Transfer Student Courses in Mathematics (Division 428)

Winter Term, 2000 (January 5 - April 26, 2000)

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

Elementary Mathematics Courses. In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics students. All courses require three years of high school mathematics; four years are strongly recommended and more information is given for some individual courses below. Students with College Board Advanced Placement credit and anyone planning to enroll in an upper-level class should consider one of the Honors sequences and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and thinking through a single course. They are neither prerequisite nor preparation for any further course. No credit will be received for the election of Math 127 or 128 if a student already has received credit for a 200- (or higher) level mathematics course.

Each of Math 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics.

The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LS&A Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. Most students electing Math 295 have completed a thorough high school calculus course. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one or two semesters of AP credit, Math 119 and Math 156. Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 156 is an honors course intended primarily for science and engineering concentrators and will emphasize both applications and theory. Interested students should consult a mathematics advisor for more details.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 295 after Math 115. A list of these and other cases of reduced credit for courses with overlapping material is available from the Department. To avoid unexpected reduction in credit, students should always consult an advisor before switching from one sequence to another. In all cases a maximum total of 16 credits may be earned for calculus courses Math 115 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

Students completing Math 116 who are principally interested in the application of mathematics to other fields may continue either to Math 215 (Analytic Geometry and Calculus III) or to Math 216 (Introduction to Differential Equations) - these two courses may be taken in either order. Students who have greater interest in theory or who intend to take more advanced courses in mathematics should continue with Math 215 followed by the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217 (or the honors version, Math 513) is required for a concentration in Mathematics; it both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations in Math 316. Math 216 is not intended for mathematics concentrators.

A maximum total of 4 credits may be earned in Mathematics courses numbered 110 and below. A maximum total of 16 credits may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

Section 170 and 173 Only by Permission of CSP

Prerequisites & Distribution: Four years of high school mathematics. See Elementary Courses above. Credit usually is granted for only one course from among Math. 112, 115, 185, and 295. No credit granted to those who have completed Math. 175. (4). (MSA). (BS). (QR/1).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.

Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions, and Graphs). Math 116 is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking Math 186. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended).

Prerequisites & Distribution: Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).

Credits: (4).

Course Homepage: No Homepage Submitted.

See Math 115 for a general description of the sequence Math 115-116-215.

Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math 186 is a somewhat more theoretical course which covers much of the same material. Math 215 is the natural sequel. A student who has done very well in this course could enter the honors sequence at this point by taking Math 285.

Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS).

Credits: (4).

Course Homepage: http://www.math.lsa.umich.edu/courses/216/

For a student who has completed the calculus sequence, there are two sequences which deal with linear algebra and differential equations, Math 216-417 (or 419) and Math 217-316. The sequence Math 216-417 emphasizes problem-solving and applications and is intended for students of engineering and the sciences. Math concentrators and other students who have some interest in the theory of mathematics should elect the sequence Math 217-316. After an introduction to ordinary differential equations, the first half of the course is devoted to topics in linear algebra, including systems of linear algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors. In the second half these tools are applied to the solution of linear systems of ordinary differential equations. Topics include: oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods. There is a weekly computer lab using MATLAB software. This course is not intended for mathematics concentrators, who should elect the sequence 217-316. Math 286 covers much of the same material in the honors sequence. The sequence Math 217-316 covers all of this material and substantially more at greater depth and with greater emphasis on the theory. Math 404 covers further material on differential equations. Math 217 and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods.

Section 001.

Prerequisites & Distribution: Math. 217. Only one credit granted to those who have completed Math. 412. (3). (Excl). (BS).

Credits: (3).

