# Spring 2006

 Math 1A - Section 1 - Calculus Instructor: Mina Aganagic Lectures: TuTh 11:00am-12:30pm, Room 155 Dwinelle Course Control Number: 54303 Office: 715 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 1B - Section 1 - Calculus Instructor: Marina Ratner Lectures: MWF 9:00-10:00am, Room 155 Dwinelle Course Control Number: 54345 Office: 827 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 1B - Section 2 - Calculus Instructor: Michael Hutchings Lectures: TuTh 3:30-5:00pm, Room 1 Pimentel Course Control Number: 54384 Office: 923 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 1B - Section 3 - Calculus Instructor: Zvezdelina Stankova Lectures: TuTh 12:30-2:00pm, Room 100 Lewis Course Control Number: 54426 Office: 713 Evans Office Hours: TuTh 2:10-3:30pm Prerequisites: Three years of high school mathematics Required Text: Stewart, Calculus, for the exact edition to be used: check with the Math Dept and/or the UCB bookstore Recommended Reading: Syllabus: To be posted on the course webpage by mid January'06. Course Webpage: http://www.math.berkeley.edu/~stankova (to be updated by mid January'06) Grading: 15% quizzes, 25% each of two midterms, 35% final. Homework: To be assigned in class. Comments: Math 16A - Section 1 - Analytical Geometry and Calculus Instructor: Hugh Woodin Lectures: TuTh 2:00-3:30pm, Room 155 Dwinelle Course Control Number: 54468 Office: 721 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 16B - Section 1 - Analytical Geometry and Calculus Instructor: Don Sarason Lectures: MWF 12:00-1:00pm, Room 2050 Valley Life Sciences Course Control Number: 54507 Office: 779 Evans Office Hours: TBA Prerequisites: Math 16A or the equivalent Required Text: Goldstein, Lay & Schneider, Calculus and its Applications, 10th edition, Prentice-Hall (You can save  by purchasing the volume containing only Chapters 7-12.) Recommended Reading: Syllabus: Functions of several variables, trigonometric functions, techniques of integration, differential equations, Taylor polynomials and infinite series, probability and calculus (Chapters 7-12 of the textbook) Course Webpage: http://math.berkeley.edu/~sarason/Class_Webpages/Spring_2006/Math16B_S1.html Grading: The course grade will be based on two midterm exams, the final exam, and section performance. Details will be provided at the first lecture. Homework: There will be regular homework assignments. Comments: This section of Math 16B will be run very much like Sarason's 16B is being run this semester. Students can get an idea of what to expect by checking the webpage for the current 16B. Math 16B - Section 2 - Analytical Geometry and Calculus Instructor: Lior Pachter Lectures: TuTh 12:30-2:00pm, Room 10 Evans Course Control Number: 54543 Office: 1081 Evans Office Hours: TBA Prerequisites: Math 16A or equivalent Required Text: Goldstein, Lay & Schneider, Calculus and its Applications, 10th edition, Prentice-Hall Recommended Reading: Syllabus: Course Webpage: http://math.berkeley.edu/~lpachter/16A/ Grading: 5% quizes, 10% homework, 25% each midterm, 35% final Homework: Homework will be assigned on the web page and is due once a week. Comments: Math 24 - Section 1 - Freshman Seminars Instructor: Jenny Harrison Lectures: F 3:00-4:00pm, Room 891 Evans Course Control Number: 54579 Office: Office Hours: Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 32 - Section 1 - Precalculus Instructor: The Staff Lectures: MWF 8:00-9:00am, Room 150 GSPP Course Control Number: 54582 Office: Office Hours: Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 39A - Section 1 - Freshman/Sophomore Seminar Instructor: Hung-Hsi Wu Lectures: Tu 4:00-6:00pm, Room 87 Evans Course Control Number: 54593 Office: 733 Evans Office Hours: TBA Required Text: None Recommended Reading: To be handed out during the seminar. Syllabus: The purpose of this seminar is to introduce the participants to life in a K-12 classroom. A main component of the seminar (but hardly the only one) is to discuss several specific mathematical topics that are known to be troublesome in the K-12 curriculum. The seminar will contrast what they learn about these topics in mathematics courses in college with how they will teach them to their students. For example, the standard algorithms for whole numbers, which are basically not taught in college but are the mainstay of the curriculum of grades 2-4. Or, the operations with fractions, which are taught in upper division classes in college but which must be taught in grades 4-6 in school. Or, polynomials, which are taught in upper division abstract algebra courses but are taught in grade 8 in school. Or, trigonometric functions. Etc. Because it is anticipated that many if not most of the participants will be freshmen who will be simultaneously helping in an elementary class, another main emphasis will be on the reports of participants on their observations in the school classroom: what the problems are and how they propose to deal with them. In particular, we will pay special attention to the problem of the wide discrepancy in the mathematics learning among elementary students, which creates almost insurmountable problems in the mathematics classrooms of middle and high schools. Other topics that will be included are: (1) Classroom management problems. (2) A proposed Intervention Program for at risk mathematics students in grades 4-7. (3) How the Mathematics Content Standards, CST, and API impact life in the school classroom. (4) How to use the Mathematics Framework of 2005. (5) The Math War of 1989-date. (6) The roles of NCTM and CMC in California's mathematics education. Course Webpage: Grading: By attendance only. Homework: There will be assignments in some weeks. Comments: Math 53 - Section 1 - Multivariable Calculus Instructor: Maciej Zworski Lectures: MWF 2:00-3:00pm, Room 155 Dwinelle Course Control Number: 54615 Office: 897 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 53 - Section 2 - Multivariable Calculus Instructor: Robion Kirby Lectures: MWF 8:00-9:00am, Room 155 Dwinelle Course Control Number: 54660 Office: 919 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math H53 - Section 1 - Honors Multivariable Calculus Instructor: Mariusz Wodzicki Lectures: TuTh 12:30-2:00pm, Room 85 Evans Course Control Number: 54699 Office: 995 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 54 - Section 1 - Linear Algebra and Differential Equations Instructor: James Sethian Lectures: TuTh 11:00am-12:30pm, Room 10 Evans Course Control Number: 54705 Office: 725 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 54 - Section 2 - Linear Algebra and Differential Equations Instructor: Jack Wagoner Lectures: MWF 10:00-11:00am, Room 155 Dwinelle Course Control Number: 54753 Office: 899 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 55 - Section 1 - Discrete Mathematics Instructor: John Strain Lectures: TuTh 2:00-3:30pm, Room 3108 Etcheverry Course Control Number: 54798 Office: 1099 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 74 - Section 1 - Transition to Upper Division Mathematics Instructor: Scott Armstrong Lectures: MWF 3:00-4:00pm, Room 75 Evans Course Control Number: 54816 Office: 1064 Evans Office Hours: M 1:30-2:30, Tu 2:30-4:30 Prerequisites: None. Required Text: Eisenberg, The Mathematical Method: A Transition to Advanced Mathematics Recommended Reading: Hartsfield and Ringel, Pearls in Graph Theory: A Comprehensive Introduction Larson, Problem-Solving Through Problems Syllabus: The course will begin with a discussion of sets, relations, functions, and an introduction to common mathematical arguments. Next, there will be an overview of mathematical induction, the pidgeonhole principle, and the well-ordering principle. The course will then move on to the study of arithmetic, including the fundamental theorem of arithmetic, Euclid‚s theorem, congruency, the chinese remainder theorem, and Fermat‚s little theorem. We will then study the theory of cardinality of sets, Cantor‚s theorem, and the Schroder-Bernstein theorem. After that, time permitting, we will discuss some elementary graph theory. Course Webpage: http://math.berkeley.edu/~sarm/math74 Grading: Grades will be assigned by the instructor after the students have been ranked according to the following scale: Homework- 30% Quizzes- 15% Midterm #1- 10% Midterm #2- 15% Final Exam- 30% Homework: There will be roughly one homework assignment each week, to be turned in each Wednesday at the beginning of class. Homework assignments and solutions will be posted on the course webpage. Comments: The primary goal of this course is to teach you to read and write mathematics. Unlike previous courses, in which you encountered new and challenging mathematical topics but which emphasized calculations and computations, in this course you will develop a rigorous understanding of topics that you may already be somewhat familiar with. You will learn the skills necessary to read and understand mathematical proofs; you will develop reasoning skills necessary to create your own mathematical arguments; and you will learn to clearly communicate your mathematical ideas so that others can understand precisely what you mean. Along the way, you will learn some interesting mathematics. Math 98 - Section 14 - Directed Group Study Instructor: Yossi Farjoun Lectures: TuTh 11:00am-12:30pm, Room B0003A Evans Course Control Number: 54858 Office: 747 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: This course will cover the bare basics of Matlab programming. It is intended for students concurrently taking Math 128a or 128b who have little or no prior experience in programming. It will cover the basics of programming (Variables, Functions, loops) while remaining focused on Matlab syntax and use. This is a 1 unit course given as pass/fail based on attendance and participation. The lectures will be given in the computer lab in Evans. Simple homework will be given and reviewed in following lecture. The course will be given over the first 5 weeks of the semester at 3 hours per week to give the students the basic grounding for doing the computer assignments of Math 128. Math 98 - Section 15 - Directed Group Study Instructor: Yossi Farjoun Lectures: WF 3:00-4:30pm, Room B0003A Evans Course Control Number: 54861 Office: 747 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: This course will cover the bare basics of Matlab programming. It is intended for students concurrently taking Math 128a or 128b who have little or no prior experience in programming. It will cover the basics of programming (Variables, Functions, loops) while remaining focused on Matlab syntax and use. This is a 1 unit course given as pass/fail based on attendance and participation. The lectures will be given in the computer lab in Evans. Simple homework will be given and reviewed in following lecture. The course will be given over the first 5 weeks of the semester at 3 hours per week to give the students the basic grounding for doing the computer assignments of Math 128. Math 104 - Section 1 - Introduction to Analysis Instructor: George Bergman Lectures: MWF 3:00-4:00pm, Room 3 Evans Course Control Number: 54876 Office: 865 Evans Office Hours: Tu 1:30-2:30, Th 10:30-11:30, F 4:15-5:15 Prerequisites: Math 53 and 54. (Math 74, which may be taken simultaneously, is also recommended for students not familiar with proofs.) Required Text: W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill Recommended Reading: None. Syllabus: We will cover Chapters 1-7 of the text. Course Webpage: None. Grading: Homework, 25%; two Midterms, 15% and 20%; Final Exam, 35%; regular submission of the daily question (see below), 5%. Homework: Weekly, generally due Wednesdays. Comments: This is the course in which the material you saw in calculus is put on a solid mathematical basis. It begins with the properties of the real numbers that underlie these results. It will be for some of you the first course in which you are expected, not to calculate answers, but to give proofs. This transition, though intellectually exciting, is difficult for many students. I will bear in mind that the function of the course is not just to teach you Real Analysis, but also to introduce you to mathematical reasoning, and that it is my job to help you with the one as much as the other. Students who think they will have particular difficulty with proofs are advised to take Math 74 simultaneously if they have not taken it earlier. I don't like the conventional 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 an assigned reading, each student is required to submit, in writing or (preferably) by e-mail, 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. Many students find Rudin a difficult text. If your response to this would be to put it aside and try to learn from the lectures alone, I advise you not to take my section. The author writes clearly, but writes as a mathematician, and in lecture I will try to help you understand him, not replace him. Math 104 - Section 2 - Introduction to Analysis Instructor: Alexander Givental Lectures: TuTh 11:00am-12:30pm, Room 71 Evans Course Control Number: 54879 Office: 701 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 104 - Section 3 - Introduction to Analysis Instructor: Michael Klass Lectures: MWF 12:00-1:00pm, Room 71 Evans Course Control Number: 54882 Office: 319 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 104 - Section 4 - Introduction to Analysis Instructor: Giulio Caviglia Lectures: TuTh 3:30-5:00pm, Room 3105 Etcheverry Course Control Number: 54885 Office: 805 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 105 - Section 1 - Analysis II Instructor: Dan Geba Lectures: MWF 3:00-4:00pm, Room 87 Evans Course Control Number: 54888 Office: 837 Evans Office Hours: MWF 2:00-3:00pm Prerequisites: Math 104 Required Text: H.S.Bear, A primer of Lebesgue integration, 2nd edition, 2002 Recommended Reading: Syllabus: Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable. Course Webpage: http://math.berkeley.edu/~dangeba/105S06 Grading: Homework 25%, Midterm 25%, Final 50% Homework: Homework is due by the end of the Friday lecture, one week after it was assigned. Late homework will not be accepted. Worst 3 grades do not count. Comments: Math 110 - Section 1 - Linear Algebra Instructor: Alberto Grünbaum Lectures: TuTh 8:00-9:30am, Room 100 GPB Course Control Number: 54891 Office: 903 Evans Office Hours: TuWTh 11:00am-12:00pm Prerequisites: Math 54, or a course with equivalent linear algebra content. (Math 74 also recommended for students not familiar with proofs) Required Text: S. H. Friedberg, A. J. Insel & L. E. Spence, Linear Algebra, 4th edition Recommended Reading: Syllabus: In this class we take a second look at linear algebra, something you have seen in Math 54. The basic issues are the same: solving linear equations, doing least square problems, finding eigenvalues and eigenvectors of a matrix, thinking of a matrix as a linear map between vector spaces, writing matrices in a more revealing form by choosing a basis appropriately, etc. We will revisit all these items form a more general viewpoint than that of Math 54, and proofs will play an important part in this class. There are TWO reasons for looking anew at these basic tasks: with the advent of the computer age there has been a revival in the search for ways of performing these basic tasks with faster and more accurate algorithms. Think of applications like signal processing, medical imaging, Pixar animation, etc. The second reason is equaly important: the ability to do abstract reasoning gives a beautiful and powerful tool not only to organize the material you already know but also to make new discoveries. If you are going to be more than just a user of "black boxes" of software (which I also love to use) you have to develop your own mental tools to judge the merit and quality of these packages. Practical advice: start doing the homework as soon as possible. Learn to walk around with these problems in your head, sometimes you will wake up with some new idea on how to solve them. Try to put together a group of two or three of you that meets regularly to discuss the material in the class; try to explain the material to each other: there is a chance that you will discover that you do not really understand it yourself. This is the first step in learning and you should repeat this till you all understand what is going on. Ask questions, propose counterexamples, challenge each other.... Course Webpage: Grading: The grade will be based on your homework (25%), two midterms (20% and 20%) and a final (35%). Homework: There will be a weekly homework assignment, with problems from the book. Comments: Math 113 - Section 1 - Introduction to Abstract Algebra Instructor: Mariusz Wodzicki Lectures: TuTh 9:30-11:00am, Room 71 Evans Course Control Number: 54915 Office: 995 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 113 - Section 2 - Introduction to Abstract Algebra Instructor: Martin Weissman Lectures: TuTh 3:30-5:00pm, Room 75 Evans Course Control Number: 54918 Office: 1067 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 113 - Section 3 - Introduction to Abstract Algebra Instructor: Kevin Woods Lectures: MWF 2:00-3:00pm, Room 71 Evans Course Control Number: 54921 Office: 867 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 113 - Section 4 - Introduction to Abstract Algebra Instructor: Giulio Caviglia Lectures: TuTh 12:30-2:00pm, Room 71 Evans Course Control Number: 54924 Office: 805 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math H113 - Section 1 - Honors Introduction to Abstract Algebra Instructor: Brian Osserman Lectures: MWF 2:00-3:00pm, Room 85 Evans Course Control Number: 54927 Office: 767 Evans Office Hours: TBA Prerequisites: Required Text: Beachy/Blair, Abstract Algebra, 2nd Edition Recommended Reading: Syllabus: We will hopefully cover groups, rings, and fields, with some exploration of Galois theory at the end to tie concepts together. Course Webpage: http://www.math.berkeley.edu/~osserman/classes/H113/ Grading: 45% homework, 15% each exam, 25% final. Homework: Homework will be assigned weekly. Comments: Math 114 - Section 1 - Abstract Algebra II Instructor: Vera Serganova Lectures: TuTh 3:30-5:00pm, Room 71 Evans Course Control Number: 54930 Office: 709 Evans Office Hours: TuTh 5:30-6:30pm Prerequisites: Math 113 Required Text: Emil Artin, Galois Theory, Dover Edition Recommended Reading: Syllabus: The course will cover Galois theory with applications: ruler and compass construction, solving cubic and quartic equations, finite fields Course Webpage: http://math.berkeley.edu/~serganov/114 Grading: 20% homework, 20% quizzes, 20% midterm, 40% final Homework: Homework will be assigned on the web every week, and due once a week. Comments: We will have a 10 minutes quiz every other week and one midterm. Math 121A - Section 1 - Mathematical Tools for the Physical Sciences Instructor: Remus Floricel Lectures: TuTh 8:00-9:30am, Room 71 Evans Course Control Number: 54933 Office: Office Hours: Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 121B - Section 1 - Mathematical Tools for the Physical Sciences Instructor: Ming Gu Lectures: TuTh 11:00am-12:30pm, Room 85 Evans Course Control Number: 54936 Office: 861 Evans Office Hours: TBA Prerequisites: Required Text: Mary Boas, Mathematical Methods in the Physical Sciences, 2nd ed., John Wiley & Sons Recommended Reading: Syllabus: This course is a continuation of Math 121A taught in the Fall 2005 by Professor Vera Serganova. We will use the same text book, Mathematical Methods in the Physical Sciences, 2nd edition, by Mary Boas and published by John Wiley & Sons. Note the 3rd edition of this book is now available, but we will only use the 2nd edition. We will cover the following chapters: Chapters 3 and 6: Vectors and vector analysis; Chapter 7: Fourier Series; Chapter 8: Ordinary differential equations; Chapter 10: Coordinate transformations and tensor analysis; Chapter 11: Special functions; and Chapter 12: Series solutions of differential equations. Time permitting, we will also discuss Chapter 15, Integral transforms. Course Webpage: Grading: Homework: Comments: Math 125A - Section 1 - Mathematical Logic Instructor: Theodore Slaman Lectures: TuTh 3:30-5:00pm, Room 9 Evans Course Control Number: 54939 Office: 719 Evans Office Hours: Tu 2:00-3:00pm, W 11:00-12:00pm Prerequisites: 113 or consent of instructor. Required Text: Slaman/Woodin lecture notes to be distributed in class. Recommended Reading: Ebbinghaus, Flum & Thomas, Mathematical Logic or Enderton, A Mathematical Introduction to Logic. Syllabus: Propositional and Predicate Logic. Tarski's definition of truth in a first-order structure, with an analysis of the expressive power of first-order definability. Goedel's Completeness Theorem, that if a sentence A is true in every model of a collection of axioms T, then there is a proof of A from T. Other topics, as time permits. Course Webpage: http://math.berkeley.edu/~slaman/courses Grading: Homework will be collected every other week, midterm examinations will be held in class during weeks 6 and 11, and a final examination will be held during finals week. The course grade will be determined according to a weighted average: homework 10%, midterms 25% each, final 40%. Homework: Comments: Math 126 - Section 1 - Introduction to Partial Differential Equations Instructor: L. Craig Evans Lectures: TuTh 9:30-11:00am, Room 70 Evans Course Control Number: 54942 Office: 907 Evans Office Hours: TBA Prerequisites: It would be good for students to have had Math 104, but this is not absolutely necessary. Required Text: Strauss, Partial Differential Equations: An Introduction (Wiley) Recommended Reading: Syllabus: Main topics: Introduction Waves and diffusion Reflections and sources Boundary problems Laplace's equation, Green's functions More on wave propagation Nonlinear PDE Course Webpage: TBA Grading: 25% homework, 25% midterm, 50% final Homework: Homework will be assigned during every class, and each assignment is due in one week). Comments: Math 127 - Section 1 - Mathematics for Computational Biology Instructor: Bernd Sturmfels Lectures: TuTh 12:30-2:00pm, Room 81 Evans Course Control Number: 54945 Office: 925 Evans Office Hours: W 8:30-11:00am or by appt. Prerequisites: Some basics of Discrete Mathematics, Statistics and Abstract Algebra. Familiarity with Molecular Biology is welcome but not necessary. Required Text: Lior Pachter & Bernd Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005 Recommended Reading: R. Durbin, S. Eddy, A. Korgh & G. Mitchison, Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids, Cambridge University Press, 1998 D. Gusfield, Algorithms on Strings, Trees, and Sequences, Cambridge University Press, 1997 Syllabus: This course is an introduction to some mathematical foundations which are relevant for computational biology. The mathematical emphasis lies on algebraic statistics and discrete algorithms. Biological applications to be discussed will center around comparative genomics. If you are uncertain whether this course is suitable for you, please talk to the instructor. Course Webpage: http://math.berkeley.edu/~bernd/math127.html Grading: The course grade will be based on the homework and the projects, with a bias towards the latter. Homework: There will be weekly homework in the first half of the course. Thereafter we shall form research teams'', and the homework will give way to the projects pursued by the teams. Comments: I expect a mix of undergraduate students and graduate students, both from mathematics and from other departments, in this class. Participants will greatly benefit from working with other members of this diverse group. Math 128A - Section 1 - Numerical Analysis Instructor: John Neu Lectures: TuTh 8:00-9:30am, Room 120 Latimer Course Control Number: 54948 Office: 1051 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Since a substantial part of the required homework in Math 128a will involve computer programming, students that have no background in programing are recommended to take Math 98. It is recommended that you take this course if you have little or no experience programming. Math 128B - Section 1 - Numerical Analysis Instructor: William Kahan Lectures: TuTh 11:00am-12:30pm, Room 51 Evans Course Control Number: 54969 Office: 863 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 135 - Section 1 - Introduction to Theory Sets Instructor: Jack Silver Lectures: TuTh 9:30-11:00am, Room 3 Evans Course Control Number: 54975 Office: 753 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 142 - Section 1 - Elementary Algebraic Topology Instructor: Robion Kirby Lectures: MWF 10:00-11:00am, Room 3 Evans Course Control Number: 54978 Office: 919 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 170 - Section 1 - Mathematical Method Optimization Instructor: Ming Gu Lectures: TuTh 2:00-3:30pm, Room 3107 Etcheverry Course Control Number: 54981 Office: 861 Evans Office Hours: TBA Prerequisites: Math 53 and 54 or equivalent. If you have doubts about them, please come talk with the instructor. Required Text: Joel Franklin, Methods of Mathematical Economics, SIAM Recommended Reading: Syllabus: In this course, we will learn some of the most basic concepts and methods in linear programming and optimization. Linear programming, which concerns optimizing a linear objective function subject to linear constraints, is one of the big magics of modern mathematics. Our main focus in this course is to develop the basic theory and the simplex algorithm for linear programs. Along the way, we will in addition discuss the Kuhn-Tucker conditions, Lagrangian multipliers, and convex analysis. Time permitting, we will also discuss quaratic programming. Course Webpage: Grading: Homework: Comments: Math 185 - Section 1 - Introduction to Complex Analysis Instructor: Leo Harrington Lectures: TuTh 9:30-11:00am, Room 75 Evans Course Control Number: 54984 Office: 711 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 185 - Section 2 - Introduction to Complex Analysis Instructor: L. Craig Evans Lectures: TuTh 12:30-2:00pm, Room 3 Evans Course Control Number: 54987 Office: 907 Evans Office Hours: TBA Prerequisites: Math 104 Required Text: Churchill & Brown, Complex Variables and Applications, 7th edition (McGraw-Hill) Recommended Reading: Syllabus: Main topics: Complex numbers Analytic functions Integrals Series Residues, poles Conformal mappings Course Webpage: TBA Grading: 20% homework, 20% midterm 1, 20% midterm 2, 40% final exam Homework: Homework will be assigned during every class, and each assignment is due in one week). Comments: Math 185 - Section 3 - Introduction to Complex Analysis Instructor: Dapeng Zhan Lectures: MWF 3:00-4:00pm, Room 71 Evans Course Control Number: 54990 Office: 873 Evans Office Hours: TBA Prerequisites: Math 104 Required Text: James Ward Brown & Ruel V. Churchill, Complex Variables and Applications, 7th edition, McGraw-Hill Recommended Reading: Syllabus: Complex Numbers, Analytic Function, Complex Integral, Cauchy Theorem, Cauchy Formula, Power Series, Laurent Series, Residues, Residue Formula, Linear Transformation, Conformal Mapping, Harmonic Functions, and etc. Course Webpage: http://math.berkeley.edu/~dapeng/ma185.html Grading: 20% homeworks, 20% each midterm, 40% final Homework: Homework will be assigned on the web every class, and due once a week. Comments: Math H185 - Section 1 - Honors Introduction to Complex Analysis Instructor: Donald Sarason Lectures: MWF 3:00-4:00pm, Room 5 Evans Course Control Number: 54993 Office: 779 Evans Hall Office Hours: TBA Prerequisites: Math 104 Required Text: Donald Sarason, Notes on Complex Function Theory, published by Henry Helson Recommended Reading: Nothing official. The books of Brown & Churchill and/or Marsden & Hoffman, to be used in the regular Math 185 sections, could be helpful. Syllabus: The field of complex numbers, differentiation of functions of a complex variable, holomorphic functions, elementary functions, power series, complex integration, Cauchy's theorem and consequences, Laurent series, Riemann mapping theorem, additional topics Course Webpage: Grading: The course grade will be based on homework, two midterm examinations, and the