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# Courses in Mathematics

This page was created at 8:15 PM on Wed, Feb 5, 2003.

## Winter Academic Term, 2003 (January 6 - April 25)

Search the LS&A Course Guide

### MATH 412. Introduction to Modern Algebra.

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

Prerequisites: MATH 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~brichert/class/412.html

This course is an introduction to Abstract Algebra, with an emphasis on the logic and mathematical techniques underlying this beautiful subject. 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: 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.

Text: Abstract Algebra, Thomas W. Hungerford, Second Edition, Harcourt College Publishers, 1997.

Coursework: There will be weekly homework assignments. We will have two in-class midterms, and a final exam.

Grades: Grades will be based on the exams and homework, although performance in class may be taken into account. Each midterm counts 20%, the final 40%, and the homework 20%.

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### MATH 412. Introduction to Modern Algebra.

#### Instructor(s): De Fernex

Prerequisites: MATH 215, 255, or 285; and 217. No credit granted to those who have completed or are enrolled in MATH 512. Students with credit for MATH 312 should take MATH 512 rather than 412. One credit granted to those who have completed MATH 312. (3). May not be repeated for credit.

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 417. Matrix Algebra I.

#### Instructor(s):

Prerequisites: Three courses beyond MATH 110. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled MATH 513. (3). May not be repeated for credit.

Credits: (3).

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

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

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

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

#### Instructor(s):

Prerequisites: Four terms of college mathematics beyond MATH 110. Credit can be earned for only one of MATH 214, 217, 417, or 419. No credit granted to those who have completed or are enrolled in MATH 513. (3). May not be repeated for credit.

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 422 / BE 440. Risk Management and Insurance.

#### Instructor(s):

Prerequisites: MATH 115, junior standing, and permission of instructor. (3). May not be repeated for credit.

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):

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2003/winter/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: No homepage submitted.

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

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

#### Instructor(s):

Prerequisites: MATH 215, 255, or 285. (3). 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): John Stembridge (jrs@umich.edu)

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

Credits: (3).

Course Homepage: http://www-personal.umich.edu/~jrs/math425.html

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

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

#### Section 002.

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

Credits: (3).

Course description: Basic concepts of probability are introduced, and applications to other sciences are noted. Most of the reasoning is rooted either in combinatorics or in calculus. The emphasis is on concepts, calculations and problem-solving, rather than on formal proofs. Serious use is made of material from Math 116 and Math 215, but no prior knowledge of combinatorics is assumed. Specific topics include methods of both discrete and continuous probability, conditional probability, independent events, random variables, jointly distributed random variables, expectations, variances, and limit laws.

Exams: There will be two midterm exams and a final exam.

Homework: Daily homework will be assigned for each section we cover. These assignments will not be collected. In addition, you will be given weekly problem sets which will be collected and graded. Grading: The course grade will be determined as follows. Homework: Exam 1: 20% Exam 2: 20% Final exam: 30%.

Text: A First Course in Probability, Sixth Edition, by Sheldon Ross, Prentice-Hall, 2002. The course covers most of Chapters 1-7, and a part of Chapter 8.

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

#### Instructor(s): Boris Kalinin (kalinin@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/~kalinin/Teach/425/425index.html

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

Text: Sheldon Ross, A First Course in Probability (6th ed.), Prentice-Hall, 2002.

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

#### Instructor(s): Burns Jr

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 450. Advanced Mathematics for Engineers I.

#### Instructor(s):

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

Credits: (4).

Course Homepage: No homepage submitted.

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

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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 452. Advanced Calculus II.

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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

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

#### Instructor(s):

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

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 462. Mathematical Models.

#### Instructor(s): David Bortz (dmbortz@umich.edu)

Prerequisites: MATH 216, 256, 286, or 316; and MATH 217, 417, or 419. Students with credit for MATH 362 must have department permission to elect MATH 462. (3). May not be repeated for credit.

Credits: (3).

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

Course Description

Introductory survey of applied mathematics with emphasis on modeling of physical and biological problems in terms of differential equations. Formulation, solution, and interpretation of the results.

Mathematical and Modeling Concepts to be covered

1. Concepts of Modelling
2. Dimensions, Units, Dimensional Analysis
3. Differential equations
4. Concepts of equilibria and stability
5. Nonlinearity, limit cycles, bifurcations
6. Asymptotics and Perturbation theory
7. Examples with partial differential equations
8. Parameter estimating techniques

Textbook: There are no required texts for this class.

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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. No credit granted to those who have completed or are enrolled in MATH 371 or 472. (3). May not be repeated for credit.

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 475. Elementary Number Theory.

#### Instructor(s):

Prerequisites: At least three terms of college mathematics are recommended. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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

#### Instructor(s):

Prerequisites: Prior or concurrent enrollment in MATH 475 or 575. (1). May not be repeated for credit.

Credits: (1).

Course Homepage: No homepage submitted.

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

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

#### Instructor(s):

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

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 486. Concepts Basic to Secondary Mathematics.

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

#### Instructor(s):

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.May not be repeated for credit.

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.

#### Instructor(s): Boris Kalinin (kalinin@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/~kalinin/Teach/490/490index.html

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

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### MATH 498. Topics in Modern Mathematics.

#### Instructor(s): Derksen

Prerequisites: Senior mathematics concentrators and Master Degree students in mathematical disciplines. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

In many areas one encounters systems of polynomial equations in several variables. There is a systematic way of solving polynomial equations which will be discussed in the course. We will introduce ideals and affine varieties. This will give the students a gentle introduction into algebraic geometry. The main tool is the so-called Buchberger algorithm for computing Groebner bases of ideals. Besides solving polynomial equations, these methods can be applied to a lot of other computational problems as well. We will discuss various applications in algebraic geometry. We will also discuss other applications, such as robotics, automated theorem proving in geometry, integer programming, algebraic geometry and satisfiability in propositional logic.

There are many computer algebra systems where the algorithms are implemented. There will be a weekly lab where students can familiarize themselves with computer algebra systems such as MAPLE. We will practice to translate problems such that they can be solved using Groebner bases packages, such as the one implemented in MAPLE.

The course is aimed at undergraduate mathematics students and graduate computer science students. Students should be familiar with linear algebra, but no knowledge of algebraic geometry is needed.

The book(s) which will be used will be announced later.

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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): Karni

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 credit for a maximum of 6 credits.

