Lower-Division

Upper-Division

Graduate

Suggested Syllabus

1A-B Pre-Calculus. Lecture, three hours; discussion, two hours.

1A (0) F, W. Basic equations and inequalities, linear and quadratic functions, and systems of simultaneous equations. Four units of workload credit only.

1B (4) F, W, S. Preparation for calculus and other mathematics courses. Exponentials, logarithms, trigonometry, polynomials, and rational functions. Satisfies no requirements other than contribution to the 180 units required for graduation. Prerequisite: Mathematics 1A, satisfactory performance on the algebra or pre-calculus placement examinations offered periodically by the Mathematics Department, or consent of instructor.

2A-B Single-Variable Calculus (4-4) F, W, S, Summer. Lecture, three hours; discussion, two hours. 2A: Introduction to derivatives, calculation of derivatives of algebraic functions, and applications of derivatives (approximations, curve plotting, related rates, maxima and minima). Indefinite integrals. Fundamental theorem of calculus. Differentiation and integration of sines and cosines. Prerequisite: pass the UCI Precalculus test no more than one year before the start of the quarter in which Mathematics 2A will be taken, or get a grade of C (2.0) or better in Mathematics 1B at UCI. 2B: Definite integrals, their applications (areas, volumes, etc.), and methods of integration. Logarithmic and exponential functions. Polar coordinates. Prerequisite for Mathematics 2B: 2A. (V)

2D-E Multivariable Calculus. Lecture, three hours; discussion, two hours.

2D (4) F, W, Summer. Differential and integral calculus of real-valued functions of several real variables, including applications. Prerequisites: Mathematics 2A-B. Mathematics 2D and H2D may not both be taken for credit. (V)

2E (4) W, S. The differential and integral calculus of vector-valued functions. Implicit and inverse function theorems. Line and surface integrals, divergence and curl, theorems of Green, Gauss, and Stokes. Prerequisite: 2D. Mathematics 2E and H2E may not both be taken for credit.

H2D-E Honors Multivariable Calculus (4-4). Lecture, three hours; discussion, two hours. Covers the same material as Mathematics 2D-E, but with a greater emphasis on the theoretical structure of the subject matter. Especially recommended for prospective Mathematics majors and others with a particular interest in mathematics. Satisfies the same requirements and prerequisites as 2D-E. Prerequisites for H2D: a grade of B (3.0) or better in Mathematics 2B or a score of 4 or 5 on the Advanced Placement Calculus BC examination; for H2E: a grade of C (2.0) or better in Mathematics H2D. Mathematics 2D-E and H2D-E may not both be taken for credit. (H2D: V)

2J Infinite Series, Complex Numbers, and Basic Linear Algebra (4). Lecture, three hours; discussion, two hours. Infinite sequences and series; complex numbers; systems of linear algebraic equations, determinants, basic matrix operations, eigenvalues, and eigenvectors. Prerequisites: Mathematics 2A-B. (V)

3A Introduction to Linear Algebra (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Vectors, matrices, linear transformations, dot products, determinants, systems of linear equations, vector spaces, subspaces, dimension. Prerequisites: Mathematics 2A-B; 2J. Only one course from Mathematics 3A, Mathematics 6C, and Physical Sciences 50A may be taken for credit.

3D Elementary Differential Equations (4) W, S. Lecture, three hours; discussion, two hours. Linear differential equations, variation of parameters, constant coefficient cookbook, systems of equations, Laplace transforms, series solutions. Further topics as time permits. Prerequisites: Mathematics 2A-B-J. Mathematics 3D and Physical Sciences 50C may not both be taken for credit.

