Spring 2016

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: Continuation of 10A

Description: Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 16A

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: Math 1A.  Further calculus is useful but not necessary.

Description: What does calculus tell us about the shape of a piece of fruit?  The mathematical field of differential geometry uses ideas from calculus to study geometric figures.  A central notion in this field is that of curvature, which measures the deviation from "straightness" in curves, surfaces, and geometric objects in higher dimensions.  (For instance, the force of gravity may be interpreted in general relativity as coming from the curvature of space time associated with the presence of matter.)  In this freshman seminar, each week will feature an investigation which will frequently involve the measurement of concrete curved objects, such as bent wires, drinking glasses, or pears.

The main reference for the course will be the book, "Exploring Curvature," by Professor James Casey of our Department of Mechanical Engineering.  

Office: 825 Evans Hall

Office Hours: to be announced

Required Text: "Exploring Curvature," by James Casey (Springer, 1996)

Recommended Reading: 

Grading: P/NP

Homework: Each student will be expected to participate (with one or two others) in one week's presentation.

Course Webpage: 

Prerequisites: Some units in a lower division Mathematics class

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam not required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: https://math.berkeley.edu/~vivek/54.html

Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended

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

Office: 1035 Evans

Office Hours: T 5:15-6:15, W 2:30-4:30.

Required Text:  Discrete Math and its Applications, 7e, Kenneth E. Rosen (UC Berkeley custom edition)

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage:  https://math.berkeley.edu/~nikhil/courses/55.s16/

Prerequisites: 

Description: Directed Group Study, topics vary with instructor.

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.

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

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

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 104

Description: 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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 54 or a course with equivalent linear algebra content

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 53 and 54

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 53 and 54

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

Office: 1083 Evans

Office Hours: W 1:30-3:30PM or by appointment.

Required Text: Mary L. Boas, Mathematical Methods in the Physical Sciences, 3e.

Grading: Letter grade. Final exam required.

Course Webpage: https://math.berkeley.edu/~linlin/121B/

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 110 and 128A

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 104 or equivalent and linear algebra.  See course webpage for specific requirements.  

Description: Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2.

Office: 

Office Hours: 

Required Text: Differential Topology, by V. Guillemin and A. Pollack

Recommended Reading: See course website.

Grading: Letter grade. Final exam required.

Homework: Weekly 

Course Webpage:  http://math.berkeley.edu/~kpmann/141differential2016.html

Prerequisites: 151, 152

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 53, 54, and 113

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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.

Office: 749 Evans

Office Hours: Tuesday 3:30-5 and Wednesday 10:30-12

Required Text: Theodore W. Gamelin, Complex Analysis.

Grading: Letter grade. Final exam required.

Homework: weekly

Course Webpage: To appear on http://math.berkeley.edu/~kozai/m185s16

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: 104

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

Prerequisites: Consent of instructor needed for students who haven't already taken 110 or H110.  (This course may substitute for the Math 110 requirement for students who haven't already taken 110 and who have been given instructor permission to enroll.)

Description: This Math 191 will use Jenny Harrison's textbook on advanced linear algebra, “Universal Linear Algebra for Students of Math and Physics.”  The course will only assume basic properties of set theory and cover the standard topics of linear algebra as well as some basic category theory, the universal property of the tensor algebra and its ramifications, and both the real and complex Jordan forms. The final chapter develops a discrete calculus relying only on linear algebra, but without limits. A further chapter might be added, showing how to define norms on Dirac chains so that limits can be taken, and the classical integral relations of calculus will drop out.  

Office:  829 Evans Hall

Office Hours:  

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam required.

Homework: 

Course Webpage: 

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

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade. Final exam not required.

Homework: 

Course Webpage: 

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

Description: Topics will vary with instructor.

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.

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

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

Homework: 

Course Webpage: 

Prerequisites: 202A and 110

Description: 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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

Prerequisites: 185 and 202A-202B or their equivalents

Description: Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course will treat the theory of holomorphic partial differential equations with applications to equations with real analytic coefficients.

Office: 801

Office Hours: TTH 11-noon

Required Text: (see the course home page)

Recommended Reading: (see the course home page)

Grading: Letter grade.