Credits: (1).

Course Homepage: No homepage submitted.

Audience: Math 501 is a required course for the AIM program, and is primarily intended for these graduate students. Other graduate students from mathematics, physics, engineering, or other applied sciences may also find this course of interest and should contact the instructor prior to registration if interested.

Background and Goals: During their first three years of study, students in the AIM graduate program are required to enroll in Math 501 in both the Fall and Winter terms. In part, this seminar course is coordinated with the Applied and Interdisciplinary Mathematics Research Seminar. The AIM Student Seminar will (i) present the background to the research to be discussed at a more advanced level in the week's AIM Research Seminar, (ii) put the work in context and enable discussion of the importance of the results, and (iii) generally provide an introduction to the topic of the research seminar. Thus students gain meaningful exposure to a broad range of problems. Through direct speaking opportunities in class, the AIM Student Seminar also teaches students to give presentations to an interdisciplinary audience. Both aspects of Math 501 listening and speaking, are vital to general interdisciplinary training, and hence Math 501 is an important part of the AIM graduate program. In Math 501, students will learn both what other students are doing and also what the current of modern research is, and in this way the course will foster interactions and camaraderie among AIM students and faculty.

Course Requirements: Vigorous, active participation is expected during class time, and in addition to participating in the student seminar, students are required to attend all AIM Research Seminars held on Friday afternoons from 3 to 4 PM, and need to allocate time for this in their schedules. This one-credit course is graded on a credit/no-credit basis.

There is no textbook for the course.

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### MATH 512. Algebraic Structures.

#### Instructor(s): Griess

Prerequisites: MATH 451 or 513. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Student Body: mainly undergrad math concentrators with a few grad students from other fields

Background and Goals: 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

Content: 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.

Alternatives: Math 412 (Introduction to Modern Algebra) is a substantially lower level course covering 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.

Subsequent Courses: 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.

#### Instructor(s): Nevins

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

Credits: (3).

Course Homepage: No homepage submitted.

This is an introduction to the theory of abstract vector spaces and linear transformations, which are fundamental structures in mathematics and form part of the basic toolkit for many areas of mathematics, science and engineering. The course will emphasize concepts and proofs and will include sufficient calculation to enable students to cement their understanding and apply the ideas of the course in a variety of fields.

Topics to be covered: Vector spaces (over arbitrary fields), linear transformations, bases, matrices, eigenvectors, bilinear and quadratic forms, and Jordan canonical form. As time permits we will cover some of the additional topics in areas to which our tools may be applied, such as differential questions and coding theory.

Text: Sheldon Axler, Linear Algebra Done Right, Springer-Verlag

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### MATH 521. Life Contingencies II.

#### Instructor(s): Huntington

Prerequisites: MATH 520. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is a continuation of Mathematics 520 (a year-long sequence). It covers the topics of reserving models for life insurance; multiple-life models including joint life and last survivor contingent insurances; multiple-decrement models including disability, retirement and withdrawal; insurance models including expenses; and business and regulatory considerations.

Text: Actuarial Mathematics (2nd Edition) by Bowers, Gerber, Hickman, Jones and Nesbitt (Society of Actuaries).

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### MATH 523. Risk Theory.

#### Instructor(s): David Schneider

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

Credits: (3).

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

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

Background and Goals: 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. It provides background for the professional exams in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society.

Content: 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. It provides background for the professional exams in Risk Theory offered by the Society of Actuaries and the Casualty Actuary Society. We intend to cover the following topics: Standard distributions used for claim frequency models and for loss variables, theory of aggregate claims, compound Poisson claims model, discrete time and continuous time models for the aggregate claims variable, the Chapman-Kolmogorov equation for expectations of aggregate claims variables, the Brownian motion process, estimating the probability of ruin, reinsurance schemes and their implications for profit and risk. Credibility theory, classical theory for independent events, least squares theory for correlated events, examples of random variables where the least squares theory is exact.

Grading: The grade for the course will be determined from performances on 8 quizzes, a midterm and a final exam. There will be 8 Problem sets. Each quiz will consist of a slightly modified homework problem. 8 quizzes: 37% of grade. midterm= 27% of grade. final= 36% of grade.

Textbook: Loss Models-from Data to Decisions by Klugman, Panjer and Willmot, Wiley 1998. The book is on reserve at the Shapiro Science Library.

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

#### Section 001.

Prerequisites: MATH 450 or 451. Students with credit for MATH 425/STATS 425 can elect MATH 525/STATS 525 for only one credit. (3). May not be repeated for credit.

Credits: (3).

This course presents a rigorous study of the mathematical theory of probability. The emphasis will be on fundamental concepts and proofs but examples and applications will also be discussed. The course covers basic results and methods of both discrete and continuous probability theory, conditional probability and conditional expectation, discrete and continuous random variables, convergence of random variables and other topics.

Exams: There will be a midterm and a cumulative final exam.

Homework, etc.: Weekly homework assignments will be collected and graded. Attendance and participation in lectures is expected. Grading: The course grade will be determined as follows. Homework: 45% Midterm Exam: 20% Final exam: 35%.

Text: Probability & Random Processes, Third Edition, by G. Grimmett and D. Stirzaker, Oxford University Press (2001).

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

#### Instructor(s): Doering

Prerequisites: MATH 525 or EECS 501. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Recommended: Good understanding of advanced calculus covering limits, series, the notion of continuity, differentiation and the Riemann integral; Linear algebra including eigenvalues and eigenfunctions.

This is a core course for the graduate program in Applied & Interdisciplinary Mathematics (AIM).

Background and Goals: The theory of stochastic processes is concerned with systems which change in accordance with probability laws. Many applications can be found in physics, engineering, computer sciences, economics, financial mathematics and biological sciences, as well as in other branches of mathematical analysis such as partial differential equations. The purpose of this course is to provide an introduction to the theory of stochastic processes. It is a second course in mathematical probability which should be of interest to students of mathematics and statistics as well as students from other disciplines in which stochastic processes have found significant applications. Special efforts will be made to both explore the rich diversity of applications as well as to make students aware of mathematical subtleties underlying stochastic processes.

Content: The material is divided between discrete and continuous time processes. In both, a general theory is developed and detailed study is made of some special classes of processes and their applications. Some specific topics include generating functions; recurrent events and the renewal theorem; random walks; Markov chains; limit theorems; Markov chains in continuous time with emphasis on birth and death processes and queueing theory; an introduction to Brownian motion; stationary processes and martingales.

Coursework: weekly (or biweekly) problem sets will count for 50% of the grade, a midterm exam will count for 20% of the grade. The final exam will count for 30%.

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

#### Instructor(s): Burns

Prerequisites: MATH 425 or MCDB 427 or BIOLCHEM 415; basic programming skills desirable. Graduate standing and permission of instructor. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 547.001.

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

#### Instructor(s): Burns

Prerequisites: MATH 425 or MCDB 427 or BIOLCHEM 415; basic programming skills desirable. Graduate standing and permission of instructor. (1). May not be repeated for credit.

Credits: (1).

Course Homepage: No homepage submitted.

This will be a computational laboratory course designed in parallel with STATS 547 / MATH 547 / BIOINF 547. Probabilistic Modeling in 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 555. Introduction to Functions of a Complex Variable with Applications.

#### Instructor(s): Stensones

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

Credits: (3).

Course Homepage: No homepage submitted.

Student Body: largely engineering and physics graduate students with some math and engineering undergrads, and graduate students in the Applied and Interdisciplinary Mathematics (AIM) program

Background and Goals: 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. This course is a core course for the Applied and Interdisciplinary Mathematics (AIM) graduate program.

Content: Differentiation and integration of complex valued functions of a complex variable, series, mappings, residues, applications. Evaluation of improper real integrals, fluid dynamics. This corresponds to Chapters 1--9 of Churchill.

Alternatives: Math 596 (Analysis I (Complex)) covers all of the theoretical material of Math 555 and usually more at a higher level and with emphasis on proofs rather than applications.

Subsequent Courses: Math 555 is prerequisite to many advanced courses in science and engineering fields.

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### MATH 558. Ordinary Differential Equations.

#### Instructor(s): Smereka

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

#### Instructor(s): Amy Ellen Mainville Cohn

Prerequisites: MATH 217, 417, or 419. (3). CAEN lab access fee required for non-Engineering students. May not be repeated for credit.

Credits: (3).

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

Course Homepage: No homepage submitted.

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

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### MATH 563. Advanced Mathematical Methods for the Biological Sciences.

#### Instructor(s): Tachette Jackson

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Math 450 or 454 or equivalent

Texts: Mathematical Biology, James D. Murray (required)

Background and Goals: This course will introduce and explore partial differential equation modeling in biological settings. Students should have some experience with solution techniques for partial differential equations as well as an interest in biomedical applications. There will also be a brief introduction to delay differential equations and age-structured models; however no previous background in these areas is required. The purpose of the course is to demonstrate the ways in which spatio-temporal modeling and analysis can assist in the understanding of important biological phenomena.

Content: Natural systems behave in a way that reflects an underlying spatial pattern. This course focuses on discovering the way in which spatial variation influences the motion, dispersion, and persistence of species. The concepts underlying spatially-dependent processes and the partial differential equations which model them will be discussed in a general manner with specific applications taken from molecular, cellular, and population biology. This course is centered around modeling in three major areas i) Models of Motion: Diffusion, Convection, Chemotaxis and Haptotaxis; ii) Biological Pattern Formation; and iii) Delay-differential Equations and Age-structured Models.

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### MATH 566. Combinatorial Theory.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/566.html

• INTRODUCTION
• Cayley's Theorem
• De Bruijn Sequences
• Hooklength formula
• ALGEBRAIC GRAPH THEORY
• Spectra of graphs
• Walks on a cube
• Sperner's theorem
• Matrix-tree theorem
• Eulerian tours
• Domino tilings
• PARTITIONS AND TABLEAUX
• Partitions. Pentagonal Number Theorem
• Young's lattice
• The Schensted correspondence
• Tableaux and involutions
• CLASSICAL ENUMERATION
• Catalan numbers
• Stirling numbers
• Inversions and major index
• q -binomial coefficients
• Rook polynomials
• Polya theory
• DISCRETE GEOMETRY
• Theorems of P.Hall and G.K&ouml;nig
• Birkhoff's theorem. The assignment polytope
• Cyclic polytopes
• Permutohedra
• The weak order of the symmetric group

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### MATH 567. Introduction to Coding Theory.

#### Instructor(s): Yu

Prerequisites: One of MATH 217, 419, 513. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is an introduction to coding theory, focusing on the mathematical background for linear error-correcting codes. It will begin with a discussion of Shannon's theorem and channel capacity. The definition of linear codes will be given along with a review of necessary tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes will be studied including the Hamming, BCH, cyclic, Melas, Reed-Muller, and Ree-Solomon codes. An introduction to the problem of decoding will be included, starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics to be included range from consideration of asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form. Student Body: Undergraduate math majors and EECS graduate students

Background and Goals: This course is designed to introduce math majors to an important area of applications in the communications industry. From a background in linear algebra it will cover the foundations of the theory of error-correcting codes and prepare a student to take further EECS courses or gain employment in this area. For EECS students it will provide a mathematical setting for their study of communications technology.

Content: Introduction to coding theory focusing on the mathematical background for error-correcting codes. Shannon's Theorem and channel capacity. Review of tools from linear algebra and an introduction to abstract algebra and finite fields. Basic examples of codes such and Hamming, BCH, cyclic, Melas, Reed-Muller, and Reed-Solomon. Introduction to decoding starting with syndrome decoding and covering weight enumerator polynomials and the Mac-Williams Sloane identity. Further topics range from asymptotic parameters and bounds to a discussion of algebraic geometric codes in their simplest form.

Alternatives: none

Subsequent Courses: Math 565 (Combinatorics and Graph Theory) and Math 556 (Methods of Applied Math I) are natural sequels or predecessors. This course also complements Math 312 (Applied Modern Algebra) in presenting direct applications of finite fields and linear algebra.

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

#### Instructor(s): Epperson

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: No homepage submitted.

Course Description: This will be an introduction to numerical linear algebra, in which we will study solution techniques for the linear systems problem (both direct and iterative), the least squares problem, and the algebraic eigenvalue problem. The text will be: Fundamentals of Matrix Computations, by David Watkins. The course will emphasize both theory and implementation of the methods, so proficiency in a computing language is necessary. A good background in linear algebra is also necessary. The course grade will be determined by weekly homework assignments, a midterm, and a final exam.

