# Spring 2004

Math 1A - Section 1 - CalculusInstructor: Zvezdelina StankovaLectures: TuTh 8:00-9:30am, Room 2050 Valley Life ScienceCourse Control Number: 54303Office: 913 Evans, e-mail: stankova [at] math [dot] berkeley [dot] eduOffice Hours: TuTh 9:40-11:00amPrerequisites: 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. However, math majors are also encouraged to take
the course. An introduction to differential and integral calculus of
functions of one variable, with applications and an introduction to
transcendental functions. Here follows a tentative Syllabus:1. Preview of Calculus. Functions and Graphs 2. Types of Functions and More on Graphs. Tangents to Graphs. 3. Limits and Limit Laws 4. Definition of Limit. Continuity 5. Continuity Laws. Inifinite Limit Laws 6. Tangents and Derivatives 7. Derivative as a Function. Derivatives of Polynomials 8. Derivative of e^x. The Product and Quotient Rules 9. Derivatives of Trigonometric Functions. The Chain Rule 10. Applications of the Chain Rule. Implicit Differentiation 11. Applications of e. Higher Derivatives. Hyperbolic Functions 12. Midterm I 13. Linear Approximations and Differentials. Applications of Derivatives 14. Maximum and Minimum Values 15. Mean Value Theorem 16. Derivatives and Graphs 17. L'Hospital's Rule 18. Slant Asymptotes 19. Optimization Problems 20. Applications to Economics. Newton's Method 21. Antiderivatives 22. Midterm II 23. Areas 24. Definite Integrals 25. Fundamental Theorem of Calculus 26. Total Change Theorem 27. Substitution Rule 28. The Logarithm Defined as an Integral. Areas Between Curves 29. Volumes 30. More Applications 31. Review for Final Exam Required Text: Stewart, Calculus: Early Transcendentals, (Brooks/Cole)Recommended Reading: Lecture and Workshop NotesCourse Webpage: http://math.berkeley.edu/~stankova/Grading: 15% quizzes, 25% each midterm, 35% finalHomework: Unless otherwise specified during the course, HWs will not be graded or collected, but will be assigned and due once a week.Comments: There will be approximately 12 quizzes in the
discussion sections. The lowest two quiz scores will be dropped when
determining a student's final grade. If you miss discussion sections
when a quiz is taken, you cannot retake the quiz in other section, and
your quiz score will be 0. Thus, when you miss discussion sessions (for
whatever reasons), keep in mind that only two quiz scores will be
dropped. The quizzes will be based on the current or previous homework
assignment.There will be two in-class midterm exams and a final exam. A substantial portion of the exams will be based on homework assignments. Math 1B - Section 1 - CalculusInstructor: M. ZworskiLectures: MWF 8:00-9:00am, Room 155 DwinelleCourse Control Number: 54339Office: 897 Evans, e-mail: zworski [at] math [dot] berkeley [dot] eduOffice Hours: MW 11:00am-12:00pm, F 2:00-3:00pm, or by appointment Required Text: J. Steward, Calculus (Early Transcendentals), 5th edition.Course Webpage: http://math.berkeley.edu/~zworski/1B/1B.htmlGrading: Homework and Quizzes 20%, Midterm #1 20%, Midterm #2
20%, Final Exam 40%. Grades will be computed in the following way. You
will be given a letter grade (+ or -, if appropriate) for each item of
work above and we will later combine these grades as indicated to obtain
the final grade for the course. The TAs will lastly identify borderline
cases, for which we will carefully look at the numerical grades on the
various tests to determine the grade. If you do not take Midterm #1, Midterm #2 will count for 40% of your grade. If you take Midterm #1 but not Midterm #2, the Final Exam will count for 60% of your grade. If you take neither Midterm #1 nor Midterm #2, you will fail the course. Consequently, please mark them in your calendars. Homework and Quizzes: There will be a weekly quizz given each
Wednesday in sections. There will be no make-up quizzes, but we will
drop the two lowest quiz scores in computing your grade. Homework from
main lecture on Monday is due on Wednesday in sections; homework from
the main lectures on Wednesday and Friday is due on Monday in sections.
The homework will be graded ``pass/fail''. Math 1B - Section 2 - CalculusInstructor: M. RatnerLectures: MWF 3:00-4:00pm, Room 2050 Valley Life ScienceCourse Control Number: 54384Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 1B - Section 3 - CalculusInstructor: N. ReshetikhinLectures: MWF 12:00-1:00pm, Room 2050 Valley Life ScienceCourse Control Number: 54420Office: 915 Evans, e-mail: reshetik [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 16A - Section 1 - Analytical Geometry and CalculusInstructor: W. H. WoodinLectures: TuTh 3:30-5:00pm, Room 2050 Valley Life ScienceCourse Control Number: 54462Office: 721 Evans, e-mail: woodin [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 16B - Section 1 - Analytical Geometry and CalculusInstructor: T. ScanlonLectures: MWF 2:00-3:00pm, Room 145 DwinelleCourse Control Number: 54501Office: 723 Evans, e-mail: scanlon [at] math [dot] berkeley [dot] eduOffice Hours: TBACourse Webpage: http://math.berkeley.edu/~scanlon/m16bs04/index.htmlMath 16B - Section 2 - Analytical Geometry and CalculusInstructor: J. H. SilverLectures: TuTh 11:00am-12:30PM, Room 2050 Valley Life ScienceCourse Control Number: 54534Office: 753 Evans, e-mail: silver [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 24 - Section 1 - Freshman SeminarsInstructor: J. HarrisonLectures: F 3:00-4:00pm, Room 891 EvansCourse Control Number: 54573Office: 851 Evans, e-mail: harrison [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 32 - Section 1 - PrecalculusInstructor: The StaffLectures: MWF 8:00-9:00am, Room 60 EvansCourse Control Number: 54576Office:Office Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 53 - Section 1 - Multivariable CalculusInstructor: T. GraberLectures: TuTh 3:30-5:00pm, Room 155 DwinelleCourse Control Number: 54606Office: 833 Evans, e-mail: graber [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 53M - Section 1 - Multivariable Calculus With ComputersInstructor: L. PachterLectures: MWF 10:00-11:00am, Room 277 CoryCourse Control Number: 54660Office: 1081 Evans, e-mail: lpachter [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 54 - Section 1 - Linear Algebra and Differential EquationsInstructor: A. K. LiuLectures: MWF 3:00-4:00pm, Room 155 DwinelleCourse Control Number: 54678Office: 905 Evans, e-mail: akliu [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 54M - Section 1 - Linear Algebra and Differential EquationsInstructor: J. SethianLectures: TuTh 8:00-9:30am, Room 10 EvansCourse Control Number: 54723Office: 725 Evans, e-mail: sethian [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math H54 - Section 1 - Linear Algebra/Differential Equations - HonorsInstructor: Mark HaimanLectures: TuTh 11:00am-12:30pm, Room 3 EvansCourse Control Number: 54717Office: 771 Evans, e-mail:
Office Hours: W 11:00am-12:30pmPrerequisites: Math 1BSyllabus: Honors version of 54. Basic linear algebra: matrix
arithmetic and determinants. Vector 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.Required Text: Hill, Elementary Linear Algebra (3rd Ed.); Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (7th Ed.).Course Webpage: http://math.berkeley.edu/~mhaiman/mathH54/index.htmlGrading: Policy to be announced.Homework: Weekly homework assignments posted on the course web page.Comments: For more information, please see the course web page.Math 55 - Section 1 - Discrete MathematicsInstructor: J. W. DemmelLectures: TuTh 2:00-3:30pm, Room 100 LewisCourse Control Number: 54756Office: 737 Soda, e-mail: demmel [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 74 - Section 1 - Transition to Upper Division MathematicsInstructor: J. BergLectures: TuTh 12:30-2:00pm, Room 110 BarrowsCourse Control Number: 54779Office: 1020 Evans, e-mail: jberg [at] math [dot] berkeley [dot] eduOffice Hours: TBARequired Text: Smith, Eggen, and St. Andre, A Transition to Advanced Mathematics, 5th edition, Brooks/ColeGrading: Homework 20%, Quizzes 20%, Midterm 25%, Final 35%Comments: The overall goal of Math 74 is two-fold: learning the
skills required to understand mathematical proofs and learning the
fundamental skills of giving (your own) mathematical proof. To this end
the course will cover:
- standard methods of mathematical proof
- basic notions from set theory
- relations
- functions
- advanced topics (decided on by the class)
