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# Courses in Mathematics

This page was created at 9:12 PM on Mon, Jan 29, 2001.

## Winter Term, 2001 (January 4 – April 26)

Open courses in Mathematics
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Wolverine Access Subject listing for MATH

Winter Term '01 Time Schedule for Mathematics.

### MATH 412. Introduction to Modern Algebra.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc. and their proofs. Math 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, complex numbers). These are then applied to the study of particular types of mathematical structures such as groups, rings, and fields. These structures are presented as abstractions from many examples such as the common number systems together with the operations of addition or multiplication, permutations of finite and infinite sets with function composition, sets of motions of geometric figures, and polynomials. Notions such as generator, subgroup, direct product, isomorphism, and homomorphism are defined and studied.

Math 312 is a somewhat less abstract course which substitutes material on finite automata and other topics for some of the material on rings and fields of Math 412. Math 512 is an Honors version of Math 412 which treats more material in a deeper way. A student who successfully completes this course will be prepared to take a number of other courses in abstract mathematics: Math 416, 451, 475, 575, 481, 513, and 582. All of these courses will extend and deepen the student's grasp of modern abstract mathematics.

Text: Abstract Algebra, An Introduction, 2nd edition Thomas Hungerford Saunders College

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### MATH 417. Matrix Algebra I.

Prerequisites: Three courses beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled Math. 513. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/courses/417/

Many problems in science, engineering, and mathematics are best formulated in terms of matrices – rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

Text: Linear Algebra with Applications, 3rd edition Otto Bretscher Prentice Hall

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### MATH 417. Matrix Algebra I.

#### Instructor(s): Allen Moy (moy@umich.edu)

Prerequisites: Three courses beyond Math. 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled Math. 513. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~moy/math417/

Many problems in science, engineering, and mathematics are best formulated in terms of matrices – rectangular arrays of numbers. This course is an introduction to the properties of and operations on matrices with a wide variety of applications. The main emphasis is on concepts and problem-solving, but students are responsible for some of the underlying theory. Diversity rather than depth of applications is stressed. This course is not intended for mathematics concentrators, who should elect Math 217 or 513 (Honors). Topics include matrix operations, echelon form, general solutions of systems of linear equations, vector spaces and subspaces, linear independence and bases, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, and similarity theory. Applications include linear networks, least squares method (regression), discrete Markov processes, linear programming, and differential equations.

Math 419 is an enriched version of Math 417 with a somewhat more theoretical emphasis. Math 217 (despite its lower number) is also a more theoretical course which covers much of the material of 417 at a deeper level. Math 513 is an Honors version of this course, which is also taken by some mathematics graduate students. Math 420 is the natural sequel but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

Textbook: Otto Bretscher, Linear algebra with applications, Prentice Hall, 1997. You should be able to find this book at Ulrich's, at the Michigan Union, and possibly elsewhere.

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### MATH 419/EECS 400/CS 400. Linear Spaces and Matrix Theory.

Prerequisites: Four terms of college mathematics beyond Math 110. Credit can be earned for only one of Math. 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in Math. 513. (3). CAEN lab access fee required for non-Engineering students.

Credits: (3).

Lab Fee: CAEN lab access fee required for non-Engineering students.

Course Homepage: No Homepage Submitted.

Math 419 covers much of the same ground as Math 417 but presents the material in a somewhat more abstract way in terms of vector spaces and linear transformations instead of matrices. There is a mix of proofs, calculations, and applications with the emphasis depending somewhat on the instructor. A previous proof-oriented course is helpful but by no means necessary. Basic notions of vector spaces and linear transformations: spanning, linear independence, bases, dimension, matrix representation of linear transformations; determinants; eigenvalues, eigenvectors, Jordan canonical form, inner-product spaces; unitary, self-adjoint, and orthogonal operators and matrices, applications to differential and difference equations.

Math 417 is less rigorous and theoretical and more oriented to applications. Math 217 is similar to Math 419 but slightly more proof-oriented. Math 513 is much more abstract and sophisticated. Math 420 is the natural sequel, but this course serves as prerequisite to several courses: Math 452, 462, 561, and 571.