Subsequent Courses: Math 572 (Numer Meth for Sci Comput II) covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.

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

#### Instructor(s): Karni

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: No homepage submitted.

Prerequisites: Solid background in advanced calculus and linear algebra is needed. 454 is not required but is good to have. Knowledge of a computing programming language such as Fortran, C or the Matlab computing environment is mandatory.

This course is an introduction to numerical methods for boundary-value and initial-value problems. The course will cover numerical methods for ordinary differential equations and for linear elliptic, parabolic and hyperbolic partial differential equations. Nonlinear hyperbolic partial differential equations may also be discussed, if time permits.

The course will focus on the derivation of methods, on their accuracy, stability and convergence properties, as well as on practical aspects of their efficient implementation. The course should be useful to students in mathematics, physics and engineering.

Text: Finite Difference Methods for Differential Equations. Notes by Randall J. LeVeque, available as a coursepack from Ulrichs.

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### MATH 582. Introduction to Set Theory.

#### Instructor(s): Peter Hinman (pgh@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-personal.umich.edu/~pgh/Math582/

Set Theory is at the same time (1) a branch of mathematics, (2) a tool used in practically every other branch of mathematics, and (3) the best medium for understanding the foundations of mathematics. This course is mainly a study of (1), but much of the motivation comes from (2) and (3) and these aspects will be covered from time to time. Everyone who has taken a course in any sort of abstract mathematics, be it algebra, analysis, or topology, has used notions such as "set", "function" "equivalence relation", "linear ordering", etc.; a low-level goal of the course is to improve familiarity and comfort with these common mathematical notions. Deeper topics are well-orderings, ordinal numbers, cardinal numbers, and their properties. Set theory as a separate discipline really began with Cantor's discovery (in the late 19th century) that infinite sets can have different sizes, and the consequences and refinements of this fact will be a centerpiece of the course. We will also discuss historically troublesome assertions such as the Axiom of Choice and the Continuum Hypothesis.

All of these will be considered from both the non-axiomatic and axiomatic perspectives. The axiomatic approach is both more necessary in set theory than is other branches of mathematics and more fruitful. It is necessary partly because of the discovery that intuitions about sets can easily go astray and lead to paradox and contradiction. It is fruitful because a relatively simple set of axioms suffices to generate all of the theorems of set theory. Since essentially all mathematical notions can be expressed in terms of sets, the axiomatization of set theory is in effect an axiomatization of all of mathematics. Hence the context of axiomatic set theory is well-suited for dealing with the philosophical issue of what it means for a mathematical assertion to be true or provable. These considerations lead to a necessarily brief discussion of consistency and independence results.

The announced prerequisites of Math 412 or 451 have more to do with general level of mathematical sophistication than specific content. The course is well-suited to math majors, honors or not, beginning graduate students, and mathematically minded students of philosophy or computer science. If you have any doubts about the level of the course, please talk with me. A course in mathematical logic is not presupposed. We will follow the book of Y.N. Moschovakis, Notes on Set Theory (Springer-Verlag, ISBN 0-387-94180-0 and 3-540-94180-0). There will be several problem assignments and perhaps a take-home final exam.

Homework sets will be assigned periodically during the term

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### MATH 592. Introduction to Algebraic Topology.

#### Instructor(s): Lott

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

Credits: (3).

Course Homepage: No homepage submitted.

This is a first course in algebraic topology. It covers the material from this area that appears on the topology qualifying review exam. The topics include the fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, and Brouwer's and Lefschetz' fixed-point theorems,

Text : "Algebraic Topology" by Allen Hatcher, Cambridge University Press

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### MATH 594. Algebra II.

#### Section 001 – Group Theory and Galois Theory.

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Knowledge of basic linear algebra over an arbitrary field, prior exposure to the concept of a finite group and polynomials in 1 variable over a field (unique factorization, division algorithm, etc.), and a capacity for abstract thought.

Course description: This course will cover the basic elements of group theory and Galois theory, as preparation for the qualifying exam in algebra. We'll begin with a tour of the standard facts from finite group theory with an emphasis on those notions which are important for more general groups (algebraic groups, Lie groups, etc.). This may include a brief discussion of some concepts in the representation theory of finite groups if time permits. Once these basics are handled, we turn out attention to the theory of fields (including characteristic p!) and the historical reason why groups were first introduced by Galois: to do Galois theory! I think that Galois theory is one of the most awe-inspiring topics in algebra. By the end of the course, we will have completely solved several classical problems, including how to determine which types of constructions are possible with a straightedge and compass, how to give an `essentially' algebraic proof of the so-called Fundamental Theorem of Algebra, and how to prove that it is impossible (in a very precise sense) to solve the general nth degree polynomial 'in radicals' when n is at least 5 (and how one can derive the classical formulas for n < 5).

Textbook: Abstract Algebra, by Dummit and Foote.

Homework/exams: There will be weekly homework and take-home exams (two midterms and a final). Late homework will not be accepted for any reason, but the two lowest homework grades will be dropped. It is your responsibility to make sure your homework is turned in on time. Your final grade will be based on 50% homeworks, 20% midterm, and 30% final exam.

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### MATH 597. Analysis II.

#### Instructor(s): Barrett

Prerequisites: MATH 451 and 513. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Topics will include: Lebesgue measure on the real line and in Rn; general measures; measurable functions; integration; monotone convergence theorem; Fatou's lemma; dominated convergence theorem; Fubini's theorem; function spaces; Holder and Minkowski inequalities; functions of bounded variation; differentiation theory; Fourier analysis. Additional topics such as Sobolev spaces to be covered as time permits.

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### MATH 604. Complex Analysis II.

#### Instructor(s): Duren

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

Credits: (3).

Course Homepage: No homepage submitted.

This is a second course in complex analysis. The prerequisite is Math 596 or equivalent. We'll begin with a review of the Riemann mapping theorem, then pass to related results in geometric function theory. Here my book "Univalent Functions" (Springer, 1983) will be a convenient reference. Then we'll treat the Dirichlet problem and use harmonic measures to construct canonical mappings of multiply connected domains. We'll also discuss extremal length and logarithmic capacity. Next we'll develop two classical topics: elliptic functions and entire functions, where Ahlfors' text "Complex Analysis" is an excellent source. We'll conclude with more modern topics: Hardy and Bergman spaces, following my book "Theory of Hp Spaces" (Dover, 2000) and the manuscript of a forthcoming book (with Alex Schuster) on Bergman spaces. There will be 5 or 6 problem sets, but no formal exams. Students will be invited to prepare short expository papers on further topics of their choice.

Outline of Topics:

1. Geometric function theory. Normal families, Riemann mapping theorem, Bergman kernel function, Koebe distortion theorem, Caratheodory convergence theorem, Schwarz-Christoffel formula. Hyperbolic geometry, pseudohyperbolic metric. Analytic continuation, monodromy theorem.
2. Potential theory. Harmonic and subharmonic functions, Poisson formula, Harnack's principle, Dirichlet problem, Poincare-Perron method. Green's function, harmonic measure, period matrix of harmonic conjugates, canonical mappings of multiply connected domains. Extremal length, logarithmic capacity.
3. Elliptic functions. Doubly periodic functions, Weierstrass P-function, elliptic integrals, Jacobi elliptic functions, elliptic modular function, mapping properties.
4. Entire functions. Jensen's theorem, growth and density of zeros, Weierstrass' theorem, canonical products, Hadamard factorization theorem, Picard's theorem.
5. Linear spaces of analytic functions. Hardy spaces, boundary values, zero-sets, Blaschke products, canonical factorization. Bergman spaces, zero-sets, sampling and interpolation problems.

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### MATH 623. Computational Finance.

#### Instructor(s): Matthias Jonsson

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/

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

Material: 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.

Textbooks:

• James and Webber: Interest Rate Modelling, Wiley, 2000.
• Tavella and Randall: Pricing Financial Instruments: the finite difference method, Wiley, 2000.
• Jaeckel: Monte Carlo Methods in Finance, Wiley, 2002.

Grading: The grade for the course will be determined from performance on 6 homework sets (50%) and a final exam (50%).

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### MATH 627 / BIOSTAT 680. Applications of Stochastic Processes I.

#### Instructor(s): Wei

Prerequisites: Graduate standing; BIOSTAT 601, 650, 602 and MATH 450. (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 632. Algebraic Geometry II.

#### Instructor(s): Hacking

Prerequisites: MATH 631. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is a continuation of 631. We describe the notion of a sheaf on a variety and introduce cohomology of sheaves as a way of passing from local data to global data. We use this theory to study curves and surfaces, giving many explicit examples.

Recommended texts: R. Hartshorne, Algebraic Geometry, Springer 1977. M. Reid, Chapters on Algebraic Surfaces, http://xxx.lanl.gov/abs/alg-geom/9602006

Hartshorne is an indispensible reference but is technically rather demanding. In particular, he considers arbitrary 'schemes', whereas I will mainly restrict myself to varieties, and defines sheaf cohomology via derived functors, whereas I will use the Cech complex. Reid's notes contain a lot of readable material which help to give a intuitive feel for the subject, without giving detailed proofs, for example his introduction to sheaves. There is much more in these notes than we will cover in the course.

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### MATH 635. Differential Geometry.

#### Instructor(s): Kleiner

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

Credits: (3).

Course Homepage: No homepage submitted.

Textbook: Riemannian Geometry, by T. Sakai.

Topics covered: Riemannian metrics, Riemannian distance functions, the Levi-Civita connection, geodesics, the first and second variation formulas, curvature, Jacobi fields, the Hopf-Rinow theorem, conjugate and cut loci, injectivity radius, comparison theorems for Jacobi fields and triangles, applications to curvature and topology.

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### MATH 636. Topics in Differential Geometry.

#### Instructor(s): Ji

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: some basic knowledge of Lie group theory.

Compactifications of locally symmetric spaces have been motivated by many different problems. For example, the Baily-Borel compactification of locally Hermitian symmetric spaces was motivated to study the field of meromorphic functions. It also turned out to be a projective variety defined over a number field and plays an important role in arithmetic algebraic geometry. To resolve the singularities of the Baily-Borel compactification, Mumford et al constructed toroidal infinitely many compactifications.

On the other hand, motivated by the study of cohomology of arithmetic groups, Borel and Serre defined the Borel-Serre compactification, a manifold with corners.

This course will discuss these and many other compactifications of both symmetric and locally symmetric spaces. Motivations and pplications of each compactifications will be explained in details.

Textbook. None required.

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### MATH 637. Lie Groups.

#### Section 001 – Theory of linear algebraic groups over algebraically closed fields.

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

Credits: (3).

Course Homepage: No homepage submitted.

The course will cover basic theory of linear algebraic groups over algebraically closed fields and also some parts of the theory over nonalgebraically closed fields. It should be of interest to any one thinking of working in Number Theory (Automorphic Forms, Langlands Program), Lie Theory, Representation Theory or Geometry (Algebraic and Differential).

It would help to know some basic Algebraic Geometry but that is not a requirement since I will carefully outline whatsoever is needed.

Textbook: Linear Algebraic Groups by A. Borel, published by Springer-Verlag, New York.

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### MATH 650. Fourier Analysis.

#### Instructor(s): Bownik

Prerequisites: MATH 596 and 602. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Fourier analysis is a subject of mathematics that originated with the study of Fourier series and integrals. Nowadays, Fourier analysis is a vast area of research with applications in various branches of science including signal analysis, tomography, partial differential equations, potential theory, mathematical physics and number theory.

A recent noteworthy area of focus in Fourier analysis is orthogonal expansions in wavelet bases. The theory of wavelets is a very active area of research with many real-world 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:

1. General properties of orthogonal systems, Riesz bases, and frames.
2. Convergence and summability of Fourier series. Intro to lacunary series.
3. Fourier transforms of L2 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, multiresolution analysis.
7. 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.

Textbooks: Y. Katznelson, An introduction to harmonic analysis (Dover Publications, 1976); P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge Univ. Press, 1997)

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

#### Instructor(s): Patrick Nelson

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/~pwn/Math651.html

The primary purpose of this class is to analyze the formulation and solution of problems that arise in the physical sciences, engineering, and medicine and are modeled by partial differential equations. The emphasis is on deriving explicit analytical results, rather than on the abstract properties of the solutions. Proofs will be omitted but the underlying concepts will be carefully explained.