Homework: 

Course Webpage: https://math.berkeley.edu/~zworski/math212.html

Prerequisites: 

Description: 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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

Prerequisites: 105 or 202A

Description: Theory of initial/boundary value problems for second-order elliptic, parabolic, and hyperbolic equations. DeGiorgi-Nash-Moser regularity theory for second-order elliptic partial differenial equations. Selected topics in nonlinear partial differential equations, including the calculus of variaitions, and optimal control and viscosity solutions. Provided there is time, we may discuss approximation schemes for viscosity solutions, and the theory of viscosity solutions for second order degenerate elliptic equations.

Office: 1053 Evans Hall

Office Hours: TBD

Required Text: Partial Differential Equations, L.C. Evans (2nd edition)

Recommended Reading:

1. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, M. Bardi and I.C. Dolcetta (1997)
2. Elliptic Partial Differential Equations of Second Order, D. Gilbarg and N.S. Trudinger (1998)
3. Elliptic Partial Differential Equations, Q. Han and F. Lin (2nd edition)

Grading: Letter grade.

Homework: 

Course Webpage: https://math.berkeley.edu/~jcalder/222BS16/

Prerequisites: 125B and 135

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

Prerequisites: 250A

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

Prerequisites: 261A, or other previous introduction to Lie groups and Lie algebras

Description: The course will cover several topics in Lie theory, including some areas with open research problems.  It should be of interest to a variety of students with some previous knowledge of Lie groups and Lie algebras, not limited to those continuing from 261A this past fall.

We will begin with the description of the semisimple Lie algebras and their representations, and Lie algebra cohomology (this is where we left off in the fall and may serve as useful review for those joining in the spring).  Next we will discuss Hopf algebras and algebraic groups, and see how to recover a linar algebraic group from its tensor category of representations; this will lead to the remarkable fact that compact real Lie groups, reductive complex Lie groups, and reductive linear algebraic groups over fields of any characteristic are classified by the same 'Cartan data' of a root system and weight lattice.

The next topic will be flag varieties G/P of a reductive group G and the Borel-Weil-Bott geometric construction of the representations of G.  I will discuss some open problems in this area, of a very concrete and combinatorial nature.

Finally we will discuss 'quantum groups,' which are non-commutative q-deformations of the Hopf algebra of functions on a reductive group, and dually, of the enveloping algebra of its Lie algebra.  This leads to the theory of canonical bases, which provide a beautiful combinatorial structure underlying the representations of reductive groups.  This structure is present even in the classical representations, even though the only known way to see it is through the q-deformation.  There are plenty of open problems in this area also.

Depending on time, possible additional topics are Kac-Moody algebras, groups, and quantum groups (especially the affine case); and Hecke algebras and Kazhdan-Lusztig theory.

Office: 855 Evans

Office Hours: TBA

Required Text: V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. This book is available online from campus computers or through the library proxy server.

Recommended Reading: see course web page

Grading: Based on homework - probably a smaller quantity than in 261A due to the more topics course flavor of the spring semester.  As an alternative, you can earn an A by working on any open research problem.  For this you don't have to solve the problem, but should make a serious attempt to understand it and work out some special cases.

Course Webpage: https://math.berkeley.edu/~mhaiman/math261-fall15-spring16/

Prerequisites: None

Description: In this seminar the following topics will be discussed:

1. Hofer Geometry and Symplectic Rigidity
2. Pseudo-Holomorphic Curves
3. Arnold Conjecture and Floer Homology

Office: 

Office Hours: 

Required Text: F. Rezakhanlou, Lectures on Symplectic Geometry

Recommended Reading: 

Grading: Offered for satisfactory/unsatisfactory grade only.

Homework: 

Course Webpage: 

Prerequisites: 

Description: 

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: 

Homework: 

Course Webpage: 

Prerequisites: Consent of instructor

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

Prerequisites: Consent of instructor

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Letter grade.

Homework: 

Course Webpage: 

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.

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Offered for pass/not pass grade only.

Homework: 

Course Webpage: 

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

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

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: Offered for satisfactory/unsatisfactory grade only.

Homework: 

Course Webpage: 

Prerequisites: 

Description: 

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: 

Homework: 

Course Webpage: 

Prerequisites: 

Description: 

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: 

Homework: 

Course Webpage: