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# Courses in Mathematics

This page was created at 4:40 PM on Fri, Mar 22, 2002.

## Winter Academic Term, 2002 (January 7 - April 26)

Open courses in Mathematics
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Wolverine Access Subject listing for MATH

Winter Academic Term '02 Time Schedule for Mathematics.

### MATH 412. Introduction to Modern Algebra.

#### Instructor(s):

Prerequisites: Math. 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in Math. 512. Students with credit for Math. 312 should take Math. 512 rather than 412. One credit granted to those who have completed Math. 312. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 416. Theory of Algorithms.

#### Instructor(s):

Prerequisites: Math. 312 or 412 or CS 203, and CS 281. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 417. Matrix Algebra I.

#### Instructor(s):

Prerequisites: 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).

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.

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### MATH 419. Linear Spaces and Matrix Theory.

#### Instructor(s):

Prerequisites: 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).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 422 / BE 440. Risk Management and Insurance.

#### Instructor(s):

Prerequisites: Math. 115, junior standing, and permission of instructor. (3).

Credits: (3).

Course Homepage: No homepage submitted.

We will explore how much insurance affects the lives of students (automobile insurance, social security, health insurance, theft insurance) as well as the lives of other family members (retirements, life insurance, group insurance). While the mathematical models are important, an ability to articulate why the insurance options exist and how they satisfy the customer's needs are equally important. In addition, there are different options available (e.g., in social insurance programs) that offer the opportunity of discussing alternative approaches.

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### MATH 423. Mathematics of Finance.

#### Instructor(s): Joseph Conlon (conlon@umich.edu)

Prerequisites: Math. 217 and 425; CS 183. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math423/index.html

Prerequisites: A solid background in probability theory at the 400 level, math 425 or equivalent.

Required Text: Options, Futures and Other Derivatives by Hull, fourth edition, Prentice Hall 1999.

Background and Goals: This course is an introduction to the mathematical models used in finance and economics with particular emphasis on models for pricing derivative instruments such as options and futures. The goal is to understand how the models derive from basic principles of economics, and to provide the necessary mathematical tools for their analysis. A solid background in basic probability theory is necessary.

Contents:

1. Forwards and Futures, Hedging using Futures, Bills and Bonds, Swaps, Perfect Hedges.
2. Options-European and American, Trading Strategies, Put-Call Parity, Black-Scholes formula.
3. Volatility, methods for estimating volatility-exponential, GARCH, maximum likelihood. (d) Dynamic Hedging, stop-loss, Black-Scholes, the Greek letters.
4. Other Options.

Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam. There will be 8 homework assignments. Each quiz will consist of a slightly modified homework problem.

• 8 quizzes= 8x10=80 points
• midterm= 60 points
• final= 80 points
• Total= 220 points

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### MATH 424. Compound Interest and Life Insurance.

#### Instructor(s):

Prerequisites: Math. 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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

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### MATH 425 / STATS 425. Introduction to Probability.

#### Instructor(s):

Prerequisites: Math. 215, 255, or 285. (3).

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.

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### MATH 425 / STATS 425. Introduction to Probability.

#### Instructor(s): Alexander Barvinok (barvinok@umich.edu)

Prerequisites: Math. 215, 255, or 285. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m425.html

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.

Text: D. Stirzaker, Probability and Random Variables. A beginner's guide, Cambridge University Press, reprinted with corrections in 2001.

Grading: The final grade will be computed from the following:

• First midterm exam: 20 %
• Second midterm exam: 20 %
• Final exam: 30 %
• Homework: 30 %

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### MATH 425 / STATS 425. Introduction to Probability.

#### Instructor(s):

Prerequisites: Math. 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.004.

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### MATH 427 / HB 603. Retirement Plans and Other Employee Benefit Plans.

#### Instructor(s):

Prerequisites: Junior standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 450. Advanced Mathematics for Engineers I.

#### Instructor(s): Brent Carswell (carswell@umich.edu)

Prerequisites: Math. 215, 255, or 285. (4).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~carswell/math450/

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### MATH 450. Advanced Mathematics for Engineers I.

#### Instructor(s): Buckley

Prerequisites: Math. 215, 255, or 285. (4).

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; and 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.

Text: Advanced Engineering Mathematics, 8th edition, Edward Kreyszig Wiley.

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### MATH 451. Advanced Calculus I.

#### Instructor(s):

Prerequisites: Math. 215 and one course beyond Math. 215; or Math. 255 or 285. Intended for concentrators; other students should elect Math. 450. (3).

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; and 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, Kenneth Ross, Springer-Verlag.

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### MATH 452. Advanced Calculus II.

#### Instructor(s):

Prerequisites: Math. 217, 417, or 419; and Math. 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course does a rigorous development of multivariable calculus and elementary function theory with some view towards generalizations. Concepts and proofs are stressed. This is a relatively difficult course, but the stated prerequisites provide adequate preparation. Topics include:

1. partial derivatives and differentiability;
2. gradients, directional derivatives, and the chain rule;
3. implicit function theorem;
4. surfaces, tangent plane;
5. max-min theory;
6. multiple integration, change of variable, etc.; and
7. Green's and Stokes' theorems, differential forms, exterior derivatives.

Math 551 is a higher-level course covering much of the same material with greater emphasis on differential geometry. Math 450 covers the same material and a bit more with more emphasis on applications, and no emphasis on proofs. Math 452 is prerequisite to Math 572 and is good general background for any of the more advanced courses in analysis (Math 596, 597) or differential geometry or topology (Math 537, 635).

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### MATH 454. Boundary Value Problems for Partial Differential Equations.

#### Instructor(s): Smereka

Prerequisites: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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

There is no textbook listed for this course.

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### MATH 454. Boundary Value Problems for Partial Differential Equations.

#### Instructor(s): Ralf Wittenberg

Prerequisites: Math. 216, 256, 286, or 316. Students with credit for Math. 354 can elect Math. 454 for one credit. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~ralf/math454/index.html

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

Text: Richard Haberman, "Elementary Applied Partial Differential Equations : with Fourier Series and Boundary Value Problems" (3rd edition), Prentice-Hall (1998).

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### MATH 462. Mathematical Models.

#### Instructor(s): Patrick Nelson (pwn@umich.edu)

Prerequisites: Math. 216, 256, 286, or 316; and 217, 417, or 419. Students with credit for Math. 362 must have department permission to elect Math. 462. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math462.html

This course will cover biological models constructed from difference equations and ordinary differential equations. Applications will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both exact solutions and simple qualitative methods for understanding dynamical systems will be stressed (anticipated text is Mathematical Models in Biology by Leah Edelstein-Keshet).

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s): Mark Dickinson (dickinsm@umich.edu)

Prerequisites: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/471w02/

Course description: The aim of this course is to present a variety of basic techniques and algorithms used to solve numerical problems arising mainly from science and engineering. We will learn to implement these methods using MATLAB, and we will also examine issues of accuracy, stability and efficiency for selected methods. The main topics to be covered in this course are: numerical linear algebra, one-dimensional root-finding, polynomial and spline interpolation, quadrature, and numerical solution of ordinary differential equations.

Prerequisites: You should be familiar with basic ideas from calculus, linear algebra and ordinary differential equations; Math 216 and Math 217 provide sufficient background for linear algebra and differential equations. You should also have had some experience of programming in a standard procedural programming language. Please talk to me if you are unsure that you satisfy all of these requirements.

Assessment: The final grade will be based on the coursework along with the midterm and final exams, in the following proportions:

• Coursework: 40%
• Midterm exam: 25%
• Final exam: 35%

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s): Ralf Wittenberg (ralf@umich.edu)

Prerequisites: Math. 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~ralf/math471/index.html

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 proven. 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, and the 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.

Text: Richard L. Burden and J. Douglas Faires, Numerical Analysis (7th edition), Brooks/Cole (2001).

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### MATH 475. Elementary Number Theory.

#### Instructor(s):

Prerequisites: At least three terms of college mathematics are recommended. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is an elementary introduction to number theory, especially congruence arithmetic. Number theory is one of the few areas of mathematics in which problems easily describable to a layman (is every even number the sum of two primes?) have remained unsolved for centuries. Recently some of these fascinating but seemingly useless questions have come to be of central importance in the design of codes and cyphers. The methods of number theory are often elementary in requiring little formal background. In addition to strictly number-theoretic questions, concrete examples of structures such as rings and fields from abstract algebra are discussed. Concepts and proofs are emphasized, but there is some discussion of algorithms which permit efficient calculation. Students are expected to do simple proofs and may be asked to perform computer experiments. Although there are no special prerequisites and the course is essentially self-contained, most students have some experience in abstract mathematics and problem solving and are interested in learning proofs. A Computational Laboratory (Math 476, 1 credit) will usually be offered as an optional supplement to this course. Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity, and quadratic fields. Math 575 moves much faster, covers more material, and requires more difficult exercises. There is some overlap with Math 412 which stresses the algebraic content. Math 475 may be followed by Math 575 and is good preparation for Math 412. All of the advanced number theory courses, Math 675, 676, 677, 678, and 679, presuppose the material of Math 575, although a good student may get by with Math 475. Each of these is devoted to a special subarea of number theory.

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### MATH 476. Computational Laboratory in Number Theory.

#### Instructor(s):

Prerequisites: Prior or concurrent enrollment in Math. 475 or 575. (1).

Credits: (1).

Course Homepage: No homepage submitted.

Students will be provided software with which to conduct numerical explorations. Students will submit reports of their findings weekly. No programming necessary, but students interested in programming will have the opportunity to embark on their own projects. Participation in the laboratory should boost the student's performance in Math 475 or Math 575. Students in the lab will see mathematics as an exploratory science (as mathematicians do). Students will gain a knowledge of algorithms which have been developed (some quite recently) for number-theoretic purposes, e.g., for factoring. No exams.

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### MATH 485. Mathematics for Elementary School Teachers and Supervisors.

#### Instructor(s): Eugene F Krause

Prerequisites: One year of high school algebra. No credit granted to those who have completed or are enrolled in Math. 385. (3). May not be included in a concentration plan in mathematics. Does not apply to any math degree programs.

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 or preparing to teach in the elementary school.

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### MATH 486. Concepts Basic to Secondary Mathematics.

#### Instructor(s): Eugene F Krause

Prerequisites: Math. 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is designed for students who intend to teach junior high or high school mathematics. It is advised that the course be taken relatively early in the program to help the student decide whether or not this is an appropriate goal. Concepts and proofs are emphasized over calculation. The course is conducted in a discussion format. Class participation is expected and constitutes a significant part of the course grade. Topics covered have included problem solving; sets, relations, and functions; the real number system and its subsystems; number theory; probability and statistics; difference sequences and equations; interest and annuities; algebra; and logic. This material is covered in the course pack and scattered points in the text book. There is no real alternative, but the requirement of Math 486 may be waived for strong students who intend to do graduate work in mathematics. Prior completion of Math 486 may be of use for some students planning to take Math 312, 412, or 425.

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### MATH 489. Mathematics for Elementary and Middle School Teachers.

#### Instructor(s): Eugene F Krause

Prerequisites: Math. 385 or 485. May not be used in any graduate program in mathematics. (3). Not to apply on any graduate program in mathematics.

Credits: (3).

Course Homepage: No homepage submitted.

All elementary teaching certificate candidates are required to take two mathematics courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall, Math 489 in the Winter. The next Spring Term offering of Math 489 will be in 2003. For further information about future course offerings, contact Prof. Krause at 763-1186 or at his office, 3086 East Hall. This course, together with its predecessor Math 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.

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### MATH 490. Introduction to Topology.

#### Instructor(s):

Prerequisites: Math. 412 or 451 or equivalent experience with abstract mathematics. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course in an introduction to both point-set and algebraic topology. Although much of the presentation is theoretical and proof-oriented, the material is well-suited for developing intuition and giving convincing proofs which are pictorial or geometric rather than completely rigorous. There are many interesting examples of topologies and manifolds, some from common experience (combing a hairy ball, the utilities problem). In addition to the stated prerequisites, courses containing some group theory (Math 412 or 512) and advanced calculus (Math 451) are desirable although not absolutely necessary. The topics covered are fairly constant but the presentation and emphasis will vary significantly with the instructor. These include point-set topology, examples of topological spaces, orientable and non-orientable surfaces, fundamental groups, homotopy, and covering spaces. Metric and Euclidean spaces are emphasized. Math 590 is a deeper and more difficult presentation of much of the same material which is taken mainly by mathematics graduate students. Math 433 is a related course at about the same level. Math 490 is not prerequisite for any later course but provides good background for Math 590 or any of the other courses in geometry or topology.

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### MATH 498. Topics in Modern Mathematics.

#### Instructor(s): Divakar Viswanath

Prerequisites: Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Markov chains form an elegant topic with numerous applications. In the first half of the course, we will discuss basic questions: What is a Markov chain? What is its eventual behaviour? How fast does it converge? The second half will make the following relatively advanced topics accessible:

1. How many times should you shuffle a deck of cards to make the deck completely random? The answer depends upon how you shuffle.
2. Can you read the encoded message (look for a flier near you) exchanged between inmates of a Texan penitentiary? Something called the Metropolis method was used to decode it.
3. Ising models and phase transitions, as when ice melts into water.
4. Random matrix theory, if time permits.

The final project has to make a connection to any area you like. Your options will be plenty: math, physics, compsci, econ, biology, and chemistry are all fair game.

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### MATH 499. Independent Reading.

#### Instructor(s):

Prerequisites: Graduate standing in a field other than mathematics. (1-4). (INDEPENDENT).

Credits: (1-4).

Course Homepage: No homepage submitted.

This course is intended for graduate students in fields other than mathematics who require mathematical skills not otherwise available though existing courses.

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### MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.

#### Instructor(s): Peter Miller (millerpd@umich.edu)

Prerequisites: At least two 300 or above level math courses, and graduate standing; Qualified undergraduates with permission of instructor only. (1). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.

Credits: (1).

Course Homepage: http://www.math.lsa.umich.edu/~millerpd/Courses/501.html

No Description Provided

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### MATH 512. Algebraic Structures.

#### Instructor(s):

Prerequisites: Math. 451 or 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 513. Introduction to Linear Algebra.

#### Instructor(s): Mahdi Asgari (asgari@umich.edu)

Prerequisites: Math. 412. Two credits granted to those who have completed Math. 214, 217, 417, or 419. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~asgari/ma513_wint02.html

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

Text: Linear Algebra, An Introductory Approach, Charles Curtis, Springer-Verlag.

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### MATH 521. Life Contingencies II.

#### Instructor(s):

Prerequisites: Math. 520. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 523. Risk Theory.

#### Instructor(s):

Prerequisites: Math. 425. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 525 / STATS 525. Probability Theory.

#### Instructor(s): Alexander Barvinok (barvinok@umich.edu)

Prerequisites: Math. 450 or 451. Students with credit for Math. 425/Stat. 425 can elect Math. 525/Stat. 525 for only one credit. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/m525.html

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.

Text:
G. Grimmett and D. Stirzaker, Probability and Random Processes, Cambridge University Press, third edition, 2001.

Grading:

The final grade will be computed from the following:

1. First midterm exam: 20 %
2. Second midterm exam: 20 %
3. Final exam: 30 %
4. Homework: 30 %

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### MATH 526 / STATS 526. Discrete State Stochastic Processes.

#### Instructor(s):

Prerequisites: Math. 525 or EECS 501. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 526 / STATS 526. Discrete State Stochastic Processes.

#### Instructor(s):

Prerequisites: Math. 525 or EECS 501. (3).

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 526.001.

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### MATH 531. Transformation Groups in Geometry.

#### Instructor(s):

Prerequisites: Math. 215, 255, or 285. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 547 / STATS 547. Probabilistic Modeling in Bioinformatics.

#### Instructor(s):

Prerequisites: Math. 425 or Biol. 427 or Biol. Chem. 415; basic programming skills desirable. Graduate standing and permission of instructor. (3).

Credits: (3).

Course Homepage: http://www.stat.lsa.umich.edu/~kshedden/Courses/Stat547/

See Statistics 547.001.

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### MATH 548 / STATS 548. Computations in Probabilistic Modeling in Bioinformatics.

#### Instructor(s): Carlos Santos

Prerequisites: Math. 425 or Biol. 427 or Biol. Chem. 415; basic programming skills desirable. Graduate standing and permission of instructor. (1).

Credits: (1).

Course Homepage: http://www.bioinformatics.med.umich.edu/%7Ebioinfo548/index.html

See Statistics 548.001.

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### MATH 550 / CMPLXSYS 510. Introduction to Adaptive Systems.

#### Instructor(s): Carl P Simon (cpsimon@umich.edu)

Prerequisites: Math. 215, 255, or 285; Math. 217; and Math. 425, and Permission of instructor. Working knowledge of calculus, probability, and matrix algebra. (3).

Credits: (3).

Course Homepage: http://precisione.physics.lsa.umich.edu/CSCS/education/CSCS-courses/cscs510-w01.html

See Complex Systems 510.001.

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### MATH 555. Introduction to Functions of a Complex Variable with Applications.

#### Instructor(s):

Prerequisites: Math. 450 or 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 557. Methods of Applied Mathematics II.

#### Instructor(s):

Prerequisites: Math. 217, 419, or 513; 451 and 555. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 558. Ordinary Differential Equations.

#### Instructor(s):

Prerequisites: Math. 450 or 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 559. Selected Topics in Applied Mathematics.

#### Instructor(s): Trachette Jackson (tjacks@umich.edu)

Prerequisites: Math. 451; and 217 or 419. (3). May be repeated for a total of six credits.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~tjacks/559f.ps

Natural systems behave in a way that reflects an underlying spatial pattern. For example, on the molecular level, rarely do reactions occur in a homogenous environment and the spatial organization does somehow influence the way in which particles interact. In this course, we will discover the way in which spatial variation influences the motion, dispersion, and persistence of species. We shall become aware of the fine balance that exists between interdependent species and demonstrate that spatial diversity can have subtle, but important effects or can lead to the emergence of remarkable spatial patterns from a previously uniform state. The concepts underlying spatially dependent processes and the partial differential equations which model them will be discussed in a general manner with examples taken from the molecular, cellular, and population levels. We will then apply these ideas to more specific cases with the aim of understanding interesting biological phenomena. Topics include: Population dispersal based on diffusion models; Cell movements (e.g., chemotaxis and haptotaxis); Growth of branching organisms; Traveling waves in microorganisms; Transport of biological substances; Models for development and pattern formation; and Age-Structured models of HIV dynamics.

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### MATH 561 / IOE 510 / SMS 518. Linear Programming I.

#### Instructor(s):

Prerequisites: Math. 217, 417, or 419. (3). 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.

No Description Provided

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### MATH 567. Introduction to Coding Theory.

#### Instructor(s): Harm Derksen (hdersken@umich.edu)

Prerequisites: One of Math. 217, 419, 513. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math567.html

No Description Provided

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### MATH 571. Numerical Methods for Scientific Computing I.

#### Instructor(s):

Prerequisites: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 572. Numerical Methods for Scientific Computing II.

#### Instructor(s): Robert Krasny (krasny@umich.edu)

Prerequisites: Math. 217, 417, 419, or 513; and one of Math. 450, 451, or 454. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math572.html

Math 572 is an introduction to numerical methods used in solving di eren- tial equations. The course will focus on finite-difference schemes for ordinary and partial differential equations. Theoretical concepts and practical computing issues will be covered.

Prerequisites.
Advanced calculus, linear algebra, complex variables. Math 571 is not a prerequisite.

Texts.
Numerical Initial Value Problems in Ordinary Differential Equations, by C. W. Gear, Prentice-Hall (available as a coursepack in the Michigan Union Bookstore)

Numerical Solution of Partial Differential Equations, by K. W. Morton and D. F. Mayers, Cambridge University Press

Syllabus.
ODEs: Euler's method, asymptotic expansion of the error, Richardson extrapolation, Runge-Kutta methods, multistep methods, leap-frog method, consistency, stability, con- vergence, root condition, absolute stability, stiff systems, A-stability PDEs: heat equation, wave equation, Laplace equation, nite-di erence schemes, artificial viscosity, Crank-Nicolson, Lax-Wendroff, operator splitting, stability analysis, maximum principle, energy method, discrete Fourier analysis, CFL condition, Lax equivalence theorem, Kriess matrix theorem, discontinuous solutions, Gibbs phenomenon, trigonometric interpolation, pseudospectral method, nonlinear equations

Course Grade.
The course grade will be based on homework (30%), a midterm exam (30%), and a final exam (40%). The homework will include programming exercises for which I recommend using Matlab.

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### MATH 582. Introduction to Set Theory.

#### Instructor(s):

Prerequisites: Math. 412 or 451 or equivalent experience with abstract mathematics. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 592. Introduction to Algebraic Topology.

#### Instructor(s):

Prerequisites: Math. 591. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 594. Algebra II.

#### Instructor(s):

Prerequisites: Math. 593. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 597. Analysis II.

#### Instructor(s):

Prerequisites: Math. 451 and 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 604. Complex Analysis II.

#### Instructor(s):

Prerequisites: Math. 596. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 615. Commutative Algebra II.

#### Instructor(s): Ben Richert (brichert@umich.edu)

Prerequisites: Math. 614 and Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~brichert/class/615.html

Course description: Commutative algebra is the study of commutative rings and their modules. As well as providing the foundation for algebraic geometry, complex analytic geometry, and algebraic number theory, this field has developed into a beautiful and deep theory in its own right, with applications for nearly every algebraist. Algebraic geometers, number theorists, algebraic combinatorialists, lie theorists, and non-commutative algebraists, among others, find it useful. Math 615 is a follow-up on the first course in commutative algebra (614), and our first order of business will be to continue with the topics basic to such a study. These topics will include depth, Cohen-Macaulay rings, Tor and Ext (briefly), Koszul complexes, injective resolutions, regular rings, Gorenstein rings, excellent rings, the structure of complete local rings, etale maps, and possibly Henselian rings. If time permits, we will also study a special topic in the latter portion of the semester. The description must necessarily be vague, but I hope to consider the theory of Hilbert functions (more carefully than in 614). This should include Macaulay's theorem (on maximal Hilbert function growth), Gotzmann's persistence theorem (relating maximal growth to depth), and explore something of the computational flavor of the theory. I also hope to present several open problems which concern questions about Hilbert functions and more generally, about free resolutions.

Prerequisites: Math 614 or consent of instructor.

Text: Cohen-Macaulay Rings, Revised Edition, Bruns and Herzog.

Coursework: Occasional homework assignments (perhaps up to 4).

Grades: Grades will be based on the homework. Performance in class may be taken into account.

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### MATH 619. Topics in Algebra.

#### Instructor(s):

Prerequisites: Math. 593. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 623. Computational Finance.

#### Instructor(s): Mattias Jonsson (mattiasj@umich.edu)

Prerequisites: Math. 316 and Math. 425 or 525. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~mattiasj/teaching/623/

This course is a self-contained introduction to mathematical finance, concentrating on pricing and hedging equity and fixed-income (interest rate) derivative securities. We will cover modeling by stochastic and partial differential equations, with emphasis on motivation, and analytical, numerical and statistical tools for calibrating and utilizing the models. Specific topics include the celebrated Black-Scholes methodology (PDE and probabilistic approaches), numerical solutions (finite-difference, binomial trees, Monte Carlo, Quasi-Monte Carlo), stochastic volatility, American style and exotic options, and modeling the yield curve. If time permits we will also look at stochastic control problems relevant for risk management and asymptotic analysis of volatility models. The course will be partially coordinated with IOE 553 taught by professor Keppo but both courses can be taken independently.

Prerequisites: Differential equations (e.g., Math 316), basic probability theory (e.g., Math 425, Stat 515) and numerical analysis (Math 471) plus basic programming (C, Matlab or Mathematica).

Texts:

1. T. Björk: Arbitrage Theory in Continuous Time, Oxford Univ. Press, 1999.
2. D. Duffie: Dynamic Asset Pricing Theory, 3rd Ed., Princeton Univ. Press, 2001.
3. P. Wilmott, S. Howison, J. Dewynne: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, 1995.

All of these books will be used to various extent in the course. The book by Björk is the most important and is quite easy to read. We will use the book by Duffie from time to time, particularly for topics that are not covered in Björk;. The last book, by Wilmott et al is great (despited being a few years old) for finite-difference methods. I strongly recommend getting all these three books. Unfortunately I have not ordered them, so you will have to get them yourself, e.g., online.

Examination: Homework (theoretical and computational) and a final exam.

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### MATH 632. Algebraic Geometry II.

#### Instructor(s): Karen E. Smith (kesmith@umich.edu)

Prerequisites: Math. 631. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~kesmith/632.ps

Algebraic geometry is one of the most highly developed and beautiful branches of mathematics. It is also one of the most central, interacting with and in uencing just about every subspeciality of algebra and of geometry. Some of the most exciting aspects of modern physics also draw heavily from algebraic geometry. This course is an introduction to schemes, the language of modern algebraic geometry as developed by Alexander Grothendieck in the 1960's. Scheme theory is so abstract and powerful that it uni es, for example, the study of number rings in algebraic number theory and the study of compact Riemann surfaces in complex geometry, by explaining each as a concrete example of a one dimensional scheme. Basically, a scheme is a topological space, together a sheaf of rings on it, which gen- eralizes in a natural way the idea of a variety together with its sheaf of regular functions. This course will devote a fair amount of time to simply developing the definition of a scheme and sheaves of modules on them, and looking at some of the major examples to see why schemes are such a natural object of study. Topics covered will include afine and projective schemes, the structure sheaf, sheaves of modules on a scheme, cohomology theory of sheaves of modules, Cech cohomology, pro- jective morphisms, differentials, the canonical module, and Serre duality. As an example of the power of cohomology theory for sheaves, we will prove the Riemann-Roch formula for curves, and, as time permit, discuss applications to the classification of curves and higher dimensional varieties. The formal prerequisites for Math 632 are the alpha series in algebra and in topol- ogy/geometry. Knowledge of the basic ideas from Math 631 is also necessary. Scheme theory is founded on commutative algebra, so familiarity with the material of Math 614 will be very helpful. There will be weekly problem sets in this course, which will be "seriously graded." There are two textbooks for this course: D . Eisenbud and J. Harris: "Schemes: the language of algebraic geometry" R . Hartshorne "Algebraic Geometry" A recommended companion text is Shafarevich's "Basic Algebraic Geometry" (Volume II). All three books are published by Springer-Verlag.

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### MATH 635. Differential Geometry.

#### Instructor(s):

Prerequisites: Math. 537 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 636. Topics in Differential Geometry.

#### Instructor(s):

Prerequisites: Math. 635. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 637. Lie Groups.

#### Instructor(s):

Prerequisites: Math. 635. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 650. Fourier Analysis.

#### Instructor(s):

Prerequisites: Math. 596 and 602. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 655. Topics in Fluid Dynamics.

#### Instructor(s):

Prerequisites: Math. 555, 556, 557, 558; Graduate standing. (3). May be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 663 / IOE 611. Nonlinear Programming.

#### Instructor(s): Romesh Saigal (rsaigal@umich.edu)

Prerequisites: IOE 510, Math. 561. Graduate standing. (3). 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.

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### MATH 665. Combinatorial Theory II.

#### Instructor(s): Sergey Fomin (fomin@umich.edu)

Prerequisites: Math. 664 or equivalent. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/665.html

This course will provide an elementary introduction to the combinatorial aspects of Schubert calculus, the part of enumerative geometry dealing with such classical varieties as Grassmannians and flag manifolds.

A classical example of a Schubert calculus question is the following: Given mp subspaces of dimension p in general position in a complex vector space of dimension m+p, how many subspaces of dimension m intersect all these mp subspaces nontrivially?

To be able to answer such questions, one needs to gain a concrete understanding of the structure of the cohomology ring of the corresponding variety (in the example above, it will be the Grassmannian Gr(m,m+p)). Computing the intersection numbers like the one defined above requires development of an extensive combinatorial machinery involving Young tableaux, Bruhat orders of finite Coxeter groups, and related aspects of the theory of symmetric functions and Schubert polynomials.

More advanced topics will include (time-permitting): intersection-theoretic computations in (partial) flag manifolds related to classical semisimple Lie groups; quantum cohomology rings and calculation of Gromov-Witten invariants; K-theoretic analogues; and real Schubert calculus.

The presentation will be essentially self-contained and elementary, and will require no special background in combinatorics, topology, algebraic geometry, commutative algebra, or Lie theory.

Course Outline
I. Schubert calculus on Grassmannian manifolds.
II. Combinatorics of Coxeter groups.
III. Schubert polynomials.
IV. Variations on the theme of Schubert.

Texts. The course will not strictly follow a particular text. Principal sources include:
[F] W.Fulton, Young tableaux, Cambridge University Press, 1997.
[H] J.E.Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1994.
[M] L.Manivel, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, AMS, 2001.

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### MATH 684. Recursion Theory.

#### Instructor(s): Peter G. Hinman (pgh@umich.edu)

Prerequisites: Math. 681. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pgh/courses/684/

The basic notion relevant to all parts of this course is that of a decidable set (of natural numbers, sentences of a formal language, etc) - one such that there is an algorithm which computes for an arbitrary given object of the right sort whether or not it belongs to the set. For example, the set of prime numbers is decidable. For a formal theory to be decidable means that there is such an algorithm to decide whether or not a given sentence is a theorem of the theory. Such a theory is in some sense trivial, since in principle there is no need to generate clever proofs to verify theorems, we can just use the algorithm. Fortunately for the profession, many common theories are {\it not} decidable; the goal of the first part of this course is to prove that.

Closely connected with undecidability are the famous Incompleteness Theorems of Goedel. A theory is complete if every sentence is provable or refutable in the theory. For example, the theory of groups is not complete, because it does not determine commutativity. If a theory has a decidable set of axioms (which any reasonable axiomatic theory does) and is complete, then it is decidable - hence if it is undecidable it is incomplete. For example, any of the usual axiomatic formulations of set theory is undecidable, hence incomplete, and that is unavoidable - even if we add an infinite (but still decidable) set of new axioms, the resulting theory remains incomplete.

The key to all of these results is a precise mathematical version of the intuitive notion of decidable set and equivalently computable function. The first part of the course develops the elementary part of the theory of recursive functions and sets and uses it to prove the undecidability and incompleteness theorems. The text for this part consists of my notes available as a coursepack.

The second half of the course will develop further the theory of recursive functions and effective computability. The basic notion is that of relative computability - a set A of natural numbers is computable relative to another set B iff membership in A can be decided algorithmically given an oracle" for deciding membership in B. The undecidability of theories provides examples of sets C which are recursively enumerable (r.e.=image of a recursive function) and such that every other r.e. set is computable relative to C. One of the main goals will be the Friedberg-Muchnik Theorem which solved in 1957 the long-open problem of Post to determine if there are weaker" r.e. sets - still non-recursive, but without this last property. The proof of this theorem introduced the priority method, which has since been exploited in many ways to gain a great deal of insight into the structure of the r.e. sets under the relation of relative computability. Other topics will include: partial recursive functions and their indices, the Recursion Theorem, degrees of unsolvability and the jump operator, and the arithmetical hierarchy. As time permits I will give a brief survey of other problems about r.e. sets and variations on the priority method. The text for this part of the course is more of my notes together with the book Recursively Enumerable Sets and Degrees, by Robert I. Soare (Springer-Verlag, ISBN 3-540-15299-7 and 0-387-15299-7).

In addition to general mathematical maturity appropriate to a 600-level course, students should have sufficient background in logic to be familiar with formal first-order languages and theories up to the Completeness Theorem. Math 681 is more than enough and a thorough understanding of Math 481 will suffice. There will be some problem sets but no exams.

Required texts:
Coursepack available from Grade A Notes in the Michigan Union book store.
Robert I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag

Homework: Four or five homework sets will be assigned during the term

Exams: None

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### MATH 696. Algebraic Topology II.

#### Instructor(s):

Prerequisites: Math. 695. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 697. Topics in Topology.

#### Instructor(s):

Prerequisites: Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 700. Directed Reading and Research.

#### Instructor(s):

Prerequisites: Graduate standing. (1-3). (INDEPENDENT).

Credits: (1-3).

Course Homepage: No homepage submitted.

Designed for individual students who have an interest in a specific topic (usually that has stemmed from a previous course). An individual instructor must agree to direct such a reading, and the requirements are specified when approval is granted.

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### MATH 704. Topics in Complex Function Theory II.

#### Instructor(s):

Prerequisites: Math. 703. Graduate standing. (3). May be taken for credit more than once.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 709. Topics in Modern Analysis I.

#### Instructor(s):

Prerequisites: Math. 597. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 710. Topics in Modern Analysis II.

#### Instructor(s):

Prerequisites: Math. 597. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 711. Advanced Algebra.

#### Instructor(s):

Prerequisites: Math. 594 or 612 and Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 715. Advanced Topics in Algebra.

#### Instructor(s):

Prerequisites: Graduate standing. (3). May be taken more than once for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 732. Topics in Algebraic Geometry II.

#### Instructor(s):

Prerequisites: Math. 631 or 731. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 756. Advanced Topics in Partial Differential Equations.

#### Instructor(s):

Prerequisites: Math. 597. Graduate standing. (3). May be taken more than once for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 775. Topics in Analytic Number Theory.

#### Instructor(s):

Prerequisites: Math. 675. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 776. Topics in Algebraic Number Theory.

#### Instructor(s):

Prerequisites: Math. 676. Graduate standing. (3).

Credits: (3).

Course Homepage: No homepage submitted.

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### MATH 797. Advanced Topics in Topology.

#### Instructor(s):

Prerequisites: Graduate standing and permission of instructor. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided

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### MATH 990. Dissertation/Precandidate.

#### Instructor(s):

Prerequisites: Election for dissertation work by doctoral student not yet admitted as a Candidate. Graduate standing. (1-8). (INDEPENDENT). May be repeated for credit.

Credits: (1-8; 1-4 in the half-term).

Course Homepage: No homepage submitted.

Election for dissertation work by doctoral student not yet admitted as a Candidate.

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### MATH 995. Dissertation/Candidate.

#### Instructor(s):

Prerequisites: Graduate School authorization for admission as a doctoral Candidate. Graduate standing. (8). (INDEPENDENT). May be repeated for credit.

Credits: (8; 4 in the half-term).

Course Homepage: No homepage submitted.

Graduate School authorization for admission as a doctoral Candidate. N.B. The defense of the dissertation (the final oral examination) must be held under a full term Candidacy enrollment period.

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### Undergraduate Course Listings for MATH.

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