Mathematical Concepts to be covered:

1. Heat Equation
• The fundamental solution
• Initial value problems in the infinite domains
• Green's functions for finite and semi-finite domains
• Higher order problems and Burger's equation
2. Laplace's Equation
• Fundamental solution; Dipole potential
• Distribution of sources and dipoles
• Green's and Neumann's functions
3. Wave Equation
• Shallow water waves
• Initial and Boundary value problems on infinite and finite domains
• Dispersive waves
• Age structure equations
4. Scalar Quasilinear First Order Equations
• Conservation laws for two variables
• Weak solutions
• Shocks, Fans, and Interfaces
5. Approximate Solutions of Perturbation Methods
• Regular perturbations
• Matched Asymptotics
• Multiple Scale Expansions
6. Delay Differential Equations
• linear DDEs
• Stability analysis

Although the subject matter of applied partial differential equations can be made rather difficult, I will attempt to present the course material in as simple a manner as possible. More theoretical aspects, such as proofs, will not be presented but I will present in detail the mathematical techniques needed for formulating the mathematical solutions. Applications will be emphasized and will come from areas of Physics, Engineering, Medicine, and the Life Sciences.

Grading: Homework assignments will count as 1/3 of grade evaluation. There will also be two quizzes and each will count for 1/3 of the grade. Each quiz will be take home and at least 48 hours will be given for completion

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### MATH 654. Introduction to Fluid Dynamics.

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

Prerequisites: MATH 450; MATH 555 or 596; and MATH 454 or 556. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~krasny/math654.html

Texts: "A Mathematical Introduction to Fluid Mechanics" by A. J. Chorin and J. E. Marsden, Springer-Verlag; "Fundamentals of Hydro- and Aeromechanics" by L. Prandtl and O. G. Tietjens, Dover Publications.

Content: Math 654 is a mathematically-oriented introduction to fluid dynamics for students in math, science, and engineering. The term fluid refers to a liquid or a gas; the key property is that a fluid deforms easily in response to an applied force. Fluid motion is described by a set of partial differential equations expressing conservation of mass, momentum, and energy. These are the Navier-Stokes equations for viscous flow, and the Euler equations for inviscid flow. The goal in solving these equations is to understand, predict, and control the fluid motion. The subject has extensive practical applications (e.g. designing the shape of an aircraft or an artificial heart valve) and there is also aesthetic appeal in many flow visualizations (e.g. clouds, ocean waves, smoke swirls). Math 654 covers basic material on compressible and incompressible flow including boundary layers, vortex dynamics, and hydrodynamic stability. The emphasis is on analytical techniques, but I will also discuss relevant experiments, numerical methods, and computer simulations.

Previous coursework in vector calculus, complex variables, and differential equations will be helpful.

There will be several homework assignments.

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### MATH 669. Topics in Combinatorial Theory.

#### Instructor(s):

Prerequisites: MATH 565, 566, or 664, and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course will be a survey of (a fraction of) the many interesting applications of Representation Theory to Combinatorics, and vice-versa. Some of the most interesting results in Combinatorics have been derived by means of representation-theoretic tools. Enumeration of plane partitions, unimodality theorems, and the Rogers-Ramanujan identities are all examples of this. In the opposite direction, the symmetry groups that occur most frequently in nature (the symmetric groups, the classical groups) have representations and characters with extensive combinatorial structure.

The course will be divided into two parts. Part I will begin with a self-contained development of the representation theory of finite groups, followed by a detailed study of the case of symmetric groups and closely related groups.

Part II will be concerned with the classical groups; primarily GL(n) and its cousins (SL(n), U(n), etc). In order to have more time to discuss the combinatorial aspects, we will probably not present detailed proofs of all of the fundamental theorems in this part of the course. (For example, existence of Haar measure will not be proved).

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### MATH 677. Diophantine Problems.

#### Instructor(s): Montgomery

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

Credits: (3).

Course Homepage: No homepage submitted.

The course will be devoted to Diophantine approximation and the geometry of numbers, with applications to Diophantine equations. In Diophantine approximation one seeks to understand how well one can approximate some class of numbers (say real numbers) by some other class (say rational numbers with denominator <= N). Here the classical theorem of Dirichlet is fundamental: For any real number r, and any given positive integer N, there is a rational number a/n, (a, n) = 1, with 1 <= n <= N, such that |r - a/n| <= 1/(n(N+1)). The geometry of numbers, largely an invention of Minkowski, deals with the existence of lattice points in various types of regions. For example, Minkowski's First Main Theorem asserts that if C is a convex body in Rn, symmetric about 0, and with volume V > 2n, then C contains a non-zero lattice point (i.e., a point with integral coordinates). This is best possible, in a sense, since the cube (-1, 1)n has volume 2n but contains no non-zero lattice point. Efficient algorithms for computation, such as the L3 algorithm, will also be discussed.

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### MATH 682. Set Theory.

#### Instructor(s): Hinman

Prerequisites: MATH 681. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This is a course on the logical foundations of Set Theory and therefore all of mathematics; the highlight is a proof of the independence of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) from the Zermelo-Fraenkel axioms (ZF). To prepare for this result, we will spend a considerable part of the semester giving a careful development of the central concepts of set theory and some other parts of mathematics from the ZF axioms. In particular we shall drive the central facts about ordinal and cardinal numbers and show how mathematical logic itself can be formalized in set theory. This leads to a proof of Goedel's Second Incompleteness Theorem for Set Theory.

After developing some general tools for proving independence, we will develop the model L of constructible sets and show that if ZF itself is consistent, then in L all of the ZF axioms together with AC and GCH hold. It follows that AC and GCH are consistent with ZF. The rest of the proof of independence consists in showing the consistency of the negations of AC and GCH with ZF. The method here is known as forcing and involves extending a model of ZF to one in which the ZF axioms still hold, but one or the other of AC or GCH fails.

The course prerequisite is a course in Mathematical Logic roughly the equivalent of Math 681. A previous course in Set Theory is helpful, but not necessary, as all of the relevant concepts will be discussed "from scratch". The text will be a course pack written by the instructor. There will be several problem sets but no exams. Hinman

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### MATH 696. Algebraic Topology II.

#### Instructor(s): Kriz

Prerequisites: MATH 695. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Recommended text: Spanier: Algebraic topology

The course will illuminate many parts of this excellent but hard reading reference.

This year, 696 will cover traditional topics of intermediate algebraic topology: Eilenberg-MacLane spaces, obstruction theory, Postnikov systems, group cohomology, cohomology operations, spectral sequences, perhaps some examples of higher homotopy groups, etc. Other similar topics may also occur as time and interest permits. This course is directed both to algebraic topology students as well as those who intend to specialize, say, in algebraic geometry or geometric topology, and need information about methods of algebraic topology.

There will be no formal requirements except for those who elect the course as a minor or cognate (those students may do some problems or give a classroom presentation).

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### MATH 697. Topics in Topology.

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

Prerequisites: Graduate standing. (2-3). May not be repeated for credit.

Credits: (2-3).

Course Homepage: No homepage submitted.

Prerequisites: 591 and 592

Recommended text: There is no recommended text, but I will give out notes and references to some papers.

The subject of the course will be 3-dimensional manifolds. I will assume some basic results in the area, for example the Sphere Theorem, but students will be able to follow the course without knowing the proofs of these basic results.

This is a subject with activity in widely different areas. The aim of the course will be to introduce the main areas of interest, and discuss some in greater detail. I will begin with some basic examples, and a survey of what is presently known and conjectured about the classification of compact 3-manifolds. After introducing Seifert fibre spaces, I will introduce the theory of least area surfaces and use this to give an approach to the theory of the characteristic submanifold of a 3-manifold.

After that I will discuss some subset of the following topics:

1. free actions of finite groups on the 3-sphere, and in particular, the recent paper by Maher and Rubinstein showing that free actions of Z3 on the 3-sphere are conjugate to linear actions.
2. cores of 3-manifolds and "wild" 3-manifolds, by which I mean 3-manifolds which are not the interior of any compact manifold.
3. the solution of the Seifert Conjecture.
4. the rigidity of Haken manifolds, hyperbolic manifolds, and cubed manifolds. Rigidity means that if M is closed and lies in one of the three named classes of 3-manifolds and if N is a 3-manifold homotopy equivalent to M, then M and N are homeomorphic.
5. Higher dimensional analogues of the characteristic submanifold.

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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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### MATH 702. Functional Analysis II.

#### Instructor(s): Taylor

Prerequisites: MATH 602; and MATH 701 is sometimes required. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

One striking aspect of the theory of functions of one complex variable is the very strong and useful connection between classical potential theory and the theory of analytic functions. The study of the analogous theory in Cn, the relationship between analytic functions and plurisubharmonic (psh) functions, has developed a lot over the past 25 years and now goes under the name "pluripotential theory". This course is intended for students with a good background in complex analysis (e.g. Math 596 and 604 or 605) who want to learn about this theory, either for its own sake or for its use as a tool in studying complex analysis and geometry.

The course will begin with the basics of the theory of psh functions, the basics of classical potential theory, a discussion of postive currents and their relationship to analytic varieties. The definition of the complex Monge-Ampere operator on psh functions will be given (and the open questions about this definition discussed), and the major results on the existence and uniqueness of solutions of the equation will be proved. While it won't be possible to give complete proofs of all the major results of this theory, we will explain as many of them as possible.

The course will then turn to special topics chosen according to the interest of the students in the class (reflecting the intention that it also serve as a service course for students in SCV or algebraic geometry who need to learn pluripotential theory as a tool for their research). Some possible topics include recent applications of pluripotential theory in complex dynamics, in complex differential geometry, and in approximation theory. Open problems abound in all of these areas, and some of these, particularly those from my current research about psh functions on analytic varieties, will be discussed.

Please contact Professor Taylor if you have questions about the course or if you would like to get a list of some of the reference material.

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

#### Instructor(s): Fornaess

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

Credits: (3).

Course Homepage: No homepage submitted.

This will be a topics course in complex dynamics in higher dimension. We will go through various papers from the recent literature. The specific results discussed will depend on the audience. The course will use significant amounts of pluripotential theory. Hence the audience is recommended to also follow the pluripotential course MATH 702.

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

#### Instructor(s): Smoller

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

Credits: (3).

Course Homepage: No homepage submitted.

Black Holes, Big-Bang, Pulsars, Quasars, Supernova Explosions, Gravitational Waves; ever wonder what these exotic, far out (pun intended) things really are? If so, then you must understand Einstein?s Theory of General Relativity. This course is intended to provide an introduction to the foundations of Einstein?s theory, and also to survey the questions it raises, its concepts, and methods.

GR is the theory of the gravitational field that was proposed by Albert Einstein in 1915. It has been universally acknowledged as being the most beautiful physical theory ever devised. It explains all of the exotic things mentioned above, and it also explains why light bends around the sun, and why the orbit of the planet Mercury precesses by a small amount each year. It is the basis for the modern theory of the "Big-Bang" and predicts the existence of Black Holes; indeed, it is now widely believed that there is a black hole at the center of our own Milky Way galaxy.

In Einstein's theory of GR, the gravitational field is simply a manifestation of the "curvature" of space-time. In this theory, the planets move around the sun because the massive sun has curved the space-time around it to the point that free falling objects, like the earth, have trajectories that have been "curved" all the way from straight lines into elliptical orbits. GR is the grandaddy of all modern field theories because it is the first physical theory in which the field (gravity) is the manifestation of curvature. All modern theories of elementary particle physics have also adopted this point of view.

In spite of the successes of the theory, GR has retained for a long time the reputation of being an esoteric science. Indeed, for about half a century, GR attracted little attention from physicists. This was perhaps because of the mathematical difficulties and the radically new concepts, and also due to the paucity of applications (say in comparison to the Quantum Theory, which came into existence at about the same time). However, the discovery of compact objects such as quasars and pulsars, as well as Black Holes, on the one hand, and the Big-Bang theory of Cosmology as well as data coming from the Hubble Space Telescope on the other hand, completely changed the picture. In addition, developments in elementary particle physics, such as predictions of the behavior of matter at the ultrahigh energies that might have prevailed in the early stages of the Big-Bang, have greatly enhanced the interest in GR. In fact the last 20 years, have seen the development of new mathematical methods for obtaining solutions, and there is now a renewed interest in the theory, and GR is becoming a hot item for mathematicians.

The language of GR is differential geometry, and the first few weeks of the term will be devoted to the basic ideas of differential geometry, which are necessary for GR: tensors, metrics, connections, and curvature. Then Einstein?s equations will be considered and we shall show how they contain Newton?s equations as a limiting case. Some important solutions will be derived, and analyzed, and their connections to such topics as stellar models, black holes, cosmological models, and the Big-Bang will be described. The initial-value problem for Einstein's equations will be discussed. We shall also consider the coupling of Einstein's equations to Maxwell's equations of electromagnetism, to the Yang/Mills equations, and to Dirac?s equation (the relativistic version of Schrodinger's equation). This final topic will bring the class up to the research level.

You, the student, will leave with an understanding of the Big-Bang, and how black holes arise in Einstein?s theory of GR, and this will surely make you a most sought-after invitee to social gatherings all over town.

Prerequisites: The course will be entirely self-contained, but some degree of mathematical maturity is necessary; for example, the completion of an alpha course will suffice. See me if you are not sure of your preparation

Grading: Decided by two or three problem sets.

Textbook: Modern Geometry, 1, (2nd edition) by Dubrovin, Fomenko, and Novikov; Springer, 1992

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### MATH 715. Advanced Topics in Algebra.

#### Instructor(s): Fulton

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

Credits: (3).

Course Homepage: No homepage submitted.

Although equivariant cohomology has existed in topology for a long time, it is only in recent years that it has become a major tool in other areas of geometry. Many fundamental spaces, such as Grassmannians, flag varieties, Schubert varieties, toric varieties, and spaces of matrices have natural group actions. Expressing classical data in terms equivariant instead of ordinary cohomology gives significantly stronger information. An underlying theme is obtaining global geometric information from behavior around the fixed points of the group action.

The course will construct and prove basic properties about equivariant cohomology and the corresponding equivariant intersection theory in algebraic geometry. The main goal is to work out the full story in interesting concrete situations.

The course will review basic intersection theory and its relation to topology. Prerequisites: first courses in algebraic topology (including homology, cohomology, and Poincar'e duality), and algebraic geometry (including definitions and basic properties of affine and projective varieties). It will be useful to know something about vector bundles and Chern classes, but this will be reviewed as necessary in the course. There are no textbooks for this course.

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

#### Instructor(s): Dolgachev

Prerequisites: MATH 631 or 731. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course will cover selected topics from classical algebraic geometry. Most of the topics will be taken from the original classical sources or research papers and cannot be found in modern books in algebraic geometry. The emphasis of the course will be on the geometry of special varieties rather than on general theory. For example, we will discuss the theory of polarity, Del Pezzo and Kummer surfaces, curves of low genus, Cremona transformations.

Some of the material covered in the course does not require any preliminary knowledge of algebraic geometry beyond Math 631. However, other parts of the course will assume a deeper background in algebraic geometry, e.g. some rudiments of the theory of algebraic surfaces.

No text-book is required. Lecture notes will be available.

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

#### Section 001 – Rigid Analytical Geometry.

Prerequisites: MATH 676. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Knowledge of basic scheme theory at the level of Hartshorne's book, basic properties of discrete valuation rings, and prior exposure to Banach spaces. No knowledge of formal schemes will be assumed. Textbooks: There is no required text, but some (expensive) books related to the course material will be kept on reserve at the library. Recommended books are Non-archimedean analysis by Bosch, Güntzer, and Remmert, and Géométric analytique rigide et applications by Frensel and van der Put.

Homework/exams: There will be no exams or homework, but students are strongly encouraged to fill in details omitted in lecture.

Course Complex analytic geometry is a powerful tool in the study of algebraic geometry over C, especially with the help of Serre's GAGA theorems. If we do algebraic geometry over other kinds of analytic fields, such as the p-adics or k((t)) (with k any field), is there a similar analytic geometry? Motivated partly by his discovery of p-adic uniformizations of (certain) elliptic curves and partly by Grothendieck's idea of associating a "generic fiber" to a formal scheme over a complete discrete valuation ring, Tate discovered how to do the impossible: analytic geometry and sheaf theory over totally disconnected ground fields. This theory of so-called rigid analytic geometry has striking similarities with algebraic and complex analytic geometry. For example, one has the GAGA theorems over any non-archimedean ground field. This theory has blossomed into a fundamental tool in the aresenal of the modern number theorist, and it deserves to be understood by a wider audience of geometers (as it provides the inspiration for work of Mumford on degenerations, etc.)

The aim of the course is to develop the basic aspects of rigid analytic geometry, startng with a discussion of affinoid algebras (these are certain Banach algebras which replace the use of finitely generated rings over a field in algebraic geometry). After we study this basic commutative algebra, we will then develop Tate's globalization of the theory to the point where we can state and prove the uniformization theorem for abelian varieties with purely toric reduction, make connections with formal schemes, and understand the significance of the work of Raynaud, Berkovich, and Huber (each of whom introduced important foundational revolutions in the theory). If time permits, we will discuss either Mumford's work on degenerations or the uniformization of more general abelian varieties.

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### MATH 777. Topics in Diophantine Problems.

#### Instructor(s): Parson

Prerequisites: MATH 677, and graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Course description: Class field theory is the study of abelian Galois extensions of the fields that arise in number theory: number fields, function fields of algebraic curves over finite fields, and local fields. It developed in the first half of the 20th century as the flower of much of the effort of 19th-century number theorists, and it remains today part of the core of number theory, underlying and motivating the Langlands Program.

The main theorems of class field theory can be formulated and proved in many different ways. Instead of focusing on the proofs of the results, this course will explain their nature, starting in the most concrete fashion and developing toward the modern formulations. The emphasis will be on examples coming from number fields, introduced in historical context to make clear why the main results answer basic questions in algebraic number theory.

Topics covered will include (as time permits)

1. Cyclotomy, cubic and biquadratic reciprocity,
2. Classical class field theory,
3. The Hilbert symbol and higher reciprocity laws,
4. Artin L-functions, Cebotarev density theorem,
5. The adelic formulation of class field theory, and
6. Local class field theory.

Depending on the background of the students, complex multiplication of elliptic curves and the classification of central simple algebras may also be discussed.

Text: There will be no required text. Recommended references include Lang's "Algebraic Number Theory," Serre's "Local Fields," and the articles in the volume "Algebraic Number Theory" edited by Cassels and Froehlich.

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### MATH 990. Dissertation/Precandidate.

#### Instructor(s):

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

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

Course Homepage: No homepage submitted.

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

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### MATH 995. Dissertation/Candidate.

#### Instructor(s):

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

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

Course Homepage: No homepage submitted.

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

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### Undergraduate Course Listings for MATH. This page was created at 8:16 PM on Wed, Feb 5, 2003.