Fall 2013

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.

Syllabus: 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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Prerequisites: 1A.

Syllabus: 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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Prerequisites: 1A.

Syllabus: Honors version of 1B. Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.

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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.

Syllabus: 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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Prerequisites: 16A.

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

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Syllabus: 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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Syllabus: 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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Prerequisites: 1B.

Syllabus: 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: 817 Evans

Office Hours: to be announced

Required Text: Stewart, Multivariable Calculus, custom UC Berkeley edition

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Grading: homework and quizzes 25%; two midterms 25% each; final exam 25%; lowest midterm can be dropped and replaced by final exam grade. There will be no make-up exams. Grading policy allows you to miss one midterm, but check your schedule to make sure you have no conflict for the final exam.

Homework: Homework will be assigned weekly.

Course Webpage: http://math.berkeley.edu/~auroux/53f13/

Prerequisites: 1B.

Syllabus: 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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Course Webpage: http://math.berkeley.edu/~grunbaum/54-2013.html

Prerequisites: 1B.

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

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Syllabus: 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. Also listed as Economics C103.

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Prerequisites: 53 and 54.

Syllabus: 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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Prerequisites: 53 and 54.

Syllabus: 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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Prerequisites: 53 and 54.

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

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

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Course Webpage: http://math.berkeley.edu/~nadler/110fall2013

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

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

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

Syllabus: 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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Prerequisites: 110 and 113, or consent of instructor.

Syllabus: 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.

Office: 785 Evans

Office Hours: tba

Required Text: "Algebra" by M. Artin.

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Course Webpage: http://math.berkeley.edu/~preygel/M114.F13/

Prerequisites: 55.

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

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Required Text: An introduction to mathematical cryptography by Hoffstein, Pipher and Silverman. 

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Course Webpage: http://math.berkeley.edu/~yxy/cryptography.html

Prerequisites: 113 or consent of instructor.

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

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Prerequisites: Math 53 and 54.

Syllabus: Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.

Office: 811 Evans

Office Hours:  TBA

Required Text: Numerical Analysis, 2nd ed, Timothy Sauer, Pearson Pub., 2011

Recommended: The MATLAB Student Version (if you have your own computer). We expect that there will be some free hours available in a computer lab equipped with MATLAB in which students can do their computer exercises, if they prefer not to use their own computer.

Course description: This is a mathematics course, and so the emphasis is on how to obtain effective methods for computation, and on analyzing when methods will, and will not, work well (in contrast to just learning methods and applying them). You will have an easier time with the course if you review Taylor's theorem and ordinary differential equations before the course begins. The programming exercises are to be done in MATLAB, but no prior knowledge of MATLAB will be assumed, and help will be provided for learning the relatively small amount of MATLAB that will be needed for the course. (But if you can learn a bit of MATLAB before the course begins, that will make the course easier.) 

Grading: There will be homework, which will count for 10% of the course grade, and there will be programming exercises, which will count for 20% of the course grade. There will be 2 midterm exams, which will each count for 15% of the course grade. There will be a final examination, which will count for 40% of the course grade. Makeup midterm exams will not be given; instead, if you tell me ahead of time that you must miss a midterm exam, then the final exam and the other midterm exam will count more to make up for it. If you do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent to try to avoid a score of 0. If you miss both midterm exams the circumstances will need to be truly extraordinary to avoid a score of 0 on at least one of them. There will be no early or make-up final examination. Students who need special accommodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each exam what specific accommodation they need, so that I will have enough time to arrange it.

Homework: Homework will be assigned at almost every class meeting, due at the section meeting the following week.

Notice: The above procedures are subject to change. This page was last updated on 06/21/2013.

Course Webpage: math.berkeley.edu/~rieffel/

Prerequisites: 104 or equivalent and linear algebra.

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

Office: 905 Evans

Office Hours: TBA

Required Text: Guillemin and Pollack, Differential Topology

Recommended Reading: Milnor, Topology from the differentiable viewpoint

Grading: (to be confirmed) 20% homework, 20% midterm, 40% final, 20% a short presentation 

Homework: Due in class, every other week.

Course Webpage: http://math.berkeley.edu/~teleman/classes/141f13

Prerequisites: 151; 54, 113, or equivalent.

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

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Prerequisites: 104.

Syllabus: 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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Prerequisites: 104.

Syllabus: 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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Prerequisites: 104.

Syllabus: 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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Prerequisites: Consent of instructor.

Syllabus:  This is Putnam Workshop. Putnam Mathematical Competition is a Mathematical Olympiad for college students. The class will function as a seminar pursuing two goals. One is, using the materials of past Putnam competitions as bait, to advance student's understanding of basic subjects of upper-division  math curriculum: first of all - linear algebra, as well as some topics in algebra, geometry and analysis. The other one is raising student's confidence in solving challenging problems. For practicing in this area, we plan to use the materials of math Olympiads of high-school level (rather than Putnam problems - which are available online anyway).

There will be no final exam, but students are expected to take part in the mock- and then real Putnam Exam of this year. Grading in the course will be based not on the results of these 'exams', but on the consistency in submitting homework and participating in classroom discussions.

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Prerequisites: 104.

Syllabus: Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.

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Prerequisites: 202A.

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

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Syllabus: 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. Also listed as Statistics C205A.

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Prerequisites: 105 or 202A.

Syllabus: 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.

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

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Prerequisites: 225B.

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

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Prerequisites: 250A or consent of instructor.

Syllabus:  Geometric and topological combinatorics, including posets, simplicial complexes, polytopes, and matroids.

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Course Webpage:  www.math.berkeley.edu/~williams/249-fall13.html

Prerequisites: 250A, or consent of the instructor.

Syllabus: An introduction to the representation theory of finite dimensional algebras, of finite groups, of quivers, and of finite dimensional semisimple Lie algebras.

Office: 873

Office Hours: MWF 10-11 AM

Required Text: None

Recommended Reading: Etingof’s “Introduction to Representation Theory” (online at arXiv:0901.0827) and the book “Representation Theory: a first course” by Fulton and Harris.

Grading: 50% weekly homeworks, 50% takehome final.

Homework: Weekly

Course Webpage: http://math.berkeley.edu/~vivek/fall2013/rt.html

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

Syllabus: 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.

Office: 855 Evans

Office Hours: WF 11-12 or by appointmen

Grading: Based on homework assigned periodically during the semester

Course Webpage: http://math.berkeley.edu/~mhaiman/math256-fall13-spring14/

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Prerequisites: Consent of instructor.

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

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Prerequisites: Consent of instructor.

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

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