# Spring 2004

Math 1A - Section 1 - Calculus
Instructor: Zvezdelina Stankova
Lectures: TuTh 8:00-9:30am, Room 2050 Valley Life Science
Course Control Number: 54303
Office: 913 Evans, e-mail: stankova [at] math [dot] berkeley [dot] edu
Office Hours: TuTh 9:40-11:00am
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. 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 Notes
Course Webpage: http://math.berkeley.edu/~stankova/
Grading: 15% quizzes, 25% each midterm, 35% final
Homework: 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 - Calculus
Instructor: M. Zworski
Lectures: MWF 8:00-9:00am, Room 155 Dwinelle
Course Control Number: 54339
Office: 897 Evans, e-mail: zworski [at] math [dot] berkeley [dot] edu
Office 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.html
Grading: 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 - Calculus
Instructor: M. Ratner
Lectures: MWF 3:00-4:00pm, Room 2050 Valley Life Science
Course Control Number: 54384
Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 1B - Section 3 - Calculus
Instructor: N. Reshetikhin
Lectures: MWF 12:00-1:00pm, Room 2050 Valley Life Science
Course Control Number: 54420
Office: 915 Evans, e-mail: reshetik [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 16A - Section 1 - Analytical Geometry and Calculus
Instructor: W. H. Woodin
Lectures: TuTh 3:30-5:00pm, Room 2050 Valley Life Science
Course Control Number: 54462
Office: 721 Evans, e-mail: woodin [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 16B - Section 1 - Analytical Geometry and Calculus
Instructor: T. Scanlon
Lectures: MWF 2:00-3:00pm, Room 145 Dwinelle
Course Control Number: 54501
Office: 723 Evans, e-mail: scanlon [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Course Webpage: http://math.berkeley.edu/~scanlon/m16bs04/index.html

Math 16B - Section 2 - Analytical Geometry and Calculus
Instructor: J. H. Silver
Lectures: TuTh 11:00am-12:30PM, Room 2050 Valley Life Science
Course Control Number: 54534
Office: 753 Evans, e-mail: silver [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 24 - Section 1 - Freshman Seminars
Instructor: J. Harrison
Lectures: F 3:00-4:00pm, Room 891 Evans
Course Control Number: 54573
Office: 851 Evans, e-mail: harrison [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 32 - Section 1 - Precalculus
Instructor: The Staff
Lectures: MWF 8:00-9:00am, Room 60 Evans
Course Control Number: 54576
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Math 53 - Section 1 - Multivariable Calculus
Instructor: T. Graber
Lectures: TuTh 3:30-5:00pm, Room 155 Dwinelle
Course Control Number: 54606
Office: 833 Evans, e-mail: graber [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 53M - Section 1 - Multivariable Calculus With Computers
Instructor: L. Pachter
Lectures: MWF 10:00-11:00am, Room 277 Cory
Course Control Number: 54660
Office: 1081 Evans, e-mail: lpachter [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 54 - Section 1 - Linear Algebra and Differential Equations
Instructor: A. K. Liu
Lectures: MWF 3:00-4:00pm, Room 155 Dwinelle
Course Control Number: 54678
Office: 905 Evans, e-mail: akliu [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 54M - Section 1 - Linear Algebra and Differential Equations
Instructor: J. Sethian
Lectures: TuTh 8:00-9:30am, Room 10 Evans
Course Control Number: 54723
Office: 725 Evans, e-mail: sethian [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math H54 - Section 1 - Linear Algebra/Differential Equations - Honors
Instructor: Mark Haiman
Lectures: TuTh 11:00am-12:30pm, Room 3 Evans
Course Control Number: 54717
Office: 771 Evans, e-mail: Office Hours: W 11:00am-12:30pm
Prerequisites: Math 1B
Syllabus: 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.html
Homework: Weekly homework assignments posted on the course web page.

Math 55 - Section 1 - Discrete Mathematics
Instructor: J. W. Demmel
Lectures: TuTh 2:00-3:30pm, Room 100 Lewis
Course Control Number: 54756
Office: 737 Soda, e-mail: demmel [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 74 - Section 1 - Transition to Upper Division Mathematics
Instructor: J. Berg
Lectures: TuTh 12:30-2:00pm, Room 110 Barrows
Course Control Number: 54779
Office: 1020 Evans, e-mail: jberg [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Required Text: Smith, Eggen, and St. Andre, A Transition to Advanced Mathematics, 5th edition, Brooks/Cole
Grading: 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)
This class will be challenging and, I hope, rewarding. This course di ers from others in that instead of introducing new challenging mathematical concepts (derivatives, linear transformations, etc.), we focus our attention on precise and clear communication about familiar and basic mathematical concepts. These skills will help you understand the new mathematical concepts you encounter in the rest of your mathematical career. Course motto: Say what you mean and mean what you say!

Math 104 - Section 1 - Introduction to Analysis
Instructor: M. Erdogan
Lectures: MW 4:00-5:30pm, Room 141 Giannini
Course Control Number: 54831
Office: 805 Evans, e-mail: burak [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Course Webpage: http://math.berkeley.edu/~burak/ma104.html
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Math 104 - Section 2 - Introduction to Analysis
Instructor: A. B. Givental
Lectures: TuTh 3:30-5:00pm, Room 71 Evans
Course Control Number: 54834
Office: 701 Evans, e-mail: givental [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 104 - Section 3 - Introduction to Analysis
Instructor: H. Wu
Lectures: TuTh 11:00am-12:30pm, Room 71 Evans
Course Control Number: 54837
Office: 733 Evans, e-mail: wu [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 Analysis
Instructor: M. J. Klass
Lectures: MWF 11:00am-12:00pm, Room 7 Evans
Course Control Number: 54840
Office: 319 Evans, e-mail: klass [at] stat [dot] berkeley [dot] edu
Office Hours: TBA
Recommended Reading: Davidson and Donsig, Real Analysis with Real Applications, Prentice Hall.

Math 104 - Section 5 - Introduction to Analysis
Instructor: Paul R. Chernoff
Lectures: MW 4:00-5:30pm, Room 71 Evans
Course Control Number: 54843
Office: 933 Evans, e-mail: chernoff [at] math [dot] berkeley [dot] edu
Office Hours: M 2:00-3:30pm, F 1:45-3:15pm
Prerequisites: Math 53 and 54
Syllabus: 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.
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 II
Instructor: Michael Christ
Lectures: MWF 2:00-3:00pm, Room 85 Evans
Course Control Number: 54846
Office: 809 Evans, e-mail: mchrist [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: Math 104
Syllabus: (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 Algebra
Instructor: J. C. Harrison
Lectures: MWF 1:00-2:00pm, Room 3109 Etcheverry
Course Control Number: 54849
Office: 851 Evans, e-mail: harrison [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 110 - Section 2 - Linear Algebra
Instructor: D. A. Geba
Lectures: TuTh 3:30-5:00pm, Room 75 Evans
Course Control Number: 54852
Office: 837 Evans, e-mail: dangeba [at] math [dot] berkeley [dot] edu
Office Hours: TuTh 2:00-3:30pm
Prerequisites: 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 forms
Required 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 Algebra
Instructor: A. Grinshpan
Lectures: MWF 3:00-4:00pm, Room 71 Evans
Course Control Number: 54855
Course Webpage: http://math.berkeley.edu/~tolya/110

Math 110 - Section 4 - Linear Algebra
Instructor: J. B. Wagoner
Lectures: TuTh 11:00am-12:30pm, Room 75 Evans
Course Control Number: 54858
Office: 899 Evans, e-mail: wagoner [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 110 - Section 5 - Linear Algebra
Instructor: A. Grinshpan
Lectures: MWF 8:00-9:00am, Room 75 Evans
Course Control Number: 54861
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Math 113 - Section 1 - Introduction to Abstract Algebra
Instructor: M. Develin
Lectures: TuTh 3:30-5:00pm, Room 141 Giannini
Course Control Number: 54864
Office:
Office Hours: TBA
Prerequisites: 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 Algebra
Recommended 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 Algebra
Instructor: R. F. Coleman
Lectures: MWF 11:00am-12:00pm, Room 9 Evans
Course Control Number: 54867
Office: 901 Evans, e-mail: coleman [at] math [dot] berkeley [dot] edu
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Math 113 - Section 3 - Introduction to Abstract Algebra
Instructor: A. Knutson
Lectures: TuTh 8:00-9:30am, Room 71 Evans
Course Control Number: 54870
Office: 1033 Evans, e-mail: allenk [at] math [dot] berkeley [dot] edu
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Math 113 - Section 4 - Introduction to Abstract Algebra
Instructor: T.-Y. Lam
Lectures: MWF 11:00am-12:00pm, Room 71 Evans
Course Control Number: 54873
Office: 871 Evans, e-mail: lam [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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-Wesley
Grading: Letter grades only, based on: 20% Homework, 30% Midterm, 50% Final.

Math 113 - Section 5 - Introduction to Abstract Algebra
Instructor: Mariusz Wodzicki
Lectures: TuTh 12:30-2:00pm, Room 4 Evans
Course Control Number: 54876
Office: 995 Evans, e-mail: wodzicki [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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 - Honors
Instructor: Vera Serganova
Lectures: TuTh 12:30-2:00pm, Room 5 Evans
Course Control Number: 54879
Office: 709 Evans, e-mail: serganova [at] math [dot] berkeley [dot] edu
Office Hours: TuTh 11:00am-12:00pm
Prerequisites: Math 53,54
Syllabus: 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% final
Homework: 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 Algebra
Instructor: Kenneth A. Ribet
Lectures: TuTh 3:30-5:00pm, Room 9 Evans
Course Control Number: 54882
Office: 885 Evans, e-mail: ribet [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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.

Math 118 - Section 1 - Wavelets and Signal Processing
Instructor: M. Rieffel
Lectures: MWF 10:00-11:00am, Room 85 Evans
Course Control Number: 54885
Office: 811 Evans, e-mail: rieffel [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 Analysis

The 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 Sciences
Instructor: F. Rezakhanlou
Lectures: MWF 2:00-3:00pm, Room 75 Evans
Course Control Number: 54888
Office: 815 Evans, e-mail: rezakhan [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 121B - Section 1 - Mathematical Tools for the Physical Sciences
Instructor: R. E. Borcherds
Lectures: TuTh 9:30-11:00am, Room 289 Cory
Course Control Number: 54891
Office: 927 Evans, e-mail: reb [at] math [dot] berkeley [dot] edu
Office Hours: WF 1:00-2:30pm
Prerequisites: Math 53 and 54
Syllabus: 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/121
Grading: 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 Logic
Instructor: J. J. Steel
Lectures: TuTh 3:30-5:00pm, Room 6 Evans
Course Control Number: 54894
Office: 717 Evans, e-mail: steel [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 126 - Section 1 - Introduction to Partial Differential Equations
Instructor: J. C. Neu
Lectures: MWF 11:00am-12:00pm, Room 3109 Etcheverry
Course Control Number: 54897
Office: 1051 Evans, e-mail: neu [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 127 - Section 1 - Mathematical and Computational Methods in Molecular Biology
Instructor: L. Pachter
Lectures: MWF 1:00-2:00pm, Room 31 Evans
Course Control Number: 54900
Office: 1081 Evans, e-mail: lpachter [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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Math 128A - Section 1 - Numerical Analysis
Instructor: P. Stinis
Lectures: TuTh 12:30-2:00pm, Room 213 Wheeler
Course Control Number: 54903
Office: 887 Evans, e-mail: stinis [at] math [dot] lbl [dot] gov
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 128A - Section 2 - Numerical Analysis
Instructor: C. K. Miller
Lectures: TuTh 2:00-3:30pm, Room 247 Cory
Course Control Number: 54909
Office: 803 Evans, e-mail: kmiller [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 Analysis
Instructor: M. Zworski
Lectures: MWF 1:00-2:00pm, Room 3 Evans
Course Control Number: 54915
Office: 897 Evans, e-mail: zworski [at] math [dot] berkeley [dot] edu
Office Hours: MW 11:00am-12:00pm, F 2:00-3:00pm, or by appointment
Prerequisites: Math 53 and 54 or equivalent
Syllabus: 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 Analysis
Instructor: W. M. Kahan
Lectures: MWF 3:00-4:00pm, Room 85 Evans
Course Control Number: 54921
Office: 863 Evans, e-mail: wkahan [at] cs [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 135 - Section 1 - Introduction to Theory Sets
Instructor: M. Ratner
Lectures: MWF 1:00-2:00pm, Room 2 Evans
Course Control Number: 54927
Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 Theory
Grading: 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 Geometry
Instructor: Bjorn Poonen
Lectures: MWF 8:00-9:00am, Room 85 Evans
Course Control Number: 54930
Office: 879 Evans, e-mail: poonen [at] math [dot] berkeley [dot] edu
Office 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.html
Grading: 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.html

Math 172 - Section 1 - Combinatorics
Instructor: A. Yong
Lectures: TuTh 3:30-5:00pm, Room 7 Evans
Course Control Number: 54933
Office: 1035 Evans, e-mail: ayong [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 I
Recommended Reading: R. P. Stanley, Enumerative Combinatorics II; D. Stanton and D. White, Constructive combinatorics
Course Webpage: http://math.berkeley.edu/~ayong/teaching_next_term.html
Grading: 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 Analysis
Instructor: George M. Bergman
Lectures: MWF 10:00-11:00am, Room 70 Evans
Course Control Number: 54936
Office: 865 Evans, e-mail: gbergman [at] math [dot] berkeley [dot] edu
Office Hours: Tu 1:30-2:30pm, Th 10:30-11:30am, F 4:00-5:00pm
Prerequisites: 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 Analysis
Instructor: Bjorn Poonen
Lectures: MWF 3:00-4:00pm, Room 213 Wheeler
Course Control Number: 54939
Office: 879 Evans, e-mail: poonen [at] math [dot] berkeley [dot] edu
Office 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. These books will be put on reserve in the math library. Course Webpage: http://math.berkeley.edu/~poonen/math185.html Grading: 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.html Math 185 - Section 3 - Introduction to Complex Analysis Instructor: A. Yong Lectures: TuTh 12:30-2:00pm, Room 9 Evans Course Control Number: 54942 Office: 1035 Evans, e-mail: ayong [at] math [dot] berkeley [dot] edu Office Hours: TBA Prerequisites: 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 Applications Recommended Reading: D. Sarason, Notes on Complex Function theory; E. B. Saff and A. D. Snider, Fundamentals of Complex analysis Course Webpage: http://math.berkeley.edu/~ayong/teaching_next_term.html Grading: 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 - Honors Instructor: M. Ratner Lectures: MWF 11:00am-12:00pm, Room 31 Evans Course Control Number: 54945 Office: 827 Evans, e-mail: ratner [at] math [dot] berkeley [dot] edu Office Hours: TBA Prerequisites: Math 104 Syllabus: 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-Verlag Grading: 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 Mechanics Instructor: A. B. Givental Lectures: TuTh 9:30-11:00am, Room 5 Evans Course Control Number: 54948 Office: 701 Evans, e-mail: givental [at] math [dot] berkeley [dot] edu Office Hours: TBA Prerequisites: Syllabus: Required Text: Recommended Reading: Course Webpage: Grading: Homework: Comments: Math 195 - Section 1 - Special Topics in Mathematics Instructor: Lester Dubins Lectures: MWF 2:00-3:00pm, Room 87 Evans Course Control Number: 55758 Office: 869 Evans, e-mail: lester [at] math [dot] berkeley [dot] edu Office Hours: TBA Syllabus: 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: TBA

Math 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 Hall
Course Control Number: 55749 (Note that this is a 1-unit class.)
Office Hours: TBA
Prerequisites: Math 1A, 1B, 53, 54
Syllabus: Seminar in applied math. There will be an hour of lecture by a speaker once a week on some topic in applied math.
Homework: 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 Analysis
Instructor: D. E. Sarason
Lectures: MWF 8:00-9:00am, Room 71 Evans
Course Control Number: 55020
Office: 779 Evans, e-mail: sarason [at] math [dot] berkeley [dot] edu
Office Hours: MW 9:30-11:30am
Prerequisites: Math 202A
Syllabus: 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, Lp 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 Analysis
Course 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 Equations
Instructor: J. C. Neu
Lectures: MWF 1:00-2:00pm, Room 47 Evans
Course Control Number: 55023
Office: 1051 Evans, e-mail: neu [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 208 - Section 1 - C*-Algebras
Instructor: M. Rieffel
Lectures: MWF 9:00-10:00am, Room 5 Evans
Course Control Number: 55026
Office: 811 Evans, e-mail: rieffel [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 Variables
Instructor: Michael Christ
Lectures: MWF 11:00am-12:00pm, Room 65 Evans
Course Control Number: 55029
Office: 809 Evans, e-mail: mchrist [at] math [dot] berkeley [dot] edu
Office Hours: TBA
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 Topology
Instructor: M. Hutchings
Lectures: TuTh 11:00am-12:30pm, Room 65 Evans
Course Control Number: 55032
Office: 923 Evans, e-mail: hutching [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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, Princeton
Recommended Reading: Bott and Tu, Differential Forms in Algebraic Topology, Springer-Verlag
Course 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 Equations
Instructor: Lawrence C. Evans
Lectures: TuTh 12:30-2:00pm, Room 31 Evans
Course Control Number: 55035
Office: 907 Evans, e-mail: evans [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: Some knowledge of Sobolev spaces.
Syllabus: 1. Second-order elliptic equations
2. Parabolic and hyperbolic equations
3. Calculus of variations methods
4. Viscosity solutions of Hamilton--Jacobi equations
Required Text: Lawrence C. Evans, Partial Differential Equations, AMS
Grading: 25% homework, 25% midterm, 50% final
Homework: I will assign a homework problem, due in one week, at the start of each class.

