Prerequisites: 1A.

Syllabus: 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.

Syllabus: 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.

Syllabus: 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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Prerequisites: 16A.

Syllabus: 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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Syllabus: 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: Some units in a lower division Mathematics class.

Syllabus: Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading.

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Prerequisites: 1B.

Syllabus: 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.

Syllabus: 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.

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Prerequisites: 1B.

Syllabus: 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.

Office: Evans 841

Office Hours:  Thursdays 9:30-11:30

Required Text: Linear Algebra and Differential Equations, second custom edition for UC Berkeley

Recommended Reading: 

Grading: (homework+quizzes) + midterm 1 + midterm 2 + (2 x final)

Homework: assigned for each class

Course Webpage: On Bspace

Prerequisites: 

Syllabus: Directed Group Study, topics vary with instructor.

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Prerequisites: 53 and 54.

Syllabus: 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.

Syllabus: 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: 851 Evans

Office Hours: T 2:30-3:30, Th 5-6

Required Text: Ross, Elementary Analysis

Recommended Reading: 

Grading: see course webpage

Homework: see course webpage

Course Webpage: http://math.berkeley.edu/~efuchs/104

Prerequisites: 53 and 54.

Syllabus (official): 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. 

Syllabus (unofficial): Proofs, mathematical thinking. Not limited to syllabus above.

Course Webpage: http://math.berkeley.edu/~yanzhang/classes/math104s14/

Prerequisites: 104.

Syllabus: 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: Evans 847

Office Hours: 

Required Text: Principles of Mathematical Analysis, 3rd Edition, Rudin

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Prerequisites: Math 54 or a course with equivalent linear algebra content.

Syllabus: 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: 811 Evans

Office Hours: TBA

Required Text: Algebra, Abstract and Concrete, edition 2.5., F. M. Goodman, available only on-line, FREE, at http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html.   The author has told me that edition 2.6 may be available (free) by the time our course begins.

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Grading: The final examination will take place on THURSDAY, MAY 15, 2014, 7-10P. The final examination will count for 50% of the course grade. There will be no early or make-up final examination. There will be two midterm examinations, which will each count for 20% of the course grade. Make-up midterm exams will not be given; instead, if you tell me ahead of time that you must miss a midterm exam, then the final exam and the other midterm exam will count more to make up for it. If you miss a midterm exam but do not tell me ahead of time, then you will need to bring me a doctor's note or equivalent in order to have the final exam and the other midterm exam count more to make up for it. If you miss both midterm exams the circumstances will need to be truly extraordinary to avoid a score of 0 on at least one of them.

Homework:  Homework will be assigned at nearly every class meeting, and be due on Tuesdays. Students are strongly encouraged to discuss the homework and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in.

Comments:  Students who need special accommodation for examinations should bring me the appropriate paperwork, and must tell me at least a week in advance of each examination what specific accommodation they need for that exam, so that I and the department will have enough time to arrange it. 

Course Webpage: http://math.berkeley.edu/~rieffel/113annS14.html

Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: 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: Evans 1053

Office Hours: Monday, 10-11am, Wednesday 11am-12pm, and by appointment

Required Text: Dummit and Foote, Abstract Algebra

Recommended Reading: Artin, Algebra and Lang, Algebra

Grading: See syllabus

Homework: 

Course Webpage: bSpace and Piazza

Prerequisites: 54 or a course with equivalent linear algebra content.

Syllabus: 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.

Syllabus: Honors section corresponding to 113. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.

Office: 883 Evans

Office Hours: MWF 10:30–11:30

Required Text: David S. Dummit and Richard M. Foote, Abstract Algebra, Third Edition, John Wiley & Sons, Inc., 2004

Recommended Reading: None

Grading: 30% homework, 15% first midterm, 20% second midterm, 35% final exam

Homework: Due weekly; see course web page for specifics.

Course Webpage: http://math.berkeley.edu/~vojta/H113.html

Prerequisites: 53 and 54.

Syllabus: 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.

Office: 785 Evans

Office Hours: Tuesday 2-4pm, Thursday 3-4pm, and by appointment

Required Text: "Mathematical methods in the physical sciences" by Mary L. Boas, 3rd edition

Course Webpage: http://math.berkeley.edu/~preygel/M121A.S14/

Prerequisites: 104.

Syllabus: Existence and uniqueness of solutions, linear systems, regular singular points. Nonlinear systems, stability and equilibria, limit sets, Sturm-Liouville theory

Office: Evans 181

Office Hours: Tuesdays 10:30-11:30 and by appointment

Required Text: Differential Equations, Dynamical Systems, and an Introduction to Chaos, third edition, Hirsch, Smale and Devaney

Recommended Reading: Introduction to Differential Equations, M.E.Taylor

Grading: homework + midterm + (2 x final)

Homework: assigned weekly

Course Webpage:  Math 123

Prerequisites: 110 and 128A.

Syllabus: Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer.

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Prerequisites: 53, 54, and 55.

Syllabus: 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.

Syllabus: The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, homology. We'll apply these invariants to study knots and links. 

Office: 921 Evans

Office Hours: MW 1:10-2pm, F 3:10-4pm

Required Text: M. A. Armstrong, Basic Topology 

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Course Webpage: BCourses

Prerequisites: 151, 152.

Syllabus: The real line and least upper bound, limit and decimal expansion of a number, differentiation and integration, Fundamental Theorem of Calculus, characterizations of sine, cosine, exp, and log.

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Prerequisites: 55.

Syllabus: This course is an introductory course to combinatorics. We will cover roughly Chapters 1 to 8 and 11 of Brualdi, that is permutations, pigeonhole principle, binomial coefficients, inclusion-exclusion principle, recurrence relations, generating functions, and basic graph theory. Then we will discuss some topics around polytopes and Ehrhart theory (time permitting).

Office: 793 Evans

Office Hours: Tuesday 10:00-11:00 and Wednesday 2:00-3:00

Required Text: R. Brualdi, Introductory Combinatorics, 5th edition, Pearson Prentice Hall, 2010

Recommended Reading: 

Grading: Midterm 1 (20%) + Midterm 2 (25%) + Final (40%) + Homework (15%) 

Course Webpage: http://math.berkeley.edu/~block/teaching/Math172.html

Prerequisites: 104.

Syllabus: 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.

Syllabus: 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: 749 Evans

Office Hours: M 4-5 W 10-11 F 12-1

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Course Webpage: http://math.berkeley.edu/~kozai/m185s14/

Prerequisites: Consent of instructor.

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

Syllabus: Topics will vary with instructor.

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Prerequisites: 202A and 110.

Syllabus: 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

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

Syllabus: 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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Prerequisites: 105 or 202A.

Syllabus: 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: 1033 Evans Hall

Office Hours: 2-3 MWF

Required Text: Evans, Partial Differential Equations (2nd edition)

Recommended Reading: 

Grading: take home midterm (25 %), homework (25 %), take home final exam (50 %)

Homework: I assign a homework problem at the start of each lecture, due in one week.

Course Webpage: 

Prerequisites: Graduate status or consent of instructor.

Syllabus: 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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Course Webpage: http://math.berkeley.edu/~strain/224b.S14/index.html

Prerequisites: 128A.

Syllabus: 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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Course Webpage: http://math.berkeley.edu/~sethian/math228b-2014.html

Prerequisites: 214.

Syllabus: The course has two parts. The first part will cover basic definitions and lemmas, corresponding to most of the catalog description (symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Lengendre transform, hamiltonian systems, lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds). The second part will introduce holomorphic curves and their applications to symplectic topology, skipping hard analysis proofs.

Office: 923 Evans.

Office Hours: 

Required Text: 

Recommended Reading: For basic definitions and lemmas, the book by Ana Cannas da Silva, "Lectures on Symplectic Geometry", available for free online, is very useful. "Introduction to symplectic topology" by McDuff and Salamon is a good introduction to some more advanced topics.

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Prerequisite: The equivalent of one of Math H113 or 114 or 250A, or consent of the instructor.

Syllabus: The theme of Math 245A, as I teach it, is universal constructions.  We begin with a well known case, the construction of free groups.  We will develop this in three quite different ways, and show why they 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 in fact 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 magnificent result of Peter Freyd, characterizing those functors between varieties of algebras that have left adjoints, and we determine, in several cases, all such examples.

Office: 865 Evans Hall

Office Hours: Tues. 2-3, Weds., 12-1, Fri. 3-4

Required Text: General Theory of Algebraic Structures, 469 pages of course notes, readable online at http://math.berkeley.edu/~gbergman/GenAlg.pdf , available softbound at $48.50 from Copy Central, 2576 Bancroft Way.

Recommended Reading: None (though some other texts useful for various purposes are discussed on pp.12-13 of the notes).

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 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 have taught the course, have developed 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 many students have acquired.  To get around this, I require each student to submit, on each class day, a question about the day's reading, if possible by e-mail at least an hour before class, so that I can prepare to address some of these questions in class.

Prerequisites: 254A.

Syllabus: 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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Course Webpage: http://math.berkeley.edu/~yxy/nt.html

Prerequisites: 250A.

Syllabus: Examples of Groups

Office: 921 Evans Hall

Office Hours: M F 10-11, W 12-1

Required Text: NA

Recommended Reading: see course webpage; Montgomery-Zippin, Topological Transformation Groups

Grading: 

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Prerequisites: Consent of instructor.

Syllabus: Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

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Prerequisites: Basic Algebra and Group Theory

Syllabus: The Langlands Program was launched by Robert Langlands in the late 1960s with the goal of relating Galois representations and Harmonic Analysis. In the last 30 years these ideas propagated to other areas of mathematics, such as geomery and representations of Lie algebras, and then to quantum physics. The goal of this introductionary course is to give a survey of recent developments, focusing on the links between the Langlands Program and quantum fieeld theory.

We will start with an overview of the "classical" Langlands correspondence, between representations of Galois groups and automorphic functions. The Galois group here is the group of automorphisms of a number field or the field of functions on a curve over a finite field. In the latter setting, one can reformulate the Langlands correspondence geometrically, using Grothendieck's dictionary linking functions and sheaves. One can then reformulate the geometric version in the case of curves over the complex field (that is, Riemann surfaces). This leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on complex algebraic curves, attached to two Langlands dual groups, which we will discuss.

The geometric form of the Langlands correspondence turns out to be closely related to two different types of quantum field theories: first, 2D conformal field theory, defined on the curve X itself, and second, 4D supersymmetric gauge theory on the product of X and another curve. I will talk about the connections to the 2D and 4D quantum field theories and what we can learn from them. Time permitting, I will also discuss the links between the 2D and 4D theories themselves, which have recently become the focus of extensive research.

Required Text:  There is no required text, but some of the lectures will follow closely these two survey articles:

Lectures on the Langlands Program and Conformal Field Theory

Gauge Theory and Langlands Duality

Recommended Reading: My book Love and Math gives an elementary introduction to many of the ideas that will be discussed in this course

Office Hours: Tu 2-3:30

Lecture Course on Functorial Field Theories and Ring Spectra

One important role of mathematics is to serve as a language for other sciences. There are many success stories, on the theoretical side general relativity (Lorentz geometry, Einstein’s equation) and quantum mechanics (functional analysis, Schrödinger’s equation) come to mind. However, new mathematical formalisms still need to be discovered for the physically relevant quantum field theories.

A successful approach, via functorial field theories, goes back to Witten, Atiyah and Segal. In this class, we will explain this approach from first principles and discuss several structural analogies to ring spectra studied in algebraic topology. For example, Sigma-models lead to spaces of field theories, deformations correspond to homotopies, central charge to the degree (or more generally, the twist) of a cohomology class. Gauge theories give equivariant cohomology classes and best of all, quantization is related to push-forward (or Gysin or wrong-way) maps. This is one way in which algebraic topology could contribute to making some of the ill-defined Feynman path integrals rigorous. 

The class will consist of survey-like parts but we'll also explain those aspects in full detail that are most interesting to the participants. This will definitely include the theory of super manifolds because without super symmetry, the spaces of field theories are often contractible and hence don't give interesting ring spectra. We will also discuss a modern point of view on ring spectra via excisive functors, as well as symmetric, orthogonal ring spectra and Gamma-spaces.  

In a seminar with Owen Gwilliams on Factorization Algebras, we'll be formalizing the perturbative aspect of quantum field theory. A close connection between this story and functorial field theories is currently under construction.

Prerequisites: Basic algebraic and differential topology: manifolds, vector bundles, homology and homotopy.

Office Hours: Tu 3:30-5:00, 703 Evans

Recommended Reading: Survey with Stephan Stolz: Supersymmetric field theories and generalized cohomology

Modern Ring Spectra: Model categories of Diagram Spectra by Mandell, May, Schwede, Shipley.

Physical background: A pseudo-mathematical pseudo-review on 4d N = 2 supersymmetric quantum field theories by Yuji Tachikawa. 

Supersymmetry for Mathematicians: An Introduction by V.S. Varadarajan. 

Course Webpage: http://people.mpim-bonn.mpg.de/teichner/Math/FTRS.html