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# Courses in Mathematics

This page was created at 6:16 PM on Wed, Jan 21, 2004.

## Winter Academic Term 2004 (January 6 - April 30)

### MATH 412. Introduction to Modern Algebra.

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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### MATH 417. Matrix Algebra I.

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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

#### Instructor(s): Curtis E Huntington

Prerequisites: MATH 115, junior standing, and permission of instructor. (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 423. Mathematics of Finance.

#### Instructor(s): Moore

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/423/001.nsf

No Description Provided. Contact the Department.

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### MATH 423. Mathematics of Finance.

#### Instructor(s): Kausch

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/423/002.nsf

No Description Provided. Contact the Department.

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

#### Instructor(s): David Towler Kausch

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/424/001.nsf

No Description Provided. Contact the Department.

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

#### Instructor(s): Mathematics faculty

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

#### Instructor(s): Statistics faculty

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

Credits: (3).

Course Homepage: No homepage submitted.

See STATS 425.

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

#### Instructor(s): Sergey Fomin (fomin@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/~fomin/425w04.html

This course introduces students to the theory of probability and to a number of applications. 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.

There will be approximately 10 problem sets. Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. pText (required): Sheldon Ross, A First Course in Probability, 6th edition, Prentice-Hall, 2002.

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

#### Instructor(s):

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

Credits: (4).

Course Homepage: No homepage submitted.

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

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): Lukas I Geyer

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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### MATH 462. Mathematical Models.

#### Instructor(s): David Bortz

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

Credits: (3).

Course Homepage: No homepage submitted.

This course will cover biological models constructed from difference equations and ordinary differential equations. Applications will be drawn from population biology, population genetics, the theory of epidemics, biochemical kinetics, and physiology. Both exact solutions and simple qualitative methods for understanding dynamical systems will be stressed.

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### MATH 471. Introduction to Numerical Methods.

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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### MATH 475. Elementary Number Theory.

#### Instructor(s): Muthukrishnan Krishnamurthy

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): Muthukrishnan Krishnamurthy

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 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: http://coursetools.ummu.umich.edu/2004/winter/math/486/001.nsf

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. (3). May not be repeated for credit. May not be used in any graduate program in mathematics. Not to apply on any graduate program in mathematics.

Credits: (3).

Course Homepage: No homepage submitted.

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

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### MATH 490. Introduction to Topology.

#### Instructor(s): Elizabeth A Burslem

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

Credits: (1).

Course Homepage: No homepage submitted.

The Applied and Interdisciplinary Mathematics (AIM) student seminar is an introductory and survey course in the methods and applications of modern mathematics in the natural, social, and engineering sciences. Students will attend the weekly AIM Research Seminar where topics of current interest are presented by active researchers (both from U-M and from elsewhere). The other central aspect of the course will be a seminar to prepare students with appropriate introductory background material. The seminar will also focus on effective communication methods for interdisciplinary research. MATH 501 is primarily intended for graduate students in the Applied & Interdisciplinary Mathematics M.S. and Ph.D. programs. It is also intended for mathematically curious graduate students from other areas. Qualified undergraduates are welcome to elect the course with the instructor's permission.

Student attendance and participation at all seminar sessions is required. Students will develop and make a short presentation on some aspect of applied and interdisciplinary mathematics.

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### MATH 512. Algebraic Structures.

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

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.

Text Book: Abstract Algebra, Second Edition by David Dummit and Richard Foote.

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### MATH 513. Introduction to Linear Algebra.

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

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Math 412 or Math 451 or permission of the instructor

Background and Goals: This is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis is on concepts and proofs with some calculations to illustrate the theory.

Content: Topics are selected from: vector spaces over arbitrary fields (including finite fields); linear transformations, bases, and matrices; eigenvalues and eigenvectors; applications to linear and linear differential equations; bilinear and quadratic forms; spectral theorem; Jordan Canonical Form. This corresponds to most of the first text with the omission of some starred sections and all but Chapters 8 and 10 of the second text.

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

Subsequent Courses: The natural sequel to Math 513 is Math 593 (Algebra I). Math 513 is also prerequisite to several other courses: Math 537, 551, 571, and 575, and may always be substituted for Math 417 or 419.

Text: Curtis: Linear Algebra, An Introductory Approach (4th edition, Springer-Verlag)

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### MATH 521. Life Contingencies II.

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

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

Credits: (3).

Course Homepage: No homepage submitted.

This course is a continuation of MATH520 (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): Conlon

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

Credits: (3).

Course Homepage: No homepage submitted.

Required Text: "Loss Models-from Data to Decisions", by Klugman, Panjer and Willmot, Wiley 1998. 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. Contents: 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 Poisson 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 homeworks, a midterm and a final exam.

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

#### Instructor(s): Gautam Bharali

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

Credits: (3).

Course Homepage: No homepage submitted.

Background: This course is a fairly rigorous study of the mathematical basis of probability theory. There is some overlap of topics with Math 425, but in Math 525, there is a greater emphasis on the proofs of major results in probability theory. This course and its sequel - Math 526 - are core courses for the Applied and Interdisciplinary Mathematics (AIM) program.

Content: The notion of a probability space and a random variable, discrete and continuous random variables, independence and expectation, conditional probability and conditional expectations, generating functions and moment generating functions, the Law of Large Numbers, and the Central Limit Theorem comprise the essential core of this course. Further topics, to be decided later (and, if feasible, selected according to audience interest), will be covered in the last month of the semester.

Alternatives: EECS 501 covers some of the above material at a lower level of mathematical rigor. Math 425 (Introduction to Probability) is recommended for students with substantially less mathematical preparation.

Text: Introduction to Probability Models by Sheldon Ross

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

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

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

Credits: (3).

Course Homepage: No homepage submitted.

See STATS 526.001.

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### MATH 528. Topics in Casualty Insurance.

#### Instructor(s): Virginia R Young

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/528/001.nsf

Risk management is of major concern to all financial institutions and is an active area of modern finance. This course is relevant for students with interests in insurance, risk management, or finance. We will cover the following topics: advanced topics in credibility theory, risk measures and premium principles, optimal (re)insurance, reinsurance products, and reinsurance pricing.

I assume that you have taken MATH 523, Risk Theory. In fact, one can think of this course as a continuation of MATH 523 with emphasis on applying the material learned in Risk Theory to more practical settings.

The official text for the course is a set of notes available at UM.CourseTools. In addition, an excellent book concerning modern reinsurance products is Integrating Corporate Risk Management by Prakash Shimpi, published by Texere. I suggest that you buy this book, but I do not require that you do so.

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### MATH 531. Transformation Groups in Geometry.

#### Instructor(s): Emina Alibegovic

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: MATH 412 or 512 would be helpful, but neither is necessary. Text required: None. Text recommended: Armstrong, Groups and Symmetry; Lyndon: Groups and Geometry. textbook comment: Your class notes and my handouts will be sufficient. The books I listed contain some of the material we will cover, but not all of it. Course description: The purpose of this course is to explore the close ties between geometry and algebra. We will study Euclidean and hyperbolic spaces and groups of their isometries. Our discussions will include, but will not be limited to, free groups, triangle groups, and Coxeter groups. We will talk about group actions on spaces, and in particular group actions on trees.

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

#### Instructor(s): Mattias Jonsson, Leonard Kofman

Prerequisites: MATH 423, IOE 452 or IOE 453; and a solid background in basic probability theory. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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

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

#### Instructor(s): Xu Meng, Jussi Samuli Keppo

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/ioe/553/001.nsf

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

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

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

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: http://www.math.lsa.umich.edu/~dburns/547/547syll.html

Probabilistic models of proteins and nucleic acids. Analysis of DNA/RNA and protein sequence data. Algorithms for sequence alignment, statistical analysis of similarity scores, hidden Markov models, neural networks training, gene finding, protein family profiles, multiple sequence alignment, sequence comparison, and structure prediction. Analysis of expression array data.

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

#### Instructor(s):

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: http://www.math.lsa.umich.edu/~dburns/547/547syll.html

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

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

#### Instructor(s): Divakar Viswanath

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

Credits: (3).

Course Homepage: No homepage submitted.

Recent Texts: Complex Variables and Applications, 6th ed. (Churchill and Brown); 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 557. Methods of Applied Mathematics II.

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

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: (1) one of the following: Math 217, 419, or 513 (i.e. a course in linear algebra); (2) one of the following: Math 216, 256, 286, 316, or 404 (i.e. a course in differential equations); (3) Math 451 (or an equivalent course in advanced calculus); (4) Math 555 (or an equivalent course in complex variables). Text: There is no required text. Lecture notes will be made available to students from the instructor's website. Recommended texts will be announced in class. Audience: Graduate students and advanced undergraduates in applied mathematics, engineering, or the natural sciences. Background and Goals: In applied mathematics, we often try to understand a physical process by formulating and analyzing mathematical models which in many cases consist of differential equations with initial and/or boundary conditions. Most of the time, especially if the equation is nonlinear, an explicit formula for the solution is not available. Even if we are clever or lucky enough to find an explicit formula, it may be difficult to extract useful information from it and in practice, we must settle for a sufficiently accurate approximate solution obtained by numerical or asymptotic analysis (or a combination of the two). This course is an introduction to the latter of these two approximation methods. The material covered in the textbook includes the nature of asymptotic approximations, asymptotic expansions of integrals and applications to transform theory (Fourier and Laplace), regular and singular perturbation theory for differential equations including transition point analysis, the use of matched expansions, and multiple scale methods. The time remaining after studying these topics will be devoted to the derivation of several famous canonical model equations of applied mathematics (e.g. the Korteweg-de Vries equation and the nonlinear Schroedinger equation) using multiscale asymptotics. Students will come to understand how these equations arise again and again from fields of study as diverse as water wave theory, molecular dynamics, and nonlinear optics. Grading: Students will be evaluated on the basis of homework assignments and also participation and lecture attendance.

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### MATH 558. Ordinary Differential Equations.

#### Instructor(s): Andrew J Christlieb

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Basic Linear Algebra, Ordinary Differential Equations (math 216), Multivariable Calculus (215) and Either Advanced Calculus (math 451) or an advanced mathematical methods course (e.g. Math 454); preferably both. Course Objective: This course is an introduction to the modern qualitative theory of ordinary differential equations with emphasis on geometric techniques and visualization. Much of the motivation for this approach comes from applications. Examples of applications of differential equations to science and engineering are a significant part of the course. There are relatively few proofs. Course Description: Nonlinear differential equations and iterative maps arise in the mathematical description of numerous systems throughout science and engineering. Such systems may display complicated and rich dynamical behavior. In this course we will focus on the theory of dynamical systems and how it is used in the study of complex systems. The goal of this course is to provide a broad overview of the subject as well as an in-depth analysis of specific examples. The course is intended for students in mathematics, engineering, and the natural sciences. Topics covered will include aspects of autonomous and driven two variable systems including the geometry of phase plane trajectories, periodic solutions, forced oscillations, stability, bifurcations and chaos. Applications to problems from physics, engineering and the natural sciences will arise in the course by way of examples in lecture ad through the homework problems. We will cover material from Chapters 1-5 and 8-13 of the text. Textbook Nonlinear Ordinary Differetial Equations, Oxford Press. by: D.W. Jordan and P. Smith References Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer. John Guckenheimer and Philip Holmes Nonlinear Differential Equations and Dynamical Systems, Springer. Ferdinand Verhulst Applications of Centre Manifold Theory, Springer. J. Carr Nonlinear Systems, Chambridge. P.G. Drazin

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

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

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

Credits: (3).

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

Course Homepage: http://www.engin.umich.edu/class/ioe510/

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

Course Objectives: To provide first-year graduate students with basic understanding of linear programming, its importance, and applications. To discuss algorithms for linear programming, available software and how to use it intelligently.

Recommended Books: K. G. Murty, Linear Programming, Wiley, 1983.
Also, R. Saigal, Linear Programming: A Modern Integrated Analysis, Kluwer, 1995, can be used as a reference book.

Course Content:

1. LP models, various applications. Separable piece-wise linear convex function minimization problems, uses in curve fitting and linear parameter estimation. Approaches for solving multiobjective linear programming models, Goal programming.
2. What useful planning information can be derived from an LP model (marginal values and their planning uses).
3. Review of Pivot operations, basic vectors, basic solutions, and bases. Brief review of polyhedral geometry.
4. Duality and optimality conditions for LP.
5. Revised primal and dual simplex methods for LP.
6. Infeasibility analysis, marginal analysis, cost coefficient and RHS constant ranging, other sensitivity analyses.
7. Algorithm for transportation models.
8. Other topics time permitting.

There will be weekly homework assignments.

Grading (approximate): Midterm Exam 20% Final Exam 50% Computational Project 15% Homework 15%

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### MATH 566. Combinatorial Theory.

#### Instructor(s): John R Stembridge

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: MATH 512 or an equivalent level of mathematical maturity.

This course will be an introduction to algebraic combinatorics. Previous exposure to combinatorics will not be necessary, but experience with proof-oriented mathematics at the introductory graduate or advanced undergraduate level, and linear algebra, will be needed.

Most of the topics we cover will be centered around enumeration and generating functions. But this is not to say that the course is only about enumeration — questions about counting are a good starting point for gaining a deeper understanding of combinatorial structure.

Some of the topics to be covered include sieve methods, the matrix-tree theorem, Lagrange inversion, the permanent-determinant method, the transfer matrix method, and ordinary and exponential generating functions. Recommended text: R. Stanley, Enumerative Combinatorics, Vol. I Cambridge Univ. Press, 1997.

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### MATH 567. Introduction to Coding Theory.

#### Instructor(s): Hendrikus Gerardus Derksen

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

Credits: (3).

Course Homepage: No homepage submitted.

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): James F 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.

This course is an introduction to numerical linear algebra, a core subject in scientific computing. Three general problems are considered: (1) solving a system of linear equations, (2) computing the eigenvalues and eigenvectors of a matrix, and (3) least squares problems. These problems often arise in applications in science and engineering, and many algorithms have been developed for their solution. However, standard approaches may fail if the size of the problem becomes large or if the problem is ill-conditioned, e.g. the operation count may be prohibitive or computer roundoff error may ruin the answer. We'll investigate these issues and study some of the accurate, efficient, and stable algorithms that have been devised to overcome these difficulties.

The course grade will be based on homework assignments, a midterm exam, and a final exam. Some homework exercises will require computing, for which Matlab is recommended.

Topics:

1. warmup: vector and matrix norms, orthogonal matrices, projection matrices, singular value decomposition (SVD);
2. least squares problems: QR factorization, Gram-Schmidt orthogonalization, Householder triangularization, normal equations;
3. backward error analysis: stability, condition number, IEEE floating point arithmetic;
4. direct methods for Ax=b: Gaussian elimination, LU factorization, pivoting, Cholesky factorization;
5. eigenvalues and eigenvectors: Schur factorization, reduction to Hessenberg and tridiagonal form, power method, inverse iteration, shifts, Rayleigh quotient iteration, QR algorithm;
6. iterative methods for Ax=b: Krylov subspace, Arnoldi iteration, GMRES, conjugate gradient method, preconditioning;
7. applications: image compression using the SVD, least squares data fitting, finite-difference schemes for a two-point boundary value problem, Dirichlet problem for the Laplace equation

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

#### Instructor(s): Divakar Viswanath

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.

Math 572 is an introduction to numerical methods for solving differential equations. These methods are widely used in science and engineering. The four main segments of the course will cover the following topics:

1. Ordinary differential equations (Runge-Kutta methods, multiple time-step methods, stiffness)

2. Finite difference methods (von Neumann stability analysis, ADI, CFL, Lax equivalence theorem)

3. Spectral methods (Fourier methods, method of lines)

4. Finite element methods (2-point boundary value problems and 2-dimensional elliptic problems) If time permits, we will go into the multigrid method for solving linear systems.

OPTIONAL TEXT: Numerical Solution Of Partial Differential Equations, K.W. Morton and D.F. Mayers, Cambridge University Press

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### MATH 592. Introduction to Algebraic Topology.

#### Instructor(s): Igor Kriz

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

Credits: (3).

Course Homepage: No homepage submitted.

