Spring 2018

Prerequisites: 1A

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

Office: 805 Evans Hall

Office Hours: M 1-2, W 3-4

Required Text: James Stewart, Single Variable Calculus: Math 1A,B at UC Berkeley, 8th Edition

Course Webpage: https://bcourses.berkeley.edu/courses/1468243

Prerequisites: Continuation of 10A

Description: Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.

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Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic exam, or 32. Consult the mathematics department for details

Description: This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions.

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Prerequisites: 16A

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

Office: 999 Evans

Office Hours: 2:30-4:00 MF

Required Text: Calculus with Applications by Lial, Greenwell, Ritchey, 11th Edition, ISBN: 9780133886832.

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Course Webpage: https://math.berkeley.edu/~yxy/math16b

Prerequisites: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or UC/CSU diagnostic examination

Description: Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences.

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Prerequisites: Mathematics 1B

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

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Prerequisites: 1B

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

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Prerequisites: 1B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Description: Directed Group Study, topics vary with instructor.

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Description: Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.

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Prerequisites: Math 53 and 54

Description: Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 53 and 54

Description: The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites: 54 or a course with equivalent linear algebra content

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

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites: 110 and 113, or consent of instructor

Description: Further topics on groups, rings, and fields not covered in Math 113. Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields.

In Spring 2018, Math 114 will be focused on Galois theory, building toward the impossibility of solving the general quintic by radicals, and proving the impossibility of some famous geometric constructions along the way.

Office: 985 Evans

Office Hours: TBA

Required Text: Galois Theory, by Ian Stewart. 4th edition, CRC Press

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Grading:  Based on weekly homework assignments, one or two midterms, and a final exam.

Homework: Weekly assignments.

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Prerequisites: 53 and 54

Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Special functions, series solutions of ordinary differential equations, partial differential equations arising in mathematical physics, probability theory.

Office: 1089 Evans

Office Hours: TBD

Required Text: Mary L. Boas, Mathematical Methods in the Physical Sciences, 3e.

Grading: Letter grade. Final exam required.

Homework: Weekly

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Prerequisites: 53 and 54

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

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Prerequisites: 53, 54, and 55

Description: Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.

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Prerequisites: 113

Description: Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.

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Prerequisites: 53, 54, and 113

Description: History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.

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Prerequisites: 55

Description: Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional topics at the discretion of the instructor.

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Prerequisites: 104

Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

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Prerequisites: 104

Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

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Prerequisites: 104

Description: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

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Prerequisites: 104

Description: Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

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Description: Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.

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Prerequisites: Math 202A and Math 110

Description: Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert 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 the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.

Office: 811 Evans

Office Hours: TBA

Recommended Text: Real and Functional Analysis 3rd ed. by Serge Lang, Springer-Verlag. Basic Real Analysis by Anthony Knapp. Advanced Real Analysis by Anthony Knapp.  My understanding is that through an agreement between UC and Springer, chapters of the Lang text and the Knapp texts are available for free download by students. See the course webpage for the links to them. For those who have used the text Real Analysis by Folland for Math 202A, that text can also be quite useful for parts of Math 202B. 

 Comments: We will continue on from wherever this Fall's Math 202A ends, to develop the theory of measure and integration.We will also develop more general topology as needed. A major further subject will be an introduction to functional analysis, which consists of methods for dealing with infinite-dimensional topological vector spaces and linear operators on them. This will use both the general topology and the measure and integration that has been covered. It has wide applications, for example to harmonic analysis, partial differential equations, analysis on manifolds, and quantum physics. Some of the items we will discuss are Banach spaces, the closed-graph theorem,  the Hahn-Banach theorem and duality, duals of classical Banach spaces, weak topologies, the Alaoglu theorem, convexity and the Krein-Milman theorem. We will also discuss further related topics if time allows. In my lectures I will try to give well-motivated careful presentations of the material. 

Grading: Grading: I plan to assign roughly-weekly problem sets. Collectively they will count for 50% of the course grade. Students are strongly encouraged to discuss the problem sets and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help. There will be a final examination on Wednesday May 9, 11:30-2:30 , which will count for 35% of the course grade. There will be a midterm exam, which will count for 15% of the course grade. There will be no early or make-up final examination. Nor will a make-up midterm exam be given; instead, if you tell me ahead of time that you must miss the midterm exam, then the final exam will count for 50% of your course grade. If you miss the midterm exam but do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent in order to try to avoid a score of 0.

Accomodations: Students who need special accomodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each exam what specific accomodation they need, so that I will have enough time to arrange it.

Course Webpage: math.berkeley.edu/~rieffel

The above information is subject to change.

Prerequisites: Because Math 206 was not offered this past Fall, the prerequisite for this course will be Math 202A-B or equivalent. (In fact, it will be reasonable to take Math 208 concurrently with Math 202B if one studies ahead of time pages 65-83 and 95-107 in the book "Real and Functional Analysis" by S. Lang.) The consequence is that during the first several weeks of the course we will develop the theory of commutative C*-algebras, a topic usually covered in Math 206. (So we will not be able to cover as much advanced material at the end of the course.)

Description: Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.

Office: 811 Evans

Office Hours:  TBA

Recommended Text: None of the available textbooks follows closely the path that 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).

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. There will be no final examination.

Comments: The theory of operator algebras grew out of the needs of quantum mechanics, but by now it 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"), and examine a number of interesting examples. We will then briefly look at "non-commutative differential geometry". Finally, time permitting, we will glance at "non-commutative vector bundles" and K-theory ("noncommutative algebraic topology") . But I will not assume any prior knowledge of algebraic topology or differential geometry, and we are unlikely to have time to go into these last topics in any depth.(For a vast panorama of the applications I strongly recommend Alain Connes' 1994 book "Noncommutative Geometry", which is available on the web as a free download. Of course much has happened since that book was written, but it is still a very good guide to a large variety of applications.)

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 parts of high-energy physics. Thus one thread that will run throughthe course will be to see what the various concepts look like for quantum tori, which are the most accessible interesting non-commutative differentiable 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: https://math.berkeley.edu/~rieffel/208ann18.html

Prerequisites: 214

Description: Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.

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Prerequisites: 105 or 202A

Description: The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor.

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Prerequisites: 125B and 135

Description: Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.

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Prerequisites: 250A

Description: Commutative algebra

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Course Webpage: https://math.berkeley.edu/~vivek/250B.html

Prerequisites: 256A

Description: Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

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Prerequisites:  Sufficient knowledge of the fundamental group and homology and cohomology.

Description: This course is meant to roughly be the old standard 215B, a successor to 215A.  It will cover some homotopy theory, some characteristic classes, some Morse/Cerf theory, some classical theorems like the s-cobordism theorem, with many examples and problems using low dimensional manifolds.  Students will be encourage to give talks.

Office:  919 Evans

Office Hours:   TBA

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Recommended Reading:  Hatcher, Algebraic Topology

Grading: Offered for satisfactory/unsatisfactory grade only.

Homework:   Yes

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Title:   The topic for this special topics course is "Complex/Kahler Geometry"  The instructor will Professor Song Sun.

Prerequisites: Math 240 or 241

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

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Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center

Description: May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.

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Grading: Offered for pass/not pass grade only.

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