Text: Linear Algebra with Applications, 3rd edition Otto Bretscher Prentice Hall

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### MATH 420. Matrix Algebra II.

#### Instructor(s): Alexander Gorokhovsky (gorokhov@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~gorokhov/math420.htm

1. Systems of Linear Equation and Gaussian Elimination: Gaussian Elimination (a.k.a. row reduction), matrices, LU factorization.
2. Vector Spaces and Subspaces: Sets of solutions of linear systems, Definition and examples of vector spaces and subspaces, Basis and dimension, Subspaces associated with a matrix, Linear transformations.
3. Inner Products and Orthogonality: Projections, Least squares solutions, Gram-Schmidt orthogonalization, Orthogonal bases, Orthogonal matrices.
4. Review of determinants
5. Eigenvalues and Eigenvectors: Diagonalization of a matrix, Application to differential and difference equations, Complex matrices, Jordan canonical form.
6. Positive definite matrices: Positive definite matrices, minimax principle, Singular value decomposition

Text: Linear Algebra and Its Applications, by G. Strang, 3rd edition.

Grades: will be based on Quizzes: 15%, First Exam: 20%, Second Exam: 25%, Final: 40%

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### MATH 422/Business Economics and Public Policy 440. Risk Management and Insurance.

#### Instructor(s): Curtis E Huntington (chunt@umich.edu), Keith J Crocker

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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### MATH 423. Mathematics of Finance.

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

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

Credits: (3).

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

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

Contents: (a) Forwards and Futures, Hedging using Futures, Bills and Bonds, Swaps, Perfect Hedges. (b) Options-European and American, Trading Strategies, Put-Call Parity, Black-Scholes formula. (c) Volatility, methods for estimating volatility-exponential, GARCH, maximum likelihood. (d) Dynamic Hedging, stop-loss, Black-Scholes, the Greek letters. (e)

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

Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam. There will be 8 homework assignments. Each quiz will consist of a slightly modified homework problem.
8 quizzes= 8x10=80 points
midterm= 60 points
final= 80 points
Total= 220 points

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

#### Instructor(s): P Jeganathan (jegan@umich.edu)

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.001.

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

#### Section 002, 003, 006.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This course introduces students to useful and interesting ideas of the mathematical theory of probability and to a number of applications of probability to a variety of fields including genetics, economics, geology, business, and engineering. The theory developed together with other mathematical tools such as combinatorics and calculus are applied to everyday problems. Concepts, calculations, and derivations are emphasized. The course will make essential use of the material of Math 116 and 215. Topics include the basic results and methods of both discrete and continuous probability theory: conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, covariances. Different instructors will vary the emphasis. Math 525 is a similar course for students with stronger mathematical background and ability. Stat 426 is a natural sequel for students interested in statistics. Math 523 includes many applications of probability theory.

Text: A First Course in Probability Sheldon Ross 5th Edition Prentice-Hall ISBN: 0-137-46314-6

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

#### Instructor(s): Yanhong Wu

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.004.

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

#### Instructor(s): P Jeganathan

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

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425.004.

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### MATH 427/Human Behavior 603 (Social Work). Retirement Plans and Other Employee Benefit Plans.

#### Section 001.

Prerequisites: Junior standing. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

An overview of the range of employee benefit plans, the considerations (actuarial and others) which influence plan design and implementation practices, and the role of actuaries and other benefit plan professionals and their relation to decision makers in management and unions. Particular attention will be given to government programs which provide the framework, and establish requirements, for privately operated benefit plans. Relevant mathematical techniques will be reviewed, but are not the exclusive focus of the course. Math 521 and/or 522 (which can be taken independently of each other) provide more in-depth examination of the actuarial techniques used in employee benefit plans. No textbook

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

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

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

Credits: (4).

Course Homepage: No Homepage Submitted.

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

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

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

#### Instructor(s): Dmitry Gokhman (gokhman@umich.edu)

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

Credits: (4).

Course Homepage: http://www.math.utsa.edu/~gokhman/courses/2001_1/math450.html

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

Book: E. Kreyszig, Advanced Engineering Mathematics, 8th ed. Wiley, ISBN: 0-471-15496-2 Material: Chapters 10-13, 16 and, time permitting, parts of 14, 15.

Exams: There will be three exams (2 midterms and a final). The exams are closed book, but standard formulas will be provided and you can bring notes on one 3" x 5" card. No make-ups. The final exam will cover all of the material. The total exam score will be the maximum of the combined percentage on the midterms and the percentage on the final.

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### MATH 451. Advanced Calculus I.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~carswell/math451/

This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc. and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions.

There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, 590, 596, and 597.

Text: Elementary Analysis, The Theory of Calculus Kenneth Ross Springer-Verlag

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### MATH 451. Advanced Calculus I.

#### Instructor(s): David Barrett (barrett@umich.edu)

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This course has two complementary goals: (1) a rigorous development of the fundamental ideas of calculus, and (2) a further development of the student's ability to deal with abstract mathematics and mathematical proofs. The key words here are "rigor" and "proof"; almost all of the material of the course consists in understanding and constructing definitions, theorems (propositions, lemmas, etc. and proofs. This is considered one of the more difficult among the undergraduate mathematics courses, and students should be prepared to make a strong commitment to the course. In particular, it is strongly recommended that some course which requires proofs (such as Math 412) be taken before Math 451. Topics include: logic and techniques of proof; sets, functions, and relations; cardinality; the real number system and its topology; infinite sequences, limits and continuity; differentiation; integration, the Fundamental Theorem of Calculus; infinite series; sequences and series of functions.

There is really no other course which covers the material of Math 451. Although Math 450 is an alternative prerequisite for some later courses, the emphasis of the two courses is quite distinct. The natural sequel to Math 451 is 452, which extends the ideas considered here to functions of several variables. In a sense, Math 451 treats the theory behind Math 115-116, while Math 452 does the same for Math 215 and a part of Math 216. Math 551 is a more advanced version of Math 452. Math 451 is also a prerequisite for several other courses: Math 575, 590, 596, and 597.

Text: Elementary Analysis, The Theory of Calculus Kenneth Ross Springer-Verlag

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### MATH 452. Advanced Calculus II.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

#### Section 001.

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

Credits: (3).

Course Homepage: https://coursetools.ummu.umich.edu/2001/winter/math/354/001.nsf

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

no textbook

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### MATH 462. Mathematical Models.

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 463. Mathematical Modeling in Biology.

#### Section 001.

Prerequisites: Math. 217, 417, or 419; 286, 256, or 316. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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, 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

How will the course be graded? I will assign projects to do throughout the term, and there will also be written assignments. The projects will require a lot of computing, and students will be able to choose between a selection of projects. There will also be a final examination, probably a take-home exam, but I have not yet decided this for sure.

Topics

• Review of phase-plane methods for ordinary differential equations
• Predator-Prey models
• Models of disease transmission. SIR models, and epidemics.
• Enzyme kinetics
• Cooperativity.
• Monod-Wyman-Changeux models.
• Cellular homeostasis
• The membrane potential. The Nernst equation. Electrodiffusion. The Goldman-Hodgkin-Katz equation. The constant field approximation.
• Osmosis
• Control of cell volume
• Sodium transport and cell volume control
• Review of limit cycles and oscillations. The Hopf bifurcation.
• Excitability
• The Hodgkin-Huxley model and action potentials
• Two-variable models. FitzHugh-Nagumo model
• Phase-plane analysis of the models
• The circulatory system
• Blood flow and compliance
• Guyton's model
• Cardiac regulation
• Fetal circulation
• The Windkessel model
• Blood
• Myoglobin and hemoglobin
• Cooperativity and oxygen transport
• Carbon dioxide transport. Bohr and Haldane effects.
• Muscle
• The Hill model
• Crossbridge theory and the Huxley model
• Discrete binding site models

The last few topics are flexible. Depending on student interest, we could cover any topic from the textbook, including the visual system, hearing, pulsatile hormone secretion, pattern formation, or wave propagation.

Text

Mathematical Physiology by Keener and Sneyd, Springer, 1998. This has a lot of additional material, but does cover the majority of the syllabus. A lot of the lecture material (but not all) will be taken from this book, as will lots of the problems.

