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# Courses in Mathematics

This page was created at 6:22 PM on Tue, Sep 23, 2003.

## Fall Academic Term 2003 (September 2 - December 19)

### MATH 404. Intermediate Differential Equations and Dynamics.

#### Instructor(s):

Prerequisites: MATH 216, 256 or 286, or 316. (3). May not be repeated for credit.

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.

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### MATH 412. Introduction to Modern Algebra.

#### Instructor(s):

Prerequisites: MATH 215, 255, or 285; and 217. (3). May not be repeated for credit. 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.

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 or 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, and 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.

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### MATH 416. Theory of Algorithms.

#### Instructor(s):

Prerequisites: MATH 312 or 412 or EECS 203, and EECS 281. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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### MATH 417. Matrix Algebra I.

#### Instructor(s):

Prerequisites: Three courses beyond MATH 110. (3). May not be repeated for credit. 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.

Credits: (3).

Course Homepage: No homepage submitted.

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 MATH 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. (3). May not be repeated for credit. 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.

Credits: (3).

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, and 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.

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### MATH 423. Mathematics of Finance.

#### Instructor(s):

Prerequisites: MATH 217 and 425; EECS 183. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/423/001.nsf

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).
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.

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

#### Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/424/002.nsf

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): Mathematics faculty

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

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. STATS 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): Statistic faculty

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.

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

#### Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

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): Michael B Woodroofe (michaelw@umich.edu)

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.003.

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

#### Instructor(s): David Schneider (daschnei@umich.edu)

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~daschnei/math425/Math425HomePage.htm

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. STATS 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): Kausch

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/425/008.nsf

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. STATS 426 is a natural sequel for students interested in statistics. MATH 523 includes many applications of probability theory.

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

#### Instructor(s): Curtis E Huntington (chunt@umich.edu)

Prerequisites: Junior standing. (3). May not be repeated for credit.

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 MATH 522 (which can be taken independently of each other) provide more in-depth examination of the actuarial techniques used in employee benefit plans. No textbook

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### MATH 431. Topics in Geometry for Teachers.

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

Prerequisites: MATH 215, 255, or 285. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math431/

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; and finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative geometry courses at this level are MATH 432 and 433. Although it is not strictly a prerequisite, MATH 431 is good preparation for MATH 531.

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### MATH 433. Introduction to Differential Geometry.

#### Instructor(s): John W Lott (lott@umich.edu)

Prerequisites: MATH 215 (or 255 or 285), and 217. (3). May not be repeated for credit.

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 (MATH 215) and linear algebra (preferably MATH 217). Some exposure to differential equations (MATH 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; and (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, and classification of minimal surfaces of revolution. MATH 537 is a substantially more advanced course which requires a strong background in topology (MATH 590), linear algebra (MATH 513), and advanced multivariable calculus (MATH 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.

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

#### Instructor(s): Paul G Federbush (pfed@umich.edu)

Prerequisites: MATH 215, 255, or 285. (4). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.

Credits: (4).

Course Homepage: No homepage submitted.

Background and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.

Content: Topics covered include; Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed.

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

#### Instructor(s): Gregory Lyng (glyng@umich.edu)

Prerequisites: MATH 215, 255, or 285. (4). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 354 or 454.

Credits: (4).

Course Homepage: Http://www.math.lsa.umich.edu/~glyng/Teaching/Math450.html

Background and Goals: This course is an introduction to some of the main mathematical techniques in engineering and physics. It is intended to provide some background for courses in those disciplines with a mathematical requirement that goes beyond calculus. Model problems in mathematical physics are studied in detail. Applications are emphasized throughout.

Content: Topics covered include; Fourier series and integrals; the classical partial differential equations (the heat, wave and Laplace's equations) solved by separation of variables; an introduction to complex variables and conformal mapping with applications to potential theory. A review of series and series solutions of ODEs will be included as needed. A variety of basic diffusion, oscillation and fluid flow problems will be discussed.

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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. No credit granted to those who have completed or are enrolled in MATH 351. (3). May not be repeated for credit.

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, and 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.

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### MATH 451. Advanced Calculus I.

#### Instructor(s): Peter L Duren

Prerequisites: MATH 215 and one course beyond MATH 215; or MATH 255 or 285. Intended for concentrators; other students should elect MATH 450. No credit granted to those who have completed or are enrolled in MATH 351. (3). May not be repeated for credit.

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, and 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.

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

#### Instructor(s): Paul G Federbush (pfed@umich.edu)

Prerequisites: MATH 216, 256, 286, or 316. (3). May not be repeated for credit. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.

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.

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

#### Instructor(s):

Prerequisites: MATH 216, 256, 286, or 316. (3). May not be repeated for credit. Students with credit for MATH 354 can elect MATH 454 for one credit. No credit granted to those who have completed or are enrolled in MATH 450.

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.

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### MATH 463. Mathematical Modeling in Biology.

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

Prerequisites: MATH 217, 417, or 419; MATH 286, 256, or 316. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/463/001.nsf

This course will concentrate on the applications of ordinary differential equations to physiological systems. Partial differential equations will not be covered in detail. Thus, a course in ODEs such as 216 or 316 will be sufficient preparation for this course.

Who could take the course? Basically anybody who is interested in applying mathematical methods to the biological sciences. For instance, students from Biology, Chemistry, Physics, Complex Systems, Biophysics, Biomedical Engineering, Mathematics, Chemical Engineering, Physiology, Microbiology, and Epidemiology.

What kind of background will you need? Basically a course in differential equations, such as 216 or 316. If you have never seen a differential equation before, you may have trouble with the course. You will also need to be familiar and comfortable with computers, as a lot of the work in the course will have to be done on a computer. You will not need to be an expert in biology, as we will learn most of what we need to know as we go.

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s):

Prerequisites: MATH 216, 256, 286, or 316; and 217, 417, or 419; and a working knowledge of one high-level computer language. (3). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.

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 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.

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s): Hans Johnston (hansjohn@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). May not be repeated for credit. No credit granted to those who have completed or are enrolled in MATH 371 or 472.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~hansjohn/m471.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.

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### MATH 481. Introduction to Mathematical Logic.

#### Instructor(s):

Prerequisites: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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. PHIL 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.

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

#### Instructor(s):

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

Credits: (3).

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.

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### MATH 497. Topics in Elementary Mathematics.

#### Section 001.

Prerequisites: MATH 489. (3). May be repeated for credit for a maximum of 6 credits.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/497/001.nsf

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 and algebra.

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#### Instructor(s):

Prerequisites: Graduate standing in a field other than mathematics. Permission of instructor required. (1-4). (INDEPENDENT). May not be repeated for credit.

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):

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

Credits: (1).

Course Homepage: No homepage submitted.

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.

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### MATH 513. Introduction to Linear Algebra.

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

Prerequisites: MATH 412. (3). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.

Credits: (3).

Course Homepage: No homepage submitted.

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 MATH 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.

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### MATH 513. Introduction to Linear Algebra.

#### Instructor(s): Howard M Thompson (hmthomps@umich.edu)

Prerequisites: MATH 412. (3). May not be repeated for credit. Two credits granted to those who have completed MATH 214, 217, 417, or 419.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~hmthomps/513.html

Student Body: a mix of math and computer science undergrads and non-math majors

Background and Goals: 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.

Content: 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.

Alternatives: MATH 419 (Lin. Spaces and Matrix Thy) covers much of the same material using the same text, but there is more stress on computation and applications. MATH 217 (Linear Algebra) is similarly proof-oriented but significantly less demanding than MATH 513. MATH 417 (Matrix Algebra I) is much less abstract and more concerned with applications.

Subsequent Courses: The natural sequel to MATH 513 is MATH 593 (Algebra I). 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: Friedberg, Insel & Spence. Linear Algebra. Fourth Edition. Springer Prentice Hall, 2003.

Course Work: There will be weekly homework assignments, an in-class midterms, and a final.

Grading: The midterm will be worth 25% of the grade, the final 45%, and the homework 30%.

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### MATH 520. Life Contingencies I.

#### Instructor(s): Curtis E Huntington (chunt@umich.edu)

Prerequisites: MATH 424 and 425. (3). May not be repeated for credit.

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. MATH 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 MATH 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 MATH 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc.

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### MATH 523. Risk Theory.

#### Instructor(s):

Prerequisites: MATH 425. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/math/523/001.nsf

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.

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

#### Instructor(s): Charles R Doering

Prerequisites: MATH 451 (strongly recommended) or 450. MATH 425 would be helpful. (3). May not be repeated for credit.

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 MATH 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, STATS 426, and the sequence STATS 510-511 are natural sequels.

An Introduction to Probability Theory and Its Applications, 3rd edition, William Feller Wiley.
recommended — Introduction to Probability Theory Hoel, Port, Stone Houghton-Mifflin.

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### MATH 537. Introduction to Differentiable Manifolds.

#### Instructor(s): Bruce Kleiner (bkleiner@umich.edu)

Prerequisites: MATH 513 and 590. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bkleiner/math537.html

Background and Goals: This course in intended for students with a strong background in topology, linear algebra, and multivariable advanced calculus equivalent to the courses MATH 513 and MATH 590. Its goal is to introduce the basic concepts and results of differential topology and differential geometry. Content: Manifolds, vector fields and flows, differential forms, Stokes' theorem, Lie group basics, Riemannian metrics, Levi-Civita connection, geodesics Alternatives: MATH 433 (Intro to Differential Geometry) is an undergraduate version which covers less material in a less sophisticated way. Subsequent Courses: MATH 635 (Differential Geometry)

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### MATH 542 / IOE 552. Financial Engineering Seminar I.

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

Prerequisites: MATH 423, IOE 452 or IOE 453. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~mattiasj/teaching/542/

Contents: The objective of the course is to present arbitrage theory and its applications to pricing for financial derivatives. The main mathematical tool used in the course is the theory of stochastic differential equations (SDEs). We treat basic SDE techniques, including martingales, Feynman-Kac representation, and the Kolmogorov equations. We also briefly consider stochastic optimal control problems. The mathematical models are applied to the arbitrage pricing of financial instruments. We consider Black-Scholes theory and its extensions, as well as incomplete markets. We cover several interest rate theories: short rates and the Heath-Jarrow-Morton framework.

Prerequisites: A solid background in basic probability theory is necessary. Also, an introductory class to finance at the level of MATH 423 is required.

Examination: Homework, a midterm exam, and a final exam.

The text book for the class is Björk: Arbitrage Theory in Continuous Time. Oxford, 1999.

Other relevant references:
Hull: Options, Futures and Other Derivatives, 5th ed, Prentice Halley, 2003.
Baxter and Rennie: Financial Calculus.

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### MATH 543 / IOE 553. Financial Engineering Seminar II.

#### Instructor(s): Jussi Samuli Keppo

Prerequisites: MATH 542. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/fall/ioe/553/001.nsf

Advanced issues in financial engineering: stochastic interest rate modeling and fixed income markets, derivatives trading and arbitrage, international finance, risk management methodologies include in Value-at-Risk and credit risk. Multivariate stochastic calculus methodology in finance: multivariate Ito's lemma, Ito's stochastic integrals, the Feynman-Kac theorem and Girsanov's theorem.

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

#### Instructor(s): Jinho Baik (baik@umich.edu)

Prerequisites: MATH 450 or 451. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~baik/Teaching/

This course is an introduction to the theory of complex valued functions of a complex variable with substantial attention to applications in science and engineering. Concepts, calculations, and the ability to apply principles to physical problems are emphasized over proofs, but arguments are rigorous. The prerequisite of a course in advanced calculus is essential. Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, and applications. Evaluation of improper real integrals and fluid dynamics. MATH 596 covers all of the theoretical material of MATH 555 and usually more at a higher level and with emphasis on proofs rather than applications. MATH 555 is prerequisite to many advanced courses in science and engineering fields.
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### MATH 556. Methods of Applied Mathematics I.

#### Instructor(s): Peter S Smereka (psmereka@umich.edu)

Prerequisites: MATH 217, 419, or 513; 451 and 555. (3). May not be repeated for credit.

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, Schrödinger's equation, and Laplace's equation.

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

#### Instructor(s): Marina A Epelman

Prerequisites: MATH 217, 417, or 419. (3). May not be repeated for credit. 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. Contact the Department.

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### MATH 562 / IOE 511 / AEROSP 577. Continuous Optimization Methods.

#### Instructor(s): Murty

Prerequisites: MATH 217, 417, or 419. (3). May not be repeated for credit. 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. Contact the Department.

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### MATH 565. Combinatorics and Graph Theory.

#### Instructor(s): Patricia Lynn Hersh (plhersh@umich.edu)

Prerequisites: MATH 412 or 451 or equivalent experience with abstract mathematics. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~plhersh/565a03.html

The syllabus will describe this more thoroughly, but this course is designed primarily for students in math, computer science, and related fields. The first half of the course is on graph theory and some complexity theory, while the second half deals with some major topics from algebraic and geometric combinatorics: partially ordered sets, simplicial complexes as they arise in combinatorics, and matroids. In the course of examining these topics, we also will briefly discuss and use a few of the major techniques from enumerative combinatorics, namely bijective proofs, generating functions and inclusion-exclusion via Möbius functions. The second half of the course will not follow the textbook quite as closely as the first half.

Textbook: "Combinatorics and Graph Theory", by Mark Skandera.

