Spring 2023

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.

Logistics: The two MATH54 classes 001 (Kelli Talaska) and 002 (Katrin Wehrheim) will be fully coordinated – same contents, structure, and assignments. We will keep what worked well in the fall 2021 structure below and make some upgrades such as mastery grading. In particular, there will be midterm and final projects, rather than timed exams, and there is no need to purchase a textbook.

Instead, regular attendance in both sections and lectures will be necessary for learning – as well as achieving a passing grade. So please don't sign up for this course (or a section) if you cannot attend it. However, you are welcome to attend either the MWF 12-1 lectures or the TR 9:30-11 lectures – no matter which class you are registered for. One caveat: If you go to the MWF12-1 lectures, then your Wednesday section needs to be after 1pm to make any sense. 

For MATH49 enrollment see here.

Prerequisites Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended

Description Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.

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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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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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Recommended Reading 

Grading 

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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.

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Prerequisites Mathematics 104

Description A rigorous development of the basics of modern probability theory based on a self-contained treatment of measure theory. The topics covered include: probability spaces; random variables; expectation; convergence of random variables and expectations; laws of large numbers; zero-one laws; convergence in distribution and the central limit theorem; Markov chains; random walks; the Poisson process; and discrete-parameter martingales.

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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 Tue 10:30am - 12:30pm

Required Text Programming for Mathematical Applications, Lecture Notes, Per-Olof Persson

Recommended Reading 

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

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 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 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 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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Prerequisites Consent of instructor

Description The topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See departmental bulletins.

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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.

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.

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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.

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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 214 and 215A

Description Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem.

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Grading Letter grade.

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Prerequisites 250A

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

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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 Math 214, 261A, Consent of instructor for undergrads.

Description: Lie Groups, Lie algebras, and integrable systems

Completely integrable systems (classical and quantum) emerged in the last 50 years as one of the areas of most powerful applications of Lie groups and Lie algebras. In this course we will use two important classes of integrable systems to illustrate the main tenets of the general theory:

--- the (quantum) Gaudin systems;

--- the (classical) infinite-dimensional soliton hierarchies of KdV type.

Though the former system can be formulated in terms of a finite-dimensional simple Lie algebra, the commuting quantum Hamiltonians of this system arise naturally from the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level. We will discuss the Bethe Ansatz method for constructing eigenvectors and eigenvalues of these Hamiltonians. The connection of these systems to moduli spaces of G-bundles on complex curves and the Langlands correspondence for complex curves will also be discussed (however, no prior knowledge of the Langlands correspondence is required).

The latter systems (and closely related to them affine Toda models) can be described in terms of a double quotient of the Kac-Moody group of affine type (it is related to the so-called infinite Grassmannian). Commuting classical flows arise from the action of a maximal commutative subalgebra of the corresponding affine Kac-Moody Lie algebra.

All necessary notions will be explained, such as affine Kac-Moody algebras and the corresponding Lie groups, their representations, vertex algebras, etc.

Finally, we will talk about the affinization of the Gaudin models in which the role of a simple Lie algebra g is played by the corresponding affine Kac-Moody algebra. It turns out that in a special case, the resulting model may also be viewed as a quantization of the corresponding KdV system. Thus, we can unify the above two classes of integrable systems.

Time permitting, we will also discuss analogues of the Gaudin models (the so-called Heisenberg XXZ spin chains and their generalizations) associated to the quantum affine algebras.

Recommended Reading: to get a general idea of some of the topics that will be discussed, see these surveys (other sources will be introduced as the course progresses):

E. Frenkel, Affine Algebras, Langlands duality and Bethe Ansatz

E. Frenkel, Gaudin Model and Opers

E. Frenkel, Five Lectures on Soliton Equations

V. Drinfeld and V. Sokolov, Lie Algebras and Equations of KdV Type

Course Webpage will be on bCourses.

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.

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