Fall 2018

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

Description: This sequence is intended for majors in engineering and the physical sciences. An introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions.

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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.

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Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry

Description: This sequence is intended for majors in the life sciences. Introduction to differential and integral calculus of functions of one variable. Representation of data, elementary probability theory, statistical models, and testing.

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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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Description: The Berkeley Seminar Program has been designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small-seminar setting. Berkeley Seminars are offered in all campus departments, and topics vary from department to department and semester to semester.

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Prerequisites: 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 or 10B. Mathematics 10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to partial differential equations.

Office: 905 Evans

Office Hours: TBA

Required Text: Linear Algebra and Differential Equations, Berkeley Custom Edition 2018

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Grading: 40% Homework + Section, 20% each midterm, 40% final with the lowest 20% exam discarded

Homework: Assigned for every lecture, due in section.

Course Webpage: in bCourses

Prerequisites: 1B or 10B. Mathematics 10B

Description: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series, application to 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. 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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Required Text: W. Rudin, Principles of Mathematical Analysis, 3rd edition

Recommended Reading: K. Ross, Elementary Analysis

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Course Webpage: https://math.berkeley.edu/~cjao/math104.html

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.

Office: 837 Evans

Office Hours: M 3-4 PM, Th 11-Noon

Required Text: Kenneth Ross, Elementary Analysis, 2nd edition

Course Webpage: https://math.berkeley.edu/~xuwenzhu/classes/Berkeley/2018Fall104/index.html

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: 55

Description: Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.

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Prerequisites: 104

Description: Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory.

Office: 1071 Evans

Office Hours: Wed 11:30 AM - 1:00 PM, Thu 11:00 AM -12:30 PM, and by appointment

Required Text: Morris W. Hirsch Stephen Smale Robert L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd Edition, Academic Press, 2012, Hardcover ISBN: 9780123820105, eBook ISBN: 9780123820112

Recommended Reading: See class web page.

Grading: See class web page.

Homework: See class web page.

Course Webpage: https://math.berkeley.edu/~simic/Math123/123.html

Prerequisites: 53 and 54

Description: Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform.

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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: 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.

Office: Evans 1055

Office Hours: Wednesday 11:00AM-12:00PM, Thursday 1:00-3:00 PM

Required Text:  T. W. Gamelin "Complex Analysis" (ISBN 978-0387-95069-3)

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Course Webpage: https://math.berkeley.edu/~art/F18-Math-185.html

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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 104 and considerable experience with other upper-division mathematics courses and with writing proofs. Math 105, 142 and 185 give especially useful preparation. I have no restrictions on enrollment by undergraduates. See Math department staff advisors for any needed enrollment codes.

Description: This course and Math 202B are "tool courses", in that they cover some basic mathematical concepts that are of importance in virtually all areas of mathematics and its applications. In Math 202A these include: Metric spaces and general topological spaces, compactness, theorems of Tychonoff, Urysohn, Tietze, locally compact spaces; an introduction to general measure spaces and integration of functions on them, with Lebesgue measure on the real line as a key example; function spaces and the very beginnings of functional analysis. In Math 202B most of these topics will be developed further, especially measure and integration, and functional analysis.

Office: 811 Evans Hall

Office Hours: TBA

Recomended Texts: Real and Functional Analysis 3rd ed. by Serge Lang, Springer-Verlag.

Basic Real Analysis by Anthony Knapp, Birkhauser.

Advanced Real Analysis by Anthony Knapp, Birkhauser.

Analysis Now by Gert K. Pedersen, Springer-Verlag,

The Lang text gives a presentation of the material that is somewhat closer to that which I will give than the other texts. My understanding is that through an agreement between UC and the publishers, chapters of the Lang text, the Knapp texts, and the Pedersen text are available for free download by students. You may need to use campus computers to authenticate yourself to gain access.  

Grading and Homework: 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, which will count for 35% of the course grade. There will be a midterm exam, at the regular class time, 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.

Course Webpage: There is a link to the course webpage on my homepage: math.berkeley.edu/~rieffel

Prerequisites: Math 202AB or equivalent is more than enough. Students who have studied only part of the material of Math 202AB and wish to enroll in Math 206 should discuss this with me.

Description: Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*-algebras of operators. Commutative C*-algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Hilbert-Schmidt operators. Fredholm operators. The Fredholm index. Selected additional topics.

Office: 811 Evans

Office Hours: TBA

Recommended Texts:  The most comprehensive text is  A Course in Functional Analysis, John B. Conway, 2nd ed., Springer-Verlag. Also useful are  Real and Functional Analysis,  Serge Lang, 3rd ed., Springer-Verlag, and Analysis Now, Gert K. Pedersen, Springer-Verlag, in part because they also contain the material from Math 202AB that is needed for Math 206. My understanding is that through an agreementbetween UC and Springer, chapters of the first edition of the Conway book, as well as the Lang and Pedersen books,are available for free download by students. You may need to use campus computers to authenticate yourself to gain access. Links to these texts are given on the course webpage. I do not plan to assign exercises from the Conway book, so the first edition should be satisfactory.

Comments: The theory of Banach algebras is a very elegant blend of algebra and topology which provides unifying principles for a number of different parts of mathematics and its applications, notably operator theory, commutative and non-commutative harmonic analysis and the theory of group representations, and the theory of functions of one and several complex variables. But at the present time probably its most extensive use is as a foundation for non-commutative measure theory (von Neumann algebras, Math 209) and non-commutative topology and geometry (C*-algebras, Math 208). These in turn provide a foundation for quantum physics, but they also have myriad applications in many other directions, including group representations and harmonic analysis, ordinary topology and geometry, and even number theory. (For a vast panorama of the applications see Alain Connes' 1994 book "Noncommutative Geometry", which is available on the web as a free download.)

I will cover the standard topics as listed in the catalog. Beyond the basic general theory of Banach algebras this will include several forms ofthe spectral theorem for self-adjoint operators on Hilbert space,compact and Fredholm operators, and group algebras and the Fourier transform and their relation to representation theory. Further topics as time permits, probably including Toeplitz operators and their index. The problem sets will involve a number of important specific examples beyond those presented in class.

Grading and homework:  Many weeks I will give out a problem set, and the course grade will be based on the work done on these. There will be no final examination.

Course Webpage: Link on math.berkeley.edu/~rieffel

Prerequisites: 113 and point-set topology (e.g. 202A)

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

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Prerequisites: 105 or 202A

Description: The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces.

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Prerequisites: 128A

Description: Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations.

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Prerequisites: 214

Description: Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.

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Prerequisites: 114 or consent of instructor

Description: Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.

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Prerequisites: 250A-250B for 256A; 256A for 256B

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.

Office: 855 Evans

Office Hours: MW 11-12 and 3-3:30. There may be many Math 55 students at my official office hours, so you might do better to look for me at other times.

Required Text / Recommended Reading: See course web page

Grading: Based on homework; see web page

Course Webpage: https://math.berkeley.edu/~mhaiman/math256-fall18-spring19

Prerequisites: 214

Description: Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.

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Prerequisites: Consent of instructor

Description: This is a graduate level course, to cover some of the analytical aspects of Mean Field Games. In the recent years, the number of areas of applications of the Mean Field Games theory have exploded, especially because the theory provides the simplest method to handle control problems with several agents. This includes communication networks, data networks, power systems, crowd motion, trade crowding and learning in Mean FieldGames. Despite the recent pioneer work by Cardialaguet–Delarue–Lasry–Lions, the theoryof Mean Field Games is not yet out of its infancy. We will briefly cover the needed stochastic analysis aspect at the undergraduate course level. Other useful geometric concepts will be briefly mentioned in order to quickly get to the heart of the matter.

This course will be taught by visiting Chancellor's Professor Wilfrid Gangbo.

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Prerequisites: Math 202A+B, or Math 222A, or consent of instructor. Can be taken concurrently with Math 222A.

Description: Semiclassical analysis is a mathematical theory underlying the classical/quantum, or particle/wave, correspondence. A key concept in semiclassical analysis is the notion of microlocalization, namely localization of functions in both position and frequency space. Another important component is propagation of singularities, which when applied to the wave equation explains why light travels along straight lines. In the recent years semiclassical analysis has found many new and exciting applications, among others to scattering theory, dynamical systems, quantum chaos, and general relativity. In this course we follow Maciej Zworski's book "Semiclassical Analysis", starting with the basic theory and ending with applications such as quantum ergodicity.

Office:  805 Evans Hall

Office Hours:   Tuesdays 2–3 and by appointment

Required Text: Maciej Zworski, "Semiclassical Analysis". The UCB Library call number is QC174.17.D54 Z96 2012 .

Grading: Letter grade based on homework.

Course Webpage: http://math.berkeley.edu/~dyatlov/279/