The purpose of this course is to introduce basic concepts of algebraic topology, in particular fundamental group, covering spaces and homology. These methods provide the first tools for proving that two topological spaces are not topologically equivalent (example: the bowling ball is topologically different from the teacup). Other simple applications of the methods will also be given, for example fixed point theorems for continuous maps. Prerequisites: basic knowledge of point set topology, such as from 590 or 591. Books: There is no ideal text covering all this material on exactly the level needed (basic but rigorous). Recommended texts include Munkres: Elements of Algebraic topology (for homology) and J.P.May: A concise course in algebraic topology (for fundamental group and covering spaces). Both texts include topics which will not be covered in 592, and are also suitable textbooks for the next course in algebraic topology, 695.

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### MATH 594. Algebra II.

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

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisite: MATH 593. I. Group theory: Group actions on sets. Linear groups. Sylow theorems. Solvable and nilpotent groups. Free groups and presentations. Linear representations of groups. Character tables. II Field extensions: Algebraic and transcendental extensions. Algebraic functions. Luroth theorem. III. Galois theory. Galois correspondence. Kummer's and Schreier's extensioins. Solutions of equations in radicals. Computation of Galois groups of equations. Text-book: M. Artin, Algebra. Prentice Hall. 1991.

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### MATH 597. Analysis II.

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

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; Hausdorff dimension; 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): Juha Heinonen (juha@umich.edu)

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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### MATH 609. Topics in Analysis.

#### Instructor(s): Jinho Baik

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Complex Analysis, Linear Algebra, basic Functional Analysis, basic Probability Description: Random matrix theory asks questions like: For a Hermitian matrix of size N whose entries are randomly chosen, what do the eigenvalues look like when N becomes large ? This theory started around 1950's in nuclear physics to describe the neutron resonances of heavy nuclei, but over the years it turned out that the eigenvalues of random matrix of large size actually describe a variety of statistical systems in physics, statistics and mathematics (famous example: statistical behavior of the zeros of the Riemann-zeta function on the critical line). On the other hand, a permutation problem that we plan to discuss in the course is the following question: If one picks a permutation of large size (say a million) at random, what is the typical length of the longest increasing subsequence ? (For example, for the permutation p=51324 of size 5, meaning p(1)=5, p(2)=1, p(3)=3, p(4)=2, p(5)=4, 124 is the longest increasing subsequence with length 3). This particular question is called the Ulam's problem', and it turned out that the answer to this problem (and more) is related to the eigenvalues of random matrix. As permutations arise naturally in any ordered objects, the above question indeed has variety applications and interpretations (like random tiling, random growth) which we will discuss over the course. The aim of this course is to show that a wide variety of problems, some classical, in combinatorics and statistical systems can be described in terms of random matrix theory. There will be both combinatorial and analytic considerations in the course, but we will not assume any combinatorial background of the students. Course lecture notes will be provided. Optional Readings: Percy Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach M.L.Mehta, Random matrices R.Stanley, Enumerative combinatorics. Vol. 2.

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

#### Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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### MATH 615. Commutative Algebra II.

#### Instructor(s): Melvin Hochster

Prerequisites: MATH 614 and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course is a topics course in the theory of commutative Noetherian rings. Some background in algebraic geometry, including courses taken concurrently, will be helpful, but is not essential. Topics to be covered include the structure of complete local rings, the structure of homomorphisms, including smooth, etale and flat homomorphisms, Zariski's main theorem, Artin approximation, homological methods including some K-theory, properties of regular, Cohen-Macaulay and Gorenstein rings, including methods for determining whether rings satisfy these conditions, and techniques for reducing the proofs of theorems about all Noetherian rings containing a field of arbitrary characteristic to the case of finitely generated algebras over a field of characteristic p > 0. The precise syllabus will depend in part on the interests and backgrounds of the students. The material should be useful to those working in commutative or non-commutative algebra, algebraic geometry, several complex variables, and algebraic number theory. There is no text: lecture notes will be provided. Grades will be based on five or six problem sets.

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### MATH 619. Topics in Algebra.

#### Instructor(s): Michael Roitman

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

Credits: (3).

Course Homepage: No homepage submitted.

PREREQUISITES: A course in linear algebra (like MATH 513 or 419) MATH 612 might be useful, but not necessary.

PHILOSOPHY: Probably the best way of learning mathematics is by organizing material in a certain direction, that should lead to a well-defined goal. In this course, the goal is to learn about rather mysterious connection, called "Moonshine", between modular functions of genus 0 and the Fischer-Griess finite simple group a.k.a. the Monster. This course should provide a shortcut from first-year graduate mathematics to the very frontier of our knowledge.

ORGANIZATION: The course is going to have the following parts

I. Kac-Moody Lie algebras. After reviewing basic Lie algebra theory, we will start with several first chapters of Kac's book [1], followed by Borcherds' characterization of generalized Kac-Moody algebras [2,3] and Weyl-Kac character formulas [1].

II. Vertex algebras. The message of this part is that the most interesting Kac-Moody algebras are those that are related with a certain new structure, called vertex algebra. We'll learn some of the general theory of vertex algebras [4], including, most importantly, the so-called Zhu's theory [5], that relates vertex algebras with modular forms. Some discussion of modular forms will be included [6]. III. The Moonshine Here we will review the construction of the Moonshine module V^\natural [7]. This is a vertex algebra with the group of automorphisms being the Monster, and the graded character being the elliptic invariant J. If time permits, we will learn Borcherds' proof [8] of the Conway-Norton conjecture, that the so-called Thompson series of V^\natural are Hauptmoduls — meromorphic functions of genus 0.

