Fall 2012

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.

Office: 883 Evans

Office Hours: TTh 1:00–2:30

Required Text: Stewart, Single Variable Calculus (Early Transcendentals for UC Berkeley, 7th edition); see the link in the Course Catalog for the exact reference.  Note: It is not the same edition as last semester's book.

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Grading: First midterm, 10%; second midterm, 20%; final exam, 45%; discussion section, 25%

Homework: Due weekly.  We will use a web-based system for homework; for details, see the course web page.

Course Webpage: http://math.berkeley.edu/~vojta/1a.html

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 or equivalent.

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. ( = Chapters 7-9, 11, 17 of the text.)

Office:  865 Evans Hall

Office Hours:  Monday, Tuesday 10:30-11:30, Friday 3:30-4:30.

Required Text:  James Stewart, Single variable calculus:  Early Transcendentals, Cengage (custom Berkeley edition for 1A/1B).  (Note: do not get the

new edition of the text, which will be used for 1A this semester and 1B in future semesters.  For continuity with last semester's 1A, this semester's (H)1B

still uses the earlier edition -- blue cover with a white-red-and-yellow framed pattern in the center.)

Grading:  Grades will be based on weekly section quizzes (25%), two Midterms (15% and 20%) a Final (35%), and regular submission of your "daily question" as described below (5%).

Homework:  Assigned weekly.  Not graded, but you are expected to work on the problems, and ask about them if you have difficulty.  The homework problems will be discussed in section, and section quizzes, which do form a part of your grade, will be closely modeled on them.

Comments:  This course is aimed at students with a strong ability and interest in mathematics.
I will follow the curriculum for Math 1B, but try to provide greater rigor (real proofs), greater insight, and more interesting exercises.  We will also go back to some of the key definitions and proofs from 1A and put them on a solid basis. To have time for this, I will spend much less time working at the blackboard computations just like those in the book than is done in non-honors courses.

I don't expect the grading scale to be either higher or lower than for 1B, but you will have to do more thinking to get a good grade; hopefully, you will enjoy this.  If you start H1B but find in a few weeks that it is not the course for you, it should be possible to transfer to regular 1B and not be at a disadvantage.

My discussion of the material in each lecture will be based on the assumption that you have done the assigned reading for that day!  You will be expected to submit a question on each day's reading by the start of class; preferably by e-mail an hour or more before.  (Details will be given in the first-day course handout.  In particular, if you understand the reading thoroughly and have no questions about it, you should submit a pro forma question about it – with the answer.)

If you are uncertain whether to take H1B, contact me, gbergman [at] math [dot] berkeley [dot] edu, and we can either discuss this by e-mail, or arrange for you to come by my office and discuss it.

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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Prerequisites: 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.

Syllabus:

"Something's going on. It has to do with that number. There's an answer in that number." Maximilian Cohen, in π (1998).

This course will offer an exploration of mathematics through the lens of a camera, the stage of a theater, and the language of a book. Can mathematics as a science, the thrill of its pursuit, or the idiosyncrasies of its practitioners be accurately portrayed in these media? Is such an accurate portrayal at all necessary or important? What societal beliefs and misconceptions are reflected in the works of literature and film dealing with mathematics? What is behind the stereotype of a crazy mathematician? How can one tell a compelling story about math to a non-mathematical audience? We will meet once a week to watch, read, argue about, and (try to) understand the mathematics within the world of literature and film. Besides reading and viewing, the students will be expected to take a very active part in class discussion and to make short presentations, which could include critique of a movie fragment, analysis of a literary text, or even a short mathematical proof. This class is intended for students with substantial interest in mathematics, film and literature.

Office: 821 Evans Hall

Office Hours: TBD

Required Text: 

Recommended Reading:

• Burkard Polster and Marty Ross.  Math Goes to the Movies. The Johns Hopkins University Press. ISBN-10: 1421404842
• James Monaco.  How to Read a Film: Movies, Media and Beyond.  Oxford University Press, USA; 4th edition. ISBN-10: 0195321057

Grading: P/NP based on class participation

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Course Webpage: http://www.cs.berkeley.edu/~oholtz/24/

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.

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Prerequisites: 1B.

Syllabus: Honors version of 53. Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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

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

Office:  855 Evans

Office Hours:  MW 10:30-11:30

Required Text: Kenneth H. Rosen, Discrete Mathematics and Its Applications, UC Berkeley Edition. The Berkeley custom edition is the same as the Seventh Edition with chapters 11 and 12 omitted. If you have a copy of the full Seventh Edition, that is fine too.

