College of LS&A

Fall '01 Graduate Course Guide

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Courses in Mathematics


This page was created at 9:22 AM on Thu, Oct 11, 2001.

Fall Academic Term, 2001 (September 5 December 21)

Open courses in Mathematics
(*Not real-time Information. Review the "Data current as of: " statement at the bottom of hyperlinked page)

Wolverine Access Subject listing for MATH

Fall Term '01 Time Schedule for Mathematics.

To see what graduate courses have been added to or changed in Mathematics this week go to What's New This Week.

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MATH 404. Intermediate Differential Equations and Dynamics.

Section 001.

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

Prerequisites: Math. 216, 256 or 286, or Math. 316. No credit granted to those who have completed Math. 256, 286, or 316. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math404.html

There are three main objectives to this class. First, we will present and explain mathematical methods for obtaining approximate analytical solutions to differential equations that cannot be solved exactly. The material in this section will mostly come from Bender and Orszag and we will present it in a very introductory manner. Second, we will introduce and explain the theories of dynamical systems, specifically bifurcation and chaos, phase portraits, linear stability analysis, and local and global behavior of both linear and non-linear differential equations. Third, we will use the computer via matlab and maple to visualize what we are learning and to gain a better understanding of the dynamics. Learning the material will be done through homework that will require both pen and paper analysis and computation support. I would expect all students to have had an introductory course in differential equations and some familiarity with matlab. There will be a special session or two on matlab for students who do not have familiarity with this computation method.

Textbooks

  • Advanced Mathematical Methods for Scientists and Engineers (required), Carl M. Bender and Steven A. Orszag, McGraw-Hill, 1978,.
    This book presents the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to 'real' differential equations that arise in physics and engineering. These equations are usually not solvable in closed form and numerical methods may not converge to useful solutions. The aim is to teach the insights that are most useful in approaching new problems and it avoids the special methods and tricks that work only for particular problems. We will use this book in an introductory manner where some of the more advanced topics will not be covered.
  • Non-linear Systems (recommended), P.G. Drazin, Cambridge texts in Applied Mathematics,1994.
  • Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields (recommended), J. Guckenheimer and P. Holmes, Springer

Mathematical concepts to be covered

  1. Ordinary differential equations
  2. Approximate solutions of linear differential equations
  3. Approximate solutions of non-linear differential equations
  4. Stability analysis, bifurcations, and limit cycles
  5. Phase portraits
  6. Perturbation theory
  7. Boundary layer theory
  8. Bessel, parabolic cylinder, and Airy functions
  9. Numerical analysis with Matlab

Learning Objectives and Instructor Expectations
The objective of this course (as worded in Bender and Orszag) is to help young and established scientists and engineers to build the skill necessary to analyze equations that they encounter in the real world. Asymptotic and perturbation analysis are some of the most useful and powerful, as well as beautiful, methods for finding approximate solutions to equations. Combining these techniques with those of dynamical systems and computation provide the student with a powerful tool for analyzing most ordinary differential equations.

Grading
Homework assignments will count as 40% of grade evaluation. There will also be two midterms worth 30% of the grade and a final that counts for 25%. The remaining 5% is student participation.

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MATH 412. Introduction to Modern Algebra.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This course is designed to serve as an introduction to the methods and concepts of abstract mathematics. A typical student entering this course has substantial experience in using complex mathematical (calculus) calculations to solve physical or geometrical problems, but is unused to analyzing carefully the content of definitions of the logical flow of ideas which underlie and justify these calculations. Although the topics discussed here are quite distinct from those of calculus, an important goal of the course is to introduce the student to this type of analysis. Much of the reading, homework exercises, and exams consists of theorems (propositions, lemmas, etc.) and their proofs. Math 217 or equivalent required as background. The initial topics include ones common to every branch of mathematics: sets, functions (mappings), relations, and the common number systems (integers, rational numbers, real numbers, 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 413. Calculus for Social Scientists.

Section 001.

Instructor(s):

Prerequisites: Not open to freshmen, sophomores or mathematics concentrators. (3). Not open to Mathematics graduate students.

Credits: (3).

Course Homepage: No homepage submitted.

A one-term course designed for students who require an introduction to the ideas and methods of the calculus. The course begins with a review of algebra and then surveys analytic geometry, derivatives, maximum and minimum problems, integrals, integration, and partial derivatives. Applications to business and economics are given whenever possible, and the level is always intuitive rather than highly technical. This course should not be taken by those who have had a previous calculus course or plan to take more than one or two further courses in mathematics. The course is specially designed for graduate students in the social sciences.

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MATH 417. Matrix Algebra I.

Instructor(s):

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

Credits: (3).

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

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

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

Text: Linear Algebra with Applications, Otto Bretscher, Prentice Hall Publishing.

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MATH 419. Linear Spaces and Matrix Theory.

Instructor(s):

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

Credits: (3).

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

Course Homepage: No homepage submitted.

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

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

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MATH 423. Mathematics of Finance.

Instructor(s): Michael G Sullivan (mikegs@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~mikegs/423/index.html

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

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

Contents:

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

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

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MATH 424. Compound Interest and Life Insurance.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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MATH 425 / STATS 425. Introduction to Probability.

Section 001.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~fomin/425.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. The material corresponds to most of Chapters 1-7 and part of 8 of Ross.

Grade will be based on two 1-hour midterm exams, 20% each; 20% homework; 40% final exam. Your lowest homework set score will be dropped.

This course will not be graded on a curve. Homework will normally be due in class on Fridays. There will be approximately 10 problem sets. The midterm exams are held in class. No makeups will be given.

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MATH 425 / STATS 425. Introduction to Probability.

Section 002, 004.

Instructor(s): Jeganathan

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

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.002.

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MATH 425 / STATS 425. Introduction to Probability.

Section 003.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~carswell/math425/

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

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MATH 425 / STATS 425. Introduction to Probability.

Section 005.

Instructor(s): Stephen S Bullock (stephns@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~stephnsb/cur425/math425005.html

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

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MATH 425 / STATS 425. Introduction to Probability.

Section 006.

Instructor(s): Amirdjanova

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

Credits: (3).

Course Homepage: No homepage submitted.

See Statistics 425.006.

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MATH 431. Topics in Geometry for Teachers.

Section 001.

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

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

Credits: (3).

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

This course is a study of the axiomatic foundations of Euclidean and non-Euclidean geometry. Concepts and proofs are emphasized; students must be able to follow as well as construct clear logical arguments. For most students this is an introduction to proofs. A subsidiary goal is the development of enrichment and problem materials suitable for secondary geometry classes. Topics selected depend heavily on the instructor but may include classification of isometries of the Euclidean plane; similarities; rosette, frieze, and wallpaper symmetry groups; tessellations; triangle groups; and finite, hyperbolic, and taxicab non-Euclidean geometries. Alternative geometry courses at this level are 432 and 433. Although it is not strictly a prerequisite, Math 431 is good preparation for 531.

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MATH 433. Introduction to Differential Geometry.

Section 001.

Instructor(s):

Prerequisites: Math. 215, or 255 or 285, and Math. 217. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course is about the analysis of curves and surfaces in 2- and 3-space using the tools of calculus and linear algebra. There will be many examples discussed, including some which arise in engineering and physics applications. Emphasis will be placed on developing intuitions and learning to use calculations to verify and prove theorems. Students need a good background in multivariable calculus (215) and linear algebra (preferably 217). Some exposure to differential equations (216 or 316) is helpful but not absolutely necessary. Topics covered include (1) curves: curvature, torsion, rigid motions, existence and uniqueness theorems; (2) global properties of curves: rotation index, global index theorem, convex curves, 4-vertex theorem; and (3) local theory of surfaces: local parameters, metric coefficients, curves on surfaces, geodesic and normal curvature, second fundamental form, Christoffel symbols, Gaussian and mean curvature, minimal surfaces, and classification of minimal surfaces of revolution. 537 is a substantially more advanced course which requires a strong background in topology (590), linear algebra (513), and advanced multivariable calculus (551). It treats some of the same material from a more abstract and topological perspective and introduces more general notions of curvature and covariant derivative for spaces of any dimension. Math 635 and Math 636 (Topics in Differential Geometry) further study Riemannian manifolds and their topological and analytic properties. Physics courses in general relativity and gauge theory will use some of the material of this course.

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MATH 450. Advanced Mathematics for Engineers I.

Section 001.

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

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

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~pwn/Math450.html

Course Description
There are three main objectives to this class. First, we will introduce the concepts of partial differential equations and complex variables and some basic techniques for analyzing these problems. Second, by studying the application of PDE's to physics, engineering, and biology, the student will begin to acquire intuition and expertise about how to use these equations to model scientific processes. Finally, by utilizing numerous numerical techniques, the student will begin to visualize, hence better understand, what a PDE is and how it can be used to study the Natural Sciences.

Textbooks

  1. Advanced Engineering Mathematics (required), Erwin Kreyszig, Wiley, 1998.
  2. Elementary Applied PDE's with Fourier Series and Boundary Value Problems (good reference), R. Haberman, Prentice-Hall, 1997.
  3. Partial Differential Equations (good reference), S.J. Farlow, Wiley ,1982.

Mathematical concepts to be covered:

  1. Review of sequences and series
  2. Fourier Series
  3. Partial Differential Equations
  4. Applications of PDE's to physics, engineering and biology
  5. Complex variables
  6. Conformal mapping
  7. Numerical analysis with Matlab

Learning Objectives and Instructor Expectations
The objective of this course is to help young and established scientists and engineers to build the skill necessary to analyze equations that they encounter in the real world. This learning will be done using homework and computer assignements as well as in class assignments that will be done in groups and presented in class. Every week on Tuesday there will be a quiz or group project to be done in class. Also, time during each class will be devoted to the discussion of homework problems. Class attendance and participation is expected and is factored into your final grade.

Grading
Homework assignments will count as 30% of grade evaluation. Quizes and group projects will count for 20%. There will also be one midterm worth 20% of the grade and a final (Fri Dec 21st, 10:30 12:30) that counts for 25%. The remaining 5% is student participation.
There will be no make-up quizzes; instead I will drop the lowest quiz score.

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MATH 450. Advanced Mathematics for Engineers I.

Section 002.

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

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

Credits: (4).

Course Homepage: No homepage submitted.

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

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

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MATH 451. Advanced Calculus I.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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

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MATH 454. Boundary Value Problems for Partial Differential Equations.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

no textbook

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MATH 463. Mathematical Modeling in Biology.

Section 001.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~tjacks/Math463.html

It is widely anticipated that Biology and Biomedical science will be the premier sciences of the 21st century. The complexity of the biological sciences makes interdisciplinary involvement essential and the increasing use of mathematics in biology is inevitable as biology becomes more quantitative. Mathematical biology is a fast growing and exciting modern application of mathematics which has gained worldwide recognition. In this course, mathematical models that suggest possible mechanisms which may underlie specific biological processes are developed and analyzed. Another major emphasis of the course is illustrating how these models can be used to predict what may follow under currently untested conditions. The course moves from classical to contemporary models at the population, organ, cellular, and molecular levels.

Textbook:
Mathematical Models in Biology, First Edition; L. Edelstein-Keshet; McGraw-Hill Publishing; 1987.

Other references:

  1. Mathematical Physiology, J. Keener & J. Sneyd, Springer, 1998.
  2. Mathematical Biology, J.D. Murray ; Springer Verlag, 1989.

Biological Topics Include

  1. Single Species and Interacting Population Dynamics
  2. Modeling Infectious and dynamic diseases
  3. The Heart, Circulation, and Blood Cell Production
  4. Regulation of Cell function
  5. Molecular Interactions and Receptor-Ligand Binding
  6. Biological oscillators: Hodgkin-Huxley theory of Nerve Membranes
  7. Intro to Reaction-diffusion and Biological Pattern Formation

Mathematical and Modeling Concepts Include

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

Learning Objectives and Instructor Expectations:
Although an interdisciplinary subject such as Mathematical Biology can be made rather difficult, I will attempt to present the course material in as simple a manner as possible. A basic knowledge of differential equations is recommended; however, I will review in detail the mathematical tool necessary to analyze the models we study. More theoretical aspects, such as proofs, will not be presented. Biological applications will be emphasized although no previous knowledge of biology is assumed. With each topic discussed I give a brief description of the biological background sufficient to understand, develop, and study the models of interest. Upon completion of this course, students will have a working knowledge of how mathematics and biology can be combined to enhance both fields.

Grading:
Homework assignments will count as 1/4 of the grade evaluation. There will also be two exams which will count as 1/4 each. The final 1/4 of the grade rests upon the completion of a substantial research paper describing a modeling project chosen with my assistance. An in-class presentation of the project is also required.

Computer Lab:
There will be a MATLAB based computer component of the course. No prior knowledge of MATLAB is required. The computer lab will be used to visualize solutions, and dynamic behavior of complex biological models.

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MATH 471. Introduction to Numerical Methods.

Section 001.

Instructor(s): Hans E Johnston (hansjohn@umich.edu)

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

Credits: (3).

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

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

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

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MATH 471. Introduction to Numerical Methods.

Section 002.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~dickinsm/471f01/main.html

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

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

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MATH 481. Introduction to Mathematical Logic.

Section 001.

Instructor(s): Andreas Blass (ablass@umich.edu)

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

Credits: (3).

Course Homepage: No homepage submitted.

All of modern mathematics involves logical relationships among mathematical concepts. In this course we focus on these relationships themselves rather than the ideas they relate. Inevitably this leads to a study of the (formal) languages suitable for expressing mathematical ideas. The explicit goal of the course is the study of propositional and first-order logic; the implicit goal is an improved understanding of the logical structure of mathematics. Students should have some previous experience with abstract mathematics and proofs, both because the course is largely concerned with theorems and proofs and because the formal logical concepts will be much more meaningful to a student who has already encountered these concepts informally. No previous course in logic is prerequisite. In the first third of the course the notion of a formal language is introduced and propositional connectives (and, or, not, implies), tautologies, and tautological consequence are studied. The heart of the course is the study of first-order predicate languages and their models. The new elements here are quantifiers ('there exists' and 'for all'). The study of the notions of truth, logical consequence, and provability lead to the completeness and compactness theorems. The final topics include some applications of these theorems, usually including non-standard analysis. Math 681, the graduate introductory logic course, also has no specific logic prerequisite but does presuppose a much higher general level of mathematical sophistication. Philosophy 414 may cover much of the same material with a less mathematical orientation. Math 481 is not explicitly prerequisite for any later course, but the ideas developed have application to every branch of mathematics.

Text: An Introduction to Mathematical Logic by Richard E. Hodel, PWS Publishing Co. 1995.

Grading: 25% homework, 30% midterm exam (Thursday 26 October 7-8:30), 45% final exam.

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MATH 485. Mathematics for Elementary School Teachers and Supervisors.

Section 001.

Instructor(s): Eugene F Krause

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

Credits: (3; 2 in the half-term).

Course Homepage: No homepage submitted.

The history, development, and logical foundations of the real number system and of numeration systems including scales of notation, cardinal numbers, and the cardinal concept; and the logical structure of arithmetic (field axioms) and relations to the algorithms of elementary school instruction. Simple algebra, functions, and graphs. Geometric relationships. For persons teaching or preparing to teach in the elementary school.

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MATH 497. Topics in Elementary Mathematics.

Section 001 Topic?

Instructor(s): Eugene F Krause

Prerequisites: Math. 489. (3). May be repeated for a total of six credits.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 497. Topics in Elementary Mathematics.

Section 531.

Instructor(s):

Prerequisites: Math. 489. (3). May be repeated for a total of six credits.

Credits: (3).

Course Homepage: No homepage submitted.

This is an elective course for elementary teaching certificate candidates that extends and deepens the coverage of mathematics begun in the required two-course sequence Math 385-489. Topics are chosen from geometry, algebra, computer programming, logic, and combinatorics. Applications and problem-solving are emphasized. The class meets three times per week in recitation sections. Grades are based on class participation, two one-hour exams, and a final exam. Selected topics in geometry, algebra, computer programming, logic, and combinatorics for prospective and in-service elementary, middle, or junior-high school teachers. Content will vary from term to term.

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MATH 499. Independent Reading.

Instructor(s):

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

Credits: (1-4).

Course Homepage: No homepage submitted.

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

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MATH 501. Applied & Interdisciplinary Mathematics Student Seminar.

Section 001.

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). Offered mandatory credit/no credit. May be repeated for a total of 6 credits.

Credits: (1).

Course Homepage: http://www.math.lsa.umich.edu/seminars/applied/index.html

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

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

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MATH 513. Introduction to Linear Algebra.

Section 001.

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

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~hderksen/math513.html

Math 513 is the Math Department's most complete and rigorous course in linear algebra. We will study in depth vector spaces and linear transformations over arbitrary fields. We will also cover bilinear and (elementary) quadratic forms and applications to differential equations. Significant applications will be an important feature of the course.

Grading: Weekly problem sets and a midterm exam will each count for 30% of the grade. The final will count for 40%.

Text: Linear Algebra, an Introductory Approach by Curtis, Springer Verlag.

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MATH 520. Life Contingencies I.

Section 001.

Instructor(s):

Prerequisites: Math. 424 and Math. 425. (3).

Credits: (3).

Course Homepage: No homepage submitted.

The goal of this course is to teach the basic actuarial theory of mathematical models for financial uncertainties, mainly the time of death. In addition to actuarial students, this course is appropriate for anyone interested in mathematical modeling outside of the physical sciences. Concepts and calculation are emphasized over proof. The main topics are the development of (1) probability distributions for the future lifetime random variable, (2) probabilistic methods for financial payments depending on death or survival, and (3) mathematical models of actuarial reserving. 523 is a complementary course covering the application of stochastic process models. Math 520 is prerequisite to all succeeding actuarial courses. Math 521 extends the single decrement and single life ideas of 520 to multi-decrement and multiple-life applications directly related to life insurance and pensions. The sequence 520-521 covers the Part 4A examination of the Casualty Actuarial Society and covers the syllabus of the Course 150 examination of the Society of Actuaries. Math 522 applies the models of 520 to funding concepts of retirement benefits such as social insurance, private pensions, retiree medical costs, etc. Recommended text: Actuarial Mathematics (Second Editions) by Bowles et al.

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MATH 523. Risk Theory.

Section 001.

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

Prerequisites: Math. 425. (3).

Credits: (3).

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

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 Brownian motion process, estimating the probability of ruin, reinsurance schemes and their implications for profit and risk. Credibility theory, classical theory for independent events, least squares theory for correlated events, examples of random variables where the least squares theory is exact.

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

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

Required Text:
Loss Models-from Data to Decisions by Klugman, Panjer and Willmot, Wiley 1998.

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MATH 524. Topics in Actuarial Science II.

Section 001.

Instructor(s):

Prerequisites: Math. 424, 425, and 520; and Stats. 426. (3). May be repeated for a total of 9 credits.

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 525 / STATS 525. Probability Theory.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

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

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

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MATH 532. Topics in Discrete and Applied Geometry.

Section 001 Topic?

Instructor(s):

Prerequisites: One of Math. 217, 417, 419 or 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

No Description Provided.

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MATH 537. Introduction to Differentiable Manifolds.

Section 001.

Instructor(s): Krishnan Shankar (shankar@umich.edu)

Prerequisites: Math. 513 and 590. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This course will be a painless introduction to differential topology and differential geometry, meaning the study of spaces and their curvatures. It is the first part of a two-semester sequence. The material in this course is crucial for students who wish to study differential geometry, topology, algebraic geometry, several complex variables, Lie groups and dynamical systems. It is also relevant for other branches of mathematics, such as partial differential equations. We'll start out by doing calculus on manifolds, introducing and using differential forms. We'll prove Stokes' theorem for compact oriented manifolds-with-boundary. We'll also define the de Rham cohomology groups of a manifold and prove their basic properties. Then I'll spend some time on Morse theory. This theory shows how, given a generic function on a manifold, one obtains a decomposition of the manifold into simple building blocks called handles. Morse theory is a basic tool in topology and was used in Smale's famous proof of the Poincare conjecture in more than four dimensions, although we will not go into this. Finally, we'll cover some basic Riemannian geometry, including Riemannian metrics, Levi-Civita connections, geodesics, and curvature. Homework assignments will be given periodically, with the frequency depending on whether or not we get a grader. There will also be a final exam.

