Fall 2017

Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic test, or 32. Consult the mathematics department for details. Students with 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.

Office: 901 Evans

Office Hours: TBA

Required Text: James Stewart, Single Variable Calculus: Early Transcendentals for UC Berkeley, 8th edition (or James Stewart, Calculus, 8th edition, early transcendentals)

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Grading: 12% homework, 8% quizzes, 20% 1st midterm, 20% 2nd midterm, 40% final

Homework: administered through WeBWorK (online homework)

Course Webpage: TBA

Prerequisites: 1A

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

Description: This sequence is intended for majors in the life sciences. Introduction to differential and integral calculus of functions of one variable. Representation of data, elementary probability theory, statistical models, and testing.

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Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic exam, or 32. 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.

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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: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or UC/CSU diagnostic examination

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

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: Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Prerequisites: 1B

Description: Honors version of 54. Basic linear algebra: matrix arithmetic and determinants. Vectors spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.

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Description: Directed Group Study, topics vary with instructor.

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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.

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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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Prerequisites: 54 or a course with equivalent linear algebra content

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

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, 54, and 55; Statistics 20 recommended

Description: Introduction to mathematical and computational problems arising in the context of molecular biology. Theory and applications of combinatorics, probability, statistics, geometry, and topology to problems ranging from sequence determination to structure analysis.

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Prerequisites: 110 and 113

Description: A critical examination of Euclid's Elements; ruler and compass constructions; connections with Galois theory; Hilbert's axioms for geometry, theory of areas, introduction of coordinates, non-Euclidean geometry, regular solids, projective 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: 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: 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.

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Prerequisites: familiarity with basic notions of Algebra (rings, modules) is helpful but not required

Description: A thoroughly modern introduction to “Differential Calculus” on general “spaces” (classical and quantum alike).

The topics to be covered:

(I) Theory of derivations: definitions, examples, the universal derivation, Bar and Hochschild complex, graded and super-derivations

(II) Infinitesimal deformations ; description of derivations from 3 fundamental algebras — tensor, symmetric, and exterior

(III) Theory of differential forms: the de Rham super-algebra, the de Rham cohomology

(IV) Connections and the curvature, characteristic classes, Chern character, K-theory

(V) Integration of differential forms, de Rham’s theorem

Office: 995 Evans Hall

Office Hours: after my lectures or by appointment

Required Text: my own notes distributed in class

Recommended Reading: announced throughout the semester

Grading: weekly homework and the final project

Homework: assigned once a week, due a week later

Course Webpage: TBA

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Description: The course is designed as a sequence with Statistics C205B/Mathematics C218B 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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Prerequisites: 125B and 135

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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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.

Office: 897 Evans

Office Hours: TBD

Required Text: "Riemann Surfaces" by Simon Donaldson and "Complex Geometry" by Daniel Huybrechts. A PDF of the latter is available for free through the library.

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Grading: Letter grade.

Homework: Weekly homework will be given.

Course Webpage: https://math.berkeley.edu/~lott/teaching.html

Prerequisites: The equivalent of one of Math H113, or 114, or 250A, or consent of the instructor.  (250A may be taken simultaneously.)

Syllabus: The theme of Math 245A, as I teach it, is universal constructions.  We begin with a well known case, free groups, which we will construct in three quite different ways, and show why these constructions come to the same thing.  We then examine a smorgasbord of other important universal constructions, noting similarities, differences, and general patterns.
     After that, we settle down to developing the tools needed to study the subject in a unified way:  Ordered sets and the axiom of choice, closure operators, category theory, and the general concept of a variety of algebras.  (We treat most of these, not merely as means to this goal, but as interesting landscapes worth lingering over.)  We find that the free group construction is a particular example of an adjoint functor (it is left adjoint to the underlying-set functor on groups), and eventually develop a marvellous result of Peter Freyd, characterizing the functors between varieties of algebras that have left adjoints.  We determine, in several cases, all such functors.

Office: 865 Evans Hall

Office Hours: TBA

Required Text: George M. Bergman, An Invitation to General Algebra and Universal Constructions, Springer Verlag, 2015; also available in a slightly different version here.

Recommended Reading: None.  (But some texts useful either for filling in background, or for pursuing some topics further, are listed in section 1.6 of the text.)

Homework and Grading: The text contains more interesting exercises than anyone could do! I will ask you to think briefly about each exercise, and select a few to work and hand in each week; more details at the first class meeting.  Grades will be based mainly on this homework.  Students wishing a reduced homework-load can enroll S/U.

Comments: My philosophy is that it does not make sense to spend the classroom time using the instructor and students as an animated copying machine.  Rather, the material that is typically delivered in a lecture should be put in duplicated notes which the students study, and class time should be devoted to the more human activities of discussing and clarifying the material, introducing some topics by the Socratic method, etc..  Such notes for Math 245, begun in Fall 1971 and reworked each time I taught the course, have evolved into the text we will use.
     There are difficulties with this way of teaching if a textbook leaves out motivation, examples, etc., that might be included in a lecture; but I have made it a point to include these in the notes.  Another problem is that doing the reading before class runs counter to the habits of many students.  To get around this, I require each student to submit, on each class day, a question about the day's reading.  If possible, these should be submitted by e-mail at least an hour before class, so that I can address some of the questions in class.

Prerequisites: 114 or consent of instructor

Description: Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.

Office: 927 Evans Hall

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Required Text: Lang, Algebra

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Grading: Letter grade.

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Course Webpage: https://bcourses.berkeley.edu/courses/1461105

Prerequisites: 250A-250B for 256A; 256A for 256B

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.

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Prerequisites: Mathematical maturity of an advanced graduate student. Interest in the MSRI program. 

Description: Our hot topic is that of the concurrent MSRI program: GEOMETRIC AND TOPOLOGICAL COMBINATORICS. We offer a lightning introduction to research themes in Geometric and Topological Combinatorics. This will enable students to participate in MSRI activities. The course consists of 14 lectures. Each lectures introduce the audience to a different text book or monograph.  I will lecture on selected topics from that book. Participants are encouraged to form study groups, work out problems, do computations, and collaborate on projects

Office: 925 Evans

Office Hours:  After class or by appointment

Reading: The 14 books listed on the course webpage.

Grading: Offered for satisfactory/unsatisfactory grade only. Students who enroll for credit will select one of our 14 books and become an expert in that book. They will receive the passing grade (S) after demonstrating that expertise, in a format to be agreed upon. 

Course Webpage: https://math.berkeley.edu/~bernd/math270.html

Prerequisites: Math 250A, Math 215A. Math 143 and Math 214 will be helpful. 

Description:

Examples of invariants of knots and links. 

Knots and links are closed embedded curves in 3-dimensional Euclidean space.

Knot theory describes the classification of knots and their relations to many

related topics. There are a plethora of invariants of knots in order to distinguish

them up to isotopy (continuous deformation preserving the embedding). The

goal of this topics class will be to investigate some of these invariants, hopefully

finding some connections between different invariants and highlighting some open problems. 

We will follow roughly the historical/topical order of introduction of these invariants. Examples

will hopefully include: 

• fundamental group
• Seifert genus
• Alexander polynomial 
• geometric decomposition
• simplicial volume
• A-polynomial
• twisted Alexander polynomials
• Jones polynomial
• other quantum polynomials (Kauffman, HOMFLYPT, colored variants)
• Vassiliev invariants
• Khovanov homology
• grid link homology

In order to understand some of these invariants, we will have to develop a bit of 

the theories of representing knots, such as projections and Reidemeister moves,

braids, and grid diagrams.  

Office: 921 Evans Hall

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Grading: Letter grade.

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 Students will be required to give a presentation at the end of the semester. 

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Prerequisites: Consent of instructor. Some linear PDE knowledge is essential.

Description: Nonlinear PDE's. This course is aimed at students with some PDE background, and will cover a selection of topics in nonlinear PDE's which are of current interest.

This may include: nonlinear elliptic and parabolic problems, nonlinear wave equations, hyperbolic systems, fluid dynmics, water waves

Required Text:  None

Recommended Reading: Will be assigned as the course progresses

Course Webpage: bcourses