Spring 2024

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 Three years of high school math, including trigonometry. Consult the mathematics department for details

Description Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions. This course is intended for business and social science majors. (See also the Math 1 sequence.)

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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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Prerequisites Three years of high school mathematics

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 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, 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 Completion of Math 1B (or 10B) and completion of or con-currently enrolled in Math 53 or 54, or 55.

Description The course will focus on reading and understanding mathematical proofs. It will emphasize precise thinking and the presentation of mathematical results, both orally and in written form. The course is intended for students who are considering majoring in mathematics but wish additional training.

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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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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 

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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 

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Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 54, or 56, or a course with equivalent linear algebra content. 55, or 74, 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 Math 53, 54, 55, or permission from instructor.

Description An introduction to computer programming with a focus on the solution of mathematical and scientific problems. Basic programming concepts such as variables, statements, loops, branches, functions, data types, and object orientation. Mathematical/scientific tools such as arrays, floating point numbers, plotting, symbolic algebra, and various packages. Examples from a wide range of mathematical applications such as evaluation of complex algebraic expressions , number theory, combinatorics, statistical analysis, efficient algorithms, computational geometry, Fourier analysis, and optimization. Mainly based on the Julia and the Mathematica programming languages.

Office 1081 Evans Hall

Office Hours TBD

Required Text Think Julia: How to Think Like a Computer Scientist, Ben Lauwens and Allen Downey.

Recommended Reading 

The official Julia documentation (latest stable version). Free online.

Insight through computing: A MATLAB introduction to computational science and engineering. Charles F. van Loan and K.-Y. Daisy Fan. SIAM, 2010. ISBN: 978-0-898716-91-7. Free online for UC Berkeley.

Grading Homework, quizzes, programming projects, midterm exam, and final exam.

Homework Weekly

Course Webpage http://persson.berkeley.edu/math124/

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 110 and 113

Description Isometries of Euclidean space. The Platonic solids and their symmetries. Crystallographic groups. Projective geometry. Hyperbolic geometry.

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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 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 

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.

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Grading Letter grade.

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Prerequisites 206

Description Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.

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Grading Letter grade.

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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.

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Grading Letter grade.

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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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Grading Letter grade.

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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 125A and 135

Description Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall.

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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

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.

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Grading Letter grade.

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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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Grading Letter grade.

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Prerequisites Consent of instructor

Description 

Office 899 Evans

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Grading Letter grade.

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Course Webpage