The textbooks will be Differential Topology by Victor Guillemin and Alan Pollack, Prentice-Hall, and Morse Theory by John Milnor, Princeton University Press. Math 591 or the equivalent is a prerequisite. I will assume a knowledge of differentiable manifold theory as covered in Sections 1.1-1.4 of the book by Guillemin and Pollack. The titles of these sections are "Definitions", "Derivatives and tangents", "The inverse function theorem and immersions" and "Submersions". If a prospective student has not seen this material before, it might be helpful to look at Chapter 1 of Guillemin and Pollack. I will review this material at the beginning of the academic term.

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

Section 001.

Instructor(s): Jeffrey Rauch (rauch@umich.edu)

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

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~rauch/courses.html

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

Text: Complex Variables with Applications, 6th edition, Brown/Churchill, McGraw Hill.

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MATH 556. Methods of Applied Mathematics I.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This is an introduction to analytical methods for initial value problems and boundary value problems. This course should be useful to students in mathematics, physics, and engineering. We will begin with systems of ordinary differential equations. Next, we will study Fourier Series, Sturm-Liouville problems, and eigenfunction expansions. We will then move on to the Fourier Transform, Riemann-Lebesgue Lemma, inversion formula, the uncertainty principle and the sampling theorem. Next we will cover distributions, weak convergence, Fourier transforms of tempered distributions, weak solution of differential equations and Green's functions. We will study these topics within the context of the heat equation, wave equation, Schrodinger's equation, and Laplace's equation.

Text: Fourier Analysis and its Applications by G.B. Folland.

Grading: homework 60%, midterm 15%, final exam 25%. Homework is key in this class. You are expected to hand in carefully completed homework.

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

Section 001.

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

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

Credits: (3).

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

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

Prerequisites: A course in linear or matrix algebra.

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

Reference Books:

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

Course Content:

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

Work:

  • Weekly Homework Assignments.
  • Midterm
  • Final Exam
  • Two Computational Projects to be solved using AMPL.

Approximate weights for determining final grade are: Homeworks (15%), Midterm (20%), Final Exam (50%), Computer Projects (15%).

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

Section 001.

Instructor(s):

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

Credits: (3).

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

Course Homepage: No homepage submitted.

Survey of continuous optimization problems. Unconstrained optimization problems: unidirectional search techniques, gradient, conjugate direction, quasi-Newtonian methods; introduction to constrained optimization using techniques of unconstrained optimization through penalty transformation, augmented Lagrangians, and others; discussion of computer programs for various algorithms.

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MATH 565. Combinatorics and Graph Theory.

Section 001.

Instructor(s):

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

Credits: (3).

Course Homepage: No homepage submitted.

This course has two somewhat distinct halves devoted to Graph Theory and Enumerative Combinatorics. Proofs, concepts, and calculations play about an equal role. Students should have taken at least one proof-oriented course. Graph Theory topics include Trees; k -connectivity; Eulerian and Hamiltonian graphs; tournaments; graph coloring; planar graphs, Euler's formula, and the 5-Color Theorem; Kuratowski's Theorem; and the Matrix-Tree Theorem. Enumeration topics include fundamental principles, bijections, generating functions, binomial theorem, Catalan numbers, tableaux, partitions and q -series, linear recurrences and rational generating functions, and Pólya theory. There is a small overlap with Math 566, but these are the only courses in combinatorics. 416 is somewhat related but much more concerned with algorithms.

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MATH 571. Numerical Methods for Scientific Computing I.

Section 001.

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

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

Credits: (3).

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

This course is a rigorous introduction to numerical linear algebra with applications to 2-point boundary value problems and the Laplace equation in two dimensions. Both theoretical and computational aspects of the subject are discussed. Some of the homework problems require computer programming. Students should have a strong background in linear algebra and calculus, and some programming experience. The topics covered usually include direct and iterative methods for solving systems of linear equations: Gaussian elimination, Cholesky decomposition, Jacobi iteration, Gauss-Seidel iteration, the SOR method, an introduction to the multigrid method, and the conjugate gradient method; finite element and difference discretizations of boundary value problems for the Poisson equation in one and two dimensions; and numerical methods for computing eigenvalues and eigenvectors. Math 471 is a survey course in numerical methods at a more elementary level. Math 572 covers initial value problems for ordinary and partial differential equations. Math 571 and 572 may be taken in either order. Math 671 (Analysis of Numerical Methods I) is an advanced course in numerical analysis with varying topics chosen by the instructor.

Text:
Introduction to Numerical Linear Algebra & Optimisation Philippe Ciarlet Cambridge.
A Multigrid Funtional William Briggs SIAM.
Numerical Linear Algebra Trefethen & Bav SIAM.

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MATH 575. Introduction to Theory of Numbers I.

Section 001.

Instructor(s): Christopher Skinner (cskinner@umich.edu)

Prerequisites: Math. 451 and 513. Students with credit for Math. 475 can elect Math. 575 for 1 credit. (1, 3).

Credits: (1, 3).

Course Homepage: No homepage submitted.

Math 575 is intended to provide graduate students with an introduction to number theory sufficient for continuing in the graduate program in number theory here at Michigan. As such, it will also provide a means for undergraduates interested in number theory to prepare for graduate study elsewhere. Graduate students not directly interested in number theory will be able to complete their distribution requirements while learning the mathematics behind such every-day applications as Public Key Cryptography.

The first half of the course will be a brisk (but thorough) discussion of the basic notions: Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, primitive roots, Public Key Cryptography, quadratic reciprocity, binary quadratic forms, and basic arithmetic functions. The second half of the course will be devoted to more advanced topics as time permits: diophantine equations, quadratic fields, p-adic numbers, diophantine approximation, arithmetic functions, continued fractions, distribution of prime numbers.

Grading: A homework assignment every two weeks, 2 in-class exams, and a take-home final exam.

Text: Introduction to the Theory of Numbers, 5th edition by Niven, Zuckerman and Montgomery.

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MATH 590. Introduction to Topology.

Section 001.

Instructor(s): Tibor Beke (tbeke@umich.edu)

Prerequisites: Math. 451. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Topology provides a foundational framework for geometry, including algebraic and differential topology. This course is an introduction to the subject and will emphasize the construction of proofs.

We will begin with metric spaces, abstract topological spaces, continuous functions, and the properties of connectedness, compactness and separability, then move on to the more geometric notions of homotopy, covering spaces and the fundamental group. Additional topics may include triangulations and the classification of surfaces. There will be problem sets, a midterm and a final exam.

Text:
(required) Munkres. Topology: a first course. Prentice-Hall, 2nd ed., 2000.
recommended) Armstrong. Basic Topology. UTM, Springer-Verlag, 1983.

Additional readings and handouts as appropriate.

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MATH 591. General and Differential Topology.

Section 001.

Instructor(s): Ralf Spatzier (spatzier@umich.edu)

Prerequisites: Math. 451. (3).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/~spatzier/591/591.html

Text:
required: Topology by Munkres, 2nd edition, Prentice Hall
optional: Differential Topology, Guillemin/Pollack, Prentice Hall

Prerequisites:
Familiarity with undergraduate analysis is recommended

Course Outline:
This course is an introduction to point set and differential topology. Specifically, we will introduce abstract topological spaces and their basic properties. Then we will discuss the properties of connectedness and compactness. We will construct new topological spaces from old ones, such as subspaces, products and quotients, group actions and orbit spaces. Finally, we will introduce manifolds and differential topology, in particular tangent spaces, the regular value theorem, Whitney's embedding theorem and transversality. Students with a strong background in point-set and differential topology may want to consider taking Math 537 instead. Math 591 will cover the prerequisites for the portions of the topology QR which concern point set topology and differential topology.

Problem Session:
TBA

Grading Policy:
homework 40%; midterm 30%; final exam 30%.

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MATH 593. Algebra I.

Section 001.

Instructor(s): Robert Lazarsfeld (rlaz@umich.edu)

Prerequisites: Math. 513. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Math 593 is the first half of the basic introductory graduate algebra course. It is designed to prepare students for the qualifying review, and to provide the foundation in algebra necessary for graduate work in mathematics.

Topics include rings and modules, Euclidean rings, principal ideal domains, classification of modules over a principal ideal domain, Jordan and rational canonical forms of matrices, structure of bilinear forms, tensor products of modules, exterior algebras.

Grading: Homework possibly done in groups will be assigned and collected regularly. There will be two in-class hour exams, and a final.

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MATH 596. Analysis I.

Section 001.

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

Prerequisites: Math. 451. Students with credit for Math. 555 may elect Math. 596 for two credits only. (3).

Credits: (3).

Course Homepage: No homepage submitted.

This is the standard beginning course in complex analysis with standard content (as described in the Department's web page, for example).

Grading: There will be regular homework assignments, midterm exam (possibly two), and a final.

Text: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable by Lars Valerian Ahlfors, (3rd Edition), McGraw-Hill Higher Education.

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MATH 602. Real Analysis II.

Section 001.

Instructor(s): Peter Duren (duren@umich.edu)

Prerequisites: Math. 590 and 597. (3).

Credits: (3).

Course Homepage: No homepage submitted.

Introduction to basic principles of functional analysis, with applications.

  • Topological vector spaces, metric spaces, Baire category theorem, contraction principle.
  • Banach spaces, dual space, Riesz representation theorems, linear operators, principle of uniform boundedness, closed graph theorem, Hahn-Banach theorem.
  • Second dual space, weak-star topology, Alaoglu's theorem, reflexive spaces, uniform convexity.
  • Local convexity, Krein-Milman theorem. Hilbert spaces, Jordan-von Neumann theorem, projection theorem, Riesz representation theorem, reproducing kernels.
  • Banach algebras, maximal ideals, multiplicative linear functionals, Gelfand transform, examples.
    As illustration of technique, frequent applications of functional analysis will be made to problems in "hard" analysis: Fourier series, approximation theory, integral representations, summability, extremal problems, complex analysis, differential equations, etc.

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    MATH 605. Several Complex Variables.

    Section 001 Analytic Geometry

    Instructor(s): Dror Varolin (varolin@umich.edu)

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

    Credits: (3).

    Course Homepage: http://www.math.lsa.umich.edu/~varolin/605.html

    I will cover some of the basic topics but not all classical topics. In fact, the main goal will be to understand and get working knowledge of H"ormander's L^2 method for solving d-bar.

    So we will study the following topics:

    Basic definitions and properties of holomorphic functions.
    Plurisubharmonic functions.
    H"ormander's L^2 theorem for solving d-bar.
    Introduction to complex and K"ahler geometry.
    L^2 theory of d-bar on vector bundle-valued forms.

    We will then look at some applications of H"ormander's technique. It is probably better to wait until the course is running before deciding for certain, but I will definitely cover applications to algebraic geometry after Demailly-Nadel-Siu (including a brief introduction to the relevant algebraic geometry).

    There will be no formal text. I will probably follow H"ormander's book and some notes from a course by Y.-T. Siu.

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    MATH 609. Topics in Analysis.

    Section 001 Complex Dynamics

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

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

    Credits: (3).

    Course Homepage: http://www-personal.umich.edu/~baporter/syl65201.html

    This will be a literature seminar. We will go through recent mathematical research papers in complex dynamics. There will be three steps in the discussion of each paper. In the first part we will discuss background material of the paper. We will motivate the questions and give examples. In the second part we will discuss the key idea of the paper itself. In the third part we will discuss questions suggested by the reading of the paper and other natural related questions and also discuss possible approaches for deciding these questions.

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    MATH 612. Lie Algebra and their Representatives.

    Section 001.

    Instructor(s): Gopal Prasad (gprasad@umich.edu)

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    In this course I will cover the standard theory of finite dimensional Lie Algebras with full proofs. The course will culminate in classification of complex semi-simple Lie algebras in terms of their root systems. Their finite dimensional representations will also be studied.

    The course should be useful for anyone interested in Lie Theory (Lie Groups, Algebraic Groups or Representation Theory), Algebraic and Differential Geometry or Combinatorics.

    Text: Introduction to Lie Algebras by J. E. Humphreys, Springer-Verlag Graduate Texts in Mathematics.

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    MATH 614. Commutative Algebra.

    Section 001.

    Instructor(s): Benjamin Richert (brichert@umich.edu)

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

    Credits: (3).

    Course Homepage: http://www.math.lsa.umich.edu/~brichert/class/614.html

    Commutative algebra is the study of commutative rings and their modules. As well as providing the foundation for algebraic geometry, complex analytic geometry, and algebraic number theory, this field has developed into a beautiful and deep theory in its own right, with applications for nearly every algebraist. Algebraic geometers, number theorists, algebraic combinatorialists, lie theorists, and non-commutative algebraists, among others, find it useful.

    This course is a basic introduction to commutative algebra. Topics covered will include localization of rings and modules, primary decomposition of ideals, integral extensions, tensor products and flatness, completions of rings, graded rings, and dimension theory for rings. Topics important in number theory, including Dedekind domains and Hensel's Lemma, will be discussed. We will also treat a number of topics essential in algebraic geometry, including the prime spectrum of a ring, Hilbert's nullstellensatz, and Noether normalization.

    Grading: bi-weekly problem sets.

    Texts:
    Introduction to Commutative Algebra by Atiyah and MacDonald (Addison-Wesley).
    Math 614 course notes by Professor Hochster.
    (recommended) Commutative Algebra with a view Towards Algebraic Geometry by D. Eisenbud.

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    MATH 619. Topics in Algebra.

    Section 001 Representation Theory of Finite Groups

    Instructor(s): Gopal Prasad (gprasad@umich.edu)

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    This will be a comprehensive course on the representation theory of finite groups based on Serre's book on this topic published in the Graduate Texts in Mathematics Series of Springer-Verlag and I.M.Isaacs' book reprinted by Dover. Some applications of the representation theory and character theory to the study of finite groups will be given.

    This course should be very useful for anyone interested in the Theory of finite groups, Lie Theory (Lie Groups, Algebraic Groups), Number Theory, Combinatorics, Physics...

    Texts:
    Linear Representations of Finite Groups by Serre; Springer Verlag Series Graduate Texts in Mathematics.
    Character Theory of Finite Groups by I.M. Isaacs, Dover.

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

    Section 001.

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

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

    Credits: (3).

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

    The goal of this course is to develop some of the major ideas of probability theory. Emphasis will be placed on specific examples and on ways to compute expectation values. We begin with the most basic of all processes, the simple random walk. This is equivalent to studying the tosses of a fair coin. We prove the strong law of large numbers, the central limit theorem, recurrence property and the law of the iterated logarithm for this system. The proofs of these theorems depend on an ability to compute expectation values of various random variables. In the next part of the course we develop systematic methods for computing expectation values. This leads to the study of finite difference equations. We construct the continuous process Brownian motion by taking a limit in which the finite difference equations become partial differential equations. The second part of the course is concerned with some ideas which have wide application. These represent an abstraction of ideas involved in studying the simple random walk. The first of them is the idea of a measure preserving transformation and the notion of ergodicity. We shall prove the vonNeumann and Birkhoff ergodic theorems. We also shall prove the Poincare recurrence theorem and show how recurrence times can be estimated from the invariant measure. The second is the idea of a Markov process. We shall discuss Markov chains on a finite state space, obtain an invariant measure for the chain and prove ergodicity.

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

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

    Grading: Grades will be based on performance in the homework sets.

    Text: Brownian Motion and Stochastic Calculus by Karatzas and Shreve, Springer 1997.

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

    Section 001.

    Instructor(s):

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    No Description Provided.

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    MATH 631. Introduction to Algebraic Geometry.

    Section 001.

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

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

    Credits: (3).

    Course Homepage: No homepage submitted.

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

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

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

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

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    MATH 636. Topics in Differential Geometry.

    Section 001 Symplectic Geometry.

    Instructor(s): Dan Burns (dburns@umich.edu)

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    We will discuss some of the basics of symplectic geometry and its relationship to complex analysis. This very beautiful line of investigation goes back more than 15 years to a paper of Gromov where he first proved results on rigidity phenomena in symplectic geometry using pseudoholomorphic curves: these are curves which are holomorphic mappings of a disk or a Riemann surface into an almost complex manifold. It turns out that such special surfaces are more plentiful than first suspected (Gromov's discovery), and that they provide rigid invariants for symplectic manifolds, which had previously been thought to be very "flabby", with virtually no invariants other than global homotopy invariants.

    The course will begin with basic symplectic geometry, as discussed, for example, in the textbook of Dusa MacDuff and Dietmar Salamon. More detailed results on the geometry of pseudoholomorphic curves can be found in the book of lectures Holomorphic Curves in Symplectic Geometry, edited by Michelle Audin and Jacques Lafontaine.

    If time permits, and we can reach advanced topics, we would like to give an introduction either to the ideas of Vafa on holomorphic disks in Calabi-Yau 3 manifolds, or the recent results of Donaldson and others showing that symplectic four manifolds are similar in many ways to algebraic surfaces. We will try to give much of the analytic background for these results, but cannot guarantee to give all details of this aspect of things.

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    MATH 638. Introduction to Representation Theory.

    Section 001.

    Instructor(s):

    Prerequisites: Math. 597. Graduate standing. (3). May be repeated for credit.

    Credits: (3).

    Course Homepage: No homepage submitted.

    No Description Provided.

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    MATH 656. Partial and Differential Equations I.

    Section 001.

    Instructor(s): Joel Smoller (smoller@umich.edu)

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    Partial differential equations are fundamental to the modeling of natural phenomena, and they arise in every field of science. Indeed, the desire to understand the properties of solutions of partial differential equations has always had a prominent place in the efforts of mathematicians, and has inspired such diverse fields as complex function theory, functional analysis, and topology. Like algebra, topology, and analysis, partial differential equations are a core area of mathematics.

    In this introductory course, we shall study the basic concepts in the field, including: characteristics and initial-value problems, the wave equation, energy integrals, and their applications, uniqueness theorems, Sobelev spaces, theory of distributions, and applications, elliptic equations, parabolic equations, and an introduction to nonlinear hyperbolic equations.

    Grading: There will be 3 or 4 homework problem sets, but no exams.
    Text: The textbook will be announced later.

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    MATH 664. Combinatorial Theory I.

    Section 001.

    Instructor(s): John Stembridge (jrs@umich.edu)

    Prerequisites: Math. 512. Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    Propaganda: Combinatorics has been in ascendance for the past twenty years or so, becoming an important part of an ever-wider range of mathematics. While this role has long been apparent in areas such as representation theory, commutative algebra, and algebraic topology, the appearance of combinatorial ideas in the work of (at least) three of the four 1998 Fields Medalists suggests that it has become even more mainstream.

    This course will be an introduction to some of the modern aspects of combinatorics, primarily enumeration and generating functions. Some specific topics to be covered include: sieve methods, partitions and q-series, Lagrange inversion, the transfer matrix method, and species. If time permits, I hope to be able to discuss more specialized topics, such as formal languages or computer proofs of combinatorial identities.

    Text: Enumerative Combinatorics, Vol. I, by Richard Stanley.
    This book is (highly) recommended, but not required.

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    MATH 675. Analytic Theory of Numbers.

    Section 001.

    Instructor(s): K Soundararajan

    Prerequisites: Math. 575 and 596. Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    This course will provide an introduction to the use of analytic methods in number theory. The subject has its origins in two famous papers by Dirichlet (on the infinitude of primes in arithmetic progressions) and Riemann (which provided a blueprint for proving the prime number theorem). Our initial motivation will be to understand the ideas in these papers, and establish the prime number theorem, and the prime number theorem for arithmetic progressions. To this end we shall develop the basic theory of Riemann's zeta function and Dirichlet's L-functions. Topics covered will include functional equations, explicit formulas connecting primes with zeros of zeta and L-functions, the classical zero free regions, Dirichlet's class number formula, and Landau-Siegel zeros. If time permits we shall discuss the large sieve and the Bombieri-Vinogradov theorem.

    Course work: There will be graded homework assignments but no exams.

    Text: Multiplicative number theory by H. Davenport. (Springer-Verlag)

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    MATH 678. Modular Forms.

    Section 001.

    Instructor(s): Brian Conrad (bdconrad@umich.edu)

    Prerequisites: Math. 575 and 596. Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    Course description: The classical theory of modular forms is a central topic in number theory for many reasons. For example, this theory provides the most basic context for constructing analytic continuation and functional equations for Dirichlet series, and it is the stepping stone for the Langlands program that links up analytic objects (automorphic forms) with representation theory (of Adele groups and Galois groups). Modular forms also play an important role in the study of quadratic forms, elliptic curves, and a host of other phenomena too.

    There are many ways to approach the theory (using complex analysis, algebraic geometry, and Adele groups), and knowledge of these different points of view is important in order to fully understand the theory. These varied points of view are extremely powerful, but unfortunately most treatments either ignore modern insights or assume a complete mastery of EGA. This makes the modern theory notoriously difficult to navigate as a beginner.

    The course will cover three general topics: the classical picture (Hecke operators, L-functions, newforms, converse theorem), the purely algebraic construction of the theory (modular curves, q-expansion principle), and the dictionary relating the analytic/geometric definitions with purely adelic constructions. This dictionary is an important step in the generalization of the theory to general number fields. Each of these three points of view (complex analytic, algebro-geometric, and adelic) has its merits, and we will see how to pass between them. Many "classical" treatments involve overwhelming amounts of tedious and rather unenlightening computations with matrices, and do not convey insights from geometry or representation theory.

    We will generally bypass such tedium and shall instead emphasize how geometric and adelic considerations enable one to think more conceptually about the theory. The central goal of the course is to discuss the geometry of modular curves and the geometric incarnation of modular forms, how this geometry informs Deligne's construction of Galois representations attached to modular forms (though we will not assume prior knowledge of scheme theory), and how this can all be reformulated in terms of adele groups. This latter topic is the starting point for the modern constructions, which relate automorphic forms and Galois representations. We will sometimes place more emphasis on an intuitive (rather than technical) understanding of the central ideas of the proofs, and such intuitive discussions should provide ample motivation to learn the scheme theory and representation theory, which underlies the technical details. Skinner's winter course will build on this material.

    Grading: There will be no exams, but there will be occasional homework.

    Textbooks: There is no required text, but some books related to the material will be kept on reserve in the library, and photocopies of important papers will be handed out.

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    MATH 681. Mathematical Logic.

    Section 001.

    Instructor(s): Andreas Blass (ablass@umich.edu)

    Prerequisites: Mathematical maturity appropriate for a 600-level course. Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    Mathematical logic begins with the study, by mathematical methods, of the process of mathematical, deductive reasoning. Among the major results are the completeness and incompleteness theorems, delineating the circumstances in which standard modes of reasoning (or indeed any modes of reasoning) are or are not adequate for their intended purposes. The completeness theorem will be covered in detail in this course; the incompleteness theorems will only be sketched here, as they are covered in detail in Math 684.

    Related topics to be covered in this course are questions of decidability, definability, and consistency. In addition, we shall see some results, like the compactness theorem, that emerge from the study of mathematical reasoning but have purely mathematical content independent of the reasoning process.

    The first half of the course will be a thorough and rigorous exposition of the completeness and compactness theorems of first-order logic. This will include the syntax of first-order languages, their interpretations in mathematical structures, and the relationship between truth and provability. The second half of the course will treat more advanced topics, such as non-standard models of arithmetic and analysis, some logical systems that go beyond first-order logic, and a sketch of the incompleteness theorems.

    Text: The text will be course notes (essentially a book) by Prof. Hinman; supplementary sources will be on reserve in the library.

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    MATH 695. Algebraic Topology I.

    Section 001.

    Instructor(s): Igor Kriz (ikriz@umich.edu)

    Prerequisites: Math. 591 and Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    This is a basic second year course in algebraic topology. It covers material needed in both topology and algebra. We will start with homology and cohomology with arbitrary coefficients, Tor and Ext groups, and duality for manifolds (Poincare duality). Later, basic techniques of homotopy theory will be covered, such as fibrations and cofibrations, higher homotopy groups, CW complexes and Whitehead's theorem.

    I do not give exams in 695, but weekly HW will be assigned. Students may get the opportunity (but will not be required) to lecture on specific topics.

    Text: (recommended) Munkres. Elements of algebraic topology.
    J.P. May. A concise course in algebraic topology.

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    MATH 697. Topics in Topology.

    Section 001 Geometric Group Theory

    Instructor(s): Richard Canary (canary@umich.edu)

    Prerequisites: Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    This course will focus on geometric group theory. In geometric group theory, one regards a group as a geometric object and attempts to bring to bear techniques and ideas from geometry and topology. Ideas from geometric group theory have been central in recent developments in geometry, topology and the study of Lie groups.

    If one is given a set of generators for a group, one may construct a graph, called the Cayley graph, whose vertices are the group elements such that two vertices are joined by an edge if and only if they differ by multiplication on the right by a generator. If one gives each edge length one, the Cayley graph is a metric space. If G is the fundamental group of a manifold M, then G "resembles in the large," i.e. is quasi-isometric to, the universal cover of M. For example, the Cayley graph of the free abelian group of rank two can be modeled by the lines on an infinite sheet of graph paper, while the universal cover of the torus is the infinite sheet of graph paper itself. One interesting theorem is that any group quasi-isometric to a free abelian group of finite rank has a finite index subgroup which is free abelian.

    In the 1980's Gromov introduced the notion of a negatively curved or hyperbolic group, i.e. a group whose Cayley graph is a "negatively curved" metric space. We will study the basic properties of Gromov hyperbolic groups and also introduce the related notions of automatic and CAT(0)-groups, whose Cayley graphs are "non-positively curved." We will also introduce hyperbolic space and hyperbolic manifolds, whose fundamental groups give the most basic examples of negatively curved groups. For example, it is known that if a finitely generated torsion-free group is quasi-isometric to hyperbolic 3-space, then it is isomorphic to the fundamental group of a hyperbolic 3-manifold. We will also study some more classical work in geometric group theory which may include the Kurosh subgroup theorem, graphs of groups and ends of groups.

    This course will be geared towards beginning graduate students with an interest in topology or geometry. In particular, we will make use of the material in MATH 592 on covering spaces, properly discontinuous group actions and the Seifert-VanKampen theorem. I will not assume any previous knowledge of Riemannian geometry.

    There will be no assignments or exams, but I will ask each student to give a presentation at some point during the semester.

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    MATH 700. Directed Reading and Research.

    Instructor(s):

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

    Credits: (1-3).

    Course Homepage: No homepage submitted.

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

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    MATH 701. Functional Analysis I.

    Section 001.

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

    Prerequisites: Math. 602. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    This course is an introduction to functional analysis with a special emphasis on Banach space theory. This course is designed for students who have already learned some functional analysis and measure theory, e.g., have taken Math 602 or its equivalent.

    Functional analysis, and in particular, Banach space theory, can be studied as an independent topic. For example, a recent spectacular progress in functional analysis is the solution of Banach's hyperplane problem by W. Gowers for which he received a Fields Medal in 1998. This topic will be discussed near the end of the semester.

    The methods created in Banach space theory can be applied in many other areas including harmonic analysis, functions of one and several complex variables, orthonormal series, approximation theory, differential equations, and probability theory. Students and researchers in these areas will find it useful to know some functional analysis and Banach space theory.

    The following topics will be covered:

    1) review of fundamentals of functional analysis, topological vector spaces, completeness, convexity, locally convex spaces, Frechet spaces, 2) weak and weak-* topologies, Mazur, Alaoglu and Goldstine theorems, reflexive spaces, 3) isomorphisms, bases and projections in Banach spaces, 4) weak compactness, Eberlein-Smulian theorem, weakly compact operators, 5) unconditionally and weakly unconditionally convergent series, Orlicz theorem and Orlicz property, unconditional bases, 6) local properties of Banach spaces, Banach-Mazur distance, Auerbach lemma, principle of local reflexivity, 7) L^p-spaces, Sobolev spaces, Bergman spaces, Kahane's inequality, Banach-Mazur theorem, 8) selected topics in L^1 spaces, C(K) spaces, and disc algebra, and 9) absolutely summing operators, Schatten-von Neumann classes and eigenvalues, Hilbert-Schmidt operators.

    Grading: There will be biweekly homework assignments. Since there will be no formal final examination, students will give an oral presentation on a subject of their choice from a list of several topics in functional analysis.

    Textbooks:
    Functional analysis by W. Rudin, Springer-Verlag, 1995.
    Banach spaces for analysts by P. Wojtaszczyk, Cambridge University Press, 1991.

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    MATH 703. Topics in Complex Function Theory I.

    Section 001 Function Theory in Several Complex Variables.

    Instructor(s): Klas Diederich

    Prerequisites: Math. 604. Graduate standing. (3). May be taken for credit more than once.

    Credits: (3).

    Course Homepage: No homepage submitted.

    The course will deal with selected topics from the modern and recent theory of Function Theory in Several Complex Variables. One of the major goals will be to study different methods for proving or disproving regularity properties of the so-called Bergman projection on different types of pseudoconvex domains. Hence the course will start with the geometry of pseudoconvex boundaries in Cn or, more generally, suitable complex manifolds like projective space etc. A particular emphasis in this chapter will lie on considering local complex submanifolds in such boundaries and different kinds of "worming." In the second part we come to solvability and regularity questions for the Cauchy-Riemann equations on such domains. The third main chapter will, finally, cover properties of the Bergman projection. The emphasis will lie on recent research in the area and we also will steadily collect important open questions for further research. The prerequisites consist of Function Theory of One Complex Variable and, if possible, some basic things from Several Complex Variables like those covered in the chapters 1-2 from Hoermanders book Introduction to Several Complex Variables.

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    MATH 709. Topics in Modern Analysis I.

    Section 001 Quasiconformal Mappings.

    Instructor(s): Pekka Koskela

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    This is an introductory course on quasiconformal mappings. Quasiconformal mappings can be described as homeomorphisms that distort the shapes of infinitesimal spheres by a bounded amount. They appear naturally in many contexts. For example, Mostow's proof for his rigidity theorem was based on quasiconformal mappings and quasiconformal surgery is by now a standard tool in complex dynamics. Quite recently, the theory and techniques have found further applications in the theory of nonlinear elasticity. In that theory, the mappings are not necessarily homeomorphisms and the bounds on the distortion of shapes are only given in an averaged sense.

    I will present an introduction to the theory of quasiconformal mappings with some emphasis on the connections with nonlinear elasticity. In the course of this, we will introduce Sobolev functions, maximal functions and other tools from real and harmonic analysis.

    Text: none

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    MATH 715. Advanced Topics in Algebra.

    Section 001 Topic?

    Instructor(s):

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

    Credits: (3).

    Course Homepage: No homepage submitted.

    No Description Provided.

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    MATH 731. Topics in Algebraic Geometry.

    Section 001 Moduli of Curves.

    Instructor(s): Angela Gibney (agibney@umich.edu)

    Prerequisites: Graduate standing. (3).

    Credits: (3).

    Course Homepage: No homepage submitted.

    The study of moduli spaces of varieties plays an important role in the understanding of the varieties themselves. Recently there has been considerable activity in and around moduli spaces of smooth algebraic curves or Riemann Surfaces of genus g.

    The primary goal of this course will be to convey a sense of the landscape of the study of moduli of curves both from the point of view of algebraic geometry and Teichmüller theory. In particular, the focus will be on the development of the subject, the kinds of questions asked and the techniques used to answer the questions. Every effort will be made to provide transparent and low-tech proofs. Material will also be drawn from various books, collections and survey articles.

    Grading: Students who give a talk will earn a grade of "A" although alternative assignments will be discussed.

    Texts:
    Geometry of Algebraic Curves, Vol I by Arbarello, Cornalba, Griffiths and Harris (Springer)
    Moduli of Curves by Joe Harris and Ian Morrison (Springer)

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    MATH 990. Dissertation/Precandidate.

    Instructor(s):

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

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

    Course Homepage: No homepage submitted.

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

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    MATH 993. Graduate Student Instructor Training Program.

    Instructor(s):

    Prerequisites: Graduate standing and appointment as GSI in Mathematics Department. (1).

    Credits: (1).

    Course Homepage: No homepage submitted.

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

    Check Times, Location, and Availability Cost: No Data Given. Waitlist Code: No Data Given.

    MATH 995. Dissertation/Candidate.

    Instructor(s):

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

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

    Course Homepage: No homepage submitted.

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

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    Undergraduate Course Listings for MATH.


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