6A Discrete Mathematics for Computer Science (4). Lecture, three hours; discussion, two hours. Covers essential tools from discrete mathematics used in computer science with an emphasis on the process of abstracting computational problems and analyzing them mathematically. Topics include: combinatorics, mathematical induction, elementary probability, and asymptotic analysis. Prerequisite: high school mathematics through trigonometry. Same as Information and Computer Science 6A. (V)

6B Discrete Mathematics: Boolean Algebra and Logic (4). Lecture, three hours; discussion, two hours. Relations and their properties; Boolean algebras, formal languages; finite automata. Prerequisite: Mathematics 6A or Information and Computer Science 6A. (V)

6C Linear Algebra (4). Lecture, three hours; discussion, two hours. Linear equations, vector spaces and subspaces, linear functions and matrices, linear codes, determinants, scalar products. Prerequisite: high school mathematics through trigonometry. Only one course from Mathematics 6C, Mathematics 3A, and Physical Sciences 50A may be taken for credit. (V)

7 Basic Statistics (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Basic inferential statistics including confidence intervals and hypothesis testing on means and proportions, t-distribution, Chi Square, regression and correlation. F-distribution and nonparametric statistics included if time permits. Only one course from Mathematics 7, Mathematics 67, and Biological Sciences 7 may be taken for credit. (V)

13 Introduction to Abstract Mathematics (4) F, S. Lecture, three hours; discussion, two hours. The style of precise definition and rigorous proof which is characteristic of modern mathematics. Topics include set theory, equivalence relations, proof by mathematical induction, and number theory. Students construct original proofs to statements. Strongly recommended for freshman and sophomore Mathematics majors as preparation for upper-division courses such as Mathematics 120 and 140.

67 Introduction to Probability and Statistics for Computer Science (4). Lecture, three hours; discussion, two hours. Introductory course focusing on basic concepts in probability and statistics with discussion of applications to computer science. Prerequisites: Mathematics 2B, 6A, and 6C or 3A. Only one course from Mathematics 7, Mathematics 67, and Biological Sciences 7 may be taken for credit.

UPPER-DIVISION

NOTE: Some of the upper-division courses listed below have one or two hours of discussion weekly in addition to the lectures. Not all courses are offered every year. Students should refer to the quarterly Schedule of Classes for specific information.

105A-B Numerical Analysis (4-4) F, W. Lecture, three hours. Introduction to the theory and practice of numerical computation. 105A: Floating point arithmetic, roundoff; solving transcendental equations; quadrature; linear systems, eigenvalues, power method. Corequisite: Mathematics 105LA if offered. Prerequisites: Mathematics 2A-B-J; some acquaintance with computer programming. Only one course from Mathematics 105A, Engineering CEE185, and Engineering MAE185 may be taken for credit. 105B: Lagrange interpolation, finite differences, splines, Padé approximations; Gaussian quadrature; Fourier series and transforms. Corequisite: Mathematics 105LB if offered. Prerequisite: Mathematics 105A.

105LA-LB Numerical Analysis Laboratory (1-1) F, W. Laboratory, two hours. Provides practical experience to complement the theory developed in Mathematics 105A-B. Corequisite: concurrent enrollment in Mathematics 105A-B.

107 Numerical Differential Equations (4) S. Lecture, three hours. Theory and applications of numerical methods to initial and boundary-value problems for ordinary and partial differential equations. Corequisite: concurrent enrollment in Mathematics 107L if offered. Prerequisites: Mathematics 2F or 3D; 105A-B.

107L Numerical Differential Equations Laboratory (1) S. Laboratory, two hours. Provides practical experience to complement the theory developed in Mathematics 107. Corequisite: concurrent enrollment in Mathematics 107.

112A-B-C Introduction to Partial Differential Equations and Applications (4-4-4). Lecture, three hours. Introduction to partial differential equations and their applications in engineering and science. Basic methods for classical PDEs (potential, heat, and wave equations). 112A: Classification of PDEs, separation of variables and series expansions, special functions, eigenvalue problems. 112B: Green functions and integral representations, method of characteristics. 112C: Galerkin method and other discretization techniques. Prerequisites for 112A: Mathematics 2D, 2J, 3D; for 112B: 2E.

114A-B Applied Complex Analysis (4-4). Lecture, three hours. Introduction to complex functions and their applications to engineering and science. 114A: Complex numbers, elementary functions; analytic functions; complex integration; power series; residue theory; conformal maps; applications. 114B: Applications to potential theory, flows; heat; Laplace transforms; asymptotic expansions. Prerequisites: for 114A: Mathematics 2D, 2J. Mathematics 2E, and 3D or 2F recommended. For 114B: Mathematics 114A. Mathematics 114A and Engineering ECE180 may not both be taken for credit.

115 Mathematical Modeling (4). Lecture, three hours. Mathematical modeling and analysis of phenomena that arise in engineering physical sciences, biology, economics, or social sciences. Corequisite or prerequisite: Mathematics 112A or Engineering MAE140. Prerequisites: Mathematics 2D; 2J or 3A or 6C; 2F or 3D.

118A-B-C Differential Equations (4-4-4). Lecture, three hours. Introductory theoretical course in ordinary and/or partial differential equations. Existence and uniqueness of solutions, methods of solution, the geometry of solutions. Prerequisites: Mathematics 2D, 2J, and 3D.

120A Introduction to Abstract Algebra: Groups (4) F, W. Lecture, three hours; discussion, two hours. Axioms for group theory; permutation groups, matrix groups. Isomorphisms, homomorphisms, quotient groups. Advanced topics as time permits. Special emphasis on doing proofs. Prerequisite: Mathematics 3A or 6C; Mathematics 13 is strongly recommended.

120B Introduction to Abstract Algebra: Rings and Fields (4) W. Lecture, three hours; discussion, two hours. Basic properties of rings; ideals, quotient rings; polynomial and matrix rings. Elements of field theory. Prerequisite: Mathematics 120A.

121A-B Linear Algebra (4-4) W, S. Lecture, three hours; discussion, two hours. Introduction to modern abstract linear algebra. Special emphasis on students doing proofs. 121A: Vector spaces, linear independence, bases, dimension. Linear transformations and their matrix representations. Theory of determinants. 121B: Canonical forms; inner products; similarity of matrices. Prerequisite: Mathematics 3A or 6C.

124 Algebra and Some Famous Impossibilities (4). Lecture, three hours. Proof of the impossibility of certain ruler-and-compass constructions (squaring the circle; trisecting angles); nonexistence of analogs to the "quadratic formula" for polynomial equations of degree 5 or higher. The necessary algebra introduced as needed. Prerequisites: Mathematics 3A or 6C; Mathematics 120A. Previous or concurrent enrollment in Mathematics 120B and 121A recommended.

130A-B-C Probability and Stochastic Processes (4-4-4). Lecture, three hours. Introductory course emphasizing applications. 130A: Bayes theorem, random variables, expectation, variance and covariance, normal distribution and limit theorems. 130B: Conditional probability and conditional expectations; Markov chains. 130C: Exponential distribution and Poisson process; Brownian motion; additional topics, such as option pricing, as time permits. Prerequisites: Mathematics 2A-B, 2J.

131A-B-C Mathematical Statistics (4-4-4) F, W, S. Lecture, three hours. Introduction to data analysis. Probability distributions, random variables, moments, estimation. Hypothesis testing and confidence intervals. Random simulations. Simple linear regression. Prerequisites: Mathematics 2A-B, 2J. For 131C: Mathematics 3A or 6C.

132A-B-C Discrete Probability and Mathematical Theory of Sample Surveys (4-4-4) F, W, S. Lecture, three hours.

132A: Introduction to discrete probability with focus on those topics required for sample survey theory, especially the case of equally likely events. Random variables. Expectation, moments of random variables, covariance and correlation. Conditional expectation. Limit theorems. Prerequisite: Mathematics 2A-B, 2J.

132B-C: Sample selection, stratification, cluster sampling, double-sampling procedures, optimal allocation, probability-proportional-to-size sampling. Applications to problems in economics, business, public health, agriculture, and the social sciences. Prerequisites: for 132B: Mathematics 132A; for 132C: Mathematics 132B.

140A-B Elementary Analysis (4-4). Lecture, three hours; discussion, two hours. Introduction to real analysis including: the real number system, convergence of sequences, infinite series, differentiation and integration, and sequences of functions. Students are expected to do proofs. Prerequisites: Mathematics 2A-B, 2D, 2J; Mathematics 13 is strongly recommended.

140C-D Analysis in Several Variables (4-4). Lecture, three hours; discussion, two hours. 140C: Rigorous treatment of multivariable differential calculus. Jacobians, Inverse and Implicit Function theorems. Prerequisites: some background in linear algebra (Mathematics 3A, 6C, or 2J), and 140B. 140D: Rigorous treatment of multivariable integral calculus. Multiple integrals in Rⁿ; iterated integrals and Fubini's theorem; change-of-variables theorem; differential forms and Stokes' theorem. Prerequisite: Mathematics 2E and 140C.

140T Topics in Analysis (4). Lecture, three hours; discussion, two hours. Additional topics in analysis. Varies from year to year. Prerequisites: Mathematics 140A-B and consent of instructor. May be repeated for credit as topics vary. Not offered every year.

141 Introduction to Topology (4). Lecture, three hours. The elements of naive set theory and the basic properties of metric spaces. Introduction to topological properties. Prerequisite: Mathematics 140A. Formerly Mathematics 141A.

146 Fourier Analysis (4) S. Lecture, three hours. Rigorous introduction to the theory of Fourier series and orthogonal expansions. Fourier transform. Prerequisites: Mathematics 3D and 140A-B. Mathematics 112A recommended.

150 Introduction to Mathematical Logic (4) F. Lecture, three hours. First-order logic through the Completeness Theorem for predicate logic. Prerequisite: consent of instructor. Only one course from Mathematics 150, Philosophy 105B, and Logic and Philosophy of Science 105B may be taken for credit.

151 Set Theory (4) W. Lecture, three hours. Axiomatic development; infinite sets; cardinal and ordinal numbers. Prerequisite: Mathematics 150. Only one course from Mathematics 151, Philosophy 105A, and Logic and Philosophy of Science 105A may be taken for credit.

152 Computability (4) S. Lecture, three hours. Computable functions; undecidability; Gödel's Incompleteness Theorem. Prerequisite: Mathematics 150. Only one course from Mathematics 152, Philosophy 105C, and Logic and Philosophy of Science 105C may be taken for credit.

162A-B Introduction to Differential Geometry (4-4) W, S. Lecture, three hours. Applications of advanced calculus and linear algebra to the geometry of curves and surfaces in space. Prerequisites: Mathematics 2A-B, 2D-E, 2J.

171A, B-C Mathematical Methods in Operations Research. Lecture, three hours.

171A Linear Programming (4). Simplex algorithm, duality, optimization in networks. Prerequisite: Mathematics 3A or 6C.

171B Nonlinear Programming (4). Conditions for optimality, quadratic and convex programming, search methods, geometric programming. Prerequisites: Mathematics 2D and either 3A or 6C.

171C Integer and Dynamic Programming (4). Multistage decision models, applications. Prerequisites: Mathematics 171B and consent of instructor.

173A-B Introduction to Cryptology (4-4). Lecture, three hours. Introduction to some of the mathematics used in the making and breaking of codes, with applications to classical ciphers and public key systems. The mathematics which is covered includes topics from number theory, probability, and abstract algebra. Prerequisites: Mathematics 2A-B; 3A or 6C.

180 Introduction to Number Theory (4). Lecture, three hours. The ring of integers. Divisibility. Prime numbers and factorization. Number-theoretic functions such as the Moebius function and the Euler function. Congruences, Moebius inversion, perfect numbers, diophantine equations, quadratic residues. Other topics as time permits. Prerequisite: Mathematics 2A-B, 2J.

182 Modern Geometry (4). Lecture, three hours. Euclidean geometry; Hilbert's axioms; absolute geometry; hyperbolic geometry; the Poincare models; geometric transformations. Prerequisites: Mathematics 2A-B; 2D, 2J, 3A, 120A.

184 History of Mathematics (4). Lecture, three hours. Topics vary from year to year. Some possible topics: mathematics in ancient times; the development of modern analysis; the evolution of geometric ideas. Students are assigned individual topics for term papers. Prerequisite: Mathematics 2A-B, 2D, 2J, 3A or 6C, 3D, 120A, 140A.

189 Special Topics in Mathematics (4). Lecture, three hours. Offered from time to time, but not on a regular basis. Content and prerequisites vary with the instructor. May be repeated for credit as topics vary.

190 Technical Writing and Communication Skills (4) F, W, S. Lecture, three hours. Workshop in writing technical reports, journal articles, proposals. Oral presentations. Communicating with the public. May not be used in satisfaction of any School or departmental requirement. Prerequisites: upper-division standing; satisfaction of the lower-division writing requirement. Open to Mathematics majors only. Same as Chemistry 139 and Physics 129.

192 Tutoring in Mathematics (2). Enrollment limited to upper-division Mathematics majors participating in the Department's Tutoring Program. Admission requires approval of Department Tutor Supervisor. For students not in the Department's specialization in Mathematics for High School Teaching, this course satisfies no requirements other than contribution to the 180 units required for graduation. Pass/Not Pass only. Prerequisites: Mathematics 2A-B; 2C or 2J; 2D; 3A or 6C or 2J; 13 or 120A or 140A. May be taken twice for credit.

194 Problem-Solving Seminar (2). Develops ability in analytical thinking and problem solving, using problems of the type found in the Mathematics Olympiad and the Putnam Mathematical Competition. Especially useful for high school mathematics teachers and for students planning to become such teachers. Pass/Not Pass only. NOTE: satisfies no requirement other than contribution to the 180 units required for graduation. May be taken twice for credit.

H195A-B Honors Seminar (4-4) W, S. A focused study of a topic which will vary from year to year, culminating in the writing of an Honors thesis. Prerequisite: enrollment in the Mathematics Honors Program or consent of instructor.

199A-B-C Special Studies in Mathematics (4-4-4) F, W, S. Supervised reading. For outstanding undergraduate mathematics majors in supervised but independent reading or research of mathematical topics. Prerequisite: consent of Department. NOTE: Cannot normally be used to satisfy departmental requirements.

GRADUATE

201A Theory of Mathematical Statistics (4) F. Lecture, three hours. Review of probability and sampling distributions. Point and interval estimation, sufficient statistics, hypothesis testing, analysis of categorical data, the multivariate normal distribution, sequential analysis. Prerequisites: Mathematics 120A, 130A, 133A-B, and 121A-B or consent of instructor. Corequisite: concurrent enrollment in Mathematics 201LA.

201B Linear Regression Analysis (4) W. Lecture, three hours. The normal linear regression model, confidence ellipsoids for regression coefficient vectors, the F-test and its applications to one- and two-way analysis of variance, analysis of covariance and a test for independence, simultaneous confidence intervals. Prerequisite: Mathematics 201A. Corequisite: concurrent enrollment in Mathematics 201LB.

201C Experimental Design (4) S. Lecture, three hours. Analysis of variance for the linear regression and other models, Latin squares, incomplete blocks, nested designs, random effects model, randomization models, confounding. Prerequisite: Mathematics 201B. Corequisite: concurrent enrollment in Mathematics 201LC.

205A-B-C Introduction to Graduate Analysis (5-5-5) F, W, S. Lecture, four hours. Construction of the real number system, topology of the real line, concepts of continuity, differential and integral calculus, sequences and series of functions, equicontinuity, metric spaces, multivariable differential and integral calculus, implicit functions, curves and surfaces. Prerequisites: Mathematics 2A-B; 2C or 2J; 2D; 2E or equivalent or consent of instructor.

206A-B-C Introduction to Graduate Algebra (5-5-5). Lecture, four hours. Introduction to abstract linear algebra, including bases, linear transformation, eigenvectors, canonical forms, inner products, symmetric operators. Introduction to groups, rings, and fields including examples of groups, group actions, Sylow theorems, modules over principal ideal domains, polynomials and Galois groups. Prerequisite: Mathematics 3A or equivalent or consent of instructor.

210A-B-C Real Analysis (4-4-4) F, W, S. Lecture, three hours. Measure theory, Lebesgue integral, signed measures, Radon-Nikodym theorem, functions of bounded variation and absolutely continuous functions, classical Banach spaces, Lp spaces, integration on locally compact spaces and the Riesz-Markov theorem, measure and outer measure, product measure spaces. Prerequisites: Mathematics 140A-B-C or consent of instructor.

211A-B-C Topics in Real Analysis (4-4-4). Lecture, three hours. A continuation of Mathematics 210A-B-C; topics selected by instructor.

218A-B-C Introduction to Manifolds and Geometry (4-4-4) F, W, S. Lecture, three hours. General topology and fundamental groups, covering space; Stokes theorem on manifolds, selected topics on abstract manifold theory. Prerequisites: Mathematics 205A-B-C or consent of instructor.

220A-B-C Analytic Function Theory (4-4-4) F, W, S. Lecture, three hours. Standard theorems about analytic functions. Harmonic functions. Normal families. Conformal mapping. Prerequisites: Mathematics 140A-B-C or equivalent or consent of instructor.

221A-B Several Complex Variables (4-4). Lecture, three hours. Introduction to the study of holomorphic functions in several complex variables. Topics include: Automorphism group of a domain, Bergman kernel function, boundary behavior of Poisson integrals, pluriharmonic functions, Hardy and Bergman spaces, Mobius invariant function spaces, subharmonicity, convexity. Prerequisites: Mathematics 210, 220, and 260.

225A-B-C Introduction to Numerical Analysis and Scientific Computing (4-4-4). Lecture, three hours. Introduction to fundamentals of numerical analysis from an advanced viewpoint. 225A: Error analysis, approximation of functions, nonlinear equations. 225B-C: Numerical linear algebra, numerical solutions of differential equations; stability. Corequisite: Mathematics 225LA-LB-LC (if offered). Prerequisites: Mathematics 3D; 105A-B or 140A-B; 121A; and Mathematics 112A or Engineering MAE140.

225LA-LB-LC Laboratory for Numerical Analysis and Scientific Computing (2-1-1). Laboratory, two hours for 225LA; one hour for 225LB and 225LC. Provides practical experience to complement the theory in Mathematics 225A-B-C. Corequisite: Mathematics 225A-B-C.

226A-B-C Computational Differential Equations (4-4-4). Lecture, three hours. Finite difference and finite element methods. Quick treatment of functional and nonlinear analysis background: weak solution, Lp spaces, Sobolev spaces. Approximation theory. Fourier and Petrov-Galerkin methods; mesh generation. Elliptic, parabolic, hyperbolic cases in 226A-B-C, respectively. Corequisite: Mathematics 226LA-LB-LC (if offered). Prerequisites: basic differential equations, such as in Mathematics 3D and either Mathematics 112A or Engineering MAE140; plus either abstract analysis (e.g., Mathematics 140A-B) or numerical analysis (Mathematics 105A-B or equivalent).

226LA-LB-LC Laboratory for Computational Differential Equations (2-1-1). Laboratory, two hours for 226LA; one hour for 226LB and 226LC. Provides practical experience to complement the theory in Mathematics 226A-B-C. Corequisite: Mathematics 226A-B-C.

230A-B-C Algebra (4-4-4) F, W, S. Lecture, three hours. Elements of the theories of groups, rings, fields, modules. Galois theory. Modules over principal ideal domains. Artinian, Noetherian, and semisimple rings and modules. Prerequisites: Mathematics 120A and 121A-B or equivalent, or consent of instructor.