Grading: 2 Midterms, Final Exam and Homework

Homework: Assigned weekly

Course Webpage: /http://math.berkeley.edu/~mhaiman/math55-fall12/

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: 53 and 54.

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

Syllabus: Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

Office:  821 Evans Hall

Office Hours: TBD

Required Text: Halmos, Paul R. Finite-dimensional vector spaces.  Reprinting of the 1958 second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1974. viii+200 pp.

Recommended Reading: Halmos, Paul R. Linear algebra problem book. (English summary)
The Dolciani Mathematical Expositions, 16. Mathematical Association of America, Washington, DC, 1995. xiv+336 pp. ISBN: 0-88385-322-1

Gelʹfand, I. M. Lectures on linear algebra. With the collaboration of Z. Ya. Shapiro. Translated from the second Russian edition by A. Shenitzer. Reprint of the 1961 translation. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, 1989. vi+185 pp. ISBN: 0-486-66082-6

Glazman, I. M.; Ljubič, Ju. I. Finite-dimensional linear analysis.  A systematic presentation in problem form. Translated from the Russian and edited by G. P. Barker and G. Kuerti. Reprint of the 1974 edition. Dover Publications, Inc., Mineola, NY, 2006. xx+520 pp. ISBN: 0-486-45332-4

Grading: This course will be taught in a highly interactive way, with student participation required during every class. This will differ substantially from the familar lecture-discussion format and will emulate the atmosphere of a research seminar. The grading scheme will be as follows: 20% class participation, 30% homework, 20% midterm, 30% final.

Homework:  A set of 8-12 problems will be assigned every week, due one week later. The difficulty level will exceed that of a regular Math 110 class.   

Course Webpage: http://www.cs.berkeley.edu/~oholtz/H110/

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: 53 and 54.

Syllabus: Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

Midterm exams: Exams will be given in class on September 20 and October 25.  Please do not plan to travel on those dates.

Office: 885 Evans Hall

Office Hours: To be announced

Required Text: An Introduction to the Theory of Numbers, Fifth Edition by Ivan Niven, H. S. Zuckerman and Hugh L. Montgomery.

Recommended Reading: If you get frustrated with the textbook, you might want to consult an old web page that lists alternative references.  Just ask me and I'll supply a few more ideas.

Grading: I will use the time-honored grading scheme that computes letter grades from a composite grade in which homework counts for 20%, the midterms count for 15% each and the final exam counts for 50%.

Homework: Assigned weekly.  Assignments will carry equal weight and each student's lowest score will be dropped.

Course Webpage: http://math.berkeley.edu/~ribet/115/ . The course will be updated before the beginning of the fall, 2012 semester.

Prerequisites: Math 113 for two reasons. There will be many algebraic examples in this course. More importantly, you are expected
to be able to write clear, precise proofs.

Syllabus: (1) will be quick, (2) and (3) are the heart of the course, (4) depends on how much time we have.

1. Sentential (aka propositional) logic, chapter 1 of Enderton.
2. Signatures and structures: bits and pieces mostly from chapter 2 of Enderton, many examples from mathematics.
3. First-order (aka predicate) logic: syntax and semantics, compactness: sections 2.0 - 2.5 of Enderton.
4. Some model theory, from the first four chapters of Marker's book.

Office: 867 Evans

Office Hours: TBA

Required Text: A Mathematical Introduction to Logic by Herbert Enderton

Recommended Reading: Model Theory : An Introduction by David Marker (electronic copy available through UCBerkeley library website)

Grading: Homeworks, 2 to 3 exams, and 0 to 1 projects will be combined in some way.

Homework: There will be weekly homework assignments. There may also be a larger collaborative project near the end of the semester.

Course Webpage: coming soon

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

Syllabus: 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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Prerequisites: 113 and 104.

Syllabus: Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. Proof by transfinite induction and definitions by transfinite recursion. Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences.

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Prerequisites: 104 or equivalent and linear algebra.

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

Office: 805 Evans

Office Hours: M 1-2, W 2-3, F 12-1

Required Text: Munkres, Analysis On Manifolds, Westview Press.

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Grading: 30% homework, 30% midterm exam, 40% final exam.

Homework: Homework will be assigned weekly. Late homework will not be accepted. The lowest two scores will be dropped.

Course Webpage: http://math.berkeley.edu/~jdahl/141/index.html

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: Consent of instructor.

Syllabus: This is Putnam Workshop. Putnam Exam is a Mathematical Olympiad for college students. The class will function as a seminar pursuing two goals. One is to give a crash course on 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 material of math olympiads of high-school level (rather than past Putnam Exams - 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 consitency 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: 113 and point-set topology (e.g. 202A).

Syllabus: Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall.