final examination. Exams will most likely be open book. More details will be provided at the first class meeting. Homework: Homework will be assigned weekly and will be carefully graded. Comments: Complex analysis can be described as calculus with complex numbers, but that description does not do the subject justice. While complex analysis is in a sense a subdiscipline within the theory of maps between Euclidean spaces (in this case two-dimensional spaces), it has a character and an elegance of its own. The notion of complex differentiability, for instance, has remarkable implications one would not expect from the study of real analysis. The elegance of the subject is typified by its main theorem, Cauchy's theorem, which devotees regard as the portal to paradise. Besides being intrinsically beautiful, complex analysis is useful - its results and ideas permeate much of modern mathematics, including applied areas. The goal of Math H185 will be to provide students with a solid grounding in the subject so that they can continue its study at a higher level or use it in other areas. Math 191 - Section 1 - Theory of Chainlets - From Discrete Calculus to Soap Films, Fractals, Manifolds, and Beyond Instructor: Jenny Harrison Lectures: MWF 1:00-2:00pm, Room 5 Evans Course Control Number: 54999 Office: 851 Evans Office Hours: TBA Prerequisites: Required Text: Bamberg & Sternberg, A Course in Mathematics for Students of Physics, Volumes 1 and 2, Cambridge Press. Recommended Reading: Syllabus: This experimental course develops a new discrete calculus at the infinitesimal level of a single point, assuming only the linear algebra of Math 110. In a process akin to taking the inverse of Xeno’s paradox, we extend by linearity this “calculus at a point” to discrete chains of finitely many points, and then to limits of these discrete chains in normed spaces. We call these limits “chainlets”. The discrete calculus extends readily to a full calculus on the space of chainlets, permitting, e.g., domains of soap films, submanifolds, and fractals. The classical theory of calculus in Euclidean space, and even smooth, abstract manifolds follows. All of the real work takes place in a single point in a normed vector space. This is a theory at the heart of multilinear algebra and standard analysis, not a work of nonstandard analysis. Cons: There is no text, only class notes. Homework assignments will be developed and sporadic. There is no history or community experience of such a course to draw upon. Pros: Students will build a mathematical umbrella from which to hang various concepts of mathematics. For example, most of the results and formulas from multivariable calculus drop out readily. The number of limits needed to develop the full calculus in minimized. The discrete theory needs no limits. Convergence to the smooth continuum of all integrals, products, and operators is guaranteed, as the distance between base points of the discrete chains tends to zero. Once the convergence theorems are established, the work remaining for both theory and applications is to find Cauchy sequences of approximators to a given chainlet domain. Throughout the course, there are no arguments using epsilons and deltas, all limits involve establishment of Cauchy sequences of domains in the norm. The ideas are based upon a blend of geometric and algebraic notions. Draft Syllabus 1. Normed vector spaces: Cauchy sequences, Banach spaces, inner product spaces. 2. Multilinear algebra: Tensors, Grassmann algebra, wedge products, symmetric algebras, superalgebras, directional derivatives of k-vectors, chains of k-vectors, pushforward and boundary operators 3. Differential forms in a normed vector space: Norms on forms, exterior derivative, directional derivatives, pullback operators, Integration over chains of k-vectors, Discrete Stokes’ theorem and change of variables through duality 4. Natural norms on chains of k-vectors: Direct limits and inverse limits. Chainlet spaces as direct limits of spaces of chains. Cochainlet spaces as inverse limits of spaces of cochains. Continuity of products, integrals and operators in the natural norms. Isomorphisms of cochains and forms, leading to generalized Stokes’ theorem, and change of variables on chainlets. 5. Examples of chainlets: polyhedral chains, manifolds, soap films, fractals, differential forms. 6. Chainlets in an inner product space: Perp operator on chainlets, Hodge star operator on forms. Divergence theorem over chainlet domains. 7. Nonsmooth forms and nonsmooth domains: how much calculus is valid as less smooth forms are taken over rough domains, what results fail and where. 8. Preintegrals of differential forms over domains where the degree of the form may not match the dimension of the domain. General Stokes’ and divergence theorems in this category. 9. Submanifolds as chainlets, abstract manifolds, Riemannian vs Finsler. The latter will not be highly developed, just the main ideas given of how to develop calculus on Riemannian and Finsler manifolds. 10. Applications (as time permits): Calculus of variations, flux across fractal domains, numerical methods, superalgebras, bosons and fermions Course Webpage: Grading: Grades will be based on written assignments, class participation, office hour discussions, a midterm and a final. Homework: Homework assignments will be developed and sporadic. Comments: Math 191 - Section 2 - Seminar for Teaching Math in Schools Instructor: Hung-Hsi Wu Lectures: Tu 4:00-6:00pm, Room 87 Evans Course Control Number: 55001 Office: 733 Evans Office Hours: TBA Prerequisites: Math 1A Required Text: None Recommended Reading: To be handed out during the seminar. Syllabus: The purpose of this seminar is to introduce the participants to life in a K-12 classroom. A main component of the seminar (but hardly the only one) is to discuss several specific mathematical topics that are known to be troublesome in the K-12 curriculum. The seminar will contrast what they learn about these topics in mathematics courses in college with how they will teach them to their students. For example, the standard algorithms for whole numbers, which are basically not taught in college but are the mainstay of the curriculum of grades 2-4. Or, the operations with fractions, which are taught in upper division classes in college but which must be taught in grades 4-6 in school. Or, polynomials, which are taught in upper division abstract algebra courses but are taught in grade 8 in school. Or, trigonometric functions. Etc. Because it is anticipated that many if not most of the participants will be freshmen who will be simultaneously helping in an elementary class, another main emphasis will be on the reports of participants on their observations in the school classroom: what the problems are and how they propose to deal with them. In particular, we will pay special attention to the problem of the wide discrepancy in the mathematics learning among elementary students, which creates almost insurmountable problems in the mathematics classrooms of middle and high schools. Other topics that will be included are: (1) Classroom management problems. (2) A proposed Intervention Program for at risk mathematics students in grades 4-7. (3) How the Mathematics Content Standards, CST, and API impact life in the school classroom. (4) How to use the Mathematics Framework of 2005. (5) The Math War of 1989-date. (6) The roles of NCTM and CMC in California's mathematics education. Course Webpage: Grading: By attendance only. Homework: There will be assignments in some weeks. Comments: Math 191 - Section 3 - Introduction to School Mathematics Instructor: Hung-Hsi Wu Lectures: TuTh 2:00-3:30pm, Room 9 Evans Course Control Number: 55821 Office: 733 Evans Office Hours: TBA Prerequisites: Math 1A, 1B, 53 Required Text: Lecture notes will be available. Recommended Reading: Syllabus: This course is for prospective high school teachers. It will take up some key topics in the mathematics of grades 7-12 and re-examine them from an advanced point of view. Like all upper division courses, the emphasis will be on the proofs of all the statements. The tentative syllabus is as follows. Fractions and rational numbers; existence and uniqueness of reduced form of a fraction; concept of limit and the least upper bound axiom; n-th root of a positive number; review of the concept of congruence; length, area and volume; the number pi; convergence of infinite series and the decimal representation of a real number; continuity, derivative, and the integral; Fundamental Theorem of Calculus; the exponential function and logarithmic function; laws of exponents. Lectures will be given in TuTh 2 - 3:30, and homework problems will be discusses each Monday 4-5. Study group for doing homework problems is required. Course Webpage: Grading: 10% first MT, 25% second midterm, 25% homework, 40% final. Homework: Homework will be assigned on the web every Wednesday, and due each Monday. Comments: Mandatory discussion section meeting Mondays 4-5pm in 75 Evans. Math 202B - Section 1 - Introduction to Topology and Analysis Instructor: Marina Ratner Lectures: MWF 12:00-1:00pm, Room 70 Evans Course Control Number: 55095 Office: 827 Evans Office Hours: TBA Prerequisites: Math 202A Required Text: Royden, Real Analysis Recommended Reading: Syllabus: Measure and integration. Product measure and Fubini type theorems. Signed measures. Hahn and Jordan decompositions. Radon-Nikodym theorem. Introduction to linear topological spaces, Banach spaces and Hilbet spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach Theorem. Duality, the dual of Lp. Measures on locally compact spaces, the dual of C(X). Weak and weak-* topologies, Banach-Alaoglu theorem. Convexity and Krein-Milman theorem. Additional topics if time allows. Course Webpage: Grading: The grade will be based 25% on weekly homework, 35% on a midterm, and 40% on a Final Exam. Homework: Comments: Math 208 - Section 1 - C*-Algebras Instructor: Marc Rieffel Lectures: MWF 9:00-10:00am, Room 5 Evans Course Control Number: 55098 Office: 811 Evans Office Hours: TBA Prerequisites: The basic theory of bounded operators on Hilbert space and of Banach algebras, especially commutative ones. (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 almost sufficient.) 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. I strongly recommend this text for its wealth of examples (and attractive exposition). 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. Course Webpage: 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. Homework: Comments: Math 214 - Section 1 - Differential Manifolds Instructor: Alexander Givental Lectures: TuTh 2:00-3:30pm, Room 81 Evans Course Control Number: 55101 Office: 701 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 215B - Section 1 - Algebraic Topology Instructor: Peter Teichner Lectures: TuTh 11:00am-12:30pm, Room 39 Evans Course Control Number: 55104 Office: 703 Evans Office Hours: TBA Prerequisites: Math 215A, see http://math.berkeley.edu/~hutching/teach/215a/index.html Required Text: Recommended Reading: G. Bredon, Geometry and Topology A. Hatcher, Algebraic Topology J. Milnor and J. Stasheff, Characteristic Classes Syllabus: We'll continue the qualitative study of topological spaces started in the fall. Topics will include fibre bundles, characteristic classes, K-theory and stable homotopy theory. Course Webpage: Under construction. Grading: From homework and course paper (in lieu of a final). Homework: Will be assigned weekly, plus a course paper. Comments: Math C218B - Section 1 - Probability Theory Instructor: David Aldous Lectures: TuTh 11:00am-12:30pm, Room 332 Evans Course Control Number: 55106 Office: 351 Evans Office Hours: TBA Prerequisites: Math C218A (stat 205A) or equivalent Required Text: Durrett, Probability: Theory and Examples Recommended Reading: Syllabus: Course Webpage: For further details see http://www.stat.berkeley.edu/users/aldous/205B/index.html Grading: Homework: Comments: Cross-listing of STAT 205B. Math 220 - Section 1 - Methods of Applied Mathematics Instructor: Alberto Grünbaum Lectures: TuTh 9:30-11:00am, Room 85 Evans Course Control Number: 55107 Office: 903 Evans Office Hours: TuWTh 11:00am-12:00pm Prerequisites: It is hard to describe exactly what are the prerequisites for this class besides a genuine interest in learning the material. You may want to try it for a few lectures and then decide if this is worth your effort. I will make every reasonable effort to start from scratch as we begin any new subject. Required Text: There is no required text but I will give pointers to the literature as we go along. Recommended Reading: Syllabus: The purpose of this class is to discuss a few topics that play in important role in several areas of "applied mathematics". All these subjects can be presented in a very detailed and technical fashion, but this is exactly what I will try to avoid. My aim will be to present these different topics ab-initio and try to emphasize their connections to each other. These topics include Random walks and Brownian motion. Birth and death processes. Recurrence, reversibility, the Ehrenfest urn model. Differential equations, coupled harmonic oscillators, some basic harmonic analysis. Evolution equations, semigroups of operators, the Feynman-Kac formula. Some prediction theory for stationary stochastic processes, the Ornstein-Uhlenbeck process. Stochastic differential equations, the harmonic oscillator driven by white noise. Invariant measures, ergodicity. Some Hamiltonian systems. Some important nonlinear equations exhibiting solitons, such as Korteweg de-Vries, non linear Schroedinger etc. The scattering transform as a nonlinear Fourier transform. Isospectral evolutions, the Toda equations. Course Webpage: Grading: Homework: Comments: Math 222B - Section 1 - Partial Differential Equations Instructor: Maciej Zworski Lectures: MWF 1:00-2:00pm, Room 35 Evans Course Control Number: 55110 Office: 897 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 224B - Section 1 - Mathematical Methods for the Physical Sciences Instructor: Jon Wilkening Lectures: MWF 