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

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

Prerequisites: MATH 217, 417, 419, or 513; and one of MATH 450, 451, or 454. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math571.html

Prerequisites: a course in linear algebra (e.g. Math 217, 417, 419, 513 or equivalent) Text: "Numerical Linear Algebra" by L. N. Trefethen and D. Bau (SIAM) This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is ill-conditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties. The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended. Topics: 1. warmup: vector and matrix norms, orthogonal matrices, projection matrices, singular value decomposition (SVD); 2. least squares problems: QR factorization, Gram-Schmidt orthogonalization, Householder triangularization, normal equations; 3. backward error analysis: stability, condition number, IEEE floating point arithmetic; 4. direct methods for Ax=b: Gaussian elimination, LU factorization, pivoting, Cholesky factorization; 5. eigenvalues and eigenvectors: Schur factorization, reduction to Hessenberg and tridiagonal form, power method, inverse iteration, shifts, Rayleigh quotient iteration, QR algorithm; 6. iterative methods for Ax=b: Krylov subspace, Arnoldi iteration, GMRES, conjugate gradient method, preconditioning; 7. applications: image compression using the SVD, least squares data fitting, finite-difference schemes for a two-point boundary value problem, Dirichlet problem for the Laplace equation krasny@umich.edu, 763-3505, 4830 EH

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### MATH 575. Introduction to Theory of Numbers I.

#### Instructor(s): Kannan Soundararajan

Prerequisites: MATH 451 and 513. (1, 3). May not be repeated for credit. Students with credit for MATH 475 can elect MATH 575 for 1 credit.

Credits: (1, 3).

Course Homepage: No homepage submitted.

Many of the results of algebra and analysis were invented to solve problems in number theory. This field has long been admired for its beauty and elegance and recently has turned out to be extremely applicable to coding problems. This course is a survey of the basic techniques and results of elementary number theory. Students should have significant experience in writing proofs at the level of MATH 451 and should have a basic understanding of groups, rings, and fields, at least at the level of MATH 412 and preferably MATH 512. Proofs are emphasized, but they are often pleasantly short. A computational laboratory (MATH 476, 1 credit) will usually be offered as a supplement to this course. Standard topics which are usually covered include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Diophantine equations, primitive roots, quadratic reciprocity and quadratic fields, application of these ideas to the solution of classical problems such as Fermat's last 'theorem'. Other topics will depend on the instructor and may include continued fractions, p-adic numbers, elliptic curves, Diophantine approximation, fast multiplication and factorization, Public Key Cryptography, and transcendence. MATH 475 is a non-Honors version of MATH 575 which puts much more emphasis on computation and less on proof. Only the standard topics above are covered, the pace is slower, and the exercises are easier. All of the advanced number theory courses (MATH 675, 676, 677, 678, and 679) presuppose the material of MATH 575. Each of these is devoted to a special subarea of number theory.

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### MATH 590. Introduction to Topology.

#### Instructor(s): Arthur G Wasserman

Prerequisites: MATH 451. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Background and Goals: Math 590 is an introduction to point set topology. It is quite theoretical and requires extensive construction of proofs. Content: Topological and metric spaces, continuous functions, homeomorphism, compactness and connectedness, covering spaces and other topics. Text: Topology, Second Edition, by Munkres, Prentice Hall The course will cover (most of) chapters 2 through 6 of the text. Grades will be based on weekly individual homework assignments, class participation, two tests and the final exam.

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### MATH 591. General and Differential Topology.

#### Instructor(s): John W Lott (lott@umich.edu)

Prerequisites: MATH 451. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Math 451 or the equivalent is a prerequisite. In addition, we will assume a knowledge of Chapter 1 of Munkres' book. Students may wish to go over this chapter before the course. This course will be a introduction to general topology and differential topology. We will spend roughly half of the semester on each of these topics. Math 591 is the first part of a two-semester sequence in topology, the sequel being Math 592. The course will be a preparation for part of the topology QR exams. Students who already have a good background in general and differential topology should consider taking Math 537 instead. We'll cover: Topological Spaces and Continuous Functions Quotient Topologies Connectedness and Local Connectedness Compactness and Local Compactness Countability and Separation Axioms Urysohn's Lemma Tychonoff's Theorem Complete Metric Spaces Manifolds and Smooth Maps Derivatives and Tangents Immersions and Submersions Transversality Homework assignments will be given periodically. There will also be a midterm exam and a final exam. Text : Topology, a First Course by James Munkres, Prentice-Hall and Differential Topology by Victor Guillemin and Alan Pollack, Prentice-Hall. We will cover roughly Chapters 2-4 of Munkres' book, parts of Chapters 5 and 7, and Chapter 1 of Guillemin and Pollack.

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### MATH 593. Algebra I.

#### Instructor(s): William E Fulton

Prerequisites: MATH 513. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: Math 513 Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Students should have had a previous course equivalent to Math 512 (Algebraic Structures). Content: Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras. Subsequent Courses: Math 594 (Algebra II) and Math 614 (Commutative Algebra I). Text: J. Rotman, "Advanced Modern Algebra", Prentice-Hall, 2002.

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### MATH 596. Analysis I.

#### Instructor(s): Mario Bonk

Prerequisites: MATH 451. (3). May not be repeated for credit. Students with credit for MATH 555 may elect MATH 596 for two credits only.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: MATH 451 or equivalent (basic principles of analysis). Background and Goals: This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. The approach is theoretical and rigorous and emphasizes abstract concepts and proofs. Content: Review of analysis in R2 including metric spaces, differentiable maps, Jacobians; analytic functions, Cauchy-Riemann equations, conformal mappings, linear fractional transformations; Cauchy's theorem, Cauchy integral formula; power series and Laurent expansions, residue theorem and applications, maximum modulus theorem, argument principle; harmonic functions; global properties of analytic functions; analytic continuation; normal families, Riemann mapping theorem. Alternatives: MATH 555 (Intro to Complex Variables) covers some of the same material with greater emphasis on applications and less attention to proofs. Subsequent Courses: MATH 597 (Analysis II (Real)), MATH 604 (Complex Analysis II), and MATH 605 (Several Complex Variables).

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### MATH 602. Real Analysis II.

#### Instructor(s): Barrett

Prerequisites: MATH 590 and 597. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Introduction to Functional Analysis Prerequisites: basic notions of linear algebra, general topology and measure theory The primary goal of the course is to cover fundamental results of functional analysis which are often cited in the general analysis literature. These include the contraction mapping theorem, the Baire category theorem, notions of compactness and convexity for function spaces, the Hahn-Banach theorem on the existence of linear functionals, dual spaces, the uniform boundedness principle, the closed graph theorem, Hilbert space, spectral analysis, and Sobolev theory. Illustrative applications of this material to such topics as Fourier analysis, complex analysis and differential equations will also be covered. Some homework will be assigned.

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### MATH 604. Complex Analysis II.

#### Instructor(s):

Prerequisites: MATH 590 and 596. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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### MATH 605. Several Complex Variables.

#### Instructor(s): John E Fornaess (fornaess@umich.edu)

Prerequisites: MATH 596 and 597. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fornaess/math605.html

Prerequisite: Math 604 or consent of instructor. Power series in several complex variables, domains of holomorphy, pseudo convexity, plurisubharmonic functions, the Levi problem. Domains with smooth boundary, tangential Cauchy-Riemann equations, the Lewy and Bochner extension theorems. The $\overlin e {\partial}$-operator and Hartog's Theorem, Dol beault-Grothendieck lemma, theorems of Runge, Mittag-Leffler and Weierstrass. Analytic continuation, monodromy theorem, uniformization and Koebe's theorem, discontinuous groups.

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### MATH 612. Lie Algebra and their Representatives.

#### Instructor(s): Robert L Griess Jr

Prerequisites: MATH 593 and 594; Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Text: Introduction to Lie Algebras and Representation Theory, by James Humphreys. The course will cover the basics of finite dimensional Lie algebra theory, including nilpotent, solvable and semisimple Lie algebras, mostly in characteristic 0. Connections with Lie group and algebraic groups, classical spaces of matrices and exceptional structures will be discussed. Root systems and Weyl groups will be described. Topics in module theory as time permits: structure of highest weight modules, character formulas, etc. Sketch of infinite dimensional Lie algebra theory, connections with finite groups, affine groups, vertex operator algebras.

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### MATH 614. Commutative Algebra.

#### Instructor(s): Melvin Hochster

Prerequisites: MATH 593. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is an introduction to commutative algebra with emphasis on the theory of commutative Noetherian rings. The prerequisite is Math 593, or some familiarity with rings and modules and some mathematical sophistication. One theme in the course will be to explain why every commutative ring is a geometric object. Another will be to explain why one can often reduce problems to the case of finite characteristic, where they may become easier: this has a flavor reminiscent of logic. Specific topics covered will include localization, integral extensions, structure of finitely generated algebras over a field (Noether normalization), Hilbert's Nullstellensatz, an introduction to affine algebraic geometry, primary decomposition, discrete valuation rings, Dedekind domains, Artin rings, dimension theory, and Hilbert functions. There may be some substitution of topics depending on the specific interests of those taking the course. This material is particularly useful to students with interests in commutative or non-commutative algebra, algebraic geometry, several complex variables, algebraic groups or Lie theory, number theory, or algebraic combinatorics. Lecture notes will be provided (my notes have been used by several instructors for this course over the years, but this will be a revised version).

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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). May not be repeated for credit.

Credits: (3).

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

Contents: This course is a course on mathematical finance with an emphasis on numerical and statistical methods. It is assumed that the student is familiar with basic theory of arbitrage pricing equity and fixed income (interest rate) derivatives in discrete and continuous time. The course will focus on numerical implementations of these models as well as statistical methods for calibration, i.e., obtaining the parameters of the models. Specific topics include finite-difference methods, trees and lattices and Monte Carlo simulations with extensions.

