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Courses

100-101-102. Essential Mathematics I, II, III. PQ: Placement recommendation. College students may not receive grades of P or N 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.

105-106. Fundamental Mathematics I, II. PQ: Adequate performance on the mathematics placement test. College students may not receive grades of P or N 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.

110-111. Studies in Mathematics I, II. PQ: Math 102 or 106 or placement into Math 131 or higher. Math 110 and 111 fulfill the Common Core requirement in mathematical sciences. This sequence covers the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. The first quarter addresses number theory, including a study of rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. The second quarter'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 sequence is at the level of difficulty of the Math 131-132-133 calculus sequence. Staff. Autumn, Winter, Spring.

131-132-133. Elementary Functions and Calculus I, II, III. PQ: Invitation based on appropriate performance on the mathematics placement test or Math 102 or 106. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. Math 131-132 fulfills the Common Core 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 131-132-133 have a command of calculus equivalent to that obtained in Math 151-152-153. Math 133 is only offered in the spring quarter. Staff. Autumn, Winter, Spring.

151-152-153. Calculus I, II, III. PQ: Superior performance on the math-ematics placement test, or Math 102, or 106. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. Math 151-152 fulfills the Common Core requirement in mathematical sciences. This is the regular calculus sequence in the department. Students entering this sequence will have mastered appropriate precalculus material and, in many cases, will have had some previous experience with calculus in high school or elsewhere. Math 151 undertakes a careful treatment of limits, the differentiation of algebraic and transcendental functions, and applications. Work in Math 152 is concerned with integration and additional techniques of integration. Math 153 deals with techniques and theoretical considerations, infinite series, and Taylor expansions. Math 151 is only offered in the autumn quarter. Staff. Autumn, Winter, Spring.

161-162-163. Honors Calculus I, II, III. PQ: Invitation 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 161-162 fulfills the Common Core requirement in mathematical sciences. Math 161-162-163 is an honors version of Math 151-152-153. 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 161-162-163. 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.

175. 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.

195-196. Mathematical Methods for Biological or Social Sciences I, II. PQ: Math 153 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.

200-201-202. Mathematical Methods for Physical Sciences I, II, III. PQ: Math 153 or equivalent. This sequence is designed for students intending to major in the physical sciences (other than mathematics). Math 200 covers partial differentiation, implicit functions, multiple integration, vectors and matrices, determinants, vector analysis (divergence and curl), and curvilinear coordinates. Math 201 deals with series, developments in power series, ordinary differential equations, and complex numbers and elementary functions of a complex variable; the course also serves as an introduction to Fourier series. Math 202 is concerned with complex variables, contour integrals, residues, and conformal mapping; Laplace transforms and Fourier transforms; and advanced matrix methods, coordinate transformations, and tensor analysis. Staff. Autumn, Winter, Spring; Winter, Spring (200-201).

203-204-205. Analysis in R[n] I, II, III. PQ: Math 133 or 153 or 163. 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 203 include the topology of R[n], compact sets, the geometry of Euclidean space, limits and continuous mappings, and partial differentiation. Math 204 deals with vector-valued functions, extrema, the inverse and implicit function theorems, and multiple integrals. Math 205 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 133 or who had a substandard performance in Math 153. This sequence is the basis for all advanced courses in analysis and topology. Staff. Autumn, Winter, Spring; Winter, Spring, Autumn.

207-208-209. Honors Analysis in R[n] 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.

211. Basic Numerical Analysis. PQ: Math 200 or 203. 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.

241. Topics in Geometry. PQ: Math 255. This course focuses on the interplay between abstract algebra (group theory, linear algebra, etc.) and geometry. Several of the following topics are covered: affine geometry, projective geometry, bilinear forms, orthogonal geometry, and symplectic geometry. Staff. Spring.

242. Algebraic Number Theory. PQ: Math 255. Factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree are covered. Not offered 1996-97; will be offered 1997-98.

250. Elementary Linear Algebra. PQ: Math 152 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.

254-255-256. Basic Algebra I, II, III. PQ: Math 133 or 153. 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 256 is offered only in spring quarter. Staff. Autumn, Winter, Spring; Winter, Spring (254-255).

257-258-259. Honors Basic Algebra I, II, III. PQ: Math 153 or 163. This is an accelerated version of Math 254-255-256. Topics covered 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 p-adic fields, and Galois theory. Staff. Autumn, Winter, Spring.

261. Set Theory and Metric Spaces. PQ: Math 254, or 203 and 250. 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.

262. Point-Set Topology. PQ: Math 203 and 254. 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 274, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Winter.

263. Introduction to Algebraic Topology. PQ: Math 262. 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.

270. Basic Complex Variables. PQ: Math 205. Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping. Staff. Autumn, Spring.

272. Basic Functional Analysis. PQ: Math 209, or 261 and 270. Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, L[p]-spaces, Fourier transforms, Plancherel's theorem for R[n], and spectral properties of bounded linear operators are some of the topics discussed. Staff. Winter.

273. Basic Theory of Ordinary Differential Equations. PQ: Math 202 or 270. 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.

274. Introduction to Differentiable Manifolds and Integration on Manifolds. PQ: Math 272. 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 262, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Spring.

275. Basic Theory of Partial Differential Equations. PQ: Math 273. 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.

277. Mathematical Logic I (=ComSci 315). PQ: Math 254. This course provides an introduction to mathematical logic. Covered 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. Staff. Autumn.

278. Mathematical Logic II. PQ: Math 277 or equivalent. Some of the topics examined are number theory, Peano arithmetic, Turing compatibility, unsolvable problems, Gödel's incompleteness theorem, undecidable theories (e.g., the theory of groups), quantifier elimination, and decidable theories (e.g., the theory of algebraically closed fields). Not offered 1996-97; will be offered 1997-98.

279. Logic and Logic Programming (=ComSci 215). PQ: Math 254, or ComSci 315, or consent of instructor. Previous programming experience 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 277; students are encouraged to take both courses. This course is offered in alternate years. Not offered 1996-97; will be offered 1997-98.

280. Introduction to Formal Languages (=ComSci 280). PQ: Math 250 or 255, and experience with mathematical proofs. Topics covered include automata theory, regular languages, CFL's, and Turing machines. This course is a basic introduction to computability theory and formal languages. This course is offered in alternate years. Staff. Autumn. Not offered 1996-97; will be offered 1997-98.

281. Introduction to Complexity Theory (=ComSci 281). PQ: ComSci 280. This course is a continuation of ComSci 280. More 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. This course is offered in alternate years. Staff. Winter.

284. Combinatorics and Probability (=ComSci 274). PQ: Math 250 or 254, 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, extremal 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. Winter.

298. Bachelor's Thesis. PQ: Open to mathematics students in the honors program in their fourth year only. Consent of departmental counselor. Students are required to submit the College Reading and Research Form. This course is designed to involve the student in mathematical research. The project is chosen in consultation with a member of the mathematics faculty and must be approved by the director and associate director of undergraduate studies. Staff. Autumn, Winter, Spring.

299. Proseminar in Mathematics. PQ: Common Core 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.

300-301. Set Theory I, II. PQ: Consent of instructor. Math 300 is a course on axiomatic set theory with applications to the undecidability of mathematical statements. Topics include axioms of Zermelo-Fraenkel (ZF) set theory; Von Neumann rank and reflection principles; the Levy hierarchy and absoluteness; inner models; Gödel's Constructible sets (L) and the consistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH); and Souslin's Hypothesis in L. Math 301 deals with models of set theory coding of syntax; Cohen's method of forcing and the unprovability of AC and GCH; Martin's axiom and the unprovability of Souslin's Hypothesis; Solovay's model in which every set of reals is Lebesgue Measurable; inaccessible and measurable cardinals; analytic determinateness, Silver indiscernibles for L (O-Sharp), larger cardinals (elementary embeddings and compactness), and the axiom of determinateness. Not offered 1996-97; will be offered 1997-98.

302-303. Theory of Recursive Functions I, II (=ComSci 380-381). PQ: Math 255 or consent of instructor. Math 302 concerns recursive (i.e., computable) functions and sets generated by an algorithm (recursively enumerable sets). Topics include various mathematical models for computations, including Turing machines and Kleene schemata; enumeration and s-m-n theorems; the recursion theorem; classification of unsolvable problems; and priority methods for the construction of recursively enumerable sets and degrees. Math 303 treats classification of sets by the degree of information they encode, algebraic structure and degrees of recursively enumerable sets, advanced priority methods, and generalized recursion theory. This course is offered in alternate years. R. Soare. Winter, Spring. Not offered 1996-97; will be offered 1997-98.

309-310. Model Theory I, II. PQ: Math 255. Math 309 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In Math 310 the following subjects are studied: saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero. R. Soare. Winter, Spring.

312-313-314. Analysis I, II, III. PQ: Math 262, 270, 272, and 274; consent of director or associate director of undergraduate studies. Topics include Lebesgue measure, abstract measure theory, and Riesz representation theorem; basic functional analysis (L[p]-spaces, elementary Hilbert space theory, Hahn-Banach, open mapping theorem, and uniform boundedness); Radon-Nikodym theorem, duality for L[p]-spaces, Fubini's theorem, differentiation, Fourier transforms, locally convex spaces, weak topologies, and convexity; compact operators; spectral theorem and integral operators; Banach algebras and general spectral theory; Sobolev spaces and embedding theorems; Haar measure; and Peter-Weyl theorem, holomorphic functions, Cauchy's theorem, harmonic functions, maximum modulus principle, meromorphic functions, conformal mapping, and analytic continuation. Staff. Autumn, Winter, Spring.

317-318-319. Topology and Geometry I, II, III. PQ: Consent of director or associate director of undergraduate studies. Math 317 covers smooth manifolds, tangent bundles, vector fields, Frobenius theorem, Sard's theorem, Whitney embedding theorem, and transversality. Math 318 considers fundamental group and covering spaces; Lie groups and Lie algebras; and principal bundles, connections, introduction to Riemannian geometry, geodesics, and curvature. Topics in Math 319 are cell complexes, homology, and cohomology; and Mayer-Vietoris theorem, Kunneth theorem, cup products, duality, and geometric applications. Staff. Autumn, Winter, Spring.

325-326-327. Algebra I, II, III. PQ: Math 254-255-256 and consent of director or associate director of undergraduate studies. Math 325 deals with groups and commutative rings. Math 326 investigates elements of the theory of fields and of Galois theory, as well as noncommutative rings. Math 327 introduces other basic topics in algebra. Staff. Autumn, Winter, Spring.

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