Course Homepage: http://www.math.lsa.umich.edu/courses/312/

One of the main goals of the course (along with every course in the algebra sequence) is to expose students to rigorous, proof-oriented mathematics. Students are required to have taken Math 217, which should provide a first exposure to this style of mathematics. A distinguishing feature of this course is that the abstract concepts are not studied in isolation. Instead, each topic is studied with the ultimate goal being a real-world application. As currently organized, the course is broken into four parts. (1) the integers "mod n" and linear algebra over the integers mod p, with applications to error correcting codes; (2) some number theory, with applications to public-key cryptography; (3) polynomial algebra, with an emphasis on factoring algorithms over various fields, and (4) permutation groups, with applications to enumeration of discrete structures "up to automorphisms" (a.k.a. Pólya Theory). Math 412 is a more abstract and proof-oriented course with less emphasis on applications. EECS 303 (Algebraic Foundations of Computer Engineering) covers many of the same topics with a more applied approach. Another good follow-up course is Math 475 (Number Theory). Math 312 is one of the alternative prerequisites for Math 416, and several advanced EECS courses make substantial use of the material of Math 312. Math 412 is better preparation for most subsequent mathematics courses.

Section 001.

Prerequisites & Distribution: Engineering 101, and Math. 216. (3). (Excl). (BS). 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.

This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra, and may provide practice in FORTRAN programming and the use of a software library subroutine. Convergence theorems are discussed and applied, but the proofs are not emphasized. Floating point arithmetic, Gaussian elimination, polynomial interpolation, spline approximations, numerical integration and differentiation, solutions to non-linear equations, ordinary differential equations, polynomial approximations. Other topics may include discrete Fourier transforms, two-point boundary-value problems, and Monte-Carlo methods. Math 471 is a similar course which expects one more year of maturity and is somewhat more theoretical and less practical. The sequence Math 571-572 is a beginning graduate level sequence which covers both numerical algebra and differential equations and is much more theoretical. This course is basic for many later courses in science and engineering. It is good background for 571-572.

Math. 424. Compound Interest and Life Insurance.

Section 001.

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Prerequisites & Distribution: Math. 215, 255, or 285. (3). (MSA). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

See Statistics 425..

Math. 450. Advanced Mathematics for Engineers I.

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Math. 452. Advanced Calculus II.

Section 001.

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Section 001.

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

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

Credits: (3).

Course Homepage: http://coursetools.ummu.umich.edu/2000/winter/lsa/math/454/001.nsf

This course covers methods of solving partial differential equations (e.g., the heat, wave, Helmholtz and Laplace equations), with specified boundary conditions in various geometries. We will cover separation of variables, Fourier series, Bessel functions, spherical harmonics, orthogonal polynomials, Sturm – Liouville theory, eigenfunctions of the Laplacian in several different coordinate systems, Fourier and Bessel transforms, conformal mapping, etc. These methods have applications in fields as diverse as mechanics, quantum mechanics, thermodynamics, aerodynamics, finance, electromagnetism, and many others, and we will take our examples from such disciplines.

Math. 471. Introduction to Numerical Methods.

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Math. 476. Computational Laboratory in Number Theory.

Section 001.

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Math. 486. Concepts Basic to Secondary Mathematics.

Section 001.

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Math. 490. Introduction to Topology.

Section 001.

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Section 001.

Instructor(s): Pasha Belorousski (belorous@umich.edu)

Prerequisites & Distribution: Math. 451 or 513. No credit granted to those who have completed or are enrolled in 412. Math. 512 requires more mathematical maturity than Math. 412. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: M. Artin, Algebra.

This course will provide a rigorous introduction to the basic structures of algebra: groups, rings and fields. These structures are central in any field of modern mathematics and are ubiquitous in its application to subjects such as crystallography, quantum mechanics, and chemistry, to name a few. The emphasis in the course will be on concepts and proofs. Concrete topics such as symmetry and ruler/compass constructions in the plane will be used to illustrate the more abstract material.

Here is a keyword outline of the route: groups, homomorphisms, quotient groups, rigid motions and symmetry of regular solids, rings, factorization, principal ideal domains, unique factorization domains, fields, algebraic extensions, finite fields, geometric constructions in the plane using ruler and compass. Time permitting, we will make further excursions into the theory of groups: class equation, Sylow theorems, simple groups.

Expected workload: Weekly homework assignments, a midterm and a final exam.

Section 001.

Instructor(s): Natasa Macura (nmacura@umich.edu)

Prerequisites & Distribution: Math. 412 or permission of honors advisor. Two credits granted to those who have completed Math. 417; one credit granted to those who have completed Math 217 or 419. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: C. Curtis, Linear Algebra, An Introductory Approach, Springer

Math 513 is an introduction to the theory of abstract vector spaces and linear transformations. The emphasis of the course will be on concepts and proofs with some calculations to illustrate the theory.

Math 513 is the linear algebra course for students in the Honors Mathematics Program. It is also appropriate for students who have completed one or more proof oriented courses such as math 412, 451, or 512 and who seek a sophisticated course in linear algebra. One of the most important topics in Math 513 is the Jordan Canonical Form, the study of which is an excellent preparation for Math 593.

The course will begin with the discussion of matrices and determinants. We shall than discuss vector spaces over arbitrary fields (including finite fields), linear transformations, diagonalization, applications to linear and linear differential equations, Jordan Canonical Form and the Spectral Theorem. If time permits, we shall study bilinear and quadratic forms.

Math. 521. Life Contingencies II.

Section 001.

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Section 001.

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

Prerequisites & Distribution: Math. 425. (3). (Excl). (BS).

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 homeworks, a midterm and a final exam.

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

Math. 526/Stat. 526. Discrete State Stochastic Processes.

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Section 001 – Biological Applications: probabilistic models of proteins and nucleic acids

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

Prerequisites & Distribution: Math. 525 or EECS 501. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

Text:

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

Section 001.

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

Prerequisites & Distribution: Math. 450 or 451. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Text: Complex Variables and Applications, 5th ed. (Churchill and Brown);

Student Body: largely engineering and physics graduate students with some math and engineering undergrads

Background and Goals: This course is an introduction to the theory of complex valued functions of a complex variable. Concepts and calculations are emphasized over proofs.

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

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

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

There will be homework, midterm and a final.

Section 001.

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

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

Credits: (3).

Course Homepage: No Homepage Submitted.

In applied mathematics, we try to understand a physical process by formulating and analyzing a mathematical model. Many models consist of a differential equation with initial and boundary conditions. Most of the time especially if the equation is nonlinear, an explicit formula for the solution is not available. Even if we are clever or lucky enough to find such a formula, it may be difficult to extract useful information from it. In practice, we must settle for a sufficiently accurate approximate solution obtained by numerical simulation or asymptotic analysis (or a combination of the two). This course is an introduction to asymptotic analysis. It is aimed at graduate and advanced undergraduate students in engineering, mathematics and science. The main prerequisite is complex analysis (e.g., Math 55 or Math 596). Math 556 is not a prerequisite. Murray’s text will occupy 2/3 of the course. In the remaining time, I will present topics in PDE and fluid dynamics.

Math. 561/SMS 518 (Business Administration)/IOE 510. Linear Programming I.

Section 001.

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

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Section 001.

Instructor(s): Thomas Storer

Prerequisites & Distribution: Math. 216, 256, 286, or 316. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Permutations, combinations, generating functions, and recurrence relations. The existence and enumeration of finite, discrete configurations. Systems of representatives, Ramsey’s theorem, and extremal problems. Construction of combinatorial designs.

Section 001.

Instructor(s): Trevor Wooley (wooley@umich.edu)

Prerequisites & Distribution: One of Math 217, 419, 513. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Discover the secrets underlying the “Digital Age''! Error Correcting Codes are fundamental to modern data transmission (telecommunications, the Internet) and data storage (compact disks, DVD's, etc.)... this is a course which introduces the mathematical theory of coding, and prepares you for the next millennium.

This course focuses on the mathematical background for linear error correcting codes. We begin with a discussion of Shannon's theorem and channel capacity. The definition of linear codes will be given along with a review of the necessary tools from linear algebra (including a review of/introduction to abstract algebra and finite fields). Basic examples of codes will be discussed, including the Hamming, BCH, cyclic, Reed-Muller and Reed-Solomon codes.

We also discuss the problem of decoding, starting with syndrome decoding and covering weight enumerator polynomials and the MacWilliams-Sloane identity. Following these basic topics, we will discuss topics of interest to the audience and instructor, amongst which may be a consideration of asymptotic parameters and bounds, algebraic-geometric codes, and a brief introduction to cryptography.

There will be at most 9 problem sets throughout the academic term, an in-class midterm, and a take-home final.

Primary Text: J. H. van Lint, Introduction to Coding Theory, 3rd Edition, Springer-Verlag GTM 86, 1999. Useful sources: "Coding Theory: the essentials", by Hoffman, Leonard, Lindner, Phelps, Rodger and Wall, Marcel Dekker, 1991. "Foundations of Coding Theory", by Jiri Adamek, Wiley, 1991.

Section 001.

Instructor(s): Alexander Kurganov (kurganov@umich.edu)

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This course is an introduction to numerical methods for solving linear systems of equations (Ax=b) and for computing eigenvalues of a matrix. Topics: singular value decomposition, QR factorization, Gram-Schmidt orthogonalization, least squares problems, condition number, Gaussian elimination, iterative methods (Arnoldi, GMRES, conjugate gradient), preconditioning, methods for computing eigenvalues (e.g., power method, inverse iteration, QR algorithm, shifts).

Text: Numerical Linear Algebra, L.N. Trefethen & D. Bau, SIAM

Prerequisites: a course in linear algebra on the level of Math 417, 419, or 513, computer programming (any language)

Section 001.

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

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

Credits: (3).

Course Homepage: No Homepage Submitted.

This is an introduction to numerical methods for initial value problems. The course will cover numerical methods for ordinary differential equations and both linear parabolic and hyperbolic partial differential equations. We also plan to discuss the numerical solution of nonlinear hyperbolic equations. This course should be useful to students in mathematics, physics, and engineering. Homework assignments will be a crucial part of the class; students must know how to program and use elementary computer graphics.

Background Required: Strong background in advanced calculus and linear algebra is needed. It would be preferable if the student has taken Math 454 or equivalent. It is mandatory that the students have a knowledge of computing programming in Fortran, c, or even Matlab.

There will be no textbook but the following would be good references:

• "Numerical solution of partial differential equations" by Morton and Mayers
• "Numerical initial value problems in ordinary differential equations" by Gear
• Finite difference schemes and partial differential equations by Strikwerda, Wadsworth and Brooks
• Numerical methods by Dahlquist, Bjork and Ande

Section 001.

Instructor(s): Peter Hinman (pgh@umich.edu)

Prerequisites & Distribution: Math. 412 or 451 or equivalent experience with abstract mathematics. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

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

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

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

Section 001.

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

Prerequisites & Distribution: Math. 591. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

This is the beginning graduate course in algebraic topology. It is an introduction to basic distinguishing characteristics between topological spaces, called topological invariants. How do we prove, for example, that the 2-dimensional sphere is topologically different from the surface of a donut? The topological invariants to be discussed in this course are the fundamental group and homology. The prerequisite for this class is Math 591. There is no single comprehensive textbook, but suggested texts are W.S. Massey: A basic course in algebraic topology, and J.R. Munkres: Elements of algebraic topology.

Section 001.

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

Prerequisites & Distribution: Math. 593. (3). (Excl). (BS).

Credits: (3).

Course Homepage: No Homepage Submitted.

Math 594 is the second half of the introductory ("alpha") graduate course in algebra. We will cover essentially the material on group theory and Galois theory that is covered in the Qualifying Review Exams. The formal text will be: I.M. Isaacs, Algebra (A Graduate Course), Brooks/Cole, although I will not necessarily follow the text very closely.