Course motto: Say what you mean and mean what you say!Math 104 - Section 1 - Introduction to AnalysisInstructor: M. ErdoganLectures: MW 4:00-5:30pm, Room 141 GianniniCourse Control Number: 54831Office: 805 Evans, e-mail: burak [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage: http://math.berkeley.edu/~burak/ma104.htmlGrading:Homework:Comments:Math 104 - Section 2 - Introduction to AnalysisInstructor: A. B. GiventalLectures: TuTh 3:30-5:00pm, Room 71 EvansCourse Control Number: 54834Office: 701 Evans, e-mail: givental [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 104 - Section 3 - Introduction to AnalysisInstructor: H. WuLectures: TuTh 11:00am-12:30pm, Room 71 EvansCourse Control Number: 54837Office: 733 Evans, e-mail: wu [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Two years of calculus, or equivalent.Syllabus: The course will try to cover the Ross text except for the starred sections. Required Text: K. A. Ross, Elementary Analysis: The Theory of Calculus, Springer.Recommended Reading: M. Rosenlicht, Introduction to Analysis, Dover.Grading: Homework 20%, First midterm 10%, Second midterm 20%, Final 50%.Homework: Homework will be assigned each week mainly from the
textbook. Students are required to form study groups to work on the
problems but not to copy from each other.Comments: The most important thing to keep in mind is that this
is a course about proofs from beginning to end, and that these are
proofs of quite subtle properties of the real numbers. For many
beginners, learning about proofs is hard work. Good work ethics is a
pre-requisite to this course, perhaps even more so than a good working
knowledge of calculus. The lectures will follow the textbook closely,
but because the details are already in the well-written text, the class
discussion can afford to put more emphasis on the main ideas and
motivations behind the details. Students are expected to read the
textbook ahead of each lecture and take the initiative to learn the
technical details; hand-holding will be in short supply.Math 104 - Section 4 - Introduction to AnalysisInstructor: M. J. KlassLectures: MWF 11:00am-12:00pm, Room 7 EvansCourse Control Number: 54840Office: 319 Evans, e-mail: klass [at] stat [dot] berkeley [dot] eduOffice Hours: TBARecommended Reading: Davidson and Donsig, Real Analysis with Real Applications, Prentice Hall.Math 104 - Section 5 - Introduction to AnalysisInstructor: Paul R. ChernoffLectures: MW 4:00-5:30pm, Room 71 EvansCourse Control Number: 54843Office: 933 Evans, e-mail: chernoff [at] math [dot] berkeley [dot] eduOffice Hours: M 2:00-3:30pm, F 1:45-3:15pmPrerequisites: Math 53 and 54Syllabus: Primarily Chapters 1-4 of Ross: the real number system;
sequences and series; continuity, convergence and uniform convergence;
sequences and series of functions, power series; introduction to metric
spaces. Brief discussion of theory of differentiation and integration.Required Text: K. Ross, Elementary Analysis: The Theory of Calculus, first edition (13th or later printing advised), published by Springer.Recommended Reading: M. Protter, Basic Elements of Real Analysis, paperback, published by Springer.Grading: 10% homework, 10% quizzes, 40% midterms, 40% final.Homework: Homework will be assigned every week, and due the following Wednesday.Comments: This is a very challenging course. Its content is very
important. Beyond that, a major goal for students is learning to
understand proofs, to create proofs, and to learn to write mathematics
clearly and concisely. I welcome questions and comments.Math 105 - Section 1 - Analysis IIInstructor: Michael ChristLectures: MWF 2:00-3:00pm, Room 85 EvansCourse Control Number: 54846Office: 809 Evans, e-mail: mchrist [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 104Syllabus: (I) Multi-variable differential calculus (following
Spivak chapter 1 and 2). Derivatives and chain rule for functions of
several variables. Implicit and inverse function theorems. (II) Measure and integration (following Stroock, chapters 2, 3, 4.1, 5.1, 6.1-2). Lebesgue measure on the real line. The Lebesgue integral. More general measures and integration. Convergence theorems. Multidimensional Lebesgue measure; Fubini's theorem. Lebesgue spaces, especially the square integrable case. Holder and Minkowski inequalities. If time permits, the course will also include a brief application to Fourier series, the divergence and Green theorems, and change of variables in multiple integrals. The first part of the course (4-5 weeks) will be a rigorous treatment of aspects of calculus in several variables. We'll begin with a review of the definition of derivative for functions beween Euclidean spaces and will formulate and prove the chain rule. This leads up to two big results, the implicit and inverse function theorems, which describe when a mapping between equidimensional spaces is locally invertible, and describes the set of all solutions of to an equation when the dimensions are unequal. The second part will develop the theory of Lebesgue integration. Lebesgue's machinery permits the integration of much more general functions than does Riemann's, and provides superior tools for working with limits, even for Riemann integrable functions. It furnishes the underlying vocabulary and conceptual foundation for probability theory. It provides a framework for the most fundamental fact of Fourier analysis. Lebesgue integration is so fundamental that a special case of it is taught to six year olds under the name ``addition''. We will focus on the basics of Lebesgue theory, emphasizing the Euclidean case and downplaying more abstract aspects. Required Text: (1) M. Spivak, Calculus on Manifolds (W.A. Benjamin, 1965)(2) D. Stroock, A Concise Introduction to the Theory of Integration (3rd edition, Birkhauser)Required Work: Final and two midterm exams. Weekly problem sets. Math 110 - Section 1 - Linear AlgebraInstructor: J. C. HarrisonLectures: MWF 1:00-2:00pm, Room 3109 EtcheverryCourse Control Number: 54849Office: 851 Evans, e-mail: harrison [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 110 - Section 2 - Linear AlgebraInstructor: D. A. GebaLectures: TuTh 3:30-5:00pm, Room 75 EvansCourse Control Number: 54852Office: 837 Evans, e-mail: dangeba [at] math [dot] berkeley [dot] eduOffice Hours: TuTh 2:00-3:30pmPrerequisites: Math 54 or a course with equivalent linear algebra content.Syllabus: Vector spaces, Linear transformations, Matrices,
Systems of linear equations, Determinants, Diagonalization, Inner
product spaces, Canonical formsRequired Text: S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed.Grading: Homework (25%), Midterm (25%), Final (50%)Homework: Assigned on Thursday, due next Thursday. The worst 3
homeworks will not count toward the final grade. No late homeworks. No
make-up exams.Math 110 - Section 3 - Linear AlgebraInstructor: A. GrinshpanLectures: MWF 3:00-4:00pm, Room 71 EvansCourse Control Number: 54855Course Webpage: http://math.berkeley.edu/~tolya/110Math 110 - Section 4 - Linear AlgebraInstructor: J. B. WagonerLectures: TuTh 11:00am-12:30pm, Room 75 EvansCourse Control Number: 54858Office: 899 Evans, e-mail: wagoner [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 110 - Section 5 - Linear AlgebraInstructor: A. GrinshpanLectures: MWF 8:00-9:00am, Room 75 EvansCourse Control Number: 54861Office:Office Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 113 - Section 1 - Introduction to Abstract AlgebraInstructor: M. DevelinLectures: TuTh 3:30-5:00pm, Room 141 GianniniCourse Control Number: 54864Office:Office Hours: TBAPrerequisites: Math 54 or equivalent knowledge of linear algebra.Syllabus: The first week of the class will consist of an
introduction to rigorous proof-writing and to basic numerical facts
about the integers. With these basics in mind, we will move on to group
theory, including factor groups and the structure theorem of Abelian
groups. From there, we tackle rings (especially commutative rings),
ideals, and fields, specifically targeting the example of polynomial
rings; this provides a natural segue into extension fields and Galois
theory. Throughout, we will present many examples of how these
fundamental structures arise throughout mathematics.Required Text: John B. Fraleigh, A First Course in Abstract AlgebraRecommended Reading: Joseph A. Gallian, Contemporary Abstract Algebra (a more example-based approach)Grading: There will be weekly problem sets, two midterms (of
which one will be take-home and one will be in-class), and a final exam.
Details will be discussed on the first day of class.Comments: An hour and a half is a long time for a class. I will
usually interpose a five-minute break in the middle of each lecture
(midterm excepted.)Math 113 - Section 2 - Introduction to Abstract AlgebraInstructor: R. F. ColemanLectures: MWF 11:00am-12:00pm, Room 9 EvansCourse Control Number: 54867Office: 901 Evans, e-mail: coleman [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 113 - Section 3 - Introduction to Abstract AlgebraInstructor: A. KnutsonLectures: TuTh 8:00-9:30am, Room 71 EvansCourse Control Number: 54870Office: 1033 Evans, e-mail: allenk [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 113 - Section 4 - Introduction to Abstract AlgebraInstructor: T.-Y. LamLectures: MWF 11:00am-12:00pm, Room 71 EvansCourse Control Number: 54873Office: 871 Evans, e-mail: lam [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: Sets, mappings, relations and equivalence relations.
Arithmetic of the integers (including prime factorizations, euclidean
algorithms, greatest common divisors and least common multiples).The concept of groups and subgroups. Additive and multiplicative groups. Cyclic groups, permutation groups and matrix groups. Orders of elements, and coset decompositions. The notions of normal subgroups and quotient groups. Basic homomorphism theorems. Elements of ring theory (mostly for commutative rings). Number rings and polynomial rings. The notions of ideals and quotient rings. Elements of field theory: field extensions and field extension degrees. Transitivity formula. Some constructions of finite fields. Required Text: Fraleigh, Abstract Algebra, 7th ed., Addison-WesleyGrading: Letter grades only, based on: 20% Homework, 30% Midterm, 50% Final.Math 113 - Section 5 - Introduction to Abstract AlgebraInstructor: Mariusz WodzickiLectures: TuTh 12:30-2:00pm, Room 4 EvansCourse Control Number: 54876Office: 995 Evans, e-mail: wodzicki [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: A standard first course in Algebra.Required Text: John B. Fraleigh, A First Course in Algebra, Seventh Edition, Addison-Wesley, 2002.Grading: Based on two midterms (February 26, and April 8; 20
percent each), the final exam (30 percent), homework (20 percent), and
occasional quizzes (10 percent).Homework: Weekly, collected every Thursday.Comments: By enrolling in this class you consent to the fact that you can take the Final Exam at the date and time prescribed by the University (May 19, 5-8pm, according to the current final exams schedule). No make-up finals will be given.Math H113 - Section 1 - Introduction to Abstract Algebra - HonorsInstructor: Vera SerganovaLectures: TuTh 12:30-2:00pm, Room 5 EvansCourse Control Number: 54879Office: 709 Evans, e-mail: serganova [at] math [dot] berkeley [dot] eduOffice Hours: TuTh 11:00am-12:00pmPrerequisites: Math 53,54Syllabus: We start with elementary number theory: fundamental
theorem of arithmetic, congruences. Then define rings and ideals, prove
unique factorization for principal ideal domains, discuss in detail the
ring of polynomials. After this we will do some group theory up to Sylow
theorems. Finally, we will study fields and Galois theory.Required Text: Hungerford, Abstract Algebra, An Introduction.Grading: 15% homework, 15% quizzes, 30% midterm, 50% finalHomework: Homework will be assigned on the web every week.Comments: Every second week you will take a 15 minute quizz, there will be one midterm in the beginning of March.Math 114 - Section 1 - Second Course in Abstract AlgebraInstructor: Kenneth A. RibetLectures: TuTh 3:30-5:00pm, Room 9 EvansCourse Control Number: 54882Office: 885 Evans, e-mail: ribet [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 113 or permission of the instructor.Syllabus: Galois theory and associated topics in field theory and group theory.Required Text: Ian Stewart, Galois Theory (third edition).Recommended Reading: Ivars Peterson, The Galois Story, and then Tony Rothman's article on Evariste Galois.Course Webpage: http://math.berkeley.edu/~ribet/114/Grading: Homework 25%, midterms 15% each, final 45%.Homework: Homework will be assigned on the web every class, and due once a week.Comments: See the instructor's course web page for more information.Math 118 - Section 1 - Wavelets and Signal ProcessingInstructor: M. RieffelLectures: MWF 10:00-11:00am, Room 85 EvansCourse Control Number: 54885Office: 811 Evans, e-mail: rieffel [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 53 and 54 or equivalent
But this is an upper-division course. Thus it will be somewhat
more theoretical and less computational than Math 53 and 54. This
reflects the fact that the subject matter is more complex, and
so requires a more solid theoretical foundation in order to
have a good understanding.Syllabus: Over the past 20 years there has been unusually rapid
development of new and more powerful methods for dealing with signals
(audio), images (pictures), changing images (video), and related types
of data. Much of this has been propelled by the emergence of digital
technology. One of the principal new tools that was discovered is that
of wavelets.In this course we will explore some of the central mathematical issues involved in these developments. We will start with classical methods of Fourier analysis (which still form the foundation for the newer methods). We will then explore the newer methods, with emphasis on wavelets. We will keep an eye on such applications as signal compression, noise reduction, image enhancement, detection of unusual events. Our aim will be to develop theoretical and algorithmic tools which are currently the basis for extensive applications in many directions. Required Text: A. Boggess and F. Narcowich, A First Course in Wavelets with Fourier AnalysisThe textbook contains the right subject matter at the right level. Unfortunately it is in many places not clearly written. One task for my lectures will be to try to give a clearer exposition of the subject matter. This means that it will often be important for the students in the class to take good notes of my lectures. Grading: The final examination will count 50% of the course
grade, the two midterm examinations will each count 20%, and the
homework 10%. (But there will be a penalty if few homework assignments
are handed in.) There will be frequent homework assignments. Students who have in mind a specific project which they would like to carry out in connection with this course are welcome to discuss the possibility of having this project count as a substantial part of the course grade, thus decreasing the weight put on the examinations. Math 121A - Section 1 - Mathematical Tools for the Physical SciencesInstructor: F. RezakhanlouLectures: MWF 2:00-3:00pm, Room 75 EvansCourse Control Number: 54888Office: 815 Evans, e-mail: rezakhan [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 121B - Section 1 - Mathematical Tools for the Physical SciencesInstructor: R. E. BorcherdsLectures: TuTh 9:30-11:00am, Room 289 CoryCourse Control Number: 54891Office: 927 Evans, e-mail: reb [at] math [dot] berkeley [dot] eduOffice Hours: WF 1:00-2:30pmPrerequisites: Math 53 and 54Syllabus: Special functions, series solutions of differential equations, partial differential equations, and probability. Required Text: M. L. Boas, Mathematical methods in the physical sciences, second edition, published by Wiley. Course Webpage: http://www.math.berkeley.edu/~reb/121Grading: 30% midterms, 40% homework, 30% final.Homework: Homework will be assigned on the web every week, and due once a week on Tuesdays in class.Comments: The course www page is http://www.math.berkeley.edu/~reb/121. Math 125A - Section 1 - Mathematical LogicInstructor: J. J. SteelLectures: TuTh 3:30-5:00pm, Room 6 EvansCourse Control Number: 54894Office: 717 Evans, e-mail: steel [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 126 - Section 1 - Introduction to Partial Differential EquationsInstructor: J. C. NeuLectures: MWF 11:00am-12:00pm, Room 3109 EtcheverryCourse Control Number: 54897Office: 1051 Evans, e-mail: neu [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Syllabus: Required Text: Recommended Reading: Course Webpage:Grading: Homework: Comments: Math 127 - Section 1 - Mathematical and Computational Methods in Molecular BiologyInstructor: L. PachterLectures: MWF 1:00-2:00pm, Room 31 EvansCourse Control Number: 54900Office: 1081 Evans, e-mail: lpachter [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Syllabus: Required Text: Recommended Reading: Course Webpage:Grading: Homework: Comments: Math 128A - Section 1 - Numerical AnalysisInstructor: P. StinisLectures: TuTh 12:30-2:00pm, Room 213 WheelerCourse Control Number: 54903Office: 887 Evans, e-mail: stinis [at] math [dot] lbl [dot] govOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 128A - Section 2 - Numerical Analysis Instructor: C. K. MillerLectures: TuTh 2:00-3:30pm, Room 247 CoryCourse Control Number: 54909Office: 803 Evans, e-mail: kmiller [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: I consider this to be a senior level analysis
course; besides sophomore calculus and linear algebra (Math53,54) it
would probably be best if you also had Math 104 and 110 (Analysis and
Advanced Linear Algebra), but these are definitely not necessary. The
computing assignments will require that you use some computer language
(Fortran, C, MatLab...); you will be at a real disadvantage if you do
not already know some computer language because of the time needed to
become computer literate.Syllabus: This course might well be called "advanced calculus
made useful in the age of computers". Ch3 polynomial interpolation, Ch4
polynomial approximation, Ch5 numerical quadrature, Ch6 solution of
ordinary differential equations, Ch1.1-8.3 direct methods for solution
of linear equations Ax = b, Ch2.2,2.10,2.11 solutions to nonlinear
equations and systems by Newton's method, etc.Required Text: Kendall Atkinson, Introduction to Numerical Analysis, 2nd edition, (1989, Wiley & Sons)Grading: Computer lab with programming assignments due
approximately every other week (20%), homework due every Thursday in
class (20%), one midterm 8th Thurs (20%), final exam (40%) exam group 2.Math 128A - Section 3 - Numerical AnalysisInstructor: M. ZworskiLectures: MWF 1:00-2:00pm, Room 3 EvansCourse Control Number: 54915Office: 897 Evans, e-mail: zworski [at] math [dot] berkeley [dot] eduOffice Hours: MW 11:00am-12:00pm, F 2:00-3:00pm, or by appointmentPrerequisites: Math 53 and 54 or equivalentSyllabus: The course will cover basic theory and practical methods for solving the fundamental problems of computational science.
- Calculus review: Taylor series with remainder. Error analysis: roundoff and truncation errors in numerical approximations and computer arithmetic. Algorithms and convergence (Chapter 1).
- Nonlinear equations: bisection, fixed point, Newton and secant methods. Zeros of polynomials and Muller's method (Chapter 2).
- Polynomial interpolation. Lagrange vs. Hermite. Divided differences. Splines (Chapter 3).
- Numerical differentiation and integration: basic rules, extrapolation, adaptive and Gaussian strategies (Chapter 4).
- Initial value problems for ordinary differential equations. Runge-Kutta and Adams methods. Introduction to stiff problems (Chapter 5).
- Matrix computations: linear systems, LU factorization (Chapter 6).
- Approximation theory: least squares, Householder QR factorization, and orthogonal polynomials (Sections 7.1, 9.3 and 8.1-2).
Required Text: R. L. Burden and J. D. Faires, Numerical Analysis, 7th edition, Brooks-Cole, 2001.Course Webpage: http://www.math.berkeley.edu/~zworski/128/Grading: Grades will be based on the best 10 problem sets (40%),
midterm exam (20%), and final exam (40%). The problem sets will be due
every Friday; late submissions cannot be accepted after solutions have
been distributed. The final exam will take place on 5/22/04, 12:30-3:30
PM, and will cover the entire course material.Math 128B - Section 1 - Numerical AnalysisInstructor: W. M. KahanLectures: MWF 3:00-4:00pm, Room 85 EvansCourse Control Number: 54921Office: 863 Evans, e-mail: wkahan [at] cs [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 135 - Section 1 - Introduction to Theory SetsInstructor: M. RatnerLectures: MWF 1:00-2:00pm, Room 2 EvansCourse Control Number: 54927Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: None, but an ability to understand mathematical proofs.Syllabus: Zermelo-Frankel axiom system, relations and functions,
the theory of natural numbers, cardinal numbers and the Axiom of Choice,
well orderings and ordinals, transfinite induction, alephs.Required Text: Enderton, Elements of Set TheoryGrading: The grade will be based 15% on the homework, 20% on quizzes, 25% on a midterm and 40% on the final.Math 140 - Section 1 - Metric Differential GeometryInstructor: Bjorn PoonenLectures: MWF 8:00-9:00am, Room 85 EvansCourse Control Number: 54930Office: 879 Evans, e-mail: poonen [at] math [dot] berkeley [dot] eduOffice Hours: (starting 1/16/04) MWF 9:30-10:30am, T 1:30-2:30pm, or by appointment (tentative)Prerequisites: Math 104 and Math 110, or permission of
instructor. (Ignore what the general catalog says.) Students having
had Math 54 but not Math 110 can still take the course, but should be
prepared to spend a little extra time learning some topics such as
diagonalization of quadratic forms.Syllabus: This course uses multivariable calculus to study the
geometry of curves and surfaces, with the emphasis on the latter. We
will cover most of Chapters 1--4 in the do Carmo text. Some of the
highlights will be Gauss's "Theorem Egregium", which states that the
Gaussian curvature of a surface is intrinsic (independent of the way the
surface is embedded), and the Gauss-Bonnet theorem, which relates area
on a surface to its curvature. In order to focus on geometric ideas, we
will not develop the technical machinery of differential forms in full
generality.Required Text: do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976.Recommended Reading: Struik, Lectures on classical differential geometry, second edition, Dover, 1988. (a little old-fashioned, but good and inexpensive)Course Webpage: http://math.berkeley.edu/~poonen/math140.htmlGrading: 35% homework, 15% first midterm, 15% second midterm, 35%
final. Each homework grade below the weighted average of your final
and midterm grades will be boosted up to that average. The course grade
will be curved.Homework: There will be weekly assignments due at the beginning of class each Monday.Comments: The up-to-date course website is at http://math.berkeley.edu/~poonen/math140.htmlMath 172 - Section 1 - CombinatoricsInstructor: A. YongLectures: TuTh 3:30-5:00pm, Room 7 EvansCourse Control Number: 54933Office: 1035 Evans, e-mail: ayong [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: No specific prerequisites, however exposure to
courses such as abstract algebra (Math 113) will be helpful. This will
be an advanced undergraduate course on combinatorics.Syllabus: Combinatorial methods are applied widely throughout
mathematics; this class will be an introduction to various aspects of
combinatorial theory. Some sample topics include: generating series,
graph theory, poset theory, combinatorics of trees (parking functions,
Catalan numbers, matrix-tree theorem), tableaux and the symmetric group
(permutation statistics, the Schensted correspondence, hook-length
formula), partitions, symmetric functions (elementary, monomial,
power-sum, Schur polynomials, Kostka coefficients, the
Littlewood-Richardson rule).Required Text: R. P. Stanley, Enumerative Combinatorica IRecommended Reading: R. P. Stanley, Enumerative Combinatorics II; D. Stanton and D. White, Constructive combinatoricsCourse Webpage: http://math.berkeley.edu/~ayong/teaching_next_term.htmlGrading: Homework (60%), Final Exam (40%)Homework: Students will be expected to do several problem sets.Exams: TBA. The final exam will be cumulative, and the date will be set by the registrar.Comments: Send me an email about you and your (mathematical) background so I can get to better know you.Math 185 - Section 1 - Introduction to Complex AnalysisInstructor: George M. BergmanLectures: MWF 10:00-11:00am, Room 70 EvansCourse Control Number: 54936Office: 865 Evans, e-mail: gbergman [at] math [dot] berkeley [dot] eduOffice Hours: Tu 1:30-2:30pm, Th 10:30-11:30am, F 4:00-5:00pmPrerequisites: Math 104 or consent of the instructor.Syllabus: We will cover the contents of Stewart and Tall (see below), and, if time allows, the Riemann Mapping Theorem as well.Required Text: Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.Recommended Reading: Donald Sarason, Notes on Complex Function Theory, pub. Henry Helson, Berkeley.Grading: Homework, 25%; two Midterms, 15% and 20%; Final Exam, 35%; regular submission of the daily question (see below), 5%.Homework: Weekly.Comments: The subject of this course is differentiable functions
of a complex variable. You might wonder: Is this anything more than
Math 1A/1B/53/104 in a new setting? In fact, we shall eventually see
that the property of differentiability on the complex plane, when
properly defined, has consequences totally unexpected from the real
case.I don't like the standard lecture system where students spend the hour copying the contents of the course from the blackboard into their notebooks. Hence I will assign readings in the text, and conduct the class on the assumption that you have done this reading and thought about the what you've read. In lecture I may go over key proofs from the reading, clarify difficult concepts, give alternative perspectives, discuss points to watch out for in the next reading, etc.. On each day for which there is a reading, each student is required to submit, by e-mail or in writing, a question
on the reading. (If there is nothing in the reading that you don't
understand, you can submit a question marked "pro forma", together with
its answer.) I try to incorporate answers to students' questions into
my lectures; when I can't do this I may instead answer your question by
e-mail. More details on this and other matters will be given on the
course handout distributed in class the first day, and available on the
door to my office thereafter.Math 185 - Section 2 - Introduction to Complex AnalysisInstructor: Bjorn PoonenLectures: MWF 3:00-4:00pm, Room 213 WheelerCourse Control Number: 54939Office: 879 Evans, e-mail: poonen [at] math [dot] berkeley [dot] eduOffice Hours: (starting 1/16/04) MWF 9:30-10:30am, T 1:30-2:30pm, or by appointment (tentative)Prerequisites: Math 104.Syllabus: This is a standard introduction to the theory of
analytic functions of one complex variable. The main topics are contour
integration, Cauchy's Theorem, power series and Laurent series
expansions of analytic functions, classification of isolated
singularities, and the residue theorem with its applications to
evaluation of definite integrals. If time permits, we will also discuss
the argument principle and Rouch'e's Theorem, analytic continuation,
harmonic functions, and conformal mapping (including fractional linear
transformations).Required Text: Sarason, Notes on complex function theory, published by Henry Helson, Berkeley.Recommended Reading: The Sarason text is concise and without many
figures or worked examples, so you are encouraged to look also at at
least one other text, such as one of the following:
- Marsden and Hoffman,
*Basic complex analysis*, 3rd edition, W. H. Freeman, 1998. (I used this book the last time I taught Math 185, but decided against it this time, since it now costs $153, nine times as much as Sarason's book!) - Stewart and Tall,
*Complex analysis*, Cambridge University Press, 1983. - Lang,
*Complex analysis*, 4th edition, Springer-Verlag.
Course Webpage: http://math.berkeley.edu/~poonen/math185.htmlGrading: 35% homework, 15% first midterm, 15% second midterm, 35%
final. Each homework grade below the weighted average of your final
and midterm grades will be boosted up to that average. The course grade
will be curved.Homework: There will be weekly assignments due at the beginning of class each Friday.Comments: The up-to-date course website is at http://math.berkeley.edu/~poonen/math185.htmlMath 185 - Section 3 - Introduction to Complex AnalysisInstructor: A. YongLectures: TuTh 12:30-2:00pm, Room 9 EvansCourse Control Number: 54942Office: 1035 Evans, e-mail: ayong [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Elementary real analysis (Math 104).Syllabus: (More details to come): Basic properties of complex
numbers, the complex plane. Power series, differentiation, integration,
Cauchy's theorem, Taylor series, residues and introduction to conformal
mappings.Required Text: J. Brown and R. Churchill, Complex Variables and ApplicationsRecommended Reading: D. Sarason, Notes on Complex Function theory; E. B. Saff and A. D. Snider, Fundamentals of Complex analysisCourse Webpage: http://math.berkeley.edu/~ayong/teaching_next_term.htmlGrading: Homework (25%), Midterm (25%), Final exam (50%)Exams: TBA. The final exam will be cumulative, and the date will be set by the registrar.Homework: Assignments will be posted weekly on the course webpage and are due in class the following week.Comments: Send me an email about you and your (mathematical) background so I can get to better know you.Math H185 - Section 1 - Complex Analysis - HonorsInstructor: M. RatnerLectures: MWF 11:00am-12:00pm, Room 31 EvansCourse Control Number: 54945Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 104Syllabus: Analytic functions, Cauchy's Integral Theorem, power
series, Laurent series. Singularities of analytic functions. The
Residue Theorem with applications to definite integrals. Open mapping
Theorem, Maximum Modulus Theorem, conformal mappings.Required Text: Conway, Functions of One Complex Variable, 2nd ed., Springer-VerlagGrading: The grade will be based 40% on the final, 25% on a midterm, 20% on quizzes and 15% on weekly homework.Homework: Weekly.Comments: This class is going to be difficult and demanding.
Only students with strong mathematical inclination and motivation should
take it.Math 189 - Section 1 - Mathematical Methods in Classical and Quantum MechanicsInstructor: A. B. GiventalLectures: TuTh 9:30-11:00am, Room 5 EvansCourse Control Number: 54948Office: 701 Evans, e-mail: givental [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 195 - Section 1 - Special Topics in MathematicsInstructor: Lester DubinsLectures: MWF 2:00-3:00pm, Room 87 EvansCourse Control Number: 55758Office: 869 Evans, e-mail: lester [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: The Problem. Imagine yourself at a casino with $1,000.