BIBLIOGRAPHY:

[1] V. Kac, Vertex algebras for beginners, Second edition, Amer. Math. Soc., Providence, RI, 1998 [2] R.E. Borcherds, Central Extensions Of Generalized Kac-Moody Algebras, J. Algebra {\bf 140} (1991), no.2, 330-335

[3] R. Borcherds, Generalized Kac-Moody algebras, J. Algebra (1988), no.2, 501-512

[4] V. G. Kac, Infinite-dimensional Lie algebras, Third edition, Cambridge Univ. Press, Cambridge, 1990 [5] Y. Zhu, Modular Invariance Of Characters Of Vertex Operator Algebras J. Amer. Math. Soc. (1996), no.1, 237-302

[6] S. Lang, Introduction To Modular Forms, Corrected reprint of the 1976 original, Springer, Berlin, 1995

[7] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras And The Monster, Academic Press, Boston, MA, 1988 [8] R.E. Borcherds, Monstrous Moonshine And Monstrous Lie superalgebras, Invent. Math. (1992), no.2, 405-444

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### MATH 623. Computational Finance.

#### Instructor(s): Yevgeny Goncharov

Prerequisites: MATH 316 and MATH 425 or 525. Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2004/winter/math/623/001.nsf

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

Prerequisites:

Differential equations (e.g. Math 316); basic probability theory (e.g,/i> 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).

Textbooks:

1. James and Webber: Interest Rate Modelling, Wiley, 2000.

2. Tavella and Randall: Pricing Financial Instruments: the finite difference method, Wiley, 2000.

3. Glasserman: "Monte Carlo methods in Financial Engineering", Springer 2003.

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

#### Instructor(s): Timothy D Johnson

Prerequisites: Graduate standing; BIOSTAT 601, 650, 602 and MATH 450. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Conditional distributions, probability generating functions, convolutions, discrete and continuous parameters, Markov chains, medical and health related applications.

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### MATH 632. Algebraic Geometry II.

#### Section 001.

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

Credits: (3).

Course Homepage: No homepage submitted.

Algebraic geometry was classically concerned with the geometric study of solutions to polynomial equations in several complex variables. In its modern reformulation based on the concept of a scheme and many brilliant ideas of Grothendieck, the subject has acquired awesome technical power and its techniques not only permit a better arsenal with which to study classical problems over an arbitrary field, but also have a vast range of applicability beyond the classical concerns: algebraic methods for studying analytic concepts (modular forms, analytic spaces, etc.), a geometric foundation that allows one to `visualize" commutative algebra and number theory, a source of important constructions and techniques in representation theory, a common framework in which one can view Galois theory and fundamental groups as the same thing, and so on ad infinitum.

I plan to cover as much of Chapters 2,3,4 of Hartshorne as I can cram into the semester. In particular, I hope to get through the proofs of some serious theorems on sheaf cohomology and its applications in order to provide some concrete payoff for the hard work. It is EXTREMELY IMPORTANT to do the homework! Modern algebraic geometry is a beautiful subject, but (in the words of Mumford) it is more widely respected and feared than understood. If you want to do more than push around abstract definitions, you must expect to devote a lot of time to this course; the payoff is worth it, since it will transform your way of thinking about many basic concepts in mathematics.

In order to cover a reasonable amount of material, I will have to assume you have mastered commutative algebra at the level of the book by Atiyah--MacDonald. The first 15 sections of the Matsumura text constitute a more sophisticated perspective on the same material, emphasizing some technical issues (like flatness) which are not so prominent in Atiyah-MacDonald. Students will be expected to gradually read the first 15 sections of Matsumura on their own as the semester progresses. I will also give an introductory lecture or two on derived functors and spectral sequences, but will expect you to teach yourself the details on this material (once we need it) by reading about it in books such as Lang's "Algebra" or Eisenbud's "Commutative algebra" (homological algebra is too dry to be learned by listening to someone else speak about it in detail at the blackboard).

Textbooks (required): Algebraic Geometry (Hartshorne), Commutative Ring Theory (Matsumura)

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### MATH 635. Differential Geometry.

#### Instructor(s): Ralf J Spatzier

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

Credits: (3).

Course Homepage: No homepage submitted.

Riemannian geometry is an active and exciting research area with close connections to other areas such as topology, algebraic geometry, dynamical systems and Lie theory. This course is an introduction to differential geometry. In the first part of the course, we will introduce the basic concepts and methods of covariant differentiation, curvature, parallel transport, geodesics, Jacobi fields, variational methods, submanifolds amongst others.

In the second part, I will emphasize global aspects, i.e. features pertaining to the Riemannian manifold as a whole. One basic theme is that curvature bounds greatly restrict the topology of the underlying manifold. As an example, the sphere theorem of Berger and Klingenberg says that every simply connected complete Riemannian manifold with curvatures strictly between 1 and 4 is homeomorphic to a sphere. We will sample the classical results and give an introduction to more recent developments in Riemannian geometry.

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### MATH 636. Topics in Differential Geometry.

#### Instructor(s): Lizhen Ji

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

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: Basically, the course is self-contained.

Motivated by the Hizrebruch signature theorem, Novikov introduced higher signatures and conjectured that these higher signatures are homotopy invariants of manifolds. The modern formulation of the Novikov conjecture is given in terms of the injectivity of the assembly map in surgery theory. There are also assembly maps in other theories such as the algebraic K-theory, C*-algebras. The assembly map in C*-algebras is closely related to the index of elliptic operators. The validity of the Novikov type conjectures allows one to compute important groups such as surgery groups, K-groups of group rings, and to understand the question of existence of metrics of positive scalar curvature etc.

Due to these connections with many different fields, the Novikov conjecture is one of the central problems in modern mathematics. This course will give an introduction to various versions of the Novikov conjectures, starting from basic definitions of K-groups, C*-algebras, and the case of discrete groups of Lie groups, in particular, arithmetic groups, will be emphasized since a lot of work on Novikov conjectures has been for such groups.

The basic reference is Novikov conjectures, Index theorems and Rigidity, ed. by Steven Freey, Andrew Ranicki, Jonathan Rosenberg, Cambridge University Press, 1995

Materials will be drawn from other sources as well.

Work expected: A talk in the class will be encouraged.

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### MATH 637. Lie Groups.

#### Section 001.

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

Credits: (3).

Course Homepage: No homepage submitted.

This will be a comprehensive introduction to the theory of Lie groups. I will prove the basic results, describe the structure of nilpotent and solvable Lie groups. I will present the results on tori in compact Lie groups and use them to describe their topological and group-theoretic structure and their classification. I will study noncompact semi-simple Lie groups in considerable detail: Look at their maximal compact subgroups, prove their conjugacy, and prove the Cartan, Iwasawa (and possibly Bruhat) decompositions.

I will assume some familiarity with differentiable manifolds. Results on Lie algebras will be used. I will give precise definitions, statements and references. Prior knowledge of Lie Algebras is therefore not required. The course should be useful for anyone interested in pursuing Differential or Algebraic Geometry, Topology, Representation Theory or Number Theory.

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### MATH 655. Topics in Fluid Dynamics.

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

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

Credits: (3).

Course Homepage: No homepage submitted.

Text: "Introduction to Hydrodynamic Stability", by P. G. Drazin, Cambridge University Press

Prerequisites: Some familiarity is assumed with fluid dynamics, e.g. the first two chapters in Chorin-Marsden (A Mathematical Introduction to Fluid Mechanics ) Springer).

Fluid flow is governed by the Navier-Stokes equations, a set of nonlinear partial differential equations expressing the conservation of mass, momentum, and energy. These equations contain a nondimensional parameter, the Reynolds number, which measures the ratio between inertial and viscous effects in a specific flow. Experiments show that if the Reynolds number is sufficiently small, the flow is laminar, but if the Reynolds number is larger than a certain critical value, the flow undergoes a transition to turbulence. The goal of hydrodynamic stability theory is to explain the details of this transition. This course is an introduction to the subject, dealing with analytical results and how they relate to experiments and computations. Thermal and centrifugal instability will be discussed but the main emphasis is on parallel shear flow, including mixing layers, jets, wakes, and boundary layers. The course will cover classical linear and nonlinear stability theory. Some key topics are: Kelvin-Helmholtz instability, Rayleigh equation, Orr-Sommerfeld equation, Landau equation, critical layer, absolute and convective instability, global modes, transient growth. There will be several homework assignments.

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

#### Instructor(s): Sijue Wu

Prerequisites: MATH 558, 596 and 597, and Graduate standing. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

Prerequisites: MATH 216, MATH 597 or equivalent, or permission of the instructor Text: L. C. Evans: Partial Differential Equations. Graduate Studies in Mathematics, 19. AMS 1998. ISBN: 0-8218-0772-2

References: G.B. Folland: Introduction to Partial Differential Equations P. Garabedian: Partial Differential Equations

F. John: Partial Differential Equations

J. Rauch: Partial Differential Equations

Partial Differential Equations are mathematical structures for models in science and technology. It is of fundamental importance in physics, biology and engineering design with connections to analysis, geometry, probability and many other subjects. The goal of this course is to introduce students (both pure and applied) to the basic concepts and methods that mathematicians have developed to understand and analyze the properties of solutions to partial differential equations.

Topics to be covered will include the Laplace, heat and wave equations, and nonlinear first order equations. The method of characteristics, energy methods, maximal principles, Fourier transform and Sobolev spaces will be introduced.

Content: Materials will be taken from Chapters 2-6 of the text.

Grading: Grades will be based on homeworks and a take home final exam.

Subsequent Courses: MATH 657 Nonlinear Partial Differential Equations.

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

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

Prerequisites: MATH 561. Graduate standing. (3). May not be repeated for credit. CAEN lab access fee required for non-Engineering students.

Credits: (3).

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

Course Homepage: http://www-personal.engin.umich.edu/~rsaigal/courses/610.w04/

A recent exciting development in linear programming is the generation of a new class of methods called Interior Point Methods. These methods solve any instance of a problem in time that is a low order polynomial of the data of the problem. These methods have been recently extended to convex programming, and solve this problem as e®ectively. An important problem in this class is the Semidefinite Programming problem, and many hard combinatorial problems like MAX CUT, quadratic assignment problem, and other integer programming problems have SDP relaxations. In addition some non-convex problems also have such relaxations.

This course will cover these recent developments. A topical list follows:

• Review of linear programming duality.
• Boundary methods for linear programming.
• Interior point methods for linear programming.
• Extensions to convex problems.
• Semidefinite programming and duality.
• Interior point methods for SDP.
• SDP relaxations of combinatorial problems and integer programs.
• Large scale implementations of these methods.
• Saddle Point Methods for Convex problems.

It will also cover the traditional topics of Large Scale Linear Programming:

• Dantzig-Wolfe Decomposition Principle.
• Compact Basis Methods.

BOOKS

1. Saigal, R, Linear Programming - a modern integrated analysis, Kluwer Academic Publishers, Norwell, MA, 1995.
2. Wright, S. J., Primal-Dual Interior Point Methods, SIAM, Philadelphia, PA, 1997.
3. Wolkowitz, H., R. Saigal and L. Vandenberghe, A Handbook of Semidefinite Programming, Kluwer Academic Publishers, Norwell, MA, 2000.

GRADING The course grade will be based on three home-works, a class presentation and a final. These will carry the following weights: Home Works* 30% Class Presentation 20% Final Exam 50%

* - Home-works will require the knowledge of MATLAB use and programming.

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

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

Prerequisites: MATH 561. Graduate standing. (3). May not be repeated for credit. CAEN lab access fee required for non-Engineering students.

Credits: (3).

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

Course Homepage: http://www-personal.engin.umich.edu/~murty/611/

Prerequisites: A course in linear programming, equivalent to IOE 510.

Course objectives: To expose the student to nonlinear models, their applications, how to construct them, and to algorithms for solving them satisfactorily

Books: M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming Theory and Algorithms, Wiley, 1993, 2nd Edition.
K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag, 1988.
R. Fletcher, Practical Methods of Optimization, Wiley-Interscience, 1987.

Contents:

1. Formulation of continuous optimization models, curve fitting, parameter estimation, and L1, L2, and L*, -measures of deviation. Difference between linear and nonlinear model building. Examples.
2. Types of problems. State of the art. What can and cannot be done efficiently? Goals for algorithms.
3. Theorems of alternatives for linear systems.
4. Convex sets, separating hyperplanes, convex and concave functions.
5. Optimality conditions.
6. Quadratic programming and complementary pivot methods.
7. Newton's method and simplicial methods for nonlinear equations.
8. Line Search methods.
9. Unconstrained minimization algorithms.
10. Constrained minimization algorithms. Penalty and barrier methods, SQP and SLP methods.

Work in the course: Homeworks every week. One midterm and final. A computer project.

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### MATH 669. Topics in Combinatorial Theory.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~barvinok/c04.html

Integer points (points with integer coordinates) play an important role in algebra, number theory, combinatorics, and optimization. In this course, I am planning to discuss some of the classical and recent results regarding integer points and lattices, such as Minkowski Theorems and their applications in number theory and analysis, sphere packing and error-correcting codes, flatness theorems and their applications in integer programming, classical results of Macdonald and McMullen on Ehrhart polynomials of integer polytopes and recent works of Brion, Lawrence and Khovanskii on rational generating functions for lattice points in polyhedra. Prerequisites: good knowledge of linear algebra Grading: we will have a number of homework problem sets Text: There is no required text. Recommended text: A. Barvinok, A Course in Convexity, Graduate Studies in Mathematics, AMS, Providence, RI, 2002. Other useful references: R.P. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth and Brooks/Cole, 1986; P.M. Gruber and C.G. Lekkerkerker, Geometry of Numbers (second edition), North Holland, 1987 and A. Schrijver, The Theory of Linear and Integer Programming, Wiley, 1986.

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### MATH 682. Set Theory.

#### Instructor(s): Peter G 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 the first 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 will derive 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 chapter of a book being written by the instructor. There will be several problem sets but no exams.

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### MATH 697. Topics in Topology.

#### Instructor(s): Richard D Canary

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

Credits: (2-3).

Course Homepage: No homepage submitted.

It has been known for many years that every surface admits a nice geometric structure, either spherical, euclidean or hyperbolic. In the 1970s Bill Thurston conjectured that every 3-manifold can be canonically cut up into pieces, each of which is either a Seifert fibre space (i.e. can be foliated by circles) or admits a hyperbolic structure. Seifert fibre spaces have been completely classified and are known to admit one of 6 possible geometric structure. In this course we will develop some of the theory of hyperbolic surfaces and 3-manifolds.

We will begin with an elementary introduction to hyperbolic space and constructions of hyperbolic manifolds. We will discuss the Teichmuller space of all hyperbolic structures on a surface, which may be thought of as the universal cover of moduli space. We will also develop the thick-thin decomposition of a hyperbolic 3-manifold and the theory of geometrically finite hyperbolic manifolds.

More advanced topics will be determined by class interest but may include the deformation theory of hyperbolic 3-manifolds, topological tameness and its implications, the complex analytic theory of Teichmuller space or applications of characteristic submanifold theory to hyperbolic 3-manifolds. I also intend to offer some survey lectures on related material that is beyond the scope of the class.

We will try to keep the presentation of the material relatively basic and self-contained. The material in 591 and 592, especially the theory of differentiable manifolds and covering spaces, should be sufficient background. We will also occasionally make use of results from complex analysis and Riemannian geometry. There will be no required text or homework, but each student will be asked to give a lecture on a topic of their choice related to the subject matter of the class.

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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 three times for credit.

Credits: (1-3).

Course Homepage: No homepage submitted.

No Description Provided. Contact the Department.

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

#### Instructor(s): Berit Stensones (berit@umich.edu)

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

Credits: (3).

Course Homepage: No homepage submitted.

This course will be a natural continuation of this falls Math.605

We will cover material from the book From Holomorphic Functions to Complex Manifolds. by Klaus Fritzsche and Hans Grauert. The topics that we will discuss are the following.

ANALYTIC SETS

Here we will do things like prove the Weierstrass preparation theorem, study branched coverings and singular locuses.