Math 224B - Section 1 - Methods of Mathematical Physics
Instructor: F. A. Grunbaum
Lectures: TuTh 8:00-9:30am, Room 31 Evans
Course Control Number: 55038
Office: 903 Evans, e-mail: grunbaum [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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 - Metamathematics
Instructor: J. Steel
Lectures: TuTh 11:00am-12:30pm, Room 72 Evans
Course Control Number: 55041
Office: 717 Evans, e-mail: steel [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 228B - Section 1 - Numerical Solution of Differential Equations
Instructor: John Strain
Lectures: TuTh 11:00am-12:30pm, Room 5 Evans
Course Control Number: 55047
Office: 1099 Evans, e-mail: strain [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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.
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 Geometry
Instructor: F. Rezakhanlou
Lectures: MWF 3:00-4:00pm, Room 5 Evans
Course Control Number: 55050
Office: 815 Evans, e-mail: rezakhan [at] math [dot] berkeley [dot] edu
Office Hours: MWF 2:00-3:00pm
Prerequisites: 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 Combinatorics
Instructor: Mark Haiman
Lectures: TuTh 2:00-3:30pm, Room 51 Evans
Course Control Number: 55053
Office: 771 Evans, e-mail: Office Hours: W 11:00am-12:30pm
Prerequisites: 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.html
Grading: Based on periodic homework assignments.

Math 250B - Section 1 - Multilinear Algebra and Further Topics
Instructor: Paul Vojta
Lectures: MWF 12:00-1:00pm, Room 5 Evans
Course Control Number: 55056
Office: 883 Evans, e-mail: vojta [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: Math 250A
Syllabus:
The course will cover the following chapters or parts of the text:
A2.1-2 Tensor products Field theory (separability and regular extensions) Homological things Localization Associated primes and primary decomposition Integral dependence and the Nullstellensatz Flatness Dimension theory Modules of differentials Regular sequences and the Koszul complex Cohen-Macaulay rings

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, Springer
Course Webpage: http://math.berkeley.edu/~vojta/250b.html
Grading: 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 weeks
• 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 Theory
Instructor: Kenneth A. Ribet
Lectures: TuTh 12:30-2:00-pm, Room 2 Evans
Course Control Number: 55059
Office: 885 Evans, e-mail: ribet [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites: 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 3-540-55640-0
Grading: Grades will be based on homework, which will be assigned periodically.

Math 256B - Section 1 - Algebraic Geometry
Instructor: A. E. Ogus
Lectures: TuTh 2:00-3:30pm, Room 85 Evans
Course Control Number: 55062
Office: 877 Evans, e-mail: ogus [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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 Groups
Instructor: N. Reshetikhin
Lectures: MWF 10:00-11:00am, Room 5 Evans
Course Control Number: 55065
Office: 915 Evans, e-mail: reshetik [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 265 - Section 1 - Differential Topology
Instructor: Rob Kirby
Lectures: TuTh 8:00-9:30am, Room 5 Evans
Course Control Number: 55068
Office: 919 Evans, e-mail: kirby [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Recommended Reading: Papers of Ozsvath and Szabo
Comments: 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 Algebra
Instructor: T. Y. Lam
Lectures: MWF 9:00-10:00am, Room 31 Evans
Course Control Number: 55071
Office: 871 Evans, e-mail: lam [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Syllabus: 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, 1982

Math 275 - Section 1 - Topics in Applied Mathematics - Scaling, Physical Similarity: Dimensional Analysis and the Renormalization Group
Instructor: Grigory Barenblatt
Lectures: TuTh 9:30-11:00am, Room 65 Evans
Course Control Number: 55074
Office: 735 Evans, e-mail: gibar [at] math [dot] berkeley [dot] edu
Office Hours: TuTh 11:15am-12:50pm
Prerequisites: 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 Varieties
Instructor: Dr. Oleg Viro (Uppsala Universitet)
Lectures: TuTh 9:30-11:00am, Room 123 Dwinelle
Course Control Number: 55076
Office:
Office Hours: TBA
Prerequisites: 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 Geometry
Instructor: A. K. Liu
Lectures: MWF 2:00-3:00pm, Room 5 Evans
Course Control Number: 55077
Office: 905 Evans, e-mail: akliu [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage:
Homework:

Math 300 - Section 1 - Teaching Workshop
Instructor: O. H. Hald
Lectures:
Course Control Number: 55668
Office: 875 Evans, e-mail: hald [at] math [dot] berkeley [dot] edu
Office Hours: TBA
Prerequisites:
Syllabus:
Required Text:
Course Webpage: