Courses in Mathematics

LOWER-DIVISION

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

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

1B (4) F, W, Summer. 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 and trigonometric functions; applications including curve sketching, related rates, and optimization. Antiderivatives. 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; the Fundamental theorem of calculus. Applications of integration including finding areas and volumes. Techniques 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, S, 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) F, W, S, Summer. 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) W, S. 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 and Basic Linear Algebra (4) F, W, S, Summer. Lecture, three hours; discussion, two hours. Systems of linear equations: matrix operations; determinants; eigenvalues, and eigenvectors. Infinite sequences and series. Complex numbers. Prerequisites: Mathematics 2A-B. (V)

3A Introduction to Linear Algebra (4) F, W, S. Lecture, three hours; discussion, two hours. Vectors, matrices, linear transformations, dot products, determinants, systems of linear equations, vector spaces, subspaces, dimension. Prerequisite: Mathematics 2J or 4. Mathematics 3A and 6G may not both be taken for credit.

3D Elementary Differential Equations (4) F, W, S, Summer. 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 2D and 2J.

4 Mathematics for Economists (4) F, S. Lecture, three hours; discussion, two hours. Topics in linear algebra and multivariable differential calculus suitable for economic applications. Prerequisites: Mathematics 2A-B. No credit for Mathematics 4 if taken after both Mathematics 2J and 2D. (V)

5 Mathematics of Compound Interest (4) F, W, S, Summer. Lecture, three hours. Introduction to fundamental concepts of financial mathematics and their application in calculating present and accumulated values for various streams of cash flows. Topics include definition and simple problems in compound interest; basic/advanced annuities; the yield curve; amortization; financial analysis. Prerequisites: Mathematics 2A-B.

6B Boolean Algebra and Logic (4) W, S, Summer. Lecture, three hours; discussion, two hours. Relations and their properties; Boolean algebras, formal languages; finite automata. Prerequisite: high school mathematics through trigonometry. Same as ICS 6B. (V)

6D Discrete Mathematics for Computer Science (4) F, S, Summer. 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: mathematical induction, combinatorics, and recurrence relations. Prerequisite: high school mathematics through trigonometry. Same as ICS 6D. Formerly Mathematics 6A. (V)

6G Linear Algebra (4) F, W, S, Summer. 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. Mathematics 6G and 3A may not both be taken for credit. Formerly Mathematics 6C. NOTE: Mathematics majors must take 3A. (V)

7 Basic Statistics (4) F, W, S, Summer. Lecture, three hours; discussion, one to two hours. Introduces 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. Same as Statistics 7. Only one course from Mathematics 7/Statistics 7, Biological Sciences 7, or Management 7 may be taken for credit. No credit for Mathematics 7/Statistics 7 if taken after Mathematics 67/Statistics 67. (V) F, W offered for seniors only.

13 Introduction to Abstract Mathematics (4) F, W, 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. Prerequisite: Mathematics 2A or Mathematics 6D/ICS 6D.

67 Introduction to Probability and Statistics for Computer Science (4) S. Lecture, three hours; discussion, two hours. Introduction to the basic concepts of probability and statistics with discussion of applications to computer science. Prerequisites: Mathematics 2B and Mathematics 6D/ICS 6D. No credit for Mathematics 7/Statistics 7, Biological Sciences 7, or Management 7 if taken after Mathematics 67/Statistics 67. Same as Statistics 67.

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.

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. Mathematics 105A and Engineering MAE185 may not both 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 3D and 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 ordinary and 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, 3D; for 112B: 2E and 112A.

114A-B Applied Complex Analysis (4-4) F, W. 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 recommended. For 114B: Mathematics 114A. Mathematics 114A may not be taken for credit after Mathematics 147. Mathematics 147 and 114B may not both be taken for credit. Mathematics 114B not offered every year.

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; 3A or 6G; 3D.

118A-B-C The Theory of Differential Equations (4-4-4). Lecture, three hours; discussion, one hour. Introductory theoretical course in ordinary and/ or partial differential equations. Existence and uniqueness of solutions, methods of solution, the geometry of solutions. Students are expected to follow and understand proofs. Prerequisites: Mathematics 3A, 3D; 140A-B or consent of instructor.

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 6G; Mathematics 13 is strongly recommended.

120B Introduction to Abstract Algebra: Rings and Fields (4) W, S. 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). 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 for 121A: Mathematics 3A or 6G.

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 6G; Mathematics 120A. Previous or concurrent enrollment in Mathematics 120B and 121A recommended.

NOTE: Only one course from Mathematics 130A, 131A, and 132A may be taken for credit. Any one of these courses, or Mathematics 67, will satisfy the prerequisite for Mathematics 130B, 131B, and 132B.

130B-C Probability and Stochastic Processes (4-4). Lecture, three hours. Introductory course emphasizing applications. 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: for 130B: Mathematics 2A-B; either 67, 130A, 131A (same as Statistics 120A), or 132A; for 130C: Mathematics 130B.

131A-B-C Introduction to Probability and Statistics (4-4-4). Lecture, three hours; discussion, one to two hours. Introductory course covering basic principles of probability and statistical inference. 131A: Axiomatic definition of probability, random variables, probability distributions, expectation. 131B: Point estimation, interval estimating, and testing hypotheses, Bayesian approaches to inference. 131C: Linear regression, analysis of variance, model checking. Prerequisites: for 131A-B: Mathematics 2A-B; 2D-2J or 4; for 131C: Mathematics 131A-B; 3A or 6G. Same as Statistics 120A-B-C.

132B-C Discrete Probability and Mathematical Theory of Sample Surveys (4-4). Lecture, three hours. 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 67 or 131A; 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 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, 6G, 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) S. 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.

147 Complex Analysis (4) W. Rigorous treatment of basic complex analysis: complex numbers, analytic functions, Cauchy integral theory and its consequences (Morera's Theorem, Argument Principle, Fundamental Theorem of Algebra, Maximum Modulus Principle, Liouville's Theorem), power series, residue calculus harmonic functions, conformal mapping. Students are expected to do proofs. Corequisite: Mathematics 140B. Prerequisite: Mathematics 140A. Mathematics 114A may not be taken for credit after 147. Mathematics 147 and 114B may not both be taken for credit.

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.

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

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 2D-E, 2J. Not offered every year.

171A-B Mathematical Methods in Operations Research. Lecture, three hours. Offered summer only.

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

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

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 6G.

174A-B Modern Graph Theory I, II. Lecture, three hours. An introduction to fundamental concepts of graph theory by developing abilities to produce examples, following and devising simple proofs, and current applications of graph theory.

174A Modern Graph Theory I (4). Topics include: graph types, matching in graphs; Menger's Theorem; Kuratowski's Theorem. Prerequisites: Mathematics 2B; Mathematics 3A or 6G; Mathematics 6D/ICS 6D.

174B Modern Graph Theory II (4). Topics include: coloring maps, plane graphs, vertices, and edges; Hadwiger's Conjecture; Hamilton Cycles; Ramsey Theory. Prerequisite: Mathematics 174A.

176 Mathematics of Finance (4). Lecture, three hours. Introduces the mathematics of finance with an emphasis on financial derivatives. After a review of certain tools from probability, statistics, and elementary differential and partial differential equations, concepts such as hedging, arbitrage, Puts, Calls, and the design of portfolios are discussed. Prerequisites: Mathematics 2A-B-J. Same as Economics 135.

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 2J.

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 2D, 2J, 3A or 6G, 3D, 120A, 140A. Not offered every year.

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.

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 2D; 2J; 3D; 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. Students taking the course in fall will prepare for and take the Putnam examination in December. Pass/Not Pass only. NOTE: satisfies no requirement other than contribution to the 180 units required for graduation. Recommended for prospective teachers. May be taken twice for credit.

H195A-B Honors Seminar (4-4) W, S. Topics vary from year to year. Provides an integrative experience, including problem-solving and oral and written presentations. Required for the Honors Program in Mathematics and open to others with 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

NOTE: Undergraduates who are not in the Honors Program and wish to take 205A or 206A should first get the consent of the instructor.

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, 2J, 2D, 2E, 3A; 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 Topics in Real Analysis (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.

227A-B Mathematics and Computational Biology (4-4). Solution techniques and numerical methods for models in biology and life sciences; dynamical systems; ordinary differential equations; reaction-diffusion equations; temporal and spatial dynamics; stability; models for genetic and developmental patterns, cancers, tissue growth, and signaling networks. 227A: Analytical methods. 227B: Computer simulations. Prerequisites: Mathematics 2A-B or equivalent.

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.

235A Mathematics of Cryptography (4). Lecture, three hours. Mathematics of public key cryptography: encryption and signature schemes; RSA; factoring; primality testing; discrete log based cryptosystems, elliptic and hyperelliptic curve cryptography. Prerequisite: consent of instructor.

236A-B Topics in the Mathematics of Cryptography (4-4). Lecture, three hours. Continuation of Mathematics 235A. Topics to be determined by the instructor. Prerequisite: consent of instructor. May be repeated for credit as topics vary.

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.

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.

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.

291A-B-C Topics in Applied and Computational Mathematics (4-4-4) F, W, S. Lecture, three hours. Topics to be determined by instructor. Prerequisite: consent of instructor. May be repeated for credit as topics vary.

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.