Other Recommended Books

Mathematical Models in Biology, by Edelstein-Keshet, McGraw-Hill,1988. Very expensive book, but covers a lot of basic material, concentrating on population biology and ecology sorts of things. Also covers a lot of basic math stuff, although not as well as many other books.

Mathematical Biology, by J. Murray, Springer, 1989. Covers a huge amount of stuff. An excellent book, but mostly too advanced for this course. Great for reference though, for all students interested in the field. One of the great books in the field.

Mathematics in Medicine and the Life Sciences, by Hoppensteadt and Peskin, Springer. A bit more elementary, but lots of good stuff.

Nonlinear Dynamics and Chaos, by S. Strogatz. This is probably the best book for revising phase plane theory, elementary bifurcations and differential equations. Has lots of applications, including some from biology.

Differential Equations and Their Applications, by Braun. This is another excellent book for learning differential equations, with applications to all kinds of things, including disease models, the theory of conflict, population models etc. Review of phase-plane methods for ordinary differential equations.

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s): Zhong-Hui Duan (zduan@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~zduan/class/471/

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

Text: An Introduction to Numerical Analysis, Kendall E. Atkinson

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### MATH 471. Introduction to Numerical Methods.

#### Section 002.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

Text: An Introduction to Numerical Analysis Kendall Atkinson Wiley

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### MATH 475. Elementary Number Theory.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

#### Section 001.

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

Credits: (1).

Course Homepage: No Homepage Submitted.

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

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This course, together with its predecessor Math 385, provides a coherent overview of the mathematics underlying the elementary and middle school curriculum. It is required of all students intending to earn an elementary teaching certificate and is taken almost exclusively by such students. Concepts are heavily emphasized with some attention given to calculation and proof. The course is conducted using a discussion format. Class participation is expected and constitutes a significant part of the course grade. Enrollment is limited to 30 students per section. Although only two years of high school mathematics are required, a more complete background including pre-calculus or calculus is desirable. Topics covered include fractions and rational numbers, decimals and real numbers, probability and statistics, geometric figures, and measurement. Algebraic techniques and problem-solving strategies are used throughout the course.

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### MATH 490. Introduction to Topology.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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### MATH 498. Topics in Modern Mathematics.

#### Instructor(s): Timothy Callahan (timcall@umich.edu)

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

Credits: (3).

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

This course is a continuation of Math 454, although the offcial prerequisite is adequate preparation. Topics to be covered (time permitting) include continuous spectra (unbounded domains and the Dirac delta function), coherent and incoherent scattering and radiation, perturbation theory, adiabatic invariants, spatially inhomogeneous media (e.g., dielectrics), anisotropic media, inhomogeneous equations and Green's functions, boundary layers, the nonlinear wave equation (characteristics and shock waves) and nonlinear PDE's. As with Math 454, this is a very applied course, with examples from electromagnetism, mechanics, aerodynamics, financial engineering, etc. There will be several homework sets, a midterm and a final.

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#### Section 001.

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

Credits: (1-4).

Course Homepage: No Homepage Submitted.

No Description Provided

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

#### Section 001.

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

Credits: (1).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 512. Algebraic Structures.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This is one of the more abstract and difficult courses in the undergraduate program. It is frequently elected by students who have completed the 295-396 sequence. Its goal is to introduce students to the basic structures of modern abstract algebra (groups, rings, and fields) in a rigorous way. Emphasis is on concepts and proofs; calculations are used to illustrate the general theory. Exercises tend to be quite challenging. Students should have some previous exposure to rigorous proof-oriented mathematics and be prepared to work hard. Students from Math 285 are strongly advised to take some 400-500 level course first, for example, Math 513. Some background in linear algebra is strongly recommended.

The course covers basic definitions and properties of groups, rings, and fields, including homomorphisms, isomorphisms, and simplicity. Further topics are selected from (1) Group Theory: Sylow theorems, Structure Theorem for finitely-generated Abelian groups, permutation representations, the symmetric and alternating groups; (2) Ring Theory: Euclidean, principal ideal, and unique factorization domains, polynomial rings in one and several variables, algebraic varieties, ideals; and (3) Field Theory: statement of the Fundamental Theorem of Galois Theory, Nullstellensatz, subfields of the complex numbers and the integers mod p.

Math 412 is a substantially lower-level course over about half of the material of Math 512. The sequence Math 593-594 covers about twice as much Group and Field Theory as well as several other topics and presupposes that students have had a previous introduction to these concepts at least at the level of Math 412. Together with Math 513, this course is excellent preparation for the sequence Math 593-594.

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### MATH 513. Introduction to Linear Algebra.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 521. Life Contingencies II.

#### Section 001.

Prerequisites: Math. 520. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

This course extends the single decrement and single life ideas of Math 520 to multi-decrement and multiple-life applications directly related to life insurance. 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. Concepts and calculation are emphasized over proof. Topics include multiple life models – joint life, last survivor, contingent insurance; multiple decrement models – disability, withdrawal, retirement, etc.; and reserving models for life insurance. Math 522 is a parallel course covering mathematical models for prefunded retirement benefit programs.

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### MATH 523. Risk Theory.

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

Prerequisites: Math. 425. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~conlon/math523/index.html

Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in finance, risk management, or insurance and provides background for the professional examinations in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. Students should have a basic knowledge of common probability distributions (Poisson, exponential, gamma, binomial, etc. and have at least junior standing. Two major problems will be considered: (1) modeling of payouts of a financial intermediary when the amount and timing vary stochastically over time; and (2) modeling of the ongoing solvency of a financial intermediary subject to stochastically varying capital flow. These topics will be treated historically beginning with classical approaches and proceeding to more dynamic models. Connections with ordinary and partial differential equations will be emphasized. Classical approaches to risk including the insurance principle and the risk-reward tradeoff. Review of probability. Bachelier and Lundberg models of investment and loss aggregation. Fallacy of time diversification and its generalizations. Geometric Brownian motion and the compound Poisson process. Modeling of individual losses which arise in a loss aggregation process. Distributions for modeling size loss, statistical techniques for fitting data, and credibility. Economic rationale for insurance, problems of adverse selection and moral hazard, and utility theory. The three most significant results of modern finance: the Markowitz portfolio selection model, the capital asset pricing model of Sharpe, Lintner and Moissin, and (time permitting) the Black-Scholes option pricing model.

Text: Loss Models – From Data to Decisions Klugman, Panjer, et al Wiley

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### MATH 525/Stat. 525. Probability Theory.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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 526/Stat. 526. Discrete State Stochastic Processes.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

The course this academic term will be about probabilistic models of proteins and nucleic acids, and their uses in molecular biology. The topics will include a review of basic concepts of probability and very rudimentary molecular biology; probability and the design of similarity scoring functions; optimal local and global alignments of sequences: dynamic programming, Smith-Waterman algorithm, other algorithms available on the Web (BLAST and FastA, etc.), probabilistic (heuristic) versus rigorous algorithms; significance of scores and simulation; dependence of scoring functions and optimal alignments on parameters, comparison of standard tables; hidden Markov models and neural network models; multiple sequence alignment methods and algorithms, families of proteins; phylogenetic tree determinations; structure of proteins and recognizable patterns in amino acid sequences (motif recognition). Guest lecturers will address the class on applications in the pharmaceutical industry, as well as some earlier examples of these techniques applied to problems in linguistics and speech recognition.

Students will be expected to complete three to four problem sets, most of which will hopefully be group projects, some of which will involve using Web-based tools. If the class demographics work out favorably, we will be mixing students with biological background and mathematical background in each group. Every effort will be made to accommodate students from diverse backgrounds.

Text:

• R. Durbin, S. Eddy, A. Krogh and G. Mitchison, Biological Sequence Analysis, Cambridge, C.U.P., 1998. [required]
• M. Waterman, Introduction to Computational Biology, London, Chapman and Hall, 1995. [optional]

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### MATH 528. Topics in Casualty Insurance.

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 531. Transformation Groups in Geometry.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~bmosher/math531/

This course lies in the intersection of group theory and geometry and will develop the foundation for many beautiful and fascinating topics. Among these topics are symmetries of polygons and polyhedra, wallpaper groups (popularized by Escher, for example), crystallographic groups, and sphere packing.

Along the way we will learn about Euclidean and hyperbolic geometry and the group theory and linear algebra that encode the symmetries of these spaces.

