Course Catalog Descriptions
Lower Division Courses

1A  Calculus [4 units]  


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: Three and onehalf years of high school math, including trigonometry and analytic geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the the Quantitative Reasoning (QR) Exam, or 32. Consult the mathematics department for details. Students with AP credit should consider choosing a course more advanced than 1A. 


Credit option: Students will receive no credit for 1A after taking 16B and 2 units after taking 16A. 


Description: This sequence is intended for majors in engineering and the physical sciences. An introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions. 


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1B  Calculus [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1A. 


Credit option: Students will receive 2 units of credit for 1B after taking 16B. 


Description: Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. Firstorder ordinary differential equations. Secondorder ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations. 


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H1B  Honors Calculus [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1A. 


Credit option: Students will receive 2 units of credit for H1B after taking 16B. 


Description: Honors version of 1B. Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. Firstorder ordinary differential equations. Secondorder ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations. 


(F) 




16A  Analytic Geometry and Calculus [3 units] 


Course Format: Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional one hour to one and onehalf hours of lecture or discussion/workshop per week. 


Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the Quantitative Reasoning (QR) Exam, or 32. Consult the mathematics department for details. 


Credit option: Students will receive no credit for 16A after taking 1A. Two units of 16A may be used to remove a deficient grade in 1A. 


Description: This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions. 


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16B  Analytic Geometry and Calculus [3 units] 


Course Format: Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional hour of lecture or discussion/workshop per week. 


Prerequisites: 16A. 


Credit option: Students will receive no credit for 16B after 1B, 2 units after 1A. Two units of 16B may be used to remove a deficient grade in 1A. 


Description: Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization. 


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24  Freshman Seminars [1 unit] 


Course Format: One hour of seminar per week. 


Credit option: Course may be repeated for credit as topic varies. 


Grading option: Sections 12 to be graded on a lettergrade basis. Sections 34 to be graded on a passed/not passed basis. 


Description: The Berkeley Seminar Program has been designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a smallseminar setting. Berkeley Seminars are offered in all campus departments, and topics vary from department to department and semester to semester. 


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32  Precalculus [4 units] 


Course Format: Two hours of lecture and two hours of discussion per week, plus, at the instructor's option, an extra hour of lecture/discussion per week. 


Prerequisites: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or the the Quantitative Reasoning (QR) Exam. 


Credit option: Students will receive no credit for 32 after taking 1A1B or 16A16B and will receive 3 units after taking 96. 


Description: Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences. 


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39A 


(24) Two to four hours of seminar per week. 




49  Supplementary Work in Lower Division Mathematics [13 units] 


Course Format: Meetings to be arranged. 


Prerequisites: Some units in a lower division Mathematics class. 


Credit option: Course may be repeated for credit. 


Description: Meetings to be arranged. Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading. 


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53  Multivariable Calculus [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1B. 


Credit option: Students will receive 1 unit of credit for 53 after taking 50B and 3 units of credit after taking 50A. 


Description: Parametric equations and polar coordinates. Vectors in 2 and 3dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes. 


(F,SP) 




H53  Honors Multivariable Calculus [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1B. 


Credit option: Students will receive 1 unit for H53 after taking 50B and 3 units after taking 50A. 


Description: Honors version of 53. Parametric equations and polar coordinates. Vectors in 2 and 3dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes. 


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54  Linear Algebra and Differential Equations [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1B. 


Credit option: Students will receive 1 unit of credit for 54 after taking 50A and 3 units of credit after taking Math 50B. 


Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; firstorder differential equations with constant coefficients. Fourier series and partial differential equations. 


(F,SP) 




H54  Honors Linear Algebra and Differential Equations [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: 1B. 


Credit option: Students will receive 1 unit for H54 after taking 50A and 3 units after taking 50B. 


Description: Honors version of 54. Basic linear algebra: matrix arithmetic and determinants. Vectors spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; firstorder differential equations with constant coefficients. Fourier series and partial differential equations. 


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55  Discrete Mathematics [4 units] 


Course Format: Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week. 


Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A1B recommended. 


Credit option: Students will receive no credit for 55 after taking Computer Science 70. 


Description: Logic, mathematical induction sets, relations,
and functions. Introduction to graphs, elementary number theory,
combinatorics, algebraic structures, discrete probability, theory, and
statistics. 


(F,SP) 




74  Transitions to Upper Division Mathematics [3 units] 


Course Format: Three hours of lecture and two hours of discussion per week. 


Grading option: 


Prerequisites: 53 and 54. 


Description: The course will focus on reading and understanding mathematical proofs. It will emphasize precise thinking and the presentation of mathematical results, both orally and in written form. The course is intended for students who are considering majoring in mathematics but wish additional training. 


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H90  Honors Undergraduate Seminar in Mathematical Problem Solving [1 units] 


Course Format: Two hours of seminar per week. 


Grading option: Course may be repeated for credit. 


Prerequisites: Consent of instructor; undergraduate standing. 


Description: This seminar is designed especially, but not exclusively, to prepare students for the annual national Putnam Mathematical Competition in December. Students will develop problem solving skills and experience by attempting the solution of challenging mathematical problems that require insight more than knowledge. 


(F) 




91  Special Topics in Mathematics [4 units] 


Course Format: Three hours of lecture/discussion per week. 


Grading option: Course may be repeated for credit. 


Prerequisites: 


Description: Topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See department bulletins. 


(F,SP) Staff 




98  Supervised Group Study [14 units] 


Course Format: 


Grading option: Must be taken on a passed/not passed basis. 


Prerequisites: 


Description: Directed Group Study, topics vary with instructor. 


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99  Supervised Independent Study [14 units] 


Course Format: Enrollment is restricted; see the Introduction to Courses and Curricula section of the catalog. Independent study, weekly meeting with faculty. 


Grading option: Course may be repeated for credit. Must be taken on a passed/not passed basis. 


Prerequisites: Restricted to freshmen and sophomores only. Consent of instructor. 


Description: Supervised independent study by academically superior, lower division students. 3.3 GPA required and prior consent of instructor who is to supervise the study. A written proposal must be submitted to the department chair for preapproval. 


(F,SP) Staff 
Upper Division Courses


C103  Introduction to Mathematical Economics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 


Description: Selected topics illustrating the application of mathematics to economic theory. This course is intended for upperdivision students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required. Also listed as Economics C103. 


Staff 




104  Introduction to Analysis [4 units] 


Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week. 


Prerequisites: 53 and 54. 


Description: The real number system. Sequences, limits, and continuous functions in R and R^{n}. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral. 


(F,SP) Staff 




H104  Introduction to Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems. 




105  Second Course in Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Differential calculus in R^{n}: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L^{2} theory. Fubini's theorem, change of variable. 


(SP) 




110  Linear Algebra [4 units] 


Course Format: Three hours of lecture per week and an additional two hours of discussion at the discretion of the instructor. 


Prerequisites: 54 or a course with equivalent linear algebra content. 


Credit option: No credit allowed after completion of Math 112 or 113B. 


Description: Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals. 


(F,SP) Staff 




H110  Linear Algebra [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 54 or a course with equivalent linear algebra content. 


Credit option: No credit allowed after completion of Math 112 or 113B. 


Description: Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems. 


(SP) 




113  Introduction to Abstract Algebra [4 units] 


Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week. 


Prerequisites: 54 or a course with equivalent linear algebra content. 


Description: Sets and relations. The integers, congruences and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions. 


(F,SP) Staff 




H113  Introduction to Abstract Algebra [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 54 or a course with equivalent linear algebra content. 


Description: Honors section corresponding to 113. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems. 


(F) 




114  Second Course in Abstract Algebra [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 110 and 113, or consent of instructor. 


Description: Further topics on groups, rings, and fields not covered in Math 113. Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields. 


(SP) Staff 




115  Introduction to Number Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems. 


(F,SP) 




116  Cryptography [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 55. 


Description: Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications. 


(F,SP) 




118  Wavelets and Signal Processing [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to onedimensional signals and multidimensional images. 


(F,SP) 




121A  Mathematical Tools for the Physical Sciences [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Rapid review of series and partial differentiation, complex variables and analytic functions, integral transforms, calculus of variations. 




121B  Mathematical Tools for the Physical Sciences [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Special functions, series solutions of ordinary differential equations, partial differential equations arising in mathematical physics, probability theory. 




123  Ordinary Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, SturmLiouville Theory. 


(F) 




125A  Mathematical Logic [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 113 or consent of instructor. 


Description: Sentential and quantificational logic. Formal grammar, semantical interpretation, formal deduction, and their interrelation. Applications to formalized mathematical theories. Selected topics from model theory or proof theory. 


(F,SP) 




126  Introduction to Partial Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53 and 54. 


Description: Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform. 


(SP) 




127  Mathematical and Computational Methods in Molecular Biology [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53, 54, and 55; Statistics 20 recommended. 