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Prerequisites: 214.

Syllabus:

Linear systems, translations on Tori, Arnold can map, Baker's transformation, Sinai's billiard, Lorentz gas,

invariant measures. ergodic theory. Kolmogorov-Sinai Entropy, Lyapunov exponents, Hyperbolic systems, Perron-Frobenius operator,

Bowen-Ruelle-Sinai measures, Pesin's theorem, Ruelle's inequality, Aubry-Mather theory.

Office: 

803 Evans

Office Hours: 

MWF, 1-2 pm

Required Text: 

F. Rezakhanlou, Lectures on Dynamical Systems,

http://math.berkeley.edu/~rezakhan/

Recommended Reading: 

Grading: Based on graded homework assignements

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Course Webpage: 

http://math.berkeley.edu/~rezakhan/math219_html/

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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Course Webpage: http://www.math.berkeley.edu/~grunbaum/224-2012.html

Prerequisites: 128A.

Syllabus: 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.

Office:  1099 Evans

Office Hours:  Tuesday 4-5 and Wednesday 2:30-4:30

Required Text: LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations

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Course Webpage: http://www.math.berkeley.edu/~strain/228a.F12/

Prerequisites: 114 or consent of instructor.

Syllabus: The official course description for this class is: 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.  More specifically, we will try to cover most of chapters I to VI from Serge Lang's Graduate Algebra, with some possible supplementary topics such as Galois cohomology if time permits.  Since we will have a lot of material to cover, it is likely that some theorems/definitions will be covered less thoroughly in class than others, but students will be expected to read up on them in the book.  A note about the book:  it is a classic text from which to learn algebra.  Some students find that it is a bit dense to learn from: if you feel this way I suggest you supplement the book with the text you used to learn undergraduate algebra or one of the other suggested texts on the course website.

Office: 851 Evans

Office Hours: MWF 2-3PM

Required Text: Serge Lang's Graduate Algebra, 3rd Edition

Recommended Reading: additional reading suggestions on course website

Grading: Based on homework (50%), one take-home midterm (15%), and a final exam (35%).

Homework: Due Fridays in class or by 2PM under office door in 851 Evans

Course Webpage: http://math.berkeley.edu/~efuchs/250A

Prerequisites: Familiarity with basic vocabulary of rings, modules, categories and functors is expected.

Syllabus: Modern Mathematics is simply unthinkable without Homological Algebra. I will give a rapid and clear introduction to all the fundamental concepts and then will proceed to cover more advanced topics: satellites, spectral sequences, exact categories, and derived categories. In the second part of the course I will provide an introduction to Homotopical Algebra and to modern theory of triangulated categories.

Required Text: Charles Weibel. An Introduction to Homological Algebra, Cambridge University Press

and

Sergei I. Gelfand, Yuri I. Manin. Methods of Homological Algebra, Springer-Verlag

Recommended Reading: Cartan, Eilenberg. Homological Algebra, Princeton 1956 (the first few chapters)

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: 927 Evans Hall

Office Hours: Tu Th 2:00-3:30

Required Text: Hartshorne, Algebraic  Geometry

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Homework: Assigned each week on the course home page

Course Webpage: http://math.berkeley.edu/~reb/256A

Prerequisites: Consent of instructor.

Syllabus: I will discuss elliptic curves, modular forms, Galois representations and related objects.  I will begin with the idea of following the textbook (see below).  The course will evolve according to the interests of the audience and the taste of the instructor.

Office: 885 Evans Hall

Office Hours: See http://math.berkeley.edu/~ribet/office_hours.txt for official hours.  You can also make an appointment.

Required Text: http://wstein.org/books/ribet-stein/main.pdf

Recommended Reading: 

Grading: As is customary in advanced topics courses, the grading will be generous.

Homework: 

Course Webpage: The page http://math.berkeley.edu/~ribet/274/ will be created when needed.

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: Basic algebraic and differential topology (214, 215a) is essential. Familiarity with more algebraic topology (215b) and symplectic geometry (242) is helpful. Background material will be reviewed as needed depending on the audience.

Syllabus: The goal of this course is to introduce different kinds of contact homology and their applications to distinguishing contact manifolds, obstructing symplectic cobordisms and symplectic embeddings, and the Weinstein conjecture and generalizations. The beginning of the course will review necessary background on Morse theory and holomorphic curves.

Office: 923 Evans

Office Hours: TBA

Required Text: None.

Recommended Reading: Lecture notes on some of the topics will be provided. References will be given for other topics.

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

Syllabus: Meeting with supervising facutly and with discussion sections. Experience in teaching under the supervision of Mathematics faculty.

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