There is no required text, but the course will follow parts of Armstrong, Groups and symmetry (Springer). The following books may pique your interest: Coxeter, Introduction to geometry; Coxeter, Regular polytopes; Cromwell, Polyhedra; Beardon, The geometry of discrete groups.

The prerequisite for the course is Math 215 or 285. We will develop the necessary algebra at a pace determined by the backgrounds of the students. Problem sets will be assigned.

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

#### Section 001.

Prerequisites: Math 425 or Biol 427 or Biochem 45; basic prgramming skills desireable, Instructor permision." Graduate standing. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

See Mathematics 547.

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

#### Section 001.

Prerequisites: "Math/Stat 425 or Biol 427 or Biochem 45; basic prgramming skills desireable, Instructor permision." Graduate standing. (1).

Credits: (1).

Course Homepage: No Homepage Submitted.

This will be a computational laboratory course designed in parallel with Math/Stat 547: Prob Mod Bioinformatics. Weekly hands-on problems will be presented on the algorithms presented in the course, the use of public sequence databases, the design of hidden markov models. Concrete examples of homology, gene finding, structure analysis.

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

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

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

Credits: (3).

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

See Complex Systems 510.001.

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 558. Ordinary Differential Equations.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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#### Instructor(s): Katta G Murty (murty@umich.edu)

Prerequisites: Math. 217, 417, or 419. (3). CAEN lab access fee required for non-Engineering students.

Credits: (3).

Lab Fee: CAEN lab access fee required for non-Engineering students.

Course Homepage: http://www-personal.engin.umich.edu/~murty/510/index.html

Background Required: Elementary matrix algebra (concept of linear independence, bases, matrix inversion, pivotal methods for solving linear equations), geometry of Rn including convex sets and affine spaces.

Reference Books:

1. K. G. Murty, Operations Research: Deterministic Optimization Models, Prentice Hall, 1995.
2. K. G. Murty, Linear Programming, Wiley, 1983.
3. M.S. Bazaraa, J. J. Jarvis, and H. D. Shirali, Linear Programming and Network Flows, Wiley, 1990.
4. R. Saigal, Linear Programming: A Modern Integrated Analysis, Kluwer, 1995.
5. D. Bertsimas and J. N. Tsitsiklis,Introduction to Linear Optimization, Athena, 1997.
6. R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Scientific Press, 1993.

Course Content:

• Linear Programming models and their various applications. Separable piece-wise linear convex function minimization problems, uses in curve fitting and linear parameter estimation. Approaches for solving multi-objective linear programming models, the Goal programming technique.
• What useful planning information can be derived from an LP model (marginal values and their planning uses).
• Pivot operations on systems of linear equations, basic vectors, basic solutions, and bases. Brief review of the geometry of convex polyhedra.
• Duality and optimality conditions for LP.
• Revised primal and dual simplex methods for LP.
• Infeasibility analysis, marginal analysis, cost coefficient and right hand side constant ranging, and other sensitivity analyses.
• Algorithm for transportation models.
• Bounded variable primal simplex method.
• Brief review of Interior point methods for LP.

Work: Weekly Homework Assignments; Midterm; Final Exam; Two Computational Projects to be solved using AMPL. Appoximate weights for determining final grade are: Homeworks(15%), Midterm(20%), Final Exam(50%), Computer Projects(15%).

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### MATH 566. Combinatorial Theory.

#### Section 001.

Prerequisites: Math. 216, 256, 286, or 316. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 567. Introduction to Coding Theory.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 582. Introduction to Set Theory.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 592. Introduction to Algebraic Topology.

#### Section 001.

Prerequisites: Math. 591. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 594. Algebra II.

#### Section 001.

Prerequisites: Math. 593. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 597. Analysis II.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 604. Complex Analysis II.

#### Section 001.

Prerequisites: Math. 596. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 609. Topics in Analysis.

#### Section 001 – Topic?

Prerequisites: Math. 451. Graduate standing. (3). May be taken for credit more than once.

Credits: (3).

Course Homepage: No Homepage Submitted.

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

#### Section 001.

Prerequisites: Math. 593 and 594; Graduate standing. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 613. Homological Algebra.

#### Section 001.

Prerequisites: Math. 590 and 594 and Graduate standing. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 615. Commutative Algebra II.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 619. Topics in Algebra.

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 623. Computational Finance.

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

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

Credits: (3).

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

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

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

Texts: T. Björk: Arbitrage Theory in Continuous Time, Oxford Univ. Press, 1999. P. Wilmott, S. Howison, J. Dewynne: The Mathematics of Financial Derivatives: A Student Introduction Cambridge University Press, 1995.

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

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### MATH 627/Biostat. 680 (Public Health). Applications of Stochastic Processes I.

#### Instructor(s): Daya Dayananda

Prerequisites: Graduate standing; Biostatistics 601, 650, 602 and Mathematics 450. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 632. Algebraic Geometry II.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 635. Differential Geometry.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 636. Topics in Differential Geometry.

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 637. Lie Groups.

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 650. Fourier Analysis.

#### Instructor(s): Marcin Bownik (marbow@umich.edu)

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

Credits: (3).

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

Fourier analysis is a subject of mathematics which originated with the study of Fourier series and integrals. Nowadays Fourier analysis is a vast area of research with applications ranging in several branches of science such as partial differential equations, potential theory, mathematical physics, number theory, signal analysis, and tomography. An important recent development in Fourier analysis is the study of a new type of orthogonal expansions in wavelet bases. Theory of wavelets has become a very active area of research with many far reaching applications.

This course is an introduction to the theory of Fourier series, Fourier integrals, wavelets, and related topics. More specifically, we are planning to cover the following topics:

1. The general properties of orthogonal systems, Riesz bases, frames.
2. Convergence and summability theory of Fourier series, lacunary series.
3. Fourier transforms of L 2 functions, inversion formula, Plancherel's theorem.
4. Multivariable Fourier series, the Poisson summation formula.
5. Theory of distributions, Fourier transforms of tempered distributions, the Paley-Wiener theorem.
6. General theory of wavelets, scaling functions, multi-resolution analysis.
7. The construction of Stromberg wavelets, Meyer wavelets, and compactly supported Daubechies wavelets.
8. Multivariable wavelets, tensor products.
9. Wavelets and Calderon-Zygmund operators in various function spaces.
10. Applications to signal processing, discrete Fourier and wavelet transforms.

Prerequisites: Math 597 and Math 602.

Grading: There will be a couple of homework assignments. Since there will be no final exam each student will give an oral presentation on a subject of his/her choice from a list of several topics in Fourier Analysis.

Textbook: P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge Univ. Press, 1997)

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### MATH 656. Partial and Differential Equations I.

#### Section 001.

Prerequisites: Math. 558, 596 and 597, and Graduate standing. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 660/IOE 610. Linear Programming II.

#### Instructor(s): Marina A Epelman (mepelman@umich.edu)

Prerequisites: Math. 561. Graduate standing. (3).

Credits: (3).

Course Homepage: http://www-personal.engin.umich.edu/~mepelman/IOE610/

Primal-dual algorithm. Resolution of degeneracy, upper bounding. Variants of simplex method. Geometry of the simplex method, application of adjacent vertex methods in non-linear programs, fractional linear programming. Decomposition principle, generalized linear programs. Linear programming under uncertainty. Ranking algorithms, fixed charge problem. Integer programming. Combinatorial problems.

This is a second-semester course in Linear Programming for graduate students. The topics we plan to cover (not necessarily in the specified order) in various depth are:

• Review of linear programming basics: polyhedral geometry, simplex method, duality.
• Variants and generalizations of simplex method.
• Complexity and polynomiality of linear programming.
• Interior point methods, including affine scaling, potential reduction, path-following techniques.
• Computational issues in implementation of simplex method and interior point method.
• Large scale techniques.

