Mathematics
College of Letters and Science
Department Office: 970 Evans Hall, (510) 642-6550
Chair: Arthur E. Ogus, PhD
Department Website: Mathematics
Majors
The department offers undergraduate major programs in mathematics and applied mathematics leading to the BA degree. These programs provide excellent preparation for advanced degrees in math, physical sciences, economics, and industrial engineering, as well as graduate study in business, education, law, and medicine. They also prepare students for post-baccalaureate positions in business, technology, industry, teaching, government, and finance. The requirements for both majors are summarized below. See the Department of Mathematics website for more information.
Students should contact an undergraduate adviser in 964 or 965 Evans Hall about requirements for admission to the major.
General Major Requirements
Both major programs require a lower-division base of Mathematics 1A-1B, 53, 54, and 55. Courses Math 16A-16B are not an acceptable alternative to Math 1A-1B. Math 1A-1B must be completed with an average grade of C or better; Math 53, 54, and 55 must be completed with minimum grades of C in each. Eight upper-division courses are required for either major. Specific course requirements follow.
Major in Mathematics
- Four core courses 104, 110, 113 and 185
- Two semi-electives: select one course from each of two of the following three subject areas: I. Computing (128A); II. Geometry (130, 140, 141, 142, 143); III. Logic and foundations (125A, 135, 136)
- Two upper division math electives. With the approval of the major adviser, students may count two mathematically theoretical courses in computer science, statistics, physics, astronomy, mathematical economics, or other sciences toward requirements for the major in mathematics.
Major in Mathematics with a Teaching Concentration
The new teaching concentration is designed to increase the number and quality of math teachers. It requires the completion of three new courses, Math 151, 152, and 153, and includes a modification to the typical major course sequence. Please see the Mathematics Department website for more information.
Major in Applied Mathematics
- 104, 110, 113, 128A, and 185
- Three additional upper division courses, approved by a major adviser, which form a coherent cluster in some applied area such as actuarial science, classical mechanics, computer science, economics, fluid mechanics, geophysics, mathematical biology, numerical analysis, operations research, probability theory, quantum mechanics, statistics, systems theory. Many other clusters are also possible.
Honors Program
In addition to completing the requirements for the major in mathematics or applied mathematics, students in the honors program must:
- Earn a GPA of at least 3.5 in upper division and graduate courses in the major and at least 3.3 in all courses taken at the University
- Complete either Math 196, in which they will write a senior honors thesis, or pass two graduate mathematics courses with a grade of at least A-
- Receive the recommendation of the Head Adviser.
Students interested in the honors program should consult with an adviser early in their program, preferably by their junior year.
The Minor Program
Students in the College of Letters and Science may complete one or more minors of their choice, normally in a field both academically and administratively distinct from their major. The minor program in the Department of Mathematics consists of the following coursework:
Prerequisites
Mathematics 1A-1B and 53 and 54 (or their equivalents). These courses must be taken for a letter grade and must be passed with average grades of C or better.
Minor Requirements
Mathematics 104, 110, 113, and 185, plus one additional upper division mathematics course. These five courses must each be taken for a letter grade, and a minimum GPA of 2.0 is required for upper division courses applied to the minor program. At least three of the five courses must be completed at Berkeley. One upper division class from your minor may overlap with your major.
For more information about this program, please contact an undergraduate adviser in 964 or 965 Evans Hall.
Preparation for Graduate Study
Students preparing for the PhD in mathematics are strongly advised to acquire a reading knowledge of one foreign language from among French, German, and Russian. Undergraduate students also often take one or more of the following introductory graduate courses: 202A-202B, 214, 225A-225B, 228A-228B, 250A-250B.
Graduate Programs
The department offers the MA degree in mathematics and PhD degrees in mathematics and applied mathematics. Detailed information concerning admission, graduate student instructorships and fellowships, and degree requirements is given in the Graduate Announcement of the Department of Mathematics, which is available online here .
MATH 1A Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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 one-half 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 UC/CSU math diagnostic test, or 32. Consult the mathematics department for details. Students with AP credit should consider choosing a course more advanced than 1A.
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.
Students will receive no credit for 1A after taking 16B and 2 units after taking 16A.
MATH 1B Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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.
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.
Students will receive 2 units of credit for 1B after taking 16B.
MATH H1B Honors Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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.
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.
Students will receive 2 units of credit for H1B after taking 16B.
MATH 10A Methods of Mathematics: Calculus, Statistics, and Combinatorics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of Lecture and 3 hours of Discussion per week for 15 weeks. 5 hours of Lecture and 5 hours of Discussion per week for 8 weeks.
Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry.
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.
Students will receive 2 units for 10A after taking 1A.
MATH 10B Methods of Mathematics: Calculus, Statistics, and Combinatorics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of Lecture and 3 hours of Discussion per week for 15 weeks. 5 hours of Lecture and 5 hours of Discussion per week for 8 weeks.
Prerequisites: Continuation of 10A.
Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.
Students will receive 2 units for 10B after taking 55.
MATH 16A Analytic Geometry and Calculus 3 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 2 hours of lecture and 1 hour of discussion/workshop per week; at the discretion of the instructor, an additional 1 hour to 1.5 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 UC/CSU math diagnostic exam, or 32. Consult the mathematics department for details.
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.
Students will receive no credit for 16A after taking 1A. Two units of 16A may be used to remove a deficient grade in 1A.
MATH 16B Analytic Geometry and Calculus 3 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 2 hours of lecture and 1 hour of discussion/workshop per week; at the discretion of the instructor, an additional hour of lecture or discussion/workshop per week.
Prerequisites: 16A.
Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization.
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.
MATH 24 Freshman Seminars 1 Unit
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: The grading option will be decided by the instructor when the class is offered.
Hours and format: 1 hour of Seminar per week for 15 weeks.
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.
Course may be repeated for credit as topic varies. Course may be repeated for credit when topic changes.
MATH 32 Precalculus 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 2 hours of lecture and 2 hours of discussion per week, plus, at the instructor's option, an extra hour of lecture/discussion per week. 5 hours of lecture and 5 hours of discussion per week for 8 weeks. 5 hours of lecture and 5 hours of discussion for 6 weeks, 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 UC/CSU diagnostic examination.
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.
Students will receive no credit for 32 after taking 1A-1B or 16A-16B and will receive 3 units after taking 96.
MATH 39A Freshman/Sophomore Seminar 2 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: Seminar format.
Prerequisites: Priority given to freshmen and sophomores.
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.
Course may be repeated for credit when topic changes.
MATH 49 Supplementary Work in Lower Division Mathematics 1 - 3 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: Meetings to be arranged.
Prerequisites: Some units in a lower division Mathematics class.
Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 53 Multivariable Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
Prerequisites: Mathematics 1B.
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.
Students will receive no credit for Mathematics 53 after completing Mathematics W53, 53M; 3 units for Mathematics 50A and 1 unit for Mathematics 50B. A deficient grade in 53 may be removed by completing Mathematics W53.
MATH H53 Honors Multivariable Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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.
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.
MATH W53 Multivariable Calculus 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Summer
Grading: Letter grade.
Hours and format: 5 hours of web-based lecture and 5 hours of web-based discussion per week for 8 weeks. This is an online course.
Prerequisites: Mathematics 1B or equivalent.
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.
Students will receive no credit for Mathematics W53 after taking Mathematics 53. A deficient grade in Mathematics W53 may be removed by completing Mathematics 53.<BR/> Instructor: Hutchings
MATH 54 Linear Algebra and Differential Equations 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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.
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.
MATH H54 Honors Linear Algebra and Differential Equations 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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.
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.
MATH 55 Discrete Mathematics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 2 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. 1A-1B recommended.
Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.
Students will receive no credit for 55 after taking Computer Science 70.
MATH 74 Transition to Upper Division Mathematics 3 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 6 hours of lecture and at the discretion of the instructor and additional 2 hours of discussion per week for 8 weeks.
Prerequisites: 53 and 54.
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.
MATH 91 Special Topics in Mathematics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 6 hours of lecture/discussion per week for 8 weeks. 3 hours of lecture/discussion per week.
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.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 96 College Algebra 2 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Summer
Grading: Letter grade.
Hours and format: 4 hours of Workshop per week for 15 weeks. 10 hours of Workshop per week for 8 weeks. 10 hours of Workshop per week for 6 weeks.
Elements of college algebra. Designed for students who do not meet the prerequisites for 32. Offered through the Student Learning Center.
Students will receive no credit for 96 after taking P, PS, or 32. Course may be repeated for credit when topic changes.
MATH 98 Supervised Group Study 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Offered for pass/not pass grade only.
Hours and format: 1 to 4 hour of Directed group study per week for 15 weeks. 1.5 to 7.5 hours of Directed group study per week for 8 weeks.
Directed Group Study, topics vary with instructor.
Course may be repeated for a maximum of 4 units.
MATH 98BC Berkeley Connect 1 Unit
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Offered for pass/not pass grade only.
Hours and format: 1 hour of discussion per week.
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.
Course may be repeated for credit when topic changes.
MATH 99 Supervised Independent Study 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Offered for pass/not pass grade only.
Hours and format: Independent study, weekly meeting with faculty. Independent study, weekly meeting with faculty.
Prerequisites: Restricted to freshmen and sophomores only. Consent of instructor.
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.
Course may be repeated for credit. Course may be repeated for credit when topic changes. Enrollment is restricted; see the Introduction to Courses and Curricula section of this catalog.
MATH C103/ECON C103 Introduction to Mathematical Economics 4 Units
Department: Mathematics; Economics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Math 53 and 54.
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.
Formerly known as 103.
MATH 104 Introduction to Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week. 6 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week.
Prerequisites: 53 and 54.
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.
MATH H104 Honors Introduction to Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.
MATH 105 Second Course in Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
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.
MATH 110 Linear Algebra 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week and an additional 2 hours of discussion at the discretion of the instructor. 6 hours of lecture per week and an additional 2 hours of discussion at the discretion of the instructor.
Prerequisites: 54 or a course with equivalent linear algebra content.
Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.
MATH H110 Honors Linear Algebra 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 54 or a course with equivalent linear algebra content.
Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
MATH 113 Introduction to Abstract Algebra 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week. 6 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week.
Prerequisites: 54 or a course with equivalent linear algebra content.
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.
MATH H113 Honors Introduction to Abstract Algebra 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 54 or a course with equivalent linear algebra content.
Honors section corresponding to 113. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.
MATH 114 Second Course in Abstract Algebra 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 110 and 113, or consent of instructor.
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.
MATH 115 Introduction to Number Theory 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week, and at the discretion of the instructor, an additional 2 hours of discussion per week. 6 hours of lecture per week, and at the discretion of the instructor, an additional 4 hours of discussion per week.
Prerequisites: 53 and 54.
Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.
MATH 116 Cryptography 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week, and at the discretion of the instructor, an additional 2 hours of discussion per week. 6 hours of lecture per week, and at the discretion of the instructor, an additional 4 hours of discussion per week.
Prerequisites: 55.
Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.
MATH 118 Fourier Analysis, Wavelets, and Signal Processing 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images.
MATH 121A Mathematical Tools for the Physical Sciences 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
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.
MATH 121B Mathematical Tools for the Physical Sciences 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
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.
MATH 123 Ordinary Differential Equations 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory.
MATH 125A Mathematical Logic 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Math 113 or consent of instructor.
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.
MATH 126 Introduction to Partial Differential Equations 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
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.
MATH 127 Mathematical and Computational Methods in Molecular Biology 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53, 54, and 55; Statistics 20 recommended.
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.
MATH 128A Numerical Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 1 hour of discussion per week. At the discretion of instructor, an additional hour of discussion/computer laboratory per week.
Prerequisites: 53 and 54.
Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.
MATH 128B Numerical Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture and 1 hour of discussion per week. At the discretion of the instructor, an additional hour of discussion/computer laboratory per week.
Prerequisites: 110 and 128A.
Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer.
MATH 130 The Classical Geometries 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 110 and 113.
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.
MATH 135 Introduction to the Theory of Sets 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 113 and 104.
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.
MATH 136 Incompleteness and Undecidability 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53, 54, and 55.
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.
MATH 140 Metric Differential Geometry 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
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.
MATH 141 Elementary Differential Topology 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104 or equivalent and linear algebra.
Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2.
MATH 142 Elementary Algebraic Topology 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104 and 113.
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.
MATH 143 Elementary Algebraic Geometry 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 113.
Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.
MATH 151 Mathematics of the Secondary School Curriculum I 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of lecture and zero to 1 hour of discussion per week.
Prerequisites: 1A-1B, 53, or equivalent.
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.
MATH 152 Mathematics of the Secondary School Curriculum II 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of lecture and zero to 1 hour of discussion per week.
Prerequisites: 151; 54, 113, or equivalent.
Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry.
MATH 153 Mathematics of the Secondary School Curriculum III 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of lecture and zero to 1 hour of discussion per week.
Prerequisites: 151, 152.
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.
MATH 160 History of Mathematics 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53, 54, and 113.
History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.
MATH 170 Mathematical Methods for Optimization 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 53 and 54.
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.
MATH 172 Combinatorics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 55.
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.
MATH 185 Introduction to Complex Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week. 6 hours of lecture per week; at the discretion of the instructor, an additional 2 hours of discussion per week.
Prerequisites: 104.
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.
MATH H185 Honors Introduction to Complex Analysis 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
MATH 189 Mathematical Methods in Classical and Quantum Mechanics 4 Units
Department: Mathematics
Course level: Undergraduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104, 110, 2 semesters lower division Physics.
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.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 191 Experimental Courses in Mathematics 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: Hours to be arranged. Hours to be arranged.
Prerequisites: Consent of instructor.
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.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 195 Special Topics in Mathematics 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: Hours to be arranged.
Prerequisites: Consent of instructor.
Lectures on special topics, which will be announced at the beginning of each semester that the course is offered.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 196 Honors Thesis 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and 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.
Independent study of an advanced topic leading to an honors thesis.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 197 Field Study 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Offered for pass/not pass grade only.
Hours and format: 5.5 hours of work per week per unit. 3 hours of work per week per unit.
Prerequisites: Upper division standing. Written proposal signed by faculty sponsor and approved by department chair.
For Math/Applied math majors. Supervised experience relevant to specific aspects of their mathmatical 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.
Course may be repeated for credit. Course may be repeated for credit when topic changes.
MATH 198 Directed Group Study 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Offered for pass/not pass grade only.
Hours and format: Group study. Group study.
Prerequisites: Must have completed 60 units and be in good standing.
Topics will vary with instructor.
MATH 198BC Berkeley Connect 1 Unit
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall and spring
Grading: Offered for pass/not pass grade only.
Hours and format: 1 hour of discussion per week.
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.
Course may be repeated for credit when topic changes.
MATH 199 Supervised Independent Study and Research 1 - 4 Units
Department: Mathematics
Course level: Undergraduate
Terms course may be offered: Fall, spring and summer
Grading: Offered for pass/not pass grade only.
Hours and format: Hours to be arranged.
Prerequisites: The standard college regulations for all 199 courses.
Course may be repeated for credit when topic changes.
MATH 202A Introduction to Topology and Analysis 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
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.
MATH 202B Introduction to Topology and Analysis 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 202A and 110.
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.
MATH 203 Asymptotic Analysis in Applied Mathematics 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
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.
MATH 204 Ordinary Differential Equations 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 104.
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.
MATH 205 Theory of Functions of a Complex Variable 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 185.
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.
MATH 206 Banach Algebras and Spectral Theory 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 202A-202B.
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.
MATH 208 C*-algebras 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 206.
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.
MATH 209 Von Neumann Algebras 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 206.
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.
MATH 212 Several Complex Variables 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 185 and 202A-202B or their equivalents.
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.
MATH 214 Differentiable Manifolds 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 202A.
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.
MATH 215A Algebraic Topology 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 113 and point-set topology (e.g. 202A).
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.
Instructors: 113C, 202A, and 214
MATH 215B Algebraic Topology 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 215A, 214 recommended (can be taken concurrently).
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.
Instructors: 113C, 202A, and 214
MATH C218A/STAT C205A Probability Theory 4 Units
Department: Mathematics; Statistics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
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.
MATH C218B/STAT C205B Probability Theory 4 Units
Department: Mathematics; Statistics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
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.
MATH 219 Dynamical Systems 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214.
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
MATH 220 Introduction to Probabilistic Methods in Mathematics and the Sciences 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Some familiarity with differential equations and their applications.
Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications.
MATH 221 Advanced Matrix Computations 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall, spring and summer
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks. 6 hours of Lecture per week for 8 weeks.
Prerequisites: Consent of instructor.
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.
MATH 222A Partial Differential Equations 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 105 or 202A.
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.
MATH 222B Partial Differential Equations 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 105 or 202A.
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.
MATH C223A/STAT C206A Advanced Topics in Probability and Stochastic Process 3 Units
Department: Mathematics; Statistics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Statistics C205A-C205B or consent of instructor.
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.
Course may be repeated for credit with a different instructor. Course may be repeated for credit when topic changes.
MATH C223B/STAT C206B Advanced Topics in Probability and Stochastic Processes 3 Units
Department: Mathematics; Statistics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
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.
Course may be repeated for credit with a different instructor. Course may be repeated for credit when topic changes.
MATH 224A Mathematical Methods for the Physical Sciences 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Graduate status or consent of instructor.
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.
Instructors: 112 or 113C; 104A and 185, or 121A-121B-121C, or 120A-120B-120C.
MATH 224B Mathematical Methods for the Physical Sciences 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Graduate status or consent of instructor.
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.
MATH 225A Metamathematics 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 125B and 135.
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.
Instructor: 125B and 135.
MATH 225B Metamathematics 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 125B and 135.
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.
Instructor: 125B and 135.
MATH 227A Theory of Recursive Functions 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring. A: (SP) B: Not offered 1984-85.
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 225B.
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.
Instructor: 225C.
MATH 228A Numerical Solution of Differential Equations 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 128A.
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.
Instructor: 128A-128B.
MATH 228B Numerical Solution of Differential Equations 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 128A.
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.
Instructor: 128A-128B.
MATH 229 Theory of Models 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 225B.
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.
MATH 235A Theory of Sets 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of lecture per week.
Prerequisites: 125A and 135.
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.
Instructor: 125A and 135.
MATH 236 Metamathematics of Set Theory 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 225B and 235A.
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.
MATH 239 Discrete Mathematics for the Life Sciences 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor.
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.
MATH C239/MCELLBI C244 Discrete Mathematics for the Life Sciences 4 Units
Department: Mathematics; Molecular and Cell Biology
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.
MATH 240 Riemannian Geometry 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214.
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.
MATH 241 Complex Manifolds 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214 and 215A.
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.
MATH 242 Symplectic Geometry 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214.
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.
MATH C243/MCELLBI C243 Seq: Methods and Applications 3 Units
Department: Mathematics; Molecular and Cell Biology
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: <BR/>
Prerequisites: Graduate standing in Math, MCB, and Computational Biology; or consent of the instructor.
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.
Instructor: Pachter
MATH 245A General Theory of Algebraic Structures 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Math 113.
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.
MATH 249 Algebraic Combinatorics 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A or consent of instructor.
(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.
MATH 250A Groups, Rings, and Fields 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 114 or consent of instructor.
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.
MATH 250B Multilinear Algebra and Further Topics 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A.
Tensor algebras and exterior algebras, with application to linear transformations. Commutative ideal theory, localization. Elementary specialization and valuation theory. Related topics in algebra.
MATH 251 Ring Theory 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A.
Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods.
MATH 252 Representation Theory 4 Units
Department: Mathematics
Course level: Graduate
Term course may be offered: Fall
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A.
Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.
MATH 253 Homological Algebra 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A.
Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules.
MATH 254A Number Theory 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring. A: Not offered 1984-85. B: (F)
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A for 254A; 254A for 254B.
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.
254B may be repeated with consent of instructor. Course may be repeated for credit when topic changes. Instructor: 250A.
MATH 254B Number Theory 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 254A.
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.
Instructor: 250A.
MATH 255 Algebraic Curves 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A-250B or consent of instructor.
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.
MATH 256A Algebraic Geometry 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A-250B for 256A; 256A for 256B.
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.
Instructor: 250A.
MATH 256B Algebraic Geometry 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 256A.
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.
Instructor: 250A.
MATH 257 Group Theory 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 250A.
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.
MATH 258 Classical Harmonic Analysis 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 206 or a basic knowledge of real, complex, and linear analysis.
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.
MATH 261A Lie Groups 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214.
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.
Instructor: 214.
MATH 261B Lie Groups 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214.
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.
Instructor: 214.
MATH 265 Differential Topology 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: 214 plus 215A or some familiarity with algebraic topology.
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.
MATH 270 Hot Topics Course in Mathematics 2 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Offered for satisfactory/unsatisfactory grade only.
Hours and format: 1.5 hours of Lecture per week for 15 weeks.
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.
Course may be repeated for credit as topic varies. Course may be repeated for credit when topic changes.
MATH 273F Topics in Numerical Analysis: Topics in Computational Physics 4 Units
Department: Mathematics
Course level: Graduate
Terms course may be offered: Fall and spring
Grading: Letter grade.
Hours and format: 3 hours of Lecture per week for 15 weeks.
Prerequisites: Consent of instructor.
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Course may be repeated for credit when topic changes.
MATH 273I Topics in Numerical Analysis: Approximation Theory 4 Units