Prerequisites: Differential equations (e.g., MATH 316); basic probability theory (e.g., MATH 425, STATS 515); numerical analysis (MATH 471); mathematical finance (MATH 423 and MATH 542/IOE 552 or permission from instructor); programming (e.g., C, Matlab, Mathematica, Java).

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

Texts:
James and Webber: Interest Rate Modelling, Wiley, 2000.
Wilmott, Howison, Dewynne: The Mathematics of Financial Derivatives, Cambridge, 1995.
Jaeckel: Monte Carlo Methods in Finance, Wiley, 2002.

Grading: The grade for the course will be determined from performance on 6 homework sets (45%) and a final exam (55%).

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### MATH 625 / STATS 625. Probability and Random Processes I.

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

Prerequisites: MATH 597. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math625.html

The goal of this course is to develop some of the major ideas of probability theory. Emphasis will be placed on specific examples and on ways to compute expectation values. We begin with the most basic of all processes, the simple random walk. This is equivalent to studying the tosses of a fair coin. We prove the strong law of large numbers, the central limit theorem, recurrence property and the law of the iterated logarithm for this system. The proofs of these theorems depend on an ability to compute expectation values of various random variables. In the next part of the course we develop systematic methods for computing expectation values. This leads to the study of finite difference equations. We construct the continuous process Brownian motion by taking a limit in which the finite difference equations become partial differential equations.

The second part of the course is concerned with some ideas which have wide application. These represent an abstraction of ideas involved in studying the simple random walk. The first of them is the idea of a measure preserving transformation and the notion of ergodicity. We shall prove the von Neumann and Birkhoff ergodic theorems. We also shall prove the Poincaré recurrence theorem and show how recurrence times can be estimated from the invariant measure. The second is the idea of a Markov process. We shall discuss Markov chains on a finite state space, obtain an invariant measure for the chain and prove ergodicity.

The final part of the course is an introduction to Ito's stochastic integration theory. We shall rigorously define a stochastic integral and prove Ito's lemma. Stochastic differential equations and their solutions will be discussed in a heuristic manner. The ideas involved will be illustrated by simple examples, in particular linear equations.

Prerequisite: Knowledge of the Lebesgue integral would be helpful in certain parts of the course.

Text: Probability by L. Breiman, SIAM reprint in classics of applied mathematics series 1992.

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### MATH 631. Introduction to Algebraic Geometry.

#### Instructor(s): Robert K Lazarsfeld

Prerequisites: MATH 594 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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 influencing the fields of commutative algebra, non-commutative algebra, representation theory, algebraic number theory, and complex analytic geometry in fundamental ways. A working knowledge of the language and ideas of algebraic geometry is also useful to mathematicians working in an even more broad range of topics including algebraic combinatorics, algebraic topology, and differential geometry. Some branches of modern physics and applied computer science also draw heavily from algebraic geometry.

This course will be a basic and broad introduction to Algebraic Geometry, for students at the "second year" level and higher. Topics covered will include affine and projective varieties, Hilbert's Nullstellensatz, the Zariski topology, the sheaf of regular functions, regular and rational maps, dimension, the Zariski tangent space, the concept of smoothness, degree, the Hilbert polynomial, blowing up, divisors, line bundles and maps to projective space, and the Riemann-Roch formula for curves. Considerable attention will be paid to the rich examples of algebraic geometry: Grassmannians, curves, Segre and Veronese maps, quadrics, and determinantal varieties.

Grading: There will be weekly problem sets plus a final project.

Text(s):
Basic Algebraic Geometry, Volume I, Shafarevich, Springer Study Edition.
An Invitation to Algebraic Geometry, Smith, Kahanpaa, Kekalainen and Traves, Springer 2000.
supplemented with Algebraic Curves, Fulton.
and Algebraic Geometry: A first course, Harris.

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### MATH 636. Topics in Differential Geometry.

#### Instructor(s): Daniel M Burns Jr

Prerequisites: MATH 635. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Symplectic geometry has its origins in Hamilton's approach to classical mechanics. A famous theorem of Darboux shows that all symplectic manifolds are locally diffeomorphic, and so there is no local symplectic geometry like the local geometry of Riemannian manifolds. Besides mechanics, the other main source of examples of symplectic geometry was Kaehler complex manifolds. In a weak sense this is always true: any symplectic manifold is an almost complex manifold in a way compatible with the symplectic structure. In the1980's Gromov and others showed that nevertheless, there are many global invariants of symplectic manifolds. In particular, Gromov and coworkers showed the power and flexibility of the method of pseudoholomorphic curves in symplectic geometry. These are holomorphic mappings of Riemann surfaces to an almost complex manifold. Floer showed that these ideas could be joined with classical ideas of Morse and others to construct homological invariants of symplectic manifolds. This technique has been extended to ever wider circles over the last fifteen years, and we want to discuss the fundamentals of the pseudoholomorphic curves technique, and begin describing some of its applications. We will start with a bit of basics about symplectic geometry, but we will mainly be treating questions about Kaehler geometry and moment maps, about the detailed analysis of pseudoholomorphic curves and at least one of the homology theories of Floer type based on them.

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### MATH 651. Topics in Applied Mathematics I.

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

Prerequisites: MATH 451, 555 and one other 500-level course in analysis or differential equations. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math431/

Various physical problems are characterized by the presence of a small disturbance which because it is active over a long period of time, has a non-negligible cumulative effect. An example would be that of a satellite which is orbiting the Earth. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually every branch of science. The aim of the course will be to survey perturbation methods as currently used in various physical, medical and engineering applications. Topics will be introduced by means of simple illustrative examples and then built up to consider more challenging problems. For a brief review of topics we will consider 1) Limit process expansions for ordinary differential equations both linear and non-linear problems. 2) The method of multiple scales for ODE's, via the method of strained coordinates and two scale expansions for the weakly non-linear autonomous oscillator. Also this method will be applied to general non-linear oscillators and systems of first order equations. 3) Limit process expansions for PDE's such as the ones used in studying viscous incompressible flow.

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### MATH 658. Ordinary Differential Equations.

#### Instructor(s): Anthony M Bloch

Prerequisites: A course in differential equations (e.g., MATH 404 or 558). Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course will discuss aspects of the modern theory of ordinary differential equations and dynamical systems, with applications to various mechanical and physical systems. Topics will include: the qualitative theory of ODE's on manifolds, nonlinear stability theory and chaos, Lagrangian and Hamiltonian mechanics, reduction and symmetries, mechanical systems with constraints including nonholonomically constrained systems, networks of oscillators and neural networks, fluids, and mechanical systems with forces and controls. The geometric underpinning of many of these concepts will be discussed. Recommended texts: The course will be drawn from several sources. The texts A. Bloch, Nonholonomic Mechanics and Control, Springer Verlag, and J. Marsden and T. Ratiu, Mechanics and Symmetry, Springer Verlag are recommended. Other books will be referenced as well as the primary mathematical literature. Grading: The course grade will be based mainly on completion of a class project or problem sets and general class participation.

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### MATH 665. Combinatorial Theory II.

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

Prerequisites: MATH 664 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/665f03.html

This is an introductory graduate course in matrix theory, emphasizing its algebraic and combinatorial aspects (as opposed to analytic and numerical). Content (tentative): Determinantal identities. Canonical forms and factorizations. Special classes of matrices. Multilinear algebra. Matrix inequalities. Nonnegative and (0,1)-matrices. Permanents. Real roots of polynomials. Total positivity. Reference texts (none required): [1] F.R.Gantmacher, The theory of matrices, vol.1-2, AMS Chelsea Publishing, Providence, RI, 1998. [2] V.V.Prasolov, Problems and theorems in linear algebra, AMS, 1994. [3] R.A.Brualdi and H.J.Ryser, Combinatorial matrix theory, Cambridge University Press, Cambridge, 1991. [4] R.B.Bapat and T.E.S.Raghavan, Nonnegative matrices and applications, Cambridge University Press, Cambridge, 1997.

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### MATH 675. Analytic Theory of Numbers.

#### Instructor(s): Hugh L Montgomery

Prerequisites: MATH 575 and 596. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This is a first course in multiplicative number theory, intended both as a foundation for students who are considering advanced study in number theory, and also for students who have no ambitions beyond seeing how the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions are proved. Another highlight of the course is Littlewood's theorem that pi(x) - li(x) changes sign infinitely many times. The text will be in the form of a course pack, a pre-publication version of a graduate text by Montgomery and Vaughan. There will be frequent homework exercises, but no exams. It will be assumed that students have had a first course in complex variables, and some familiarity with elementary number theory (e.g., the Chinese remainder theorem, primitive roots).

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### MATH 679. Arithmetic of Elliptic Curves.

#### Section 001.

Prerequisites: MATH 594. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Math 592, 596, 676 or familiarity with basic properties of p-adic fields and number fields This course will cover basic aspects of the theory of elliptic curves, including uniformization over C, theory over an arbitrary field, and special results over finite fields (Riemann Hypothesis estimates), number fields (Mordell-Weil theorem), and p-adic fields (Tate uniformization). If time permits, there will be discussion of more advanced topics (abelian varieties, modular curves, etc.) without proofs.

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### MATH 681. Mathematical Logic.

#### Instructor(s): Andreas Blass

Prerequisites: Mathematical maturity appropriate for a 600-level MATH course. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Mathematical logic begins with the study, by mathematical methods, of the process of mathematical, deductive reasoning. Among the major results are the completeness and incompleteness theorems, delineating the circumstances in which standard modes of reasoning (or indeed any modes of reasoning) are or are not adequate for their intended purposes. The completeness theorem will be covered in detail in this course; the incompleteness theorems will only be sketched here, as they are covered in detail in Math 684. Related topics to be covered in this course are questions of decidability, definability, and consistency. In addition, we shall see some results, like the compactness theorem, that emerge from the study of mathematical reasoning but have purely mathematical content independent of the reasoning process. The first half of the course will be a thorough and rigorous exposition of the completeness and compactness theorems of first-order logic. This will include the syntax of first-order languages, their interpretations in mathematical structures, and the relationship between truth and provability. The second half of the course will treat more advanced topics, such as non-standard models of arithmetic and analysis, some logical systems that go beyond first-order logic, and a sketch of the incompleteness theorems. There are no specific prerequisites, but mathematical sophistication appropriate to a 600-level course is expected. The text will be course notes (essentially a book) by Prof. Hinman; supplementary sources will be on reserve in the library.

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### MATH 694. Differential Topology II.

#### Instructor(s): Igor Kriz

Prerequisites: MATH 537 and 591 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

In addition to its central role in modern theoretical physics, string theory has been a rich source of inspiration in mathematics. Despite of that, much of the mathematical language used in string theory is metaphorical. The aim of this course is to bridge the distance between physical phenomenology of string theory and current approaches to its rigorous mathematical foundations. With this goal in mind, we will give an introduction to the five fundamental superstring theories, and also extended objects such as D-branes.

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### MATH 695. Algebraic Topology I.

#### Instructor(s): G Peter Scott

Prerequisites: MATH 591 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Textbook: For the first part of the course, the book by J. R. Munkres entitled (approximately) Algebraic Topology. I will recommend reading for the second part of the course later. This course is a continuation of Math 592. Together with the material of that course, it provides most of the basic algebraic topology needed by an intending geometric topologist. For intending algebraic topologists, this course is just the beginning. The first two-thirds of the course will be largely homological algebra, which should be of interest to some students specialising in algebra. The following are the main topics to be covered: - Singular homology and cohomology with abelian group coefficients, and the Eilenberg-Steenrod axioms. - The universal coefficient theorems, including the homological algebra of Ext and Tor. - Cup and cap products. Poincaré duality for the homology and cohomology of manifolds. - The last third of the course will be less algebraic. It will cover higher homotopy groups, the Hurewicz Theorem and the Whitehead theorems. - The homology and cohomology of groups will be discussed briefly at some point in the course.

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### MATH 700. Directed Reading and Research.

#### Instructor(s):

Prerequisites: Graduate standing. Permission of instructor required. (1-3). (INDEPENDENT). May be elected up to five times for credit. May be elected more than once in the same term.

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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#### Section 001.

Prerequisites: MATH 594 or 612 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This will be a continuation of the course 637 on Algebraic Groups which I am giving in this (spring) semester. In this course I will prove the main theorems on the structure and classification of reductive groups over algebraically closed fields and also present the Borel-Tits theory of reductive groups over non-algebraically closed fields. Classification of semi-simple groups over such fields will also be discussed at length and basic results on Galois cohomology will be given. If time permits, I will outline the Bruhat-Tits theory of reductive groups over nonarchimedean local fields. The course should be useful for anyone interested in Number Theory, Langlands Program, Lie Theory, Representation Theory and Algebraic and Differential Geometry. In the beginning of the course, I will use the book of Armand Borel on Linear Algebraic Groups (publisher: Springer Verlag, New York).

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### MATH 725 / STATS 725. Topics in Advanced Probability I.

#### Instructor(s): Anna Amirdjanova

Prerequisites: MATH 626. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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### MATH 731. Topics in Algebraic Geometry.

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

Prerequisites: Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

The course will cover some topics from classical algebraic geometry. This may include such topics as Fano varieties, the theory of apolarity, Cremona transformations of projective spaces of dimension greater than 2, special subvarieties of Grassmann varieties, linear systems of quadrics, and Kummer surfaces. This course can be considered as a continuation of my Winter 2003 course MATH 732, however the knowledge of the topics covered in this course is helpful but will not be assumed. Although some of the topics will require only the knowledge of basic algebraic geometry as, for example, covered in the first two chapter of Shafarevich's book, a good knowledge of Hartshorne's book, including Chapter 5, will be assumed.

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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 993. Graduate Student Instructor Training Program.

#### Instructor(s):

Prerequisites: Graduate standing and appointment as GSI in Mathematics Department. (1). May not be repeated for credit.

Credits: (1).

Course Homepage: No homepage submitted.

A seminar for all beginning graduate student instructors, consisting of a two day orientation before the term starts and periodic workshops/meetings during the Fall Term. Beginning graduate student instructors are required to register for this course.

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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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