COMPLEX MANIFOLDS: Material covered in this section will be things like holomorphic mappings, complex fiber bundels and so on. Also, we will study examples of complex manifolds like Complex Tori, Hoph manifolds and Complex Projective Spaces

STEIN MANIFOLDS: One of topics that we will cover here is the fact that every Stein Manifold can be realiced as a closed submanifold of Complex Eucledean Space.

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

#### Instructor(s): Juha Heinonen (juha@umich.edu)

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

Credits: (3).

Course Homepage: No homepage submitted.

This is a course on ANALYSIS ON SINGULAR SPACES. Development of first order differential analysis on non-Riemannian spaces has been extensive in recent years. The need for such analysis arises in many areas of mathematics: in Riemannian and metric geometry, partial differential equations, function theory, combinatorial group theory. The topic is also of intrinsic interest, as we are dealing with questions like: what does differentiability mean in general situations, and what spaces allow for calculus? It turns out that there are fascinating examples of spaces, that go well beyond the standard Euclidean or Riemannian cases, where answer to the second question is yes.

We will cover the following topics: the basic theory of metric spaces and Lipschitz functions on them, the Gromov-Hausdorff convergence, Cheeger's differentiability theorem for Lipschitz functions on metric spaces (following S. Keith's thesis), Sobolev spaces on metric measure spaces, plus other topics related to the theme and depending on the interests of the audience.

I will attempt to make this course appealing to a diverse audience. To this end, many different examples will be studied. Please contact me if you have any questions about the course.

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### MATH 715. Advanced Topics in Algebra.

#### Instructor(s): Stephen M DeBacker

Prerequisites: MATH 594 and 612. Graduate standing. (3). May be elected more than once for credit.

Credits: (3).

Course Homepage: No homepage submitted.

This course will serve as an introduction to the representation theory of reductive p-adic groups. The main goal of the course is to present the Bernstein decomposition of the category of smooth representations of a p-adic group. This is a very beautiful result which uses most of the basic ideas in this branch of representation theory. We shall begin the course by working out the representation theory of the (non-reductive) Heisenberg group and of GL_1. We shall then present some structure theory results (sticking to the example of GL_n). After this, we shall really start: we'll discuss the nature of supercuspidal representations, parabolic induction, Jacquet restriction, and the relations among these concepts. After presenting the Bernstein decomposition, we shall turn our attention to the Bernstein center, the Langlands classification, and various other topics, time permitting.

The course should be useful for anyone interested in representation theory or number theory. Moreover, it will be accessible to most everyone that has done well in algebra (though it might help to recall what exactly a p-adic field is -- I won't dwell on that for long).

There is no text which covers this material.

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

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

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

Credits: (3).

Course Homepage: No homepage submitted.

In recent years, two topics of somewhat classical origin have emerged as being of fundamental importance in the study of algebraic varieties.

On the one hand, the study of arcs on a variety has led to some surprising insights and results. Here the new technique of motivic integration, due to Kontsevich, Denef and Loeser (among others), plays a central role. It links geometric, topological and arithmetic ideas.

In another direction, the (derived) category of sheaves on a variety has proven to have surprising geometric content. It is the subject of a number of interesting results and conjectures in higher dimensional geometry. The goal of the course is to work through some of this material. In order to better understand the genesis of the results, my plan is to proceed in a somewhat historical fashion. I expect to be able to cover a substantial part of the natural material involving arcs, but we may only get through the first steps with sheaves.

This is intended to be a research-level course, so I will assume a good working knowledge of basic complex algebraic geometry. However where possible I will try to provide heuristic explanations of some of the less familiar techniques.
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### MATH 775. Topics in Analytic Number Theory.

#### Instructor(s): Soundararajan

Prerequisites: MATH 675. (3). May not be repeated for credit.

Credits: (3).

Course Homepage: No homepage submitted.

In this course I will discuss several topics related to the distribution of zeros of $L$-functions and their applications to arithmetic questions. The fundamental questions which motivate our discussions are the Riemann hypothesis (which predicts that all non-trivial zeros of $L$-functions lie on the critical line) and the Birch-Swinnerton-Dyer conjectures (which predict that the order of vanishing of $L$-functions at a special point has arithmetical implications).

I will begin by recalling the ideas behind the classical zero free regions for $L$-functions (for $\zeta(s)$, Dirichlet $L$-functions and $L$-functions attached to newforms). Then I will develop the mollifier method, exhibiting its connections to Selberg's sieve method, and use it to obtain "zero density results". These results show that $L$-functions don't have too many zeros away from the critical line: such results are often powerful enough to substitute for the Riemann hypothesis in applications (e.g., the Bombieri-Vinogradov theorem, and Linnik's theorem on the least prime in an arithmetic progression). The mollifier method is also useful in obtaining non-vanishing results for $L$-functions at special points, which is a topic of great current interest.

In the second half of the course, I will consider the emerging (conjectural) connections between the distribution of zeros of $L$-functions and the distribution of eigenvalues of large random matrices. I will first describe the "pair correlation" work of Montgomery, which first pointed to this connection, and then sketch the "n-level correlation" results of Rudnick-Sarnak. I will also describe the random matrix theory aspect (work of Dyson, Gaudin and Mehta). Lastly, I'll discuss the recent work of Katz and Sarnak on the distribution of "low lying" zeros in families of $L$-functions, and its implications to (for example) the distribution of ranks of elliptic curves.

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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. This course has a grading basis of "S" or "U."

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 (Prerequisites enforced at registration). (8). (INDEPENDENT). May be repeated for credit. This course has a grading basis of "S" or "U."

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

Course Homepage: No homepage submitted.

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

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### Undergraduate Course Listings for MATH.

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