Fall 2016

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

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 1A

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry

Description: 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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Grading: Letter grade. Final exam required.

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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

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

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Grading: Letter grade. Final exam required.

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Description: The Berkeley Seminar Program has been designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a 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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Grading: The grading option will be decided by the instructor when the class is offered. Final exam required.

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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

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

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Grading: Letter grade. Final exam required.

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Prerequisites: Mathematics 1B

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 1B

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 1B

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

Office: 883 Evans

Office Hours: Mondays, 11:30–12:30, Wednesdays and Fridays, 11–12

Required Text: Lay, Nagle, Saff, & Snider, Linear Algebra and Differential Equations (UC Berkeley custom edition). See the bookstore web site for exact details.

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: Due weekly, to be handed in during discussion sections.

Course Webpage: https://math.berkeley.edu/~vojta/H54.html

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Description: Directed Group Study, topics vary with instructor.

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Grading: Offered for pass/not pass grade only. Final exam not required.

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Description: Directed Group Study, topics vary with instructor.

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Grading: Offered for pass/not pass grade only. Final exam not required.

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Description: 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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Grading: Offered for pass/not pass grade only. Final exam not required.

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Description: 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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Grading: Offered for pass/not pass grade only. Final exam not required.

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Prerequisites: 53 and 54

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 53 and 54

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 53 and 54

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 53 and 54

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: 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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Grading: Letter grade. Final exam required.

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Prerequisites: 54 or a course with equivalent linear algebra content

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 54 or a course with equivalent linear algebra content

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 54 or a course with equivalent linear algebra content

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 104

Description: 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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Grading: Letter grade. Final exam required.

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

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 110 and 113

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

Office: 933 Evans

Office Hours: TBA

Required Text: The Four Pillars of Geometry, by John Stillwell

Grading: Letter grade. Final exam required.

Course Webpage: https://math.berkeley.edu/~conway/Teaching/fall2016-130/fall2016-130.html

Prerequisites: 113

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 104

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

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Grading: Letter grade. Final exam required.

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Prerequisites: 104

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

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Grading: Letter grade. Final exam required.

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Prerequisites: Consent of instructor

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

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Grading: Letter grade. Final exam required.

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Description: 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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Offered for pass/not pass grade only. Final exam not required.

Homework: 

Course Webpage: 

Prerequisites: 

Description: 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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Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Offered for pass/not pass grade only. Final exam not required.

Homework: 

Course Webpage: 

Prerequisites: 104. 110, 113, and 185 are strongly recommended.

Syllabus: Brief introduction to metric spaces.Measures (sigma--algebras, outer measures, Borel measures). Lebesgue integration. Integrals and limits (Monotone and Dominated Convergence theorems; Fatou's lemma). Signed measures, and differentiation of measures (Radon-Nikodym theorem,Lebesgue's differentiation theorem). Functions of bounded variation. Point set topology (metric spaces, completion, topological spaces, compactness, separation properties, local compactness, Weierstrass and Stone-Weierstrass theorems, Tychonoff product theorem). As time permits: Baire's theorem. Product measures. L^p spaces (completeness and the Holder and Minkowski inequalities, duality).

Office: 809 Evans Hall.

Required Text: Real Analysis (2nd edition) by Gerald B. Folland.

Please see more detailed announcement at https://math.berkeley.edu/~mchrist/Courses/202Aannounce.pdf .

Prerequisites: 202A-202B or equivalent is more than enough. Students who have studied only part of the material of Math 202AB and wish to enroll in Math 206 should discuss this with me.

Syllabus: 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.

Office: 811 Evans

Office Hours: M 1-2,  W 11-12,  F 8:45-9:45

Required Text: 

Recommended Reading: A Course in Functional Analysis, John B. Conway, 2nd ed., Springer-Verlag.

My understanding is that through an agreement between UC and Springer, chapters of the first edition of this book are available for free download by students. They should be satisfactory - I do not plan to assign exercises from the Conway book.

Course description: 

The theory of Banach algebras is a very elegant blend of algebra and topology which provides unifying principles for a number of different parts of mathematics and its applications, notably operator theory, commutative and non-commutative harmonic analysis and the theory of group representations, and the theory of functions of one and several complex variables. But at the present time probably its most extensive use is as a foundation for non-commutative measure theory (von Neumann algebras, Math 209) and non-commutative topology and geometry (C*-algebras, Math 208). These in turn provide a foundation for quantum physics, but they also have myriad applications in many other directions, including group representations and harmonic analysis, ordinary topology and geometry, and even number theory. (For a vast panorama of the applications see Alain Connes' 1994 book "Noncommutative Geometry", which is available on the web as a free download.) I will cover the standard topics as listed in the catalog. Beyond the basic general theory of Banach algebras this will include several forms of the spectral theorem for self-adjoint operators on Hilbert space, compact and Fredholm operators, and group algebras and the Fourier transform and their relation to representation theory. Further topics as time permits, probably including Toeplitz operators and their index. The problem sets will involve a number of important specific examples beyond those presented in class.

Homework: Many weeks I will give out a problem set, and the course grade will be based on the work done on these. There will be no final examination.

Course Webpage: https://math.berkeley.edu/~rieffel/206annF16.html

Prerequisites: 214

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

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Grading: Letter grade.

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Prerequisites: 105 or 202A

Description: 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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Prerequisites: Graduate status or consent of instructor.

Description: The course will survey basic theory and practical methods for solving the fundamental problems of mathematical physics. Topics covered will include Hilbert spaces, orthogonal polynomials, approximation theory, the Fourier transform, Green's functions, Sturm-Liouville theory, and operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves.

Office: 1051 Evans

Office Hours: Wed 2:30-3:30, Thurs 3-4

Required Text: see course webpage

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Course Webpage: https://math.berkeley.edu/~wilken/224A.F16/

Prerequisites: Math 128A or equivalent knowledge of undergraduate numerical analysis, MATLAB or equivalent programming experience.

Description: Theory and practical methods for numerical solution of differential equations. Ordinary differential equations: Runge-Kutta and multistep methods, stability theory, stiff equations, boundary value problems. Partial differential equations: Finite difference and spectral methods for elliptic, parabolic and hyperbolic equations, stability, accuracy and convergence, von Neumann analysis and CFL conditions.

Office: 1089 Evans

Office Hours: Mon 12:30pm - 2pm and Fri 11am - 12:30pm in 1089 Evans.

Required Text: R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM 2007.

Recommended Reading: A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Second Edition, Cambridge University Press 2008, E. Hairer and G. Wanner, Solving ordinary differential equations II, Springer 2010.

Grading: Letter grade.

Homework: 7 extensive problem sets.

Course Webpage: https://bcourses.berkeley.edu/courses/1453861

Prerequisites: 250A or consent of instructor

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

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Grading: Letter grade.

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Prerequisites: 250A

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

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Grading: Letter grade.

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

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

Office: 883 Evans

Office Hours: Mondays, 11:30–12:30; Wednesdays and Fridays, 11–12

Required Text: Hartshorne, Algebraic Geometry

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Grading: Letter grade.

Homework: Due every one or two weeks

Course Webpage: https://math.berkeley.edu/~vojta/256a.html

Prerequisites: 202AB and 206. A basic knowledge of real, complex, and linear analysis.

Description: Please see the course announcement posted at https://math.berkeley.edu/~mchrist/Courses/258announce.pdf .

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Grading: Letter grade.

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

Description: 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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Grading: Offered for pass/not pass grade only.

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

Description: 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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Grading: Offered for pass/not pass grade only.

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