Fall 2019

Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry. Students with high school exam credits (such as 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 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 and one-half years of high school math, including trigonometry and analytic geometry. Students who have not had calculus in high school are strongly advised to take the Student Learning Center's Math 98 adjunct course for Math 10A; contact the SLC for more information

Description: The sequence Math 10A, Math 10B is intended for majors in the life sciences. Introduction to differential and integral calculus of functions of one variable, ordinary differential equations, and matrix algebra and systems of linear equations.

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Prerequisites: Three years of high school math, including trigonometry. 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.

Exams: Please note this section of Math 16A has an evening midterm on Monday, October 28, between 8 and 10pm.  Do not schedule conflicting activities at this time.

Office: 785 Evans

Office Hours: TBD

Required Text: 

Calculus with Applications
Lial, Greenwell, and Ritchey 
11th edition, ISBN: 9780321979421

Use other editions at your own risk. Please note the department has negotiated a greatly reduced price for this textbook, so the campus bookstore will most likely be the cheapest place to buy it and possibly to rent it. The campus bookstore price is about $95, so you should not pay more than that. You are NOT required to have an online access code.

Homework: Lots

Instructor's Webpage: https://math.berkeley.edu/~talaska/index.php

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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.

Office: 911 Evans Hall

Office Hours: Tu 5-6:30pm (outside 155 Dwinelle), Wed 1:30-3pm 911 Evans

Required Text: Stewart's "Multivariable Calculus, Math 53 UC Berkeley" 8th edition

Recommended Reading: Stewart's "Multivariable Calculus, Math 53 UC Berkeley" 8th edition

Grading: 10% Homework, 10% Quizzes, 25% Each Midterm Exam (2 Midterms), 30% Final.

Homework: posted on bCourses

Course Webpage: bCourses

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, application to partial differential equations.

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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, application to partial differential equations.

Course Webpage: https://math.berkeley.edu/~katrin/teach/54/

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: 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.

The bigger events, joining all sections of Berkeley Connect in Math, will be on Tuesdays, Wednesdays, or Thursdays at 5 or 6pm, and might not be at your regular scheduled time. There will be only four of those events: the opening meeting and three panels.

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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 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 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 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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Required Text: Elementary Analysis by Kenneth Ross

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Prerequisites: 53 and 54

Description: Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.

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Prerequisites: 54 or a course with equivalent linear algebra content

Description: Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.

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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: 53 and 54

Description: Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Rapid review of series and partial differentiation, complex variables and analytic functions, integral transforms, calculus of variations.

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Prerequisites: Math 113 or consent of instructor

Description: Sentential and quantificational logic. Formal grammar, semantical interpretation, formal deduction, and their interrelation. Applications to formalized mathematical theories. Selected topics from model theory or proof theory.

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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: 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: 104 and 113

Description: The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, plus further topics at the discretion of the instructor.

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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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Description: This class will function as a problem-solving seminar intended, in the first place, for those who plan to take the Putnam Exam, the US and Canada math Olympiad for college students, on the first Saturday of December 2019. Our objectives will be to build participant's confidence in solving challenging math problems and, more importantly, to learn interesting mathematics, using materials of various Mathematical Circles and Olympiads as motivation.

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Prerequisites: Must have completed 60 units and be in good standing

Description: Topics will 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.

The bigger events, joining all sections of Berkeley Connect in Math, will be on Tuesdays, Wednesdays, or Thursdays at 5 or 6pm, and might not be at your regular scheduled time. There will be only four of those events: the opening meeting and three panels.

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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: Because Math 206 was not offered recently, the prerequisite for this course will be Math 202A-B or equivalent. (For those who may not have background fully equivalent to Math 202A-B it will be reasonable to enroll in Math 208  if one studies ahead of time chapters I to V of the book "Real and Functional Analysis" by S. Lang, or chapters 1 and 2, and sections 3.1 and 3.2 of the book "Analysis Now" by G. K. Pedersen. 

Brief Description: 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 Reading: 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).

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 happened since 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, group representations 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: Letter grade.

Homework: 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: math.berkeley.edu/~rieffel

Prerequisites: fluency in basic abstract algebra and point-set topology + appropriate level of "mathematical maturity". Experience from Math 214 could be useful.   Instructions for undergraduate students to request an enrollment permission can be found here:  https://tinyurl.com/F19GradEnrollPolicies 

Description: In this first semester of the Algebraic Topology sequence, we will follow the book "Homotopical topology" by A. Fomenko and D. Fuchs. The book is freely available in electronic form to UC Berkeley library patrons. The main difference of this text from Hatcher's (usually used for this course) is that it begins with homotopy groups, and introduces (co)homology only afterwards, mostly as a tool for computing the former. This seems conceptually more appropriate.  

The plan is to cover Introduction, Chapter 1, and the most of Chapter 2 - which means we will move quickly. To help with digesting the fair amount of material, I plan, if there is enough interest, to open a seminar which can function as both discussion section and office hours. 

I recommend those who plan to take this course to check their readiness by trying to read "Introduction" in advance, and to do exercises from it.

Also, you need to be willing to abandon the familiar mathematical environment, and immerse into some alien extra-terrestial universe (depicted by Fomenko's drawings found in the book) from which there might be no return.          

Office: 701 Evans

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Required Text: see description above

Recommended Reading: yes, it is highly recommended

Grading: Letter grade.

Homework: yes

Course Webpage:  TBD

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: Graduate status or consent of instructor

Description: Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.

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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: 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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Prerequisites: 250A for 254A; 254A for 254B

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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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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Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center

Description: May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.

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Grading: Offered for pass/not pass grade only.

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