Mathematics (MATH)

Courses

MATH 1A Calculus 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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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MATH 1B Calculus 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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MATH H1B Honors Calculus 4 Units

Terms offered: Fall 2015, Fall 2014, Fall 2013
Honors version of 1B. Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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MATH 10A Methods of Mathematics: Calculus, Statistics, and Combinatorics 4 Units

Terms offered: Fall 2017, Summer 2017 8 Week Session, Fall 2016
This sequence is intended for majors in the life sciences. Introduction to differential and integral calculus of functions of one variable. Representation of data, elementary probability theory, statistical models, and testing.

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MATH 10B Methods of Mathematics: Calculus, Statistics, and Combinatorics 4 Units

Terms offered: Spring 2018, Summer 2017 8 Week Session, Spring 2017
Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.

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MATH 16A Analytic Geometry and Calculus 3 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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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MATH 16B Analytic Geometry and Calculus 3 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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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MATH 24 Freshman Seminars 1 Unit

Terms offered: Spring 2018, Fall 2017, Spring 2017
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 small-seminar setting. Berkeley Seminars are offered in all campus departments, and topics vary from department to department and semester to semester.

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MATH 32 Precalculus 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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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MATH N32 Precalculus 4 Units

Terms offered: Prior to 2007
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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MATH 39A Freshman/Sophomore Seminar 2 - 4 Units

Terms offered: Spring 2018, Spring 2010, Spring 2009
Freshman and sophomore seminars offer lower division students the opportunity to explore an intellectual topic with a faculty member and a group of peers in a small-seminar setting. These seminars are offered in all campus departments; topics vary from department to department and from semester to semester.

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MATH 49 Supplementary Work in Lower Division Mathematics 1 - 3 Units

Terms offered: Spring 2017, Spring 2016, Fall 2015
Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading.

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MATH 53 Multivariable Calculus 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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MATH H53 Honors Multivariable Calculus 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
Honors version of 53. Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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MATH W53 Multivariable Calculus 4 Units

Terms offered: Summer 2017 8 Week Session, Summer 2016 10 Week Session, Summer 2016 8 Week Session
Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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MATH 54 Linear Algebra and Differential Equations 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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MATH H54 Honors Linear Algebra and Differential Equations 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2016
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; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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MATH 55 Discrete Mathematics 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

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MATH 74 Transition to Upper Division Mathematics 3 Units

Terms offered: Spring 2009, Fall 2008, Summer 2008 8 Week Session
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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MATH 91 Special Topics in Mathematics 4 Units

Terms offered: Spring 2016, Fall 2012, Spring 2012
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.

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MATH 96 College Algebra 2 Units

Terms offered: Summer 2017 8 Week Session, Summer 2015 10 Week Session, Summer 2014 10 Week Session
Elements of college algebra. Designed for students who do not meet the prerequisites for 32. Offered through the Student Learning Center.

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MATH 98 Supervised Group Study 1 - 4 Units

Terms offered: Fall 2017, Spring 2017, Fall 2016
Directed Group Study, topics vary with instructor.

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MATH 98BC Berkeley Connect 1 Unit

Terms offered: Spring 2018, Fall 2017, Spring 2017
Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department
faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.
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MATH 99 Supervised Independent Study 1 - 4 Units

Terms offered: Spring 2017, Spring 2016, Fall 2015
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 pre-approval.

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MATH C103 Introduction to Mathematical Economics 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2016
Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required.

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MATH 104 Introduction to Analysis 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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MATH H104 Honors Introduction to Analysis 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.

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MATH 105 Second Course in Analysis 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Differential calculus in Rn: 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, L2 theory. Fubini's theorem, change of variable.

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MATH 110 Linear Algebra 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.

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MATH H110 Honors Linear Algebra 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

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MATH 113 Introduction to Abstract Algebra 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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.

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MATH H113 Honors Introduction to Abstract Algebra 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Honors section corresponding to 113. Recommended for students who enjoy mathematics and are willing to work hard in order to understand the beauty of mathematics and its hidden patterns and structures. Greater emphasis on theory and challenging problems.

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MATH 114 Second Course in Abstract Algebra 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
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.

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MATH 115 Introduction to Number Theory 4 Units

Terms offered: Spring 2018, Summer 2017 8 Week Session, Spring 2017
Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

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MATH 116 Cryptography 4 Units

Terms offered: Fall 2015, Fall 2014, Fall 2013
Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.

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MATH 118 Fourier Analysis, Wavelets, and Signal Processing 4 Units

Terms offered: Fall 2017, Spring 2017, Spring 2016
Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images.

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MATH 121A Mathematical Tools for the Physical Sciences 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2016
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.

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MATH 121B Mathematical Tools for the Physical Sciences 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
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.

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MATH 123 Ordinary Differential Equations 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2016
Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory.

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MATH 125A Mathematical Logic 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
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.

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MATH 126 Introduction to Partial Differential Equations 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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.

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MATH 127 Mathematical and Computational Methods in Molecular Biology 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2016
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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MATH 128A Numerical Analysis 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.

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MATH 128B Numerical Analysis 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
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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MATH 130 The Classical Geometries 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
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, non-Euclidean geometry, regular solids, projective geometry.

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MATH 135 Introduction to the Theory of Sets 4 Units

Terms offered: Fall 2017, Spring 2017, Spring 2016
Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. 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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MATH 136 Incompleteness and Undecidability 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.

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MATH 140 Metric Differential Geometry 4 Units

Terms offered: Fall 2017, Spring 2017, Fall 2015
Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet-Von Dyck Theorem.

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MATH 141 Elementary Differential Topology 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2016
Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2.

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MATH 142 Elementary Algebraic Topology 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
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.

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MATH 143 Elementary Algebraic Geometry 4 Units

Terms offered: Spring 2018, Fall 2016, Fall 2015
Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.

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MATH 151 Mathematics of the Secondary School Curriculum I 4 Units

Terms offered: Spring 2017, Spring 2016, Spring 2015
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.

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MATH 152 Mathematics of the Secondary School Curriculum II 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry.

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MATH 153 Mathematics of the Secondary School Curriculum III 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
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.

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MATH 160 History of Mathematics 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.

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MATH 170 Mathematical Methods for Optimization 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
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.

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MATH 172 Combinatorics 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
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.

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MATH 185 Introduction to Complex Analysis 4 Units

Terms offered: Spring 2018, Fall 2017, Summer 2017 8 Week Session
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.

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MATH H185 Honors Introduction to Complex Analysis 4 Units

Terms offered: Spring 2018, Spring 2016, Spring 2015
Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

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MATH 189 Mathematical Methods in Classical and Quantum Mechanics 4 Units

Terms offered: Fall 2015, Fall 2014, Fall 2013
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.

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MATH 191 Experimental Courses in Mathematics 1 - 4 Units

Terms offered: Spring 2018, Fall 2017, Spring 2017
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.

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MATH 195 Special Topics in Mathematics 4 Units

Terms offered: Spring 2011, Spring 2004, Spring 2003
Lectures on special topics, which will be announced at the beginning of each semester that the course is offered.

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MATH 196 Honors Thesis 4 Units

Terms offered: Spring 2017, Spring 2016, Spring 2015
Independent study of an advanced topic leading to an honors thesis.

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MATH 197 Field Study 1 - 4 Units

Terms offered: Spring 2016, Spring 2015, Spring 2014
For Math/Applied math majors. Supervised experience relevant to specific aspects of their mathematical emphasis of study in off-campus organizations. Regular individual meetings with faculty sponsor and written reports required. Units will be awarded on the basis of three hours/week/unit.

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MATH 198 Directed Group Study 1 - 4 Units

Terms offered: Spring 2017, Fall 2016, Spring 2016
Topics will vary with instructor.

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MATH 198BC Berkeley Connect 1 Unit

Terms offered: Spring 2018, Fall 2017, Spring 2017
Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department
faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.
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MATH 199 Supervised Independent Study and Research 1 - 4 Units

Terms offered: Fall 2017, Spring 2017, Fall 2016

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MATH 202A Introduction to Topology and Analysis 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
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; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure
on the line and Rn. Construction of the integral. Dominated convergence theorem.
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MATH 202B Introduction to Topology and Analysis 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact
spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.
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MATH 203 Asymptotic Analysis in Applied Mathematics 4 Units

Terms offered: Fall 2011, Spring 2011, Spring 2010
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.

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MATH 204 Ordinary Differential Equations 4 Units

Terms offered: Fall 2016, Spring 2016, Fall 2014
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, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.

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MATH 205 Theory of Functions of a Complex Variable 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2015
Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued 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.

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MATH 206 Banach Algebras and Spectral Theory 4 Units

Terms offered: Fall 2016, Fall 2015, Fall 2014
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 self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Hilbert-Schmidt operators. Fredholm operators. The
Fredholm index. Selected additional topics.
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MATH 208 C*-algebras 4 Units

Terms offered: Spring 2018, Spring 2015, Spring 2013
Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.

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MATH 209 Von Neumann Algebras 4 Units

Terms offered: Spring 2017, Spring 2014, Spring 2012
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.

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MATH 212 Several Complex Variables 4 Units

Terms offered: Spring 2016, Fall 2014, Spring 2012
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.

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MATH 214 Differentiable Manifolds 4 Units

Terms offered: Fall 2017, Spring 2017, Fall 2015
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.

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MATH 215A Algebraic Topology 4 Units

Terms offered: Fall 2017, Spring 2017, Fall 2015
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.

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MATH 215B Algebraic Topology 4 Units

Terms offered: Spring 2016, Spring 2015, Spring 2014
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.

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MATH C218A Probability Theory 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
The course is designed as a sequence with Statistics C205B/Mathematics C218B with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion.

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MATH C218B Probability Theory 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion.

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MATH 219 Dynamical Systems 4 Units

Terms offered: Spring 2018, Fall 2016, Spring 2015
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.

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MATH 220 Introduction to Probabilistic Methods in Mathematics and the Sciences 4 Units

Terms offered: Spring 2012, Spring 2011, Spring 2010
Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications.

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MATH 221 Advanced Matrix Computations 4 Units

Terms offered: Spring 2018, Fall 2016, Spring 2016
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.

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MATH 222A Partial Differential Equations 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces.

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MATH 222B Partial Differential Equations 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor.

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MATH C223A Advanced Topics in Probability and Stochastic Process 3 Units

Terms offered: Fall 2016, Fall 2015, Fall 2014, Fall 2013
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.

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MATH C223B Advanced Topics in Probability and Stochastic Processes 3 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.

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MATH 224A Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Fall 2016, Fall 2014, Fall 2013
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 non-linear waves. Sequence begins fall.

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MATH 224B Mathematical Methods for the Physical Sciences 4 Units

Terms offered: Spring 2015, Spring 2014, Spring 2013
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 non-linear waves. Sequence begins fall.

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MATH 225A Metamathematics 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
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.

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MATH 225B Metamathematics 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
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.

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MATH 227A Theory of Recursive Functions 4 Units

Terms offered: Fall 2015, Fall 2013, Spring 2012
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.

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MATH 228A Numerical Solution of Differential Equations 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Ordinary differential equations: Runge-Kutta and predictor-corrector 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.

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MATH 228B Numerical Solution of Differential Equations 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Ordinary differential equations: Runge-Kutta and predictor-corrector 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.

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MATH 229 Theory of Models 4 Units

Terms offered: Spring 2015, Spring 2013, Spring 2012
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 first-order.

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MATH 235A Theory of Sets 4 Units

Terms offered: Spring 2014, Fall 2011, Spring 2008
Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall.

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MATH 236 Metamathematics of Set Theory 4 Units

Terms offered: Fall 2014, Fall 2010, Spring 2009
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.

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MATH 239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2011, Fall 2008, Spring 2008
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.

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MATH C239 Discrete Mathematics for the Life Sciences 4 Units

Terms offered: Spring 2013
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.

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MATH 240 Riemannian Geometry 4 Units

Terms offered: Fall 2016, Fall 2015, Fall 2014
Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.

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MATH 241 Complex Manifolds 4 Units

Terms offered: Fall 2017, Fall 2014, Spring 2013
Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem.

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MATH 242 Symplectic Geometry 4 Units

Terms offered: Fall 2017, Fall 2015, Spring 2014
Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds.

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MATH C243 Seq: Methods and Applications 3 Units

Terms offered: Spring 2015, Spring 2014
A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof.

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MATH 245A General Theory of Algebraic Structures 4 Units

Terms offered: Fall 2017, Fall 2015, Spring 2014
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 subalgebra-lattices, or category-theory and adjoint functors, or other aspects.

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MATH 249 Algebraic Combinatorics 4 Units

Terms offered: Fall 2017, Fall 2016, Spring 2015
(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.

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MATH 250A Groups, Rings, and Fields 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Group theory, including the Jordan-Holder 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.

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MATH 250B Multilinear Algebra and Further Topics 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Tensor algebras and exterior algebras, with application to linear transformations. Commutative ideal theory, localization. Elementary specialization and valuation theory. Related topics in algebra.

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MATH 251 Ring Theory 4 Units

Terms offered: Fall 2016, Spring 2013, Fall 2009
Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods.

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MATH 252 Representation Theory 4 Units

Terms offered: Fall 2015, Fall 2014, Fall 2013
Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.

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MATH 253 Homological Algebra 4 Units

Terms offered: Fall 2016, Fall 2014, Summer 2014 10 Week Session
Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules.

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MATH 254A Number Theory 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.

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MATH 254B Number Theory 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2015
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.

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MATH 255 Algebraic Curves 4 Units

Terms offered: Fall 2014, Fall 2011, Spring 2009
Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch 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.

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MATH 256A Algebraic Geometry 4 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

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MATH 256B Algebraic Geometry 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

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MATH 257 Group Theory 4 Units

Terms offered: Spring 2018, Spring 2014, Fall 2011
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.

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MATH 258 Harmonic Analysis 4 Units

Terms offered: Fall 2016, Spring 2015, Spring 2012
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.

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MATH 261A Lie Groups 4 Units

Terms offered: Spring 2017, Fall 2015, Fall 2013
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple 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.
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MATH 261B Lie Groups 4 Units

Terms offered: Fall 2017, Spring 2016, Spring 2014
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple 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.
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MATH 265 Differential Topology 4 Units

Terms offered: Spring 2011, Fall 2008, Fall 2004
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.

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MATH 270 Hot Topics Course in Mathematics 2 Units

Terms offered: Spring 2018, Fall 2017, Spring 2017
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.

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MATH 273 Topics in Numerical Analysis 4 Units

Terms offered: Spring 2016, Spring 2014
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 274 Topics in Algebra 4 Units

Terms offered: Spring 2018, Spring 2017, Fall 2016
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 275 Topics in Applied Mathematics 4 Units

Terms offered: Spring 2017, Spring 2014, Fall 2013
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 276 Topics in Topology 4 Units

Terms offered: Fall 2017, Spring 2016, Spring 2015
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 277 Topics in Differential Geometry 4 Units

Terms offered: Spring 2018, Spring 2017, Spring 2016
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 278 Topics in Analysis 4 Units

Terms offered: Spring 2018, Fall 2015, Spring 2015
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 279 Topics in Partial Differential Equations 4 Units

Terms offered: Fall 2017, Spring 2017, Spring 2016
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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MATH 290 Seminars 1 - 6 Units

Terms offered: Spring 2017, Spring 2015, Fall 2014
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.

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MATH 295 Individual Research 1 - 12 Units

Terms offered: Summer 2016 10 Week Session, Spring 2016, Fall 2015
Intended for candidates for the Ph.D. degree.

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MATH 299 Reading Course for Graduate Students 1 - 6 Units

Terms offered: Fall 2017, Fall 2016, Fall 2015
Investigation of special problems under the direction of members of the department.

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MATH 301 Undergraduate Mathematics Instruction 1 - 2 Units

Terms offered: Fall 2017, Spring 2017, Fall 2016
May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.

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MATH 302 Teaching Workshop 1 Unit

Terms offered: Summer 2002 10 Week Session, Summer 2001 10 Week Session
Mandatory for all graduate student instructors teaching summer course for the first time in the Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis, classroom visitations by senior faculty member.

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MATH 303 Professional Preparation: Supervised Teaching of Mathematics 2 - 4 Units

Terms offered: Spring 2017, Spring 2016, Fall 2015
Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty.

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MATH 375 Teaching Workshop 4 Units

Terms offered: Spring 2018, Fall 2017, Spring 2017
Mandatory for all graduate student instructors teaching for the first time in the Mathematics Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis of videotapes, reciprocal classroom visitations, and an individual project.

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MATH 600 Individual Study for Master's Students 1 - 6 Units

Terms offered: Summer 2006 10 Week Session, Fall 2005, Spring 2005
Individual study for the comprehensive or language requirements in consultation with the field adviser.

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MATH 602 Individual Study for Doctoral Students 1 - 8 Units

Terms offered: Fall 2016, Fall 2015, Spring 2015
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.

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