The primary source of reading will be the text "Introduction to Linear Optimization," by Dimitris Bertsimas and John Tsitsiklis, and the (occasional) handouts with supplemental material. We will also work through several journal articles. Additional recommended books:

• R. Saigal, "Linear Programming: A Modern Integrated Analysis," Kluwer Academic Publishers, 1995.
• L. Lasdon, "Optimization Theory for Large Scale Systems," Macmillian, 1970.
• A. George and J. W. H. Liu, "Computer Solution of Large Sparse and Positive Definite Systems," Prentice Hall, 1981.
• R.J. Vanderbei, "Linear Programming: foundations and extensions," Kluwer Academic Publishers, 1997.
• S.J. Wright, "Primal-Dual Interior Point Methods," SIAM, 1997

Grading The grading scheme is not finalized yet, but expect the following components to be weighted at about 25% each: Homeworks (appr. weekly); Two take-home exams; An in-class presentation (details to follow).

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

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

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

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 669. Topics in Combinatorial Theory.

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

Prerequisites: Math. 565, 566, or 664, and Graduate standing. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/c01.html

This is an interdisciplinary course in convexity and its numerous applications. Convexity, while very simple to define formally and to understand intuitively, plays a key role in number theory, algebraic geometry, analysis, optimization, control theory, combinatorics and mathematical economics. The goal of the course is to demonstrate the great power of convexity, which is based on a small number of unifying principles, through a variety of across-the-board applications. We will start with "classical convexity". theorems of Radon, Caratheodory, Helly, Krein-Milman, polyhedral combinatorics and duality theory. Applications to number theory (around Minkowski's Theorem), analysis (Chebyshev approximations, "double precision" formulas for numerical integration, non-negative polynomials), coding (lattice packings) and combinatorics (combinatorial applications of network flows and semidefinite programming) will follow. Finally, we discuss some infinite-dimensional convexity and, possibly, geometric probability (around Crofton's formula and Buffon's needle).

Grading: we will have a number of homework problem sets

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### MATH 682. Set Theory.

#### Section 001.

Prerequisites: Math. 681. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 696. Algebraic Topology II.

#### Section 001.

Prerequisites: Math. 695. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 697. Topics in Topology.

#### Section 001 – Topic?

Credits: (3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 700. Directed Reading and Research.

Credits: (1-3).

Course Homepage: No Homepage Submitted.

No Description Provided

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### MATH 702. Functional Analysis II.

#### Instructor(s): Alexander Gorokhovsky (gorokhov@umich.edu)

Prerequisites: Math. 602; and Math. 701 is sometimes required. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~gorokhov/courses/courses.htm

No Description Provided

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

#### Instructor(s): John Erik Fornæss (fornæss@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fornaess/math704.html

This will be a literature seminar. We will go through recent mathematical research papers. There will be three steps in the discussion of each paper. In the first part we will discuss background material of the paper. We will motivate the questions and give examples. In the second part we will discuss the key idea of the paper itself. In the third part we will discuss questions suggested by the reading of the paper and other natural related questions and also discuss possible approaches for deciding these questions. Our main focus will be on papers in several complex variables and complex dynamics. The seminar is intended to be a warm-up for the upcoming Fred and Lois Gehring special year in Complex Analysis here in 2001/2002.

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

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

Credits: (3).

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

A quiver is just a directed graph where we interpret the vertices as vector spaces and the arrows as linear maps. The representation theory of quivers plays an important role in the study of noncommutative algebras, but it also has interesting relations to other areas such as root systems of Lie algebras, the representation theory of GL(n) and algebraic geometry (moduli spaces).

The course is an opportunity to learn about various topics such as invariant theory, (elementary) homological algebra, representation theory, etc. This course will be interesting for students with some background in commutative or noncommutative algebra, algebraic geometry or representation theory.

The course does not follow any book. However, I intend to write down notes or at least summaries of all the lectures. Each week there will be 2 lectures and 1 meeting is dedicated to exercises which participants are expected to work on.

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### MATH 715. Advanced Topics in Algebra.

#### Section 001 – Topic?

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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

#### Section 001 – Topic?

Prerequisites: Math 631 or 731. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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

#### Section 001 – Topic?

Prerequisites: Math 675. (3).

Credits: (3).

Course Homepage: No Homepage Submitted.

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

#### Section 001.

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

Credits: (3).

Course Homepage: No Homepage Submitted.

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### MATH 990. Dissertation/Precandidate.

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

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

Course Homepage: No Homepage Submitted.

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

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### MATH 995. Dissertation/Candidate.

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.

This page was created at 9:12 PM on Mon, Jan 29, 2001.