10:00-11:00am, Room 39 Evans Course Control Number: 55113 Office: 1091 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 225B - Section 1 - Metamathematics Instructor: Thomas Scanlon Lectures: MWF 12:00-1:00pm, Room 72 Evans Course Control Number: 55116 Office: 723 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 228B - Section 1 - Numerical Solution of Differential Equations Instructor: John Strain Lectures: TuTh 11:00am-12:30pm, Room 5 Evans Course Control Number: 55119 Office: 1099 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 235A - Section 1 - Theory of Sets Instructor: Hugh Woodin Lectures: TuTh 11:00am-12:30pm, Room 7 Evans Course Control Number: 55122 Office: 721 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 250B - Section 1 - Commutative Algebra Instructor: Bernd Sturmfels Lectures: TuTh 8:00-9:30am, Room 5 Evans Course Control Number: 55128 Office: 925 Evans Office Hours: W 8:30-11:00am or by appt. Prerequisites: Math 250a or equivalent. Familiarity with algebraic geometry at the undergraduate level [Cox-Little-O'Shea] is helpful. Required Text: Gert-Martin Greuel & Gerhard Pfister, A SINGULAR Introduction to Commutative Algebra, Springer Verlag, 2002 Recommended Reading: David Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer Graduate Text, 1995 Martin Kreuzer & Lorenzo Robbiano, Computational Commutative Algebra, Volumes 1 and 2, Springer Verlag, 2000 and 2005 Syllabus: This course is an introduction to commutative algebra and its computational and applied aspects. In parallel to our study of the mathematical concepts (local rings, primary decomposition, modules and their free resolutions, integral closure, normalization etc.), we shall learn how to use the computer algebra system SINGULAR. I intend to follow the Greuel-Pfister book, but I will place more emphasis on Grobner bases in polynomial rings instead of standard bases in local rings. Course Webpage: http://math.berkeley.edu/~bernd/math250b.html Grading: The course grade will be based on weekly homework and a take home final exam. Homework: There will be weekly homework in the first half of the course. A fraction of it will require the use of SINGULAR. Comments: Anyone interested in learning algebraic tools for solving systems of polynomial equations is more than welcome to check out this course. Math 251 - Section 1 - Ring Theory Instructor: Tsit-Yuen Lam Lectures: MWF 3:00-4:00pm, Room 45 Evans Course Control Number: 55131 Office: 871 Evans Office Hours: TBA Prerequisites: Required Text: T. Y. Lam, A First Course in Noncommutative Rings, 2nd ed. Graduate Texts in Math., Vol. 131, Springer, 2001. Recommended Reading: T. Y. Lam, Exercises in Classical Ring Theory, 2nd ed. Problem Books in Math., Springer, 2003. Syllabus: This course will be an introduction to basic ring theory, mostly from a noncommutative perspective. The core material consists of things that a good student in algebra should know, including, e.g. semisimple rings and Artin-Wedderburn theorem, radical theory and Jacobson density theorem, prime and semiprime rings, local and semilocal rings, Krull-Schmidt-Azumaya theorem, etc. I'll try to cover a little more than half of my book. Since everything is written down already in the text, I will not repeat too many proofs, but will instead count on the students to read them at home. In lectures, I will only present selected proofs, try to do some exercises (or "volunteer" my students to do them!), and discuss motivation and perspectives. Course Webpage: Grading: There will be no finals and students will be graded based on their homework and participation. Those less prepared to participate actively are encouraged to take the course on S-NS basis. Homework: Comments: The main text (2nd ed.) is available from local bookstores as well as from Springer (or amazon.com). Students should have a copy for reading, and for doing the exercises. The "Problem Book" contains solutions to all exercises in the text and more, and has also come out in a new edition in 2003. This Problem Book is optional for the course, but is highly recommended to all students who want to have a broader and deeper understanding of the course material. Math 254B - Section 1 - Number Theory Instructor: Robert Coleman Lectures: MWF 1:00-2:00pm, Room 2 Evans Course Control Number: 55134 Office: 901 Evans Office Hours: TBA Prerequisites: Required Text: Koblitz, P-adic Numbers, p-adic Analysis, and Zeta-Functions Recommended Reading: J. W. S. Cassels, Algebraic Number Theory, A., Frs Frohlich (Editor) Jean-Pierre Serre, Local Fields Syllabus: Applications of p-adic Analysis, including Lubin-Tate Theory and rationality of the zeta function Course Webpage: Grading: Homework: Comments: Math 256B - Section 1 - Algebraic Geometry Instructor: Mark Haiman Lectures: MWF 2:00-3:00pm, Room 39 Evans Course Control Number: 55137 Office: 771 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: Math 261A - Section 1 - Lie Groups Instructor: Nicolai Reshetikhin, Vera Serganova, Richard Borcherds Lectures: TuTh 12:30-2:00pm, Room 7 Evans Course Control Number: 55139 Office: 917 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Fulton and Harris, Representation theory, Springer-Verlag, 1991. Among other useful books are J. P. Serre, Complex semisimple Lie algebras, 1987; A. Borel, Linear Algebraic Groups, Second Edition, Graduate Texts in Mathematics 126, Springer-Verlag, 1991; J. F. Adams, Lectures on Lie Groups, The University of Chicago Press, 1982. Syllabus: This is the first semester of the year-long course on Lie groups and Lie algebras. During this semester basic facts about Lie groups and Lie algebras, their structural theory, the classification of simple Lie algebras and the beginning of the representation theory of Lie groups and Lie algebras will be covered. More detail syllabus for the first third of the semester can be found at: http://www.math.berkeley.edu/~reshetik/Lie/Lie-06.pdf Course Webpage: Grading: The final grades will be assign on the base of homework and of the take-home final examination. Homework: Comments: Math 274 - Section 1 - Topics in Algebra Instructor: Arthur Ogus Lectures: MWF 11:00am-12:00pm, Room 71 Evans Course Control Number: 55142 Office: 877 Evans Office Hours: TBA Prerequisites: Math 250, Math 256A Required Text: Recommended Reading: Syllabus: Logarithmic geometry was originated in the eighties by Deligne, Faltings, Illusie, Fontaine and Kato. It provides a powerful conceptual framework for dealing with two fundamental related problems in algebraic and arithmetic geometry: compactification and degeneration. Among the classical techniques for dealing with these problems are: toroidal stacks and toroidal embeddings, semistable reduction, logarithmic differentials, and regular singular points of differential equations. Logarithmic geometry puts all these into a common unified geometric setting. As Kato has said, a logarithmic structure is a “something”—a “magic powder”—which when added to a “bad” situation can often transform it into a “good” one. The basic notion of a logarithmic scheme is appealingly simple, but working out the foundations thoroughly does pose some serious challenges which I will try to address. Although this material is fairly new, the course will be elementary (foundational), and I will try to make it accessible to students who have completed Math 256A. After a brief motivational introduction, I will discuss the foundations of toric geometry (the geometry of commutative monoids) which form the technical underpinnings of log geometry. Then I will discuss log structures, log schemes and morphisms of log schemes. We will especially discuss log smoothness and log differentials. As time permits I will discuss topological invariants of log schemes, including their fundamental groups and cohomology theories, including singular , de Rham , and crystalline cohomologies, with a sketch of the main application to p-adic Hodge theory. Course Webpage: http://math.berkeley.edu/~ogus/Math_274 Grading: Homework: Comments: Math 275 - Section 1 - Topics in Applied Mathematics: Flow, Deformation and Fracture Instructor: Grigori Barenblatt Lectures: TuTh 9:30-11:00am, Room 41 Evans Course Control Number: 55143 Office: 735 Evans Office Hours: TuTh 11:15am-12:50pm Prerequisites: No special knowledge of advanced mathematics and continuum mechanics will be assumed - all needed concepts and methods will be explained on the spot. Required Text: Landau, L. D. and Lifshits, E. M., Fluid Mechanics (Pergamon Press, London, New York 1987) Landau, L. D. and Lifshits, E. M., Theory of Elasticity (Pergamon Press, London, New York, 1986) Chorin, A. J. and Marsden, J. E., A Mathematical Introduction to Fluid Mechanics (Springer, 1990) Barenblatt, G. I., Scaling (Cambridge University Press, 2003) Recommended Reading: Batchelor, G. K., An Introduction to Fluid Dynamics (Cambridge University Press, 1998) Syllabus: Fluid Mechanics and Mechanics of Deformable Solids, including Fracture Mechanics are (fundamental) disciplines, playing an important and ever-growing role in applied mathematics, including computing, and also physics, and engineering science. The models of fluid flow, deformation and fracture of solids under various conditions are in all branches of applied mathematics, engineering science and many branches of physical science. Among the problems of these sciences which are under current active study there are great scientific challenges of our time such as turbulence, fracture and fatigue of metals, and damage accumulation. The proposed course will present the basic ideas and methods of fluid mechanics, including turbulence, mechanics of deformable solids, including fracture as a unified mathematical, physical and engineering discipline. The possibility of such a unified presentation is based on the specific `intermediate-asymptotic approach’ which allows the explanation of the main ideas simultaneously for the problems of fluid mechanics and deformable solids. Course Webpage: Grading: Homework: There will be no systematic homework. Some problems will be presented shortly at the lectures, their solutions will be outlined, and interested students will be offered the opportunity to finish the solutions. This will not be related to the final exams. 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 should be able to answer questions concerning the details of these topics. After that general questions (without details) will be asked concerning the other parts of the course. Math 275 - Section 2 - Topics in Applied Mathematics Instructor: Nicolai Reshetikhin Lectures: MWF 12:00-1:00pm, Room 39 Evans Course Control Number: 55146 Office: 917 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Statistical mechanics is one of the most fascinating areas in mathematical physics. Its aim is to describe "infinitely large" (order of 1020) systems of atoms. In this course we will study the models in statistical mechanics which describe interacting atoms placed on a graph (on a simplicial complex, a "crystal"). Many of these models are related to very interesting combinatorial and geometric structures. In this course we will focus on dimer models also known as weighted perfect matchings on a graph, or as tilings of a plane by polygons, or spanning trees of a graph, or discrete Dirac operators etc. It turns out that some of these models admit "exact solution", which usually means that certain algebraic structure emerges and as a result many characteristics of the system can be computed exactly. In many cases this brings very interesting representation theory into the picture. Perhaps most known example of this is the conformal symmetry of critical systems, which bring the representation theory of the Virasoro algebra into the picture. We will discuss this if time will permit. It will be shown how the representation theory of infinite general linear Lie algebra can be used in dimer models, how the quantized universal enveloping algebra of affine sl2 appear in the 6-vertex model (this models can be regarded as a model of "interacting dimers"). The details of the syllabus can be found in http://math.berkeley.edu/~reshetik/topics/dimers-syllab.pdf Course Webpage: Grading: Homework: Comments: Math 279 - Section 1 - Conformally Invariant Critical Systems of Statistical Physics and SLEs Instructor: Fraydoun Rezakhanlou Lectures: TuTh 12:30-2:00pm, Room 39 Evans Course Control Number: 55149 Office: 815 Evans Office Hours: TuTh 2:30-4:00pm Prerequisites: Some familiarity with stochastic calculus Required Text: Recommended Reading: Syllabus: One of the fundamental goal of statistical physics is the understanding of the large scale behavior of microscopic models. Recently, a new method for identifying the scaling limit of various conformally invariant two dimensional critical systems has been developed. The primary goal of this course is to study a number of lattice models in statistical physics at such as percolation, stochastic Ising model, self-avoiding random walks, loop-erased walks, and diffusion limited aggregation. It is conjectured that these models (at criticality) are macroscopically described by various members of a continuous family of random processes known as Schramm-Loewner evolutions (SLE). This conjectured has been established by Schramm for loop-erased random walk and by Smirnov for the critical percolation on the triangular lattice. Here is an outline of the course: 1. Some discrete models of equilibrium statistical physics. 2. Loop erased random walks and its conformal invariant continuum limit. 3. Riemann mapping theorem, capacity and Loewner’s differential equation. 4. Schramm-Lowener evolution. 5. Cardy’s formula and Smirnov’s theorem for critical percolation model. Hand-written notes will be provided during the semester. Course Webpage: Grading: Homework: There will be some homework assignments. Comments: Math 300 - Section 1 - Teaching Workshop Instructor: Hung-Hsi Wu Lectures: TBA Course Control Number: 55761 Office: 733 Evans Office Hours: TBA Prerequisites: Required Text: Recommended Reading: Syllabus: Course Webpage: Grading: Homework: Comments: