During 1999-2000 returning students may choose whether to fulfill their requirements for physical, biological, and mathematical sciences according to the new rules or according to those in force prior to this year. To satisfy the old requirements, students must take two quarters of an approved mathematical sciences sequence. If they elect to take the new curriculum, they must take one quarter of an approved mathematical sciences course. In addition, a sixth quarter of an approved course in physical, biological, or mathematical sciences must be taken to complete the general education requirements. NOTE: In order to get general education credit for calculus, two quarters must be taken. This will count as two quarters toward fulfilling the science general education requirement.

10000-10100-10200. Essential Mathematics I, II, III. PQ: Placement recommendation. Students may not receive grades of P/N or P/F in this sequence. Students who place into this course must take it as first-year students. The autumn quarter in this sequence is concerned with topics in arithmetic, elementary algebra, and geometry necessary to proceed to precalculus topics. The winter quarter continues with elements of algebra and coordinate geometry. In the spring quarter, algebraic, circular, and exponential functions are covered. Staff. Autumn, Winter, Spring.

10500-10600. Fundamental Mathematics I, II. PQ: Adequate performance on the mathematics placement test. Students may not receive grades of P/N or P/F in this sequence. Students who place into this course must take it as first-year students. This two-course sequence covers basic precalculus topics. The autumn quarter course is concerned with elements of algebra, coordinate geometry, and elementary functions. The winter quarter course continues with algebraic, circular, and exponential functions. Staff. Autumn, Winter.

11200. Studies in Mathematics. PQ: Math 10200 or 10600, or placement into Math 13100 or higher. This course meets the general education requirement in mathematical sciences. This course covers the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. The first part addresses number theory, including a study of the rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. The second part's main topic is symmetry and geometry, including a study of polygons, Euclidean construction, polyhedra, group theory, and topology. The course emphasizes the understanding of ideas and the ability to express them through mathematical arguments. The course is at the level of difficulty of the Math 13100-13200-13300 calculus sequence. Staff. Autumn, Spring.

13100-13200-13300. Elementary Functions and Calculus I, II, III. PQ: Invitation only based on appropriate performance on the mathematics placement test or Math 10200 or 10600. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. Math 13100-13200 meets the general education requirement in mathematical sciences. This sequence provides the opportunity for students who are somewhat deficient in their precalculus preparation to complete the necessary background and cover basic calculus in three quarters. This is achieved through three regular one-hour class meetings and two mandatory one-and-one-half hour tutorial sessions each week. A class is divided into tutorial groups of about eight students each, and these meet with an undergraduate junior tutor for problem solving related to the course. The autumn quarter component of this sequence covers real numbers (algebraic and order properties), coordinate geometry of the plane (circles and lines), and real functions, and introduces the derivative. Topics examined in the winter quarter include differentiation, applications of the definite integral and the fundamental theorem, and antidifferentiation. In the spring quarter, subjects include exponential and logarithmic functions, trigonometric functions, more applications of the definite integral, and Taylor expansions. Students are expected to understand the definitions of key concepts (limit, derivative, and integral) and to be able to apply definitions and theorems to solve problems. In particular, all calculus courses require students to do proofs. Students completing Math 13100-13200-13300 have a command of calculus equivalent to that obtained in Math 15100-15200-15300. Math 13300 is only offered in the spring quarter. Staff. Autumn, Winter, Spring.

15100-15200-15300. Calculus I, II, III. PQ: Superior performance on the mathematics placement test, or Math 10200 or 10600. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. Math 15100-15200 meets the general education requirement in mathematical sciences. This is the regular calculus sequence in the department. Students entering this sequence are to have mastered appropriate precalculus material and, in many cases, have had some previous experience with calculus in high school or elsewhere. Math 15100 undertakes a careful treatment of limits, the differentiation of algebraic and transcendental functions, and applications. Work in Math 15200 is concerned with integration and additional techniques of integration. Math 15300 deals with techniques and theoretical considerations, infinite series, and Taylor expansions. Math 15100 is offered only in the autumn quarter. Staff. Autumn, Winter, Spring.