Description: Introduction to mathematical and computational problems arising in the context of molecular biology. Theory and applications of combinatorics, probability, statistics, geometry, and topology to problems ranging from sequence determination to structure analysis. 


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128A  Numerical Analysis [4 units] 


Course Format: Three hours of lecture and one hour of discussion per week. At the discretion of instructor, an additional hour of discussion/computer laboratory per week. 


Prerequisites: 53 and 54. 


Description: Programming for numerical calculations, roundoff error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer. 


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128B  Numerical Analysis [4 units] 


Course Format: Three hours of lecture and one hour of discussion per week. At the discretion of the instructor, an additional hour of discussion/computer laboratory per week. 


Prerequisites: 110 and 128A. 


Description: Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer. 


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130  The Classical Geometries [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 110 and 113. 


Description: A critical examination of Euclid's Elements; ruler and compass constructions; connections with Galois theory; Hilbert's axioms for geometry, theory of areas, introduction of coordinates, nonEuclidean geometry, regular solids, projective geometry. 


(F,SP) 




135  Introduction to the Theory of Sets [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 113 and 104. 


Description: Settheoretical paradoxes and means of avoiding them. Sets, relations, functions, order and wellorder. Proof by transfinite induction and definitions by transfinite recursion. Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences. 


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136  Incompleteness and Undecidability [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53, 54, and 55. 


Description: Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, manyone reductions. Selfreferential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories. 


(F,SP) 




140  Metric Differential Geometry [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the GaussBonnetVon Dyck Theorem. 


(F,SP) 




141  Elementary Differential Topology [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104 or equivalent and linear algebra. 


Description: Manifolds in ndimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact onemanifolds, transversality and intersection modulo 2. 




142  Elementary Algebraic Topology [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104 and 113. 


Description: The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, plus further topics at the discretion of the instructor. 


(F) 




143  Elementary Algebraic Geometry [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 113. 


Description: Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties. 


(F,SP) 




151  Mathematics of the Secondary School Curriculum I [4 units] 


Course Format: Three hours of lecture and zero to one hour of discussion per week. 


Prerequisites: 1A1B, 53, or equivalent. 


Description: Theory of rational numbers based on the number line, the Euclidean algorithm and fractions in lowest terms. The concepts of congruence and similarity, equation of a line, functions, and quadratic functions. 


(F,SP) Staff 




152  Mathematics of the Secondary School Curriculum II [4 units] 


Course Format: Three hours of lecture and zero to one hour of discussion per week. 


Prerequisites: 151; 54, 113, or equivalent. 


Description: Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry. 


(F,SP) Staff 




153  Mathematics of the Secondary School Curriculum III [4 units] 


Course Format: Three hours of lecture and zero to one hour of discussion per week. 


Prerequisites: 151, 152. 


Description: The real line and least upper bound, limit and decimal expansion of a number, differentiation and integration, Fundamental Theorem of Calculus, characterizations of sine, cosine, exp, and log. 


(F,SP) Staff 




160  History of Mathematics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53, 54, and 113. 


Description: History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history. 


(SP) 




170  Mathematical Methods for Optimization [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 53and 54. 


Description: Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry, polyhedral geometry, the calculus of variations, and control theory. 


(F,SP) Staff 




172  Combinatorics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 55. 


Description: Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional topics at the discretion of the instructor. 


(F,SP) Staff 




185  Introduction to Complex Analysis [4 units] 


Course Format: Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week. 


Prerequisites: 104. 


Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping. 


(F,SP) Staff 




H185  Introduction to Complex Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems. 


(SP) 




189  Mathematical Methods in Classical and Quantum Mechanics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104, 110, 2 semesters lower division Physics. 


Credit option: Course may be repeated for credit. 


Description: Topics in mechanics presented from a mathematical viewpoint: e.g., hamiltonian mechanics and symplectic geometry, differential equations for fluids, spectral theory in quantum mechanics, probability theory and statistical mechanics. See department bulletins for specific topics each semester course is offered. 


(SP) 




191  Experimental Courses in Mathematics [14 units] 


Course Format: Hours to be arranged. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: The topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See departmental bulletins. 


(F,SP) 




195  Special Topics in Mathematics [4 units] 


Course Format: Hours to be arranged. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Lectures on special topics, which will be announced at the beginning of each semester that the course is offered. 




196  Honors Thesis [4 units] 


Course Format: Hours to be arranged. 


Prerequisites: Admission to the Honors Program; an overall GPA of 3.3 and a GPA of 3.5 in the major. 


Credit option: Course may be repeated for credit. 


Description: Independent study of an advanced topic leading to an honors thesis. 


(F,SP) 




197  Field Study [14 units] 


Course Format: Three hours of work per week per unit. 


Prerequisites: Upper division standing. Written proposal signed by faculty sponsor and approved by department chair. 


Credit option: Must be taken on a passed/not passed basis. 


Description: For Math/Applied math majors. Supervised experience relevant to specific aspects of their mathematical emphasis of study in offcampus organizations. Regular individual meetings with faculty sponsor and written reports required. Units will be awarded on the basis of three hours/week/unit. 


(F,SP) Staff 




198  Directed Group Study [14 units] 


Course Format: Group study. 


Prerequisites: Must have completed 60 units and be in good standing. 


Credit option: Must be taken on a passed/not passed basis. 


Description: Topics will vary with instructor. 


(F,SP) Staff 




199  Supervised Independent Study and Research [14 units] 


Course Format: Hours to be arranged. 


Prerequisites: The standard college regulations for all 199 courses. 


Grading option: Must be taken on a passed/not passed basis. 


Description: 


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Graduate Courses


202A  Introduction to Topology and Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; ArzelaAscoli and StoneWeierstrass theorems. Partitions of unity. Locally compact spaces; onepoint compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and R^{n}. Construction of the integral. Dominated convergence theorem. 


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202B  Introduction to Topology and Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 202A and 110. 


Description: Measure and integration. Product measures and Fubinitype theorems. Signed measures; Hahn and Jordan decompositions. RadonNikodym theorem. Integration on the line and in R^{n}. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. BanachSteinhaus theorem; closed graph theorem. HahnBanach theorem. Duality; the dual of L^{p}. Measures on locally compact spaces; the dual of C(X). Weak and weak* topologies; BanachAlaoglu theorem. Convexity and the KreinMilman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations. 


(F,SP) 




203  Asymptotic Analysis in Applied Mathematics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Asymptotic methods for differential equations, with emphasis upon many physical examples. Topics will include matched asymptotic expansions, Laplace's method, stationary phase, boundary layers, multiple scales, WKB approximations, asymptotic Lagrangians, bifurcation theory. 


(F,SP) 




204  Ordinary Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 104. 


Description: Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefﬁcients and Floquet Theory, Green's functions, eigenvalue problems, SturmLiouville theory, phase plane analysis, PoincareBendixon Theorem, bifurcation, chaos. 


(F,SP) 




205  Theory of Functions of a Complex Variable [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 185. 


Description: Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiplevalued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem. 




206  Banach Algebras and Spectral Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 202A202B. 


Description: Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*algebras of operators. Commutative C*algebras. Spectral theorem for bounded selfadjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. HilbertSchmidt operators. Fredholm operators. The Fredholm index. Selected additional topics. 


(F) 




208  C*algebras [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 206. 


Description: Basic theory of C*algebras. Positivity, spectrum, GNS construction. Group C*algebras and connection with group representations. Additional topics, for example, C*dynamical systems, Ktheory. 




209  Von Neumann Algebras [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 206. 


Description: Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability. 




212  Several Complex Variables [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 185 and 202A202B or their equivalents. 


Description: Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohomology and Stein manifolds, or the theory of analytic subvarieties and spaces. 




214  Differentiable Manifolds [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 202A. 


Description: Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. Morse functions, differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor. 


(F,SP) 




215A215B  Algebraic Topology [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 113 and pointset topology (e.g. 202A). 


Description: Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall. 




C218A  Probability Theory [4 units] 


Course Format: Three hours of lecture per week. 


Description: Some knowledge of real analysis and metric spaces, including compactness, Riemann integral. Knowledge of Lebesgue integral and/or elementary probability is helpful, but not essential, given otherwise strong mathematical background. Measure theory concepts needed for probability. Expectation, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations; martingales and theory convergence. Markov chains. Stationary processes. Also listed as Statistics C205A. 


Staff 




C218B  Probability Theory [4 units] 


Course Format: Three hours of lecture per week. 


Description: Some knowledge of real analysis and metric spaces, including compactness, Riemann integral. Knowledge of Lebesgue integral and/or elementary probability is helpful, but not essential, given otherwise strong mathematical background. Measure theory concepts needed for probability. Expectation, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations; martingales and theory convergence. Markov chains. Stationary processes. Also listed as Statistics C205B. 


Staff 




219  Dynamical Systems [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214. 


Description: Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor. 


(F) 




220  Introduction to Probabilistic Methods in Mathematics and the Sciences [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Some familiarity with differential equations and their applications. 


Description: Brownian motion, Langevin and FokkerPlanck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications. 


(F,SP) 




221  Advanced Matrix Computations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Description: Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions. 


(F,SP) 




222A  Partial Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 105 or 202A. 


Description: The theory of initial value and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear firstorder equations, conservation laws, HamiltonJacobi equations, Fourier transform, Sobolef spaces. 


(F) 




222B  Partial Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 105 or 202A. 


Description: The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Secondorder elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor. 


(F) 




C223A  Stochastic Processes [3 units] 


Course Format: Three hours of work per week per unit. 


Credit option: Course may be repeated for credit with a different instructor. 


Description: The content of this course changes from year to year. Course topics will be selected from the general theory of processes, sample function properties, weak convergence, Brownian motion, diffusions, Levy processes, Markov processes, martingales, Gaussian processes, and further topics. Also listed as Statistics C206A. 


(F,SP) Staff 




C223B  Stochastic Processes [3 units] 


Course Format: Three hours of work per week per unit. 


Credit option: Course may be repeated for credit with a different instructor. 


Description: The content of this course changes from year to year. Course topics will be selected from the general theory of processes, sample function properties, weak convergence, Brownian motion, diffusions, Levy processes, Markov processes, martingales, Gaussian processes, and further topics. Also listed as Statistics C206B. 


(F,SP) Staff 




224A224B  Mathematical Methods for the Physical Sciences [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Graduate status or consent of instructor. 


Description: Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and nonlinear waves. Sequence begins fall. 


(F,SP) 




225A225B  Metamathematics [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 125B and 135. 


Description: Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall. 




227A  Theory of Recursive Functions [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 225B. 


Description: Recursive and recursively enumerable sets of natural numbers; characterizations, significance, and classification. Relativization, degrees of unsolvability. The recursion theorem. Constructive ordinals, the hyperarithmetical and analytical hierarchies. Recursive objects of higher type. Sequence begins fall. 




228A228B  Numerical Solution of Differential Equations [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 128A. 


Description: Ordinary differential equations: RungeKutta and predictorcorrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations. 




229  Theory of Models [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 225B. 


Description: Syntactical characterization of classes closed under algebraic operations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than firstorder. 




235A  Theory of Sets [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 125A and 135. 


Description: Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, wellordering, and wellfounded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Settheoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall. 




236  Metamathematics of Set Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 225B and 235A. 


Description: Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity. 




239  Discrete Mathematics for the Life Sciences [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor. 


Description: Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. 


(F,SP) Staff 




240  Riemannian Geometry [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214. 


Description: Riemannian metric and LeviCivita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and CartanHadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the GaussBonnet theorem, and characteristic classes. 


(SP) 




241  Complex Manifolds [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214 and 215A. 


Description: Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical RiemannRoch theorem, theorem of AbelJacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem. 


(SP) 




242  Symplectic Geometry [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214. 


Description: Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Lengendre transform, hamiltonian systems, lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds. 


(F,SP) 




245A  General Theory of Algebraic Structures [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 113 and 135. 


Description: Structures defined by operations and/or relations, and their homomorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence and subalgebralattices, or categorytheory and adjoint functors, or other aspects. 




249  Algebraic Combinatorics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A or consent of instructor. 


Description: (I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor. 


(F,SP) Staff 




250A  Groups, Rings, and Fields [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 114 or consent of instructor. 


Description: Group theory, including the JordanHolder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree. 


(F) 




250B  Multilinear Algebra and Further Topics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A. 


Description: Tensor algebras and exterior algebras, with application to linear transformations. Commutative ideal theory, localization. Elementary specialization and valuation theory. Related topics in algebra. 


(SP) 




251  Ring Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A. 


Description: Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods. 




252  Representation Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A. 


Description: Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups. 


(F) 




253  Homological Algebra [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A. 


Description: Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules. 




254A254B  Number Theory [4;4 units] 


Course Format: Three hours of lecture per week. 


Credit option: 254B may be repeated with consent of instructor. 


Prerequisites: 250A for 254A; 254A for 254B. 


Description: Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zetafunctions and Lseries, distribution of primes, modular forms, quadratic forms, diophantine equations, Padic analysis, and transcendental numbers. Sequence begins fall. 




255  Algebraic Curves [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A250B or consent of instructor. 


Description: Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. RiemannRoch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis. 


(F,SP) 




256A256B  Algebraic Geometry [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A250B for 256A; 256A for 256B. 


Description: Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. RiemannRoch theorem and selected applications. Sequence begins fall. 




257  Group Theory [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 250A. 


Description: Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohomological methods. 




258  Classical Harmonic Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 206 or a basic knowledge of real, complex, and linear analysis. 


Description: Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor. 




261A261B  Lie Groups [4;4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214. 


Description: Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semisimple Lie groups; classification theory and representation theory of semisimple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall. 




265  Differential Topology [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: 214 plus 215A or some familiarity with algebraic topology. 


Description: Approximations, degrees of maps, vector bundles, tubular neighborhoods. Introduction to Morse theory, handlebodies, cobordism, surgery. Additional topics selected by instructor from: characteristic classes, classification of manifolds, immersions, embeddings, singularities of maps. 




270  Hot Topics Course in Mathematics [2 units] 


Course Format: One and onehalf hours of lecture per week. 


Credit option: Course may be repeated for credit as topic varies. Must be taken on a satisfactory/unsatisfactory basis. 


Description: This course will give introductions to current research developments. Every semester we will pick a different topic and go through the relevant literature. Each student will be expected to give one presentation. 


(F,SP) Staff 




273  Topics in Numerical Analysis 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 


(F,SP) 




273F  Topics in Computational Physics [4 units] 




273I  Approximation Theory [4 units] 




274  Topics in Algebra [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




275  Topics in Applied Mathematics [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




276  Topics in Topology [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




277  Topics in Differential Geometry [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




278  Topics in Analysis [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




279  Topics in Partial Differential Equations [4 units] 


Course Format: Three hours of lecture per week. 


Prerequisites: Consent of instructor. 


Credit option: Course may be repeated for credit. 


Description: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars. 




290  Seminars [16 units] 


Course Format: Hours to be arranged. 


Credit option: Course may be repeated for credit. 


Description: Topics in foundations of mathematics, theory of numbers, numerical calculations, analysis, geometry, topology, algebra, and their applications, by means of lectures and informal conferences; work based largely on original memoirs. 


(F,SP) 




295  Individual Research [112 units] 


Course Format: Hours to be arranged. 


Credit option: Course may be repeated for credit. 


Grading option: Sections 130 to be graded on a lettergrade basis. Sections 3160 to be graded on a satisfactory/unsatisfactory basis. 


Description: Intended for candidates for the Ph.D. degree. 


(F,SP) 




299  Reading Course for Graduate Students [16 units] 


Course Format: Hours to be arranged. 


Credit option: Course may be repeated for credit. 


Grading option: Sections 130 to be graded on a lettergrade basis. Sections 3160 to be graded on a satisfactory/unsatisfactory basis. 


Description: Investigation of special problems under the direction of members of the department. 


(F,SP) 
Professional Courses


300  Teaching Workshop [3 units] 


Course Format: Two hours of lecture per week, plus class visits. 


Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor. 


Grading option: Must be taken on a satisfactory/unsatisfactory basis. 


Description: Mandatory for all graduate student instructors teaching for the first time in the department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and selfanalysis of videotapes, reciprocal classroom visitations, and an individual project. 


(F,SP) Staff 




301  Undergraduate Mathematics Instruction [12 units] 


Course Format: Three hours of seminar and four hours of tutorial per week. 


Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center. 


Credit option: Course may be repeated once for credit. 


Grading option: Must be taken on a passed/not passed basis. 


Description: May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program. 


(F,SP) 




303  Professional Preparation: Supervised Teaching of Mathematics [24 units] 


Course Format: No formal meetings. 


Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor. 


Credit option: Course may be repeated four times for credit. 


Grading option: Must be taken on a satisfactory/unsatisfactory basis. 


Description: Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty. 


(F,SP) Staff 
Special Study for Master's Exam


600  Individual Study for Master's Students [16 units] 


Course Format: Hours to be arranged. 


Prerequisites: For candidates for master's degree. 


Credit option: Course may be repeated for credit. Course does not satisfy unit or residence requirements for master's degree. 


Grading option: Must be taken on a satisfactory/unsatisfactory basis. 


Description: Individual study for the comprehensive or language requirements in consultation with the field adviser. 


(F,SP) 
Special Study for Doctoral Qualifying Exam


602  Individual Study for Doctoral Students [18 units] 


Course Format: 


Prerequisites: For qualified graduate students. 


Credit option: Course may be repeated for credit. 


Grading option: Must be taken on a satisfactory/unsatisfactory basis. 


Description: Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree. 


(F,SP) Staff 