For some reason, you desperately need $10,000 by morning; anything less
is worth nothing for your purpose. What should you do? The only thing
possible is to gamble away your last cent, if need be, in an attempt to
reach the target sum of $10,000. We have all been taught how wrong and
futile it is to gamble, especially when short of funds. The question is
how to play, not whether.Any policy of compounding bets that are subfair to you must decrease your expected wealth. Consequently, no matter how you play, your chance of converting $1,000 into $10,000 will be less than 1/10. How close to 1/10 can you make it and by what strategy? That is the sort of problem we will attack. It is the instructor’s wish that the course be conducted in the spirit of a seminar, indeed a research seminar, and that there be considerable student participation. Preferably, each student will, in consultation with the instructor, prepare some material for presentation to the class. Required Text: TBAMath 198 - Section 3 - Undergraduate Seminar in Applied Mathematics (1 Unit Class)Instructor: L. C. Evans/Frances Hammock (For questions, contact Frances at hammockf [at] hotmail [dot] com.)Lectures: Th 4:00-5:00pm, Room 247 Cory HallCourse Control Number: 55749 (Note that this is a 1-unit class.)Office Hours: TBAPrerequisites: Math 1A, 1B, 53, 54Syllabus: Seminar in applied math. There will be an hour of lecture by a speaker once a week on some topic in applied math.Grading: P/NPHomework: One hour of reading per week.Comments: Ever wondered what you can do with a background in
applied mathematics? Come find out from some of the leading minds in
biotechnology, computer science, economics, astrophysics....the list
goes on! Math is used everywhere - to learn more about it come check out
this seminar.Speakers from around campus and from LBL will be giving undergraduate level talks about their work. Some speakers in the past have included: Stefano Dellavi, Economics Richard Plant, Agriculture Richard Muller, Physics Math 202B - Section 1 - Introduction to Topology and AnalysisInstructor: D. E. SarasonLectures: MWF 8:00-9:00am, Room 71 EvansCourse Control Number: 55020Office: 779 Evans, e-mail: sarason [at] math [dot] berkeley [dot] eduOffice Hours: MW 9:30-11:30amPrerequisites: Math 202ASyllabus: MEASURE AND INTEGRATION. Sigma-rings, monotone
families, product measures, theorems of Tonelli and Fubini, signed
measures, Hahn and Jordan decompositions, absolute continuity of
measures, Radon-Nikodym theorem, Lebesgue decomposition, Lebesgue
differentiation theorem, L^{p} spaces and their duality,
measures on locally compact spaces, Riesz representation theorem.
FUNCTIONAL ANALYSIS. Topological vector spaces, Banach spaces, principle
of uniform boundedness, linear transformations, open mapping theorem,
dual spaces, Hahn-Banach theorem, weak and weak-star topologies,
Alaoglu's theorem, Krein-Milman theorem, possible additional topics.Recommended Reading: H. L. Royden, Real Analysis; W. Rudin, Functional AnalysisCourse Webpage: http://www.math.berkeley.edu/~sarason/Class_Webpages/Math202B_S1.html. Grading: There will be no exams. The course grade will be based on homework.Homework: Homework will be assigned weekly and will be carefully graded.Comments: The lectures will not follow the texts closely. They
will be self-contained except for routine details. The syllabus near
the beginning could be modified slightly, depending on what is covered
in 202A in the fall. Math 204B - Section 1 - Ordinary and Partial Differential EquationsInstructor: J. C. NeuLectures: MWF 1:00-2:00pm, Room 47 EvansCourse Control Number: 55023Office: 1051 Evans, e-mail: neu [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 208 - Section 1 - C*-AlgebrasInstructor: M. RieffelLectures: MWF 9:00-10:00am, Room 5 EvansCourse Control Number: 55026Office: 811 Evans, e-mail: rieffel [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: The basic theory of bounded operators on Hilbert
space and of commutative Banach algebras. (Math 206 is more than
sufficient. Self-study of sections 3.1-2, 4.1-4 of "Analysis Now" by G.
K. Pedersen will be sufficient.)Syllabus: The theory of operator algebras grew out of the needs
of quantum mechanics, but by now also has strong interactions with many
other areas of mathematics. Operator algebras are very profitably
viewed as "non-commutative (algebras "of functions" on) spaces", thus
"quantum spaces". As a rough outline, we will first develop the basic
facts about C*-algebras ("non-commutative locally compact spaces") We
will then briefly look at "non-commutative vector bundles" and K-theory
("noncommutative algebraic topology"). Finally we will glance at
"non-commutative differential geometry" (e.g. cyclic homology as
"noncomutative deRham cohomology"). But I will not assume prior
knowledge of algebraic topology or differential geometry, and we are
unlikely to have time to go into these last topics in any depth.I will discuss a variety of examples, drawn from dynamical systems, group representations and mathematical physics. But I will somewhat emphasize examples which go in the directions of my current research interests, which involve certain mathematical issues which arise in string theory and related high-energy physics. Thus one thread which will run through the course will be to see what the various concepts look like for quantum tori, which are the most accessible non-commutative differential manifolds. In spite of what is written above, the style of my lectures will be to give motivational discussion and complete proofs for the central topics, rather than just a rapid survey of a large amount of material. Required Text: Recommended Reading: None of the available textbooks follows closely the path which I will take through the material. The closest is probably:"C*-algebras by Example", K. R. Davidson, Fields Institute Monographs, A. M. S. It does discuss a great collection of important examples. Grading: I plan to assign several problem sets. Grades for the
course will be based on the work done on these. But students who would
like a different arrangement are very welcome to discuss this with me.Math 212 - Section 1 - Complex VariablesInstructor: Michael ChristLectures: MWF 11:00am-12:00pm, Room 65 EvansCourse Control Number: 55029Office: 809 Evans, e-mail: mchrist [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: 202AB ( graduate real analysis) and 185 (undergraduate complex analysis) or equivalents. Syllabus: The course will be an introduction to multivariable
complex analysis, suitable for nonspecialists, with emphasis on
scalar-valued holomorphic functions, pseudoconvexity/domains of
holomorphy, plurisubharmonic functions and their uses, solution of the
Cauchy-Riemann equations, and Cousin problems. The syllabus will be
roughly Chapters I-IV and VI of Hormander's standard text, plus
additional topics. Coherent analytic sheaves will be introduced but not
discussed in detail. If time permits, we'll briefly discuss the Bergman projection for bounded domains, and the Szego projection operators associated to ample line bundles over complex manifolds, at the end of the term. Hormander's text concisely treats the basics but cannot begin to survey all aspects of a large subject. Other sources will be placed on reserve for consultation. Required Text: L. Hormander, An Introduction to Complex Analysis in Several Variables (3rd edition)Required Work: Work problem sets, or read and write a brief report on an article or book chapter.Math 215B - Section 1 - Algebraic TopologyInstructor: M. HutchingsLectures: TuTh 11:00am-12:30pm, Room 65 EvansCourse Control Number: 55032Office: 923 Evans, e-mail: hutching [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: In 215a we studied the fundamental group, homology, and
cohomology, from Chapters 1-3 of Hatcher's book "Algebraic Topology".
(We skipped over some of what Hatcher did, and we did a few things which
Hatcher didn't, particularly in connection with differential topology.
Bredon's book "Topology and Geometry" is a good reference for those
additional topics.)With the basics of algebraic topology established, in 215b we will study a selection of more advanced topics, which nonetheless are in my opinion very important and useful in geometry and topology. I will emphasize applications to the geometry of smooth manifolds. We will do as much of the following as time permits (not in exactly this order): - Higher homotopy groups and obstruction theory.
- Bundles and characteristic classes.
- Basic Morse theory and applications to differential topology.
- Spectral sequences.
- As time permits, possible additional topics to be determined, such as:
- Secondary invariants such as Reidemeister torsion, Alexander polynomial, Massey products, etc.
- Equivariant cohomology.
Required Text: Milnor and Stasheff, Characteristic Classes, PrincetonRecommended Reading: Bott and Tu, Differential Forms in Algebraic Topology, Springer-VerlagCourse Webpage: http://math.berkeley.edu/~hutching/teach/215b/Homework: I will suggest some homework exercises; these will not be graded, although I am happy to discuss how to solve them.Comments: The only course requirement will be to write an
expository paper on some topic of interest in algebraic topology. This
should be about 10 pages. Inspired by Prof. Weinstein, I will ask
students to referee each other's papers and then revise them.Math 222B - Section 1 - Partial Differential EquationsInstructor: Lawrence C. Evans Lectures: TuTh 12:30-2:00pm, Room 31 EvansCourse Control Number: 55035Office: 907 Evans, e-mail: evans [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Some knowledge of Sobolev spaces.Syllabus: 1. Second-order elliptic equations2. Parabolic and hyperbolic equations 3. Calculus of variations methods 4. Viscosity solutions of Hamilton--Jacobi equations Required Text: Lawrence C. Evans, Partial Differential Equations, AMSGrading: 25% homework, 25% midterm, 50% finalHomework: I will assign a homework problem, due in one week, at the start of each class.Math 224B - Section 1 - Methods of Mathematical PhysicsInstructor: F. A. GrunbaumLectures: TuTh 8:00-9:30am, Room 31 EvansCourse Control Number: 55038Office: 903 Evans, e-mail: grunbaum [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: The beauty and the challenge of this class is that of
presenting material that has been around in one form or another for a
long time in a way that is fresh and attractive to students. I will try
hard to walk this narrow line.In the first semester I will try to stick to the material in the book: intuitive ideas about Green functions, Fourier analysis and distribution theory, one dimensional boundary value problems, Hilbert spaces, Operator theory and integral equations. Spectral theory of second order differential operators. In the second semester, I will have to do some unfinished material from the first one, and then we will see how this classical stuff plays a crucial role in problems of current interest. Among the topics that I would like to touch upon, either during the main part of the course or towards the end are Tridiagonal matrices, i.e., discrete versions of second order differential operators. The classical orthogonal polynomials: Hermite, Laguerre, Bessel and Jacobi. Spectral theory for finite, semiinfinite and doubly infinite tridiagonal matrices. The Toda equation for coupled unharmonic oscillators. Beyond orthogonal polynomials. The relativistic Toda chain and Laurent orthogonal polynomials. Birth and death processes. Uses of the spectral theorem in examples. The Schroedinger equation, including the main examples: the free particle, the harmonic oscillator, the hydrogen atom etc. to give a concrete discussion of H. Weyl's limit point-limit circle classification of separated boundary conditions. Regular and irregular singular points for linear differential equations in the complex plane. Gauss' linear second order hypergeometric equation. The Bessel equation. Sampling, aliasing and all that. The relation between the Fourier transform, the Fourier series and the DFT. Heisenberg's principle. The problem of double concentration. Some basic ideas behind the construction of "wavelet basis". Random walks on integer lattices, discrete time and continuous time. The issue of recurrence in different dimensions. Brownian motion. Integration in function space, the Feynman-Kac formula. Paul Levy's arcsine law. The scattering transform as an important nonlinear version of the Fourier transform. Reflectionless potentials, solitons, recovering a potential from scattering data, the Korteweg-de Vries equation. Some matrix valued versions of the Schroedinger equation. Required Text: I. Stakgold, Green's functions and boundary value problems. (I consider this book as required, since it contains a lot of the basic material.)Recommended Reading: The following is a partial list of recommended books, covering different aspects of the class.G. Lamb, Elements of soliton theory.M. Toda, Theory of nonlinear lattices.S. Karlin and H. Studden, A second course in Stochastic Processes.Grading: It will be based solely on homework.Homework: A weekly assignment.Math 225B - Section 1 - MetamathematicsInstructor: J. SteelLectures: TuTh 11:00am-12:30pm, Room 72 EvansCourse Control Number: 55041Office: 717 Evans, e-mail: steel [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 228B - Section 1 - Numerical Solution of Differential EquationsInstructor: John StrainLectures: TuTh 11:00am-12:30pm, Room 5 EvansCourse Control Number: 55047Office: 1099 Evans, e-mail: strain [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: 128A or equivalent.Syllabus: Math 228B will survey the theory and practice of finite
difference methods for parabolic, hyperbolic and elliptic partial
differential equations.Topics will include: Basic linear partial differential equations and schemes. Convergence, stability and consistency. Practical stability analysis. ADI schemes. Numerical boundary conditions. GKSO theory. Dispersion and dissipation. Theory of nonlinear hyperbolic conservation laws. Entropy conditions and TVD schemes. Relaxation and multigrid for linear elliptic equations. Required Text: 1. J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, 1995.2. J. W. Thomas, Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations, Springer, 1999.Course Webpage: http://math.berkeley.edu/~strain/228b.S04/ Grading: Based on weekly homework and one or two projects.Homework: Will be posted on the class web site, and due once a week.Math 242 - Section 1 - Symplectic GeometryInstructor: F. RezakhanlouLectures: MWF 3:00-4:00pm, Room 5 EvansCourse Control Number: 55050Office: 815 Evans, e-mail: rezakhan [at] math [dot] berkeley [dot] eduOffice Hours: MWF 2:00-3:00pmPrerequisites: Some familiarity with differential forms and manifolds.Syllabus: Hamiltonian systems appear in conservative problems in
mechanics as in celestial mechanics but also in the statistical
mechanics governing the motion of atoms and molecules in matter. The
discoveries of last century have opened up new perspectives for the old
field of Hamiltonian systems and let to the formation of the new field
of symplectic geometry. In the course, I will give a detailed acount of
some basic methods and results in symplectic geometry and its
application to physics and other fields of mathematics.Here is an outline of the course: - Sympletic linear algebra. Quadratic Hamiltonians.
- Symplectic manifolds, cotangent bundles, Kahler manifold. Darboux's theorem. Contact manifolds.
- Variational problems. Generation functions. Lagrange submaifolds. Minimax principle.
- Nother principle. Arnold-Jost-Liouville Theorem. Action-angles coordinates.
- Hamiltonian mechanics. Momentum map.
- Gromov's squeezing theorem. Viterbo's existence of periodic orbits. Sympletic capacities.
Grading: There will be some homework assignments.Math 249 - Section 1 - Algebraic CombinatoricsInstructor: Mark HaimanLectures: TuTh 2:00-3:30pm, Room 51 EvansCourse Control Number: 55053Office: 771 Evans, e-mail:
Office Hours: W 11:00am-12:30pmPrerequisites: Math 250A or equivalent algebra background.Syllabus: Introduction to combinatorics at the graduate level,
covering four general areas: (I) enumeration (ordinary and exponential
generating functions), (II) order (posets, lattices, incidence algebas),
(III) geometric combinatorics (hyperplane arrangements, simplicial
complexes, polytopes), (IV) symmetric functions, tableaux and
representation theory.Required Text: Richard P. Stanley, Enumerative Combinatorics, Vols. I & II. Cambridge Univ. Press 1999, 2000.Recommended Reading: William Fulton, Young Tableaux. London Math. Soc. Student Texts, Vol. 35, Cambridge Univ. Press 1997.Course Webpge: http://math.berkeley.edu/~mhaiman/math249/index.htmlGrading: Based on periodic homework assignments.Comments: For more information, please see the course web page.Math 250B - Section 1 - Multilinear Algebra and Further TopicsInstructor: Paul VojtaLectures: MWF 12:00-1:00pm, Room 5 EvansCourse Control Number: 55056Office: 883 Evans, e-mail: vojta [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 250ASyllabus:The course will cover the following chapters or parts of the text:
I haven't taught from this book before, so I don't know whether the above is too ambitious or not. Required Text: Eisenbud, Commutative algebra with a view toward algebraic geometry, SpringerCourse Webpage: http://math.berkeley.edu/~vojta/250b.htmlGrading: Grades will be based on homework assignments, including a final problem set in place of a final exam.Homework: Homework will be assigned approximately every two weeksComments:
- The course title does not accurately reflect recent practice in teaching this course. There really isn't much (if any) multilinear algebra taught in 250b anymore.
- As part of Springer's Yellow Sale, the text is available for $33.50 through December 31. The regular price is $49.95.
Math 254B - Section 1 - Number TheoryInstructor: Kenneth A. RibetLectures: TuTh 12:30-2:00-pm, Room 2 EvansCourse Control Number: 55059Office: 885 Evans, e-mail: ribet [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites: Math 254A, although usually required, will not be an absolute prerequisite this year.Required Text: Henri Cohen, A course in computational number theory, Graduate Texts in Mathematics volume 138, ISBN 3-540-55640-0Grading: Grades will be based on homework, which will be assigned periodically.Math 256B - Section 1 - Algebraic GeometryInstructor: A. E. OgusLectures: TuTh 2:00-3:30pm, Room 85 EvansCourse Control Number: 55062Office: 877 Evans, e-mail: ogus [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: This semester's course will continue the study of the
foundations of the theory of schemes. We should also have time to
illustrate the general theory with applications to the study of curves
and surfaces. Most of this material can be found in selected sections of
Chapters 3, 4 and 5 of Hartshorne's book. The main new technical tool
we shall need is cohomology, and we will begin with by following, more
or less, Hartshorne's treatment in Chapter III. Students need not have
taken my own version of 256A, but will need to know the language of
schemes and should have a solid foundation in linear algebra (i.e, the
theory of R-modules.)Math 261B - Section 1 - Lie GroupsInstructor: N. ReshetikhinLectures: MWF 10:00-11:00am, Room 5 EvansCourse Control Number: 55065Office: 915 Evans, e-mail: reshetik [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 265 - Section 1 - Differential TopologyInstructor: Rob KirbyLectures: TuTh 8:00-9:30am, Room 5 EvansCourse Control Number: 55068Office: 919 Evans, e-mail: kirby [at] math [dot] berkeley [dot] eduOffice Hours: TBARecommended Reading: Papers of Ozsvath and SzaboComments: The course will have as its goal the Heegaard Floer
homology of 3-manifolds discovered by Ozsvath and Szabo during the last 3
years. This goal will justify excursions into various topics in low
dimensional manifold theory, including some contact and symplectic
topology.Math 274 - Section 1 - Topics in AlgebraInstructor: T. Y. LamLectures: MWF 9:00-10:00am, Room 31 EvansCourse Control Number: 55071Office: 871 Evans, e-mail: lam [at] math [dot] berkeley [dot] eduOffice Hours: TBASyllabus: This course will be an introduction to the algebraic
theory of quadratic forms, based on the the text I wrote on the subject
in the Benjamin Series. The course will be pretty self-contained; a
thorough grounding in Math 250AB is probably all that is required for
following the course.The algebraic theory of quadratic forms was created by E. Witt in his influential 1937 paper in the Crelle Journal. Witt's idea was to use the geometric language of symmetric inner product spaces to study quadratic forms over arbitrary fields, at least in the case of characteristic not 2. More importantly, Witt introduced the viewpoint of dealing with the totality of nonsingular quadratic forms as a whole, which was a substantial departure from earlier treatments of "one form at a time" by his predecessors in quadratic form theory. The legacy of Witt's 1937 paper is the famous "Witt ring" W(F) of quadratic forms over a field F. It is of historic interest to note that Witt's formation of W(F) preceded by about 20 years the Atiyah-Hirzebruch formation of the K-group of vector bundles over a space, and Grothendieck general formation of the Grothendieck group of an abstract category. The course builds on Witt's theory, and combines it with the classical Artin-Schreier theory of formally real fields to yield the Local-Global Principle of quadratic forms over such fields. The theory of Pfister forms will be presented in the second half of the course: we'll learn, for instance, how to prove that the product of two sums of 1024 squares over a field is still a sum of 1024 squares, and to prove that, if -1 is a sum of 1023 squares in a field, then it is actually a sum of 512 squares in the same field! Of course, we will not be able to reach the "high end" of this theory, such as Voevodsky's Fields Medal work on Milnor's Conjecture in Milnor's K-theory of fields, or even the proof of the Merkuryev-Suslin Theorem on norm residue symbols. However, the course will provide firm foundations for some of these modern developments. There will be occasional homework assignments, but there will be no exams. Students are encouraged to take the course on N/NP basis if possible. Required Text: T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin-Addison Wesley, 1982Math 275 - Section 1 - Topics in Applied Mathematics - Scaling,
Physical Similarity: Dimensional Analysis and the Renormalization GroupInstructor: Grigory BarenblattLectures: TuTh 9:30-11:00am, Room 65 EvansCourse Control Number: 55074Office: 735 Evans, e-mail: gibar [at] math [dot] berkeley [dot] eduOffice Hours: TuTh 11:15am-12:50pmPrerequisites: No special knowledge of advanced mathematics
and/or continuum mechanics will be assumed -- all needed concepts and
methods will be explained on the spot.Syllabus: Similarity methods play an important and ever growing
role in applied mathematics, including computing, and engineering
science. Sometimes a simple application of dimensional analysis leads
to results of extreme importance (e.g., Taylor-von Neumann scaling laws
for intense blast waves; Kolmogorov scaling laws in turbulence). More
recent concepts such as the renormalization group, fractals, etc., are
in fact closely related to dimensional analysis and physical similarity.The proposed course will give a systematic presentation of similarity methods, including dimensional analysis, physical similarity, complete and incomplete similarity, intermediate asymptotics, the renormalization group and various types of self-similar solutions and scaling laws. The presentation will be illustrated by many examples, including the examples from turbulence and fracture. The typical difficulties arising in using similarity methods will be illustrated by examples. Required Text: Barenblatt, G. I., Scaling (Cambridge University Press, 2003)Recommended Reading: 1. Barenblatt, G. I., Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, 1996)2. Goldenfeld, N. D., Lectures on Phase Transitions and Renormalization Group (Addison-Wesley, 1992) 3. Chorin, A. J., and Marsden, J. E., A Mathematical Introduction to Fluid Mechanics (Springer, 1990)4. Landau, L.D. and Lif*bleep*z, E.M., Theory of Elasticity (Pergamon Press, London, New York, 1986)5. Landau, L.D. and Lif*bleep*z, E.M., Fluid Mechanics (Pergamon Press, London, New York, 1987)Homework: There will be no systematic homework.Comments: In the end of the course the instructor will give a
list of 10 topics. Students are expected to come to the exam having an
essay (5-6 pages) concerning one of these topics which they have chosen.
They have to answer the detailed questions concerning this topic.
After that general questions without details concerning the course will
be asked.Math 276 - Section 1 - Topology of Real Algebraic VarietiesInstructor: Dr. Oleg Viro (Uppsala Universitet)Lectures: TuTh 9:30-11:00am, Room 123 DwinelleCourse Control Number: 55076Office:Office Hours: TBAPrerequisites: A familiarity with elementary homology theory and Poincare duality.Syllabus: We will study real algebraic varieties from the most
qualitative viewpoint, that of topology. Together with topology of the
set of real points, we are forced to study topology of the set of
complex points, and the way the former sits in the latter. We will learn
to look at the complexification from the real point of view and think
about the "real stuff" in the framework of its complexification. Even
the simplest varieties, such as circles and lines, when treated in this
way, may have surprises.We will start with plane curves and study restrictions on the topology of the set of real points for curves of a given degree and its position in the complexification. Then a similar investigation will be undertaken for surfaces and varieties of higher dimensions. We will study and apply tropical geometry for the explicit construction of varieties with interesting topology. No preliminary knowledge of algebraic geometry is expected; all the required geometric objects will be considered from scratch. However, a familiarity with elementary homology theory and Poincare duality would be helpful for understanding of some of the proofs. The lectures are intended to provide a broad introduction to a nice research area which will be the topic of the Spring 2004 MSRI program: Topological Aspects of Real Algebraic Geometry. Math 277 - Section 1 - Topics in Differential GeometryInstructor: A. K. LiuLectures: MWF 2:00-3:00pm, Room 5 EvansCourse Control Number: 55077Office: 905 Evans, e-mail: akliu [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments:Math 300 - Section 1 - Teaching WorkshopInstructor: O. H. HaldLectures: Course Control Number: 55668Office: 875 Evans, e-mail: hald [at] math [dot] berkeley [dot] eduOffice Hours: TBAPrerequisites:Syllabus:Required Text:Recommended Reading:Course Webpage:Grading:Homework:Comments: |