16100-16200-16300. Honors Calculus I, II, III. PQ: Invitation only based on an outstanding performance on the mathematics placement test or a creditable performance on the optional calculus placement test. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. Math 16100-16200 meets the general education requirement in mathematical sciences. Math 16100-16200-16300 is an honors version of Math 15100-15200-15300. A student with a strong background in the problem-solving aspects of one-variable calculus may, by suitable achievement on the calculus placement test, be permitted to register for Math 16100-16200-16300. This sequence emphasizes the theoretical aspects of one-variable analysis and, in particular, the consequences of completeness in the real number system. Staff. Autumn, Winter, Spring.

17500. Elementary Number Theory. PQ: Two quarters of calculus. This course covers basic properties of the integers following from the division algorithm, primes and their distribution, congruences, existence of primitive roots, arithmetic functions, quadratic reciprocity, and other topics. Some transcendental numbers are covered. The subject is developed in a leisurely fashion, with many explicit examples. Staff. Winter.

19500-19600. Mathematical Methods for Biological or Social Sciences I, II. PQ: Math 15300 or equivalent. This sequence includes some linear algebra and three-dimensional geometry, a review of one-variable calculus, ordinary differential equations, partial derivatives, multiple integrals, partial differential equations, sequences, and series. Staff. Summer; Autumn, Winter; Winter, Spring.

20000-20100-20200. Mathematical Methods for Physical Sciences I, II, III. PQ: Math 15300 or equivalent. Entering students who have placement for Math 15100-15200 may begin Math 20000; such students have the requirement for Math 15300 waived, but do not receive placement for Math 15300. This sequence is designed for students intending to major in the physical sciences (other than mathematics). Math 20000 covers linear algebra and multivariable calculus. Topics include linear systems of equations, vector spaces, matrices, eigenvalue problems, partial derivatives, minimum and maximum problems, coordinate transformations, and multiple integrals. Math 20100 deals with vector differential calculus, line integrals, theorems of Green, Gauss, and Stokes, complex numbers, introduction to ordinary differential equations, Fourier series, and partial differential equations. Math 20200 is concerned with functions of a complex variable, Laplace and Fourier transforms, and an introduction to tensor calculus. Staff. Autumn, Winter, Spring; Winter, Spring.

20300-20400-20500. Analysis in Rn I, II, III. PQ: Math 13300 or 15300 or 16300. This three-course sequence is for students who intend to concentrate in mathematics or who require a rigorous treatment of analysis in several dimensions. Here, both the theoretical and problem-solving aspects of multivariable calculus are treated carefully. Topics covered in Math 20300 include the topology of Rn, compact sets, the geometry of Euclidean space, limits and continuous mappings, and partial differentiation. Math 20400 deals with vector-valued functions, extrema, the inverse and implicit function theorems, and multiple integrals. Math 20500 is concerned with line and surface integrals, and the theorems of Green, Gauss, and Stokes. One section of this course is intended for students who have taken Math 13300 or who had a substandard performance in Math 15300. This sequence is the basis for all advanced courses in analysis and topology. Staff. Autumn, Winter, Spring; Winter, Spring, Autumn.

20700-20800-20900. Honors Analysis in Rn I, II, III. PQ: Invitation only. This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. Staff. Autumn, Winter, Spring.

21100. Basic Numerical Analysis. PQ: Math 20000 or 20300. This course covers direct and iterative methods of solution of linear algebraic equations and eigenvalue problems. Topics include numerical differentiation and quadrature for functions of a single variable; approximation by polynomials and piece-wise polynomial functions; approximate solution of ordinary differential equations; and solution of nonlinear equations. Staff. Spring.

24100. Topics in Geometry. PQ: Math 25500. This course focuses on the interplay between abstract algebra (group theory, linear algebra, and the like) and geometry. Several of the following topics are covered: affine geometry, projective geometry, bilinear forms, orthogonal geometry, and symplectic geometry. Not offered 1999-2000; will be offered 2000-2001.

24200. Algebraic Number Theory. PQ: Math 25500. Factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree are covered. Staff. Spring.

25000. Elementary Linear Algebra. PQ: Math 15200 or equivalent. This course takes a concrete approach to the subject and includes some applications in the physical and social sciences. Topics covered in the course include the theory of vector spaces and linear transformations, matrices and determinants, and characteristic roots and similarity. Staff. Autumn, Spring.

25400-25500-25600. Basic Algebra I, II, III. PQ: Math 13300 or 15300. This sequence covers groups, subgroups, and permutation groups; rings and ideals; some work on fields; vector spaces, linear transformations and matrices, and modules; and canonical forms of matrices, quadratic forms, and multilinear algebra. Math 25600 is offered only in spring quarter. Staff. Autumn, Winter, Spring; Winter, Spring (Math 25400-25500).

25700-25800-25900. Honors Basic Algebra I, II, III. PQ: Math 15300 or 16300. This is an accelerated version of Math 25400-25500-25600. Topics include the theory of finite groups, commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms. The course also covers basic field theory, the structure of padic fields, and Galois theory. Staff. Autumn, Winter, Spring.

261. Set Theory and Metric Spaces. PQ: Math 25400, or 20300 and 25000. This course covers sets, relations, and functions; partially ordered sets; cardinal numbers; Zorn's lemma, well-ordering, and the axiom of choice; metric spaces; and completeness, compactness, and separability. Staff. Autumn.

26200. Point-Set Topology. PQ: Math 20300 and 25400. This course examines topology on the real line, topological spaces, connected spaces and compact spaces, identification spaces and cell complexes, and projective and other spaces. With Math 27400, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Winter.

26300. Introduction to Algebraic Topology. PQ: Math 26200. Some of the topics covered are the fundamental group of a space; Van Kampen's theorem; covering spaces and groups of covering transformation; existence of universal covering spaces built up out of cells; and theorems of Gauss, Brouwer, and Borsuk-Ulam. Staff. Spring.

27000. Basic Complex Variables. PQ: Math 20500. Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping. Staff. Autumn, Spring.

27200. Basic Functional Analysis. PQ: Math 20900, or 26100 and 27000. Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, Lp-spaces, Fourier transforms, Plancherel's theorem for Rn, and spectral properties of bounded linear operators are some of the topics discussed. Staff. Winter.

27300. Basic Theory of Ordinary Differential Equations. PQ: Math 20200 or 27000. This course covers first-order equations and inequalities, Lipschitz condition and uniqueness, properties of linear equations, linear independence, Wronskians, variation-of-constants formula, equations with constant coefficients and Laplace transforms, analytic coefficients, solutions in series, regular singular points, existence theorems, theory of two-point value problem, and Green's functions. Staff. Winter.

27400. Introduction to Differentiable Manifolds and Integration on Manifolds. PQ: Math 27200. Topics include exterior algebra, differentiable manifolds and their basic properties, differential forms, integration on manifolds, Stoke's theorem, DeRham's theorem, and Sard's theorem. With Math 26200, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Spring.

27500. Basic Theory of Partial Differential Equations. PQ: Math 27300. This course covers classification of second-order equations in two variables, wave motion and Fourier series, heat flow and Fourier integral, Laplace's equation and complex variables, second-order equations in more than two variables, Laplace operators, spherical harmonics, and associated special functions of mathematical physics. Staff. Spring.

27700. Mathematical Logic I (=ComSci 315, Math 27700). PQ: Math 25400. This course provides an introduction to mathematical logic. Topics include propositional and predicate logic, natural deduction systems, models, and the syntactic notion of proof versus the semantic notion of truth, including soundness and completeness. The incompleteness theorems are also covered. A. Nies. Autumn.

27800. Mathematical Logic II. PQ: Math 27700 or equivalent. Some of the topics examined are number theory, Peano arithmetic, Turing compatibility, unsolvable problems, Gödel's incompleteness theorem, undecidable theories (for example, the theory of groups), quantifier elimination, and decidable theories (for example, the theory of algebraically closed fields). Not offered 1999-2000; will be offered 2000-2001.

27900. Logic and Logic Programming (=ComSci 215, Math 27900). PQ: Math 25400, or ComSci 315, or consent of instructor. Programming knowledge not required. Predicate logic is a precise logical system developed to formally express mathematical reasoning. Prolog is a computer language intended to implement a portion of predicate logic. This course covers both predicate logic and Prolog, presented to complement each other and to illustrate the principles of logic programming and automated theorem proving. Topics include syntax and semantics of propositional and predicate logic, tableaux proofs, resolution, Skolemization, Herbrand's theorem, unification, refining resolution, and programming in Prolog, including searching, backtracking, and cut. This course overlaps only slightly with Math 27700; students are encouraged to take both courses. R. Soare. Spring.

28000. Introduction to Formal Languages (=ComSci 280, Math 28000). PQ: Math 25000 or 25500, and experience with mathematical proofs. Topics include automata theory, regular languages, CFL's, and Turing machines. This course is a basic introduction to computability theory and formal languages. Not offered 1999-2000; will be offered 2000-2001.

28100. Introduction to Complexity Theory (=ComSci 281, Math 28100). PQ: Math 25000 or 25500, and experience with mathematical proofs. Computability topics are discussed, including the s-m-n theorem and the recursion theorem. We also discuss resource-bounded computation. This course introduces complexity theory. Relationships between space and time, determinism and nondeterminism, NP-completeness, and the P versus NP question are investigated. Not offered 1999-2000; will be offered 2000-2001.

28400. Combinatorics and Probability (=ComSci 274, Math 28400). PQ: Math 25000 or 25400, or ComSci 275, or consent of instructor. Some experience with mathematical proofs. Problems and methods of enumeration, construction, and existence of discrete structures are discussed in conjunction with the basic concepts of probability theory over a finite sample space. Enumeration techniques are applied to the calculation of probabilities; conversely, probabilistic arguments are used in the analysis of combinatorial structures. Topics include permutations, combinations, linear recurrences, generating functions, principle of inclusion and exclusion, external set systems, coloring graphs and set systems, random variables, independence, expected value, standard deviation, Chebyshev's and Chernoff's inequalities, the structure of random graphs and tournaments, and probabilistic proofs of existence. L. Babai. Spring.

29000. Fourier Analysis. PQ: Working knowledge of the Lebesgue integral and Lp-spaces such as is found in Math 20700-20800-20900 or 27200. This is a course on the convergence properties of Fourier Series and Integrals, and on the real variable theory. We consider such operators as the Hardy-Littlewood Maximal Operator and the Hilbert Transform, and give an introduction to classical Calderon-Zygmund theory. If time permits, we also discuss some Tauberian theorems and Lacunary series (and their relation to probability theory), as well as other topics from the classical theory of harmonic analysis. Not offered 1999-2000; will be offered 2000-2001.

29200. Chaos, Complexity, and Computers (=ComSci 279, Math 29200, Phys 251). PQ: One year of calculus and two quarters of physics at any level. Knowledge of computer programming not required. In this course we use the computer to investigate the question of how patterns and complexity arise in nature. The systems studied are drawn from physics, biology, and other areas of science. This course also is intended to be an introduction to the use of computers in the physical sciences. S. Coppersmith. Winter. L.

29700. Proseminar in Mathematics. PQ: General education mathematics sequence. Consent of instructor and departmental counselor. Open to mathematics concentrators only. Students are required to submit the College Reading and Research Course Form. Must be taken for a letter grade. Staff. Autumn, Winter, Spring.