinary Mathematics Student Seminar. #### Section 001. #### Instructor(s): Peter S Smereka (psmereka@umich.edu) Prerequisites & Distribution: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). (Excl). Offered mandatory credit/no credit. May be repeated for a total of 6 credits. Credits: (1). Course Homepage: http://www.math.lsa.umich.edu/seminars/applied/index.html The Applied and Interdisciplinary Mathematics(AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. Math 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission. Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 513. Introduction to Linear Algebra. #### Instructor(s): Carolyn Dean (cdean@umich.edu) Prerequisites & Distribution: Math. 412. Two credits granted to those who have completed Math. 214, 217, 417, or 419. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. Math 513 is the Math Department's most complete and rigorous course in linear algebra. The formal prerequisite is Math 412; however, it is recommended that students have some experience with other, more demanding, proof oriented courses. Examples would include Math 451, Math 525, Math 531, or any higher level course. COMPLETION OF THE 90'S SEQUENCE IS ITSELF AN EXCELLENT QUALIFICATION FOR MATH 513. The student body is usually a fairly even mix of Honors Math and CS undergraduates and graduate students from math-related fields. Math 513 is also good for Master's students in Math. The text will be, as usual, Linear Algebra, an Introductory Approach, by Curtis. We will study in depth vector spaces and linear transformations over arbitrary fields. We will also cover bilinear and (elementary) quadratic forms and applications to differential equations. Significant applications will be an important feature of the course. Weekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 520. Life Contingencies I. #### Instructor(s): Curtis Huntington (chunt@umich.edu) Prerequisites & Distribution: Math. 424 and Math. 425. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc. Recommended text: Actuarial Mathematics (Second Editions) by Bowles et al.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 523. Risk Theory. #### Instructor(s): Joseph Conlon (conlon@umich.edu) Prerequisites & Distribution: Math. 425. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc. and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the risk-reward tradeoff. Review of probability. Bachelier and Lundberg models of investment and loss aggregation. Fallacy of time diversification and its generalizations. Geometric Brownian motion and the compound Poisson process. Modeling of individual losses which arise in a loss aggregation process. Distributions for modeling size loss, statistical techniques for fitting data, and credibility. Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory. The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the Black-Scholes option pricing model.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 525/Stat. 525. Probability Theory. #### Instructor(s): Prerequisites & Distribution: Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 532. Topics in Discrete and Applied Geometry. #### Section 001 – Crystals and Quasicrystals #### Instructor(s): Dan Burns (dburns@umich.edu) Prerequisites & Distribution: One of Math 217, 417, 419 or 513. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. Crystalline patterns have always been observed in nature. This kind of regularity has been expressed mathematically in terms of symmetry groups operating on space preserving (periodic) lattice configurations, e.g., of molecule sites. In the mid-1980's the adequacy of this point of view was challenged by the discovery of materials (dubbed "quasicrystals". which appeared crystalline in nature, but which violated certain basic restrictions derived from the group-symmetry paradigm (existence of "forbidden" five-fold point symmetry). In this course we propose to study first the basics of symmetry groups in geometry and especially crystallographic groups and their relationship to physical crystals, in particular, the symmetry restrictions on lattices in Euclidean three-space, and at least an introduction to the classification of crystallographic symmetries in two and three space dimensions. The middle third of the course, roughly, will treat the rudiments of Fourier analysis and the theory of x-ray crystallography. We will use these in order to examine and evaluate computer simulations of diffraction patterns and experiments. Time permitting, we will also discuss some relatively current issues in x-ray crystallography related to protein crystallography. Finally we will discuss aperiodic phenomena in tilings, examine what their regularities are and discuss some very open issues: Are these really a candidate for the modeling of quasicrystals in nature? Are there finite sets of local rules by which one may know how to build up a Penrose pattern to cover the whole plane? How does one classify the set of Penrose tilings geometrically and what information does such a tiling carry? The course will involve lectures, regular problem sets and a term project, but no exams. In addition, there will be computer simulations to do (packages provided). The texts will be: "Groups and Symmetry", M.A. Armstrong (Springer), "Quasicrystals and Geometry", by M. Senechal (Cambridge), "Miles of Tiles", by C. Radin (American Mathematical Society), as well as some notes on Fourier analysis, and the book "Principles of Protein X-Ray Crystallography", by J. Drenth (Springer) [not required]. Web-based materials will also be used. Background prerequisites will be flexible. The course is suitable for undergraduates with a background of calculus, linear algebra and, perhaps, the rudiments of groups, and graduate students in mathematics or areas of possible application: chemistry, physics, engineering and biology. If in doubt, contact the instructor.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 537. Introduction to Differentiable Manifolds. #### Instructor(s): John Lott (lott@umich.edu) Prerequisites & Distribution: Math. 513 and 590. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. This course will be a painless introduction to differential topology and differential geometry, meaning the study of spaces and their curvatures. It is the first part of a two-semester sequence. The material in this course is crucial for students who wish to study differential geometry, topology, algebraic geometry, several complex variables, Lie groups and dynamical systems. It is also relevant for other branches of mathematics, such as partial differential equations. We'll start out by doing calculus on manifolds, introducing and using differential forms. We'll prove Stokes' theorem for compact oriented manifolds-with-boundary. We'll also define the de Rham cohomology groups of a manifold and prove their basic properties. Then I'll spend some time on Morse theory. This theory shows how, given a generic function on a manifold, one obtains a decomposition of the manifold into simple building blocks called "handles''. Morse theory is a basic tool in topology and was used in Smale's famous proof of the Poincare conjecture in more than four dimensions, although we will not go into this. Finally, we'll cover some basic Riemannian geometry, including Riemannian metrics, Levi-Civita connections, geodesics and curvature. Homework assignments will be given periodically, with the frequency depending on whether or not we get a grader. There will also be a final exam. The textbooks will be "Differential Topology" by Victor Guillemin and Alan Pollack, Prentice-Hall, and "Morse Theory" by John Milnor, Princeton University Press. Math 591 or the equivalent is a prerequisite. I will assume a knowledge of differentiable manifold theory as covered in Sections 1.1-1.4 of the book by Guillemin and Pollack. The titles of these sections are "Definitions", "Derivatives and tangents", "The inverse function theorem and immersions" and "Submersions". If a prospective student has not seen this material before, it might be helpful to look at Chapter 1 of Guillemin and Pollack. I will review this material at the beginning of the academic term.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 555. Introduction to Functions of a Complex Variable with Applications. #### Instructor(s): Berit Stensønes (berit@umich.edu) Prerequisites & Distribution: Math. 450 or 451. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. Text: Complex Variables and Applications, 5th ed. (Churchill and Brown); Student Body: largely engineering and physics graduate students with some math and engineering undergrads Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable. Concepts and calculations are emphasized over proofs. Content: Differentiation and integration of complex valued functions of a complex variable, series mappings, residues, applications. Evaluation of improper real integrals. This corresponds to Chapters 1-9 of Churchill. Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than calculations. Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields. There will be homework, midterm and a final.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 556. Methods of Applied Mathematics I. #### Instructor(s): Charles Doering (doering@umich.edu) Prerequisites & Distribution: Math. 217, 419, or 513; 451 and 555. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. This is an introduction to analytical methods for initial value problems and boundary value problems. This course should be useful to students in mathematics, physics, and engineering. We will begin with systems of ordinary differential equations. Next, we will study Fourier Series, Sturm-Liouville problems, and eigenfunction expansions. We will then move on to the Fourier Transform, Riemann-Lebesgue Lemma, inversion formula, the uncertainty principle and the sampling theorem. Next we will cover distributions, weak convergence, Fourier transforms of tempered distributions, weak solution of differential equations and Green's functions. We will study these topics within the context of the heat equation, wave equation, Schrodinger's equation, Laplace's equation. Text: Fourier Analysis and its Applications by G.B. Folland Grading: homework 60%, midterm 15%, final exam 25%. Homework is key in this class. You are expected to hand in carefully completed homework.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I. #### Section 001. Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 562/IOE 511/Aero. 577. Continuous Optimization Methods. #### Section 001. Prerequisites & Distribution: Math. 217, 417, or 419. (3). (Excl). (BS). CAEN lab access fee required for non-Engineering students. Credits: (3). Lab Fee: CAEN lab access fee required for non-Engineering students. Course Homepage: No Homepage Submitted. 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.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given. ### Math. 565. Combinatorics and Graph Theory. #### Instructor(s): Sergey Fomin (fomin@umich.edu) Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS). Credits: (3). Course Homepage: http://www.math.lsa.umich.edu/~fomin/565.html Student work expected: several problem sets. Synopsis: The ultimate fun course, with a focus on problem solving, showcasing the gems of enumerative and algebraic combinatorics. The course will cover a dozen of virtually independent topics, chosen solely on the basis of their beauty. Topics will include generating functions, algebraic graph theory, partially ordered sets, combinatorics of polytopes, matching theory, enumeration of tilings, partitions, and Young tableaux. Reference texts (none required): [BS] A.Bjorner and R.P.Stanley, A combinatorial miscellany, Cambridge University Press, to appear. [GS] I.Gessel and R.P.Stanley, Algebraic enumeration, in Handbook of Combinatorics, MIT Press, 1995. [vW] J.H. van Lint and R.M.Wilson, A course in combinatorics , Cambridge University Press, 1996. [EC] R.P.Stanley, Enumerative combinatorics, vol.1-2, Cambridge University Press, 1997-1999. Potential topics to be covered:  Hooklength formula. De Bruijn sequences. Enumeration of trees. Stirling numbers. Spectra of graphs. Walks on a cube. Sperner theory. Inversions and major index. q-binomial coefficients. Distributive lattices. Gaussian coefficients. Tableaux and involutions. Schensted's correspondence. Catalan numbers. Matrix-tree theorem. Eulerian tours. Domino tilings. Polya theory. The marriage theorem. Assignment polytope. Cyclic polytopes. Permutohedra.  Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: 2 ### Math. 571. Numerical Methods for Scientific Computing I. #### Instructor(s): Smadar Karni (karni@umich.edu) Prerequisites & Distribution: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3). (Excl). (BS). Credits: (3). Course Homepage: No Homepage Submitted. Prerequisites: linear algebra on the level of Math 419 or Math 513, working knowledge of computer programming (any language). Background: This course deals with numerical methods for solving linear systems of equations ($Ax{=}b$) and the eigenvalue problem ($Ax{=}\lambda x\$). A main motivation will be systems that arise from discretizing elliptic boundary value problems by finite-difference and variational methods, in one and two space dimensions. Material will include chapters 1-6 of Ciarlet's book and Briggs' notes. Additional topics (e.g., least squares problem, conjugate gradient method, GMRES) may be presented if there is enough time.

Alternatives: There is no real alternative. Math 471 (Introduction to numerical methods) covers a small part of the same material at a lower level. Math 571 and 572 may be taken in either order.

Subsequent Courses: Math 671 (Analysis of numerical methods I) is an advanced topics course in numerical analysis. Topics vary.

Texts:

1. Introduction to Numerical Linear Algebra and Optimisation, by P. G. Ciarlet, Cambridge University Press.
2. A Multigrid Tutorial, by W. L. Briggs, SIAM.
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### Math. 575. Introduction to Theory of Numbers I.

#### Instructor(s): Trevor Wooley (wooley@umich.edu)

Prerequisites & Distribution: Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (1, 3). (Excl). (BS).

Credits: (1, 3).

Course Homepage: No Homepage Submitted.

Prerequisites: Basic analysis and modern algebra (equivalent to the level of Math 451 and 412)

Text: Class notes and problem sheets will be self-contained and comprehensive. The standard source should be: an Introduction to the Theory of Numbers (Niven, Zuckerman and Montgomery, 5th edition, Wiley, 1991), or failing that: Introduction to Number Theory (L.-K. Hua, Springer-Verlag, 1982).

Interested in graduate study in Number Theory? Heard about Public Key Cryptosystems and want to find out what makes them tick? Intrigued by number theory after hearing about Fermat's Last Theorem? Then Math 575 is the course for you!

Math 575 has recently been revised with the intention of providing graduate students with a solid introduction to Number Theory suitable for continuing through the subsequent courses in the graduate program in Number Theory here at Michigan. As such, it will also provide a means for undergraduates interested in Number Theory to prepare for graduate study elsewhere. Graduate students not directly interested in Number Theory will be able to complete their renaissance education by hearing one of the epic tales of mathematical conquest on a Homeric scale.

Students wishing to take the course should already have significant experience in writing proofs, and should have a basic understanding of analysis and abstract algebra (groups, rings, fields).

Content: We begin with a reasonably brisk discussion of the basic notions: Euclidean algorithm, primes and unique factorisation, congruences, Chinese Remainder Theorem (Public Key Cryptosystems), primitive roots, quadratic reciprocity and binary quadratic forms. The second half of the course is devoted to topics (as time permits) which lead to substantial lines of research covered in subsequent graduate courses: diophantine equations, quadratic fields, p-adic numbers, elliptic curves, diophantine approximation and transcendence, arithmetic functions, continued fractions, distribution of prime numbers.

Coursework: Approximately one assignment every two weeks, containing both easy and challenging questions, together with 2 straightforward in-class midterms and a take-home final exam (4 days available to complete this). Anyone wishing to discuss the course, or Number Theory in general is welcome to talk with me.

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### Math. 590. Introduction to Topology.

#### Instructor(s): Bryan Mosher (bmosher@umich.edu)

Prerequisites & Distribution: Math. 451. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math590/

Text: Sieradski, An Introduction to Topology and Homotopy, PWS-Kent.

Topology provides a foundation for many areas of mathematics and is itself an active area of research. This course is an introduction to the subject and will emphasize the construction of proofs.

Topics include metric spaces, abstract topological spaces, continuous functions, connectedness, compactness, the fundamental group and surfaces.

The student will engage in brief presentations, problem sets, a midterm and a final exam.

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### Math. 591. General and Differential Topology.

#### Instructor(s): Peter Scott (pscott@umich.edu)

Prerequisites & Distribution: Math. 451. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pscott/Math591F00.html

This course will cover the prerequisites for the portions of the topology qualifying examination, which involve point-set and differential topology. We will start by introducing abstract topological spaces and their basic properties. We will examine in detail the properties of connectedness and compactness. Then we will focus on the quotient topology, group actions and orbit spaces. The course will end by studying manifolds and differential topology, where topics covered will include tangent spaces, the regular value theorem. Whitney's embedding theorem and transversality. Students with a strong background in point-set and differential topology may want to consider taking Math 537 instead.

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### Math. 593. Algebra I.

#### Instructor(s): Igor Dolgachev (idolga@umich.edu)

Prerequisites & Distribution: Math. 513. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~idolga/teach.html

Prerequisites: First courses in abstract algebra and linear algebra (Math 513 or 419 and Math 512).

This is the first part of a two-semester course in basic algebra. The main topic is linear algebra of modules over any ring. We shall classify modules over principal ideal domains and deduce from this the theory of Jordan forms of matrices over a field. Other topics include multi-linear algebra (tensors and exterior algebra), structure of symmetric bilinear forms over arbitrary fields, orthogonal groups, Clifford algebras, elements of homological algebra.

The work will be evaluated on the basis of homework problem solutions and one final exam.

Textbook: S. Lang Algebra, 3d edition. Addison-Wesley 1993. We plan to cover Chapters 3,4,13,14,15,16,19,20 from this book.

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### Math. 596. Analysis I.

#### Instructor(s): Dan Burns (dburns@umich.edu)

Prerequisites & Distribution: Math. 451. Students with credit for Math. 555 may elect Math 596 for two credits only. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Complex numbers, geometric properties, stereographic projection, basic spherical geometry. Complex functions, differentiability and the Cauchy-Riemann equations, the Laplace operator, elementary analytic functions and linear fractional transformations, construction of conformal mappings.

Contour integrals, Cauchy's theorem and the Cauchy integral formula, Taylor series and Laurent expansions, Liouville's theorem, unique continuation, Morera's theorem. Residue theorem and applications. Analytic continuation and Schwarz reflection principle.

Argument principle, Rouche's theorem, Hurwitz's theorem, local univalence. Maximum modulus theorem, the Schwarz lemma and some generalizations. Harmonic functions, Poisson formula and Jensen's thoerem.

Meromorphic functions, Mittag-Leffler's theorem, infinite products, Weierstrass' theorem, removeable singularities, Caserati-Weierstrass theorem.

Normal families, Montel's theorem, Riemann mapping theorem. There will be approximately weekly problem sets, two mid-term exams and a final. A written term project can be done in lieu of the second midterm exam.

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