232A-B-C Algebraic Number Theory (4-4-4) F, W, S. Lecture, three hours. Prime number theorem, quadratic reciprocity, Gauss sums, diophantine equations, zeta functions over finite fields. Algebraic integers, prime ideals, class groups, Dirichlet unit theorem, localization, completion, Galois extensions, Chebatarev density theorem. Representations of finite groups, L-functions, Hecke L-functions. Introduction to class field theory. Prerequisites: Mathematics 206A-B-C or consent of instructor.

233A-B-C Algebraic Geometry (4-4-4). Lecture, three hours. Basic commutative algebra and classical algebraic geometry. Algebraic varieties, morphisms, rational maps, blow ups. Theory of schemes, sheaves, divisors, cohomology. Algebraic curves and surfaces, Riemann-Roch theorem, Jocobian classification of curves and surfaces.

234A-B-C Topics in Algebra (4-4-4). Lecture, three hours. Group theory, homological algebra, and other selected topics. Prerequisites: Mathematics 230A-B-C or consent of instructor.

237A-B Homological Algebra (4-4). Lecture, three hours. Categories and functors, including the category of modules over a (possibly noncommutative) ring; direct sums and products, direct and projective limits, tensor products and Hom; image, kernal, complexes, homology and exact sequences. Applications. Prerequisites: Mathematics 230A-B-C or consent of instructor.

240A-B-C Differential Geometry (4-4-4). Lecture, three hours. Riemannian manifolds, connections, curvature and torsion. Submanifolds, mean curvature, Gauss curvature equation. Geodesics, minimal submanifolds, first and second fundamental forms, variational formulas. Comparison theorems and their geometric applications. Hodge theory applications to geometry and topology. Prerequisites: Mathematics 141A-B or consent of instructor.

245A-B-C Topics in Differential Geometry (4-4-4). Lecture, three hours. Continuation of Mathematics 240A-B-C. Topics to be determined by the instructor. Prerequisites: Mathematics 240A-B-C or consent of instructor. May be repeated for credit as topics vary.

250A-B-C Algebraic Topology (4-4-4). Lecture, three hours. Provides fundamental materials in algebraic topology: fundamental group and covering space, homology and cohomology theory, and homotopy group. Prerequisites: Mathematics 230A and 141A-B, or equivalent, or consent of instructor.

255A-B-C Topics in Algebraic Topology (4-4-4). Lecture, three hours. Continuation of Mathematics 250A-B-C. Topics to be determined by the instructor. Prerequisite: 250A-B-C or consent of instructor. May be repeated for credit as topics vary.

260A-B-C Functional Analysis (4-4-4). Lecture, three hours. Normed linear spaces, Hilbert spaces, Banach spaces, Stone-Weierstrass Theorem, locally convex spaces, bounded operators on Banach and Hilbert spaces, the Gelfand-Neumark Theorem for commutative C*-algebras, the spectral theorem for bounded self-adjoint operators, unbounded operators on Hilbert spaces. Prerequisites: Mathematics 210A-B-C and 220A-B-C or consent of instructor.

268A-B-C Topics in Functional Analysis (4-4-4). Lecture, three hours. Selected topics such as spectral theory, abstract harmonic analysis, Banach algebras, operator algebras. Prerequisite: consent of instructor.

270A-B-C Probability (4-4-4). Lecture, three hours. Probability spaces, distribution and characteristic functions. Strong limit theorems. Limit distributions for sums of independent random variables. Conditional expectation and martingale theory. Stochastic processes. Prerequisites: Mathematics 130A-B-C and 210A-B-C or consent of instructor.

271A-B-C Stochastic Processes (4-4-4). Lecture, three hours. Processes with independent increments, Wiener and Gaussian processes, function space integrals, stationary processes, Markov processes. Prerequisites: Mathematics 210A-B-C or consent of instructor.

274 Topics in Probability (4-4-4). Lecture, three hours. Selected topics, such as theory of stochastic processes, martingale theory, stochastic integrals, stochastic differential equations. Prerequisites: Mathematics 270A-B-C or consent of instructor. May be repeated for credit as topics vary.

277A-B-C Topics in Mathematical Physics (4-4-4). Lecture, three hours. Topics to be determined by the instructor. Prerequisite: consent of instructor. May be repeated for credit as topics vary.

280A-B-C Mathematical Logic (4-4-4). Lecture, three hours. Basic set theory; models, compactness, and completeness; basic model theory; Incompleteness and Gödel's Theorems; basic recursion theory; constructible sets. Prerequisite: consent of instructor.

281A-B-C Set Theory (4-4-4). Lecture, three hours. Ordinals, cardinals, cardinal arithmetic, combinatorial set theory, models of set theory, Gödel's constructible universe, forcing, large cardinals, iterate forcing, inner model theory, fine structure. Prerequisites: Mathematics 280A-B-C or consent of instructor.

282A-B-C Model Theory (4-4-4). Lecture, three hours. Languages, structures, compactness and completeness. Model-theoretic constructions. Omitting types theorems. Morley's theorem. Ranks, forking. Model completeness. O-minimality. Applications to algebra. Prerequisites: Mathematics 280A-B-C.

285A-B-C Topics in Mathematical Logic (4-4-4). Lecture, three hours. Continuation of Mathematics 280A-B-C. Topics to be conducted by the instructor. Prerequisite: Mathematics 280A-B-C or consent of instructor. May be repeated for credit as topics vary.

290A-B-C Methods in Applied Mathematics (4-4-4). Lecture, three hours. Introduction to ODEs and dynamical systems: existence and uniqueness. Equilibria and periodic solutions. Bifurcation theory. Perturbation methods: approximate solution of differential equations. Multiple scales and WKB. Matched asymptotic. Calculus of variations: direct methods, Euler-Lagrange equation. Second variation and Legendre condition.

292A-B-C Applied Mathematics (4-4-4) F, W, S. Lecture, three hours. Mathematical techniques and methods applied to specific questions in physics, chemistry, and engineering. Background material in science and mathematics introduced as needed. Prerequisites: Mathematics 140A-B-C or consent of instructor. May be repeated for credit.

294A, B, C Applied Nonlinear Analysis (4, 4, 4). Lecture, three hours. Methods for nonlinear problems in mathematics, science, and engineering. Includes perturbation techniques, variational methods, bifurcation, degree theory, Newton's methods, implicit functions, minimax theorems, optimal control. Background material presented as needed. Each quarter may be taken independently. Prerequisite: Mathematics 210A or consent of instructor.

295A-B-C Partial Differential Equations (4-4-4). Lecture, three hours. Theory and techniques for linear and nonlinear partial differential equations. Local and global theory of partial differential equations: analytic, geometric, and functional analytic methods. Prerequisites: Mathematics 112A-B-C, 210A-B-C or equivalent, or consent of instructor.

296 Topics in Partial Differential Equations (4). Lecture, three hours. Continuation of Mathematics 295A-B-C. Topics to be determined by the instructor. Prerequisites: Mathematics 295A-B-C or consent of instructor. May be repeated for credit as topics vary.

297 Mathematics Colloquium (1). Weekly colloquia on topics of current interest in mathematics. Satisfactory/Unsatisfactory Only. May be repeated for credit.

298A-B-C Seminar (1 to 3) F, W, S. Seminars organized for detailed discussion of research problems of current interest in the Department. The format, content, frequency, and course value are variable. Prerequisite: consent of the Department. May be repeated for credit.

299A-B-C Supervised Reading and Research (2 to 12) F, W, S. May be repeated for credit.

399 University Teaching (1 to 4) F, W, S. Limited to Teaching Assistants. Does not satisfy any requirements for the Master's degree. Satisfactory/Unsatisfactory Only. May be repeated for credit.