Spring 2021

Prerequisites 1A or N1A

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 Continuation of 10A

Description The sequence Math 10A, Math 10B is intended for majors in the life sciences. Elementary combinatorics and discrete and continuous probability theory. Representation of data, statistical models and testing. Sequences and applications of linear algebra.

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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.

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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 or N1B

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, N1B, 10B, or N10B

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.

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Prerequisites 

Description Directed Group Study, topics vary with instructor.

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Prerequisites 

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.

Office 

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Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

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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Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

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 

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Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

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 

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Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs is recommended

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. 55 or an equivalent exposure to proofs

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. 55 or an equivalent exposure to proofs

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. 55 or an equivalent exposure to proofs

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. 55 or an equivalent exposure to proofs

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.

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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. Chapters 11, 12, 13 and 15 of the textbook.

Office Hours Via Zoom, probably M 10:30-11:30; W 2-3:30; F 10:15-10:45 

Required Text Mathematical Methods in the Physical Sciences, 3rd ed, by Mary L. Boas, Wiley Pub

Lectures There will be a bCourses site for the course. The lectures (via Zoom) will be recorded and posted on that site.

Examinations The format and mechanics of the examinations will be posted on the bCourses site for this course.  They will probably involve GradeScope.

The final examination will take place on Monday, May 10, 7-10 PM.    There will be no early or make-up final examination. 

There will be two midterm examinations. They will take place on Wednesday February 17  and Wednesday March 31. (Those dates could change if important reasons for a change arise.)   

Most weeks there will be one short quiz. 

Grading The final examination will count for 30% of the course grade.

Each midterm exam will each count for 25% of the course grade. Makeup midterm exams will not be given; instead, if you tell me ahead of time that you must miss one of the midterm exams, then the final exam and the other components will count more to make up for it. If you do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent to try to avoid a score of 0. If you miss both midterm exams, you will need a truly extraordinary documented reason in order to avoid a score of 0 on at least one of them.

The combined quiz scores will count for 20% of the course grade. The two lowest quiz scores will be dropped. Makeup quizzes will not be given. 

Homework There will be weekly homework assignments due the Tuesday of the following week. (That could be changed to Thursday if there is a strong preference for that.) The homework will not be graded, but submitted homework will be registered and may affect your grade by a few points. Late homework will not be accepted. 

Comment Students are strongly encouraged to discuss the course material and homework with each other (remotely).

Accommodation Students who need special accommodation for examinations should bring me the appropriate paperwork, and must tell me between one and two weeks in advance of each exam what specific accommodation they need, so that I will have enough time to arrange it.

Course Webpage Link at:  http://math.berkeley.edu/~rieffel

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 110 and 128A

Description Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer.

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Prerequisites 104

Description Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet-Von Dyck Theorem.

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Prerequisites 151; 54, 113, or equivalent

Description Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry.

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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 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 Math 53, 54.  Math 170 would be useful, but is not required.

Description Topic:  Dynamic Optimization.  This course will discuss in detail the theory and applications of ``dynamic optimization'', primarily in the context of the calculus of variations and optimal control theory. Topics will include careful derivation of Euler-Lagrange equations, second variation calculations, the Pontryagin maximum principle and dynamic programming, with many interesting examples from pure and applied mathematics.    For the complete list of topics see this handout.

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Prerequisites 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 185

Description Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem.

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Grading Letter grade.

Homework 

Course Webpage 

Prerequisites 202A

Description Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. Morse functions, differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.

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Grading Letter grade.

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Prerequisites 

Description The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion.

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Grading Letter grade.

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Course Webpage 

Prerequisites 222A

Description: The first part of the course will cover Sobolev spaces, Gagliardo--Nirenberg--Sobolev and Morrey inequalties, Schauder estimates and some applications. We will continue with basic microlocal analysis with applications to hyperbolic equations (the Cauchy problem via energy estimates and the parametrix constructions), unique continuation and other topics.  Basic Fourier analysis and theory of distributions (as covered in 222A in the Fall of 2020) are the only prerequisites.

Office Hours: Wed 2-4 PM (via Zoom)

Required Text:  "Partial Differential Equations" by LC Evans (available on-line to UC students at https://bookstore.ams.org/ ) and  "Microlocal Analysis for Differential Operators" by A Grigis and J Sjostrand (available on-line to UC students at https://www.cambridge.org/ )

Grading Letter grade.

Homework weekly homework assignments (total of 8 assignments)

Course Webpage  https://math.berkeley.edu/~zworski/222B_21.html

Prerequisites 125A and (135 or 136)

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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Grading Letter grade.

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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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Grading Letter grade.

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Prerequisites 250A or consent of instructor

Description (I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor.

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Grading Letter grade.

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Prerequisites 254A and 256A

Description Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.

Office 883 Evans

Office Hours TBD

Required Text None (notes will be provided)

Recommended Reading None

Grading Letter grade.

Homework Weekly or biweekly, with a final homework assignment due during finals week

Course Webpage https://math.berkeley.edu/~vojta/254b.html

Prerequisites 250A

Description We will start with the braid groups, an excellent case study for infinite discrete groups, which can be defined by generators and relations as well as geometrically, as fundamental (or mapping class) groups. Symmetric groups and Hecke algebras appear naturally from braid groups. As an application, we will discuss the Jones polynomial and its generalizations. After a brief survey of simple Lie groups, will look at groups from the Hopf algebra perspective and use it to go from groups to quantum groups. Next, the Tannakian formalism, which enables one to reconstruct an algebraic group from the tensor category of its representations. Time permitting, we will also discuss loop groups, affine Grassmannians, and the geometric Satake correspondence.

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Recommended Reading  We will use these books, available electronically from Springer Link (via a UCB login):

Christian Kassel and Vladimir Turaev, Braid Groups, Springer GTM 247

Christian Kassel, Quantum Groups, Springer GTM 155

as well as other materials that will be made available by the instructor in due time.

Grading Letter grade.

Homework 

Course Webpage 

Prerequisites Consent of instructor

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

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Grading Letter grade.

Homework 

Course Webpage 

Prerequisites Consent of instructor

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

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Grading Letter grade.

Homework 

Course Webpage 

Prerequisites 

Description A classical subject in the mathematical theory of hydrodynamics consists in studying the evolution of the free surface separating the air from a perfect incompressible fluid. We will examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. They are of different nature, dispersive versus parabolic, but we will see that they can be studied by related tools.

These courses are intended for graduate students with a general interest in analysis and no pre-requisites about any advanced theory is required. A large part of the courses will consist of short self-contained introductions to the following topics: Paradifferential calculus, the fractional Laplacian and the multiplier methods of Morawetz and Lions. These lectures also aim to give a self-contained introduction to certain aspects at the cutting edge of research. I will give a detailed analysis of the Cauchy problem for the water wave, Hele-Shaw and Muskat equations.

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Office Hours 

Required Text  None.  Professor Alazard will type lecture notes and post them every week on his webpage.

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage