Spring 2022

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

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 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 

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 104

Description 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.

Office 

Office Hours 

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

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.

Office 

Office Hours 

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

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.

Office 

Office Hours 

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

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.

Office 

Office Hours 

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

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent in discrete math

Description The optimistic plan is to complete by the spring recess the material of the group and ring theory standard for the honors version of Math 113, and spend April studying Galois theory of finite field extensions up to Gauss' solution of the problem about straightedge-and-compass constructibility of regular polygons and Abel's (un)solvability theorem of polynomial equations in radicals. To what extent the plan is going to be realized will partly depend on the students.    

Office 701 Evans

Office Hours TBD

Required Text  We will closely follow the yet unpublished "Lectures on Groups, Rings, and Fields" of the instructor's own writing, which will be provided online in several installments during the semester  

Recommended Reading A more standard text "Topics in Abtract Algebra" by Herstein is also recommended; it contains most of the needed material, but we will not really follow it in any detail

Grading  should be normally based on homework, quizzes, and the final, but COVID constraints may require some changes, especially concerning quizzes 

Homework Weekly

Course Webpage Math H113, Spring'22

Prerequisites Math 55 is recommended

Description Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 53 and 54. In particular, Math 121A is not a prerequisite, and we will not assume knowledge of material from that course. However, we can expect that many of the students enrolled in 121B will have taken 121A (and others may have taken other upper-division mathematics courses). As a consequence, students for whom 121B is their first upper division mathematics course should expect to struggle much more then the more mathematically experienced students in the course in dealing with the greater level of abstraction and emphasis on theory that is characteristic of upper division mathematics courses.  

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 811 Evans

Office Hours TBA

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

Grading The final examination will take place on Tuesday, May 10, 3-6 PM. There will be no early or make-up final examination.

The final examination will count for 35% of the course grade. 

There will be two midterm examinations. They will take place on Wednesday February 16  and Wednesday March 30. (Those dates could change if important reasons for a change arise.)  

Each midterm exam will 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 persuasive 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. 

 Students who need special accommodation for examinations should have the appropriate paperwork sent to me, 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.

Homework  There will be weekly homework assignments. The homework will count for 15% of the course grade. The two lowest homework scores will be dropped. Late homework will not be accepted. Homework will be submitted by uploading to GradeScope.

Students are strongly encouraged to discuss the course material and homework with each other, but each student should write up their own homework solutions, reflecting their own understanding of the material, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this collaboration or help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help. 

Course Webpage Link at 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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites Math 104 and 113 or consent of instructor

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 113

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

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

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 

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 

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 202A and 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 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

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 sections3.1-2, 4.1-4 of "Analysis Now" by G. K. Pedersen would be sufficient.It is my understanding that through an agreement between UC and the publisher, the Pedersen text can be downloaded at no cost through the campus library system.

Syllabus 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).UCB students may be able to freely downloadthis book through the campus library system.

Description  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 can be freely downloaded from the web. Of course much has happenedsince that book was written, but it is still a very good guide to the very large variety of applications.)

I will discuss a variety of examples, drawn from dynamical systems, grourepresentations 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 through the 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.

Grading I plan to assign problem sets roughly every other week. 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.

Course Webpage link at  math.berkeley.edu/~rieffel

Prerequisites 215A, 214 recommended (can be taken concurrently)

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

Prerequisites 214

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

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

Prerequisites 250A

Description Development of the main tools of commutative and homological algebra applicable to algebraic geometry, number theory and combinatorics.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

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.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage 

Prerequisites 

Description This course will give introductions to current research developments. Every semester we will pick a different topic and go through the relevant literature. Each student will be expected to give one presentation.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading Offered for satisfactory/unsatisfactory grade only.

Homework 

Course Webpage 

Prerequisites 222A/B or equivalent; 222B may be taken concurently; instructor's permission is required for undergraduate students.

Description  Free boundary problems arise whenever a pde evolution happens in a domain whose boundary is freely moving.  This may occur in elliptic, parabolic as well as in hyperbolic problems.

Well-known examples are the Stefan problem for ice melting, water waves or compressible gases (e.g. a gaseous star).  The aim of the course will be to describe several of these problems, along with the pde

and microlocal analysis tools needed to approach them. Some preference will be given to problems arising in fluid dynamics.

Office

Office Hours 

Required Text 

Recommended Reading 

Grading Letter grade.

Homework 

Course Webpage https://bcourses.berkeley.edu/courses/1513090/  (ask to be added to bcourses if needed)  The course will start on zoom and then switch to the in class mode when possible.