Fall 2015

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.

Syllabus: 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.

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: 1A.

Syllabus: Honors version of 1B. 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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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: Mathematics 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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Lecture room (as of 8/7/15): WHEELER AUDITORIUM

For enrollment questions or issues: Please speak with the registrar or an academic advisor.

The math department undergraduate advisors are:

Thomas Brown, 965 Evans, and Jennifer Sixt, 964 Evans.

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.

Course Webpage: https://math.berkeley.edu/~nadler/54fall2015.html

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

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

Office: 1071 Evans Hall

Office Hours: TBA

Required Text: Discrete Mathematics and its Applications, (7th Edition, custom UC Berkeley paperback edition.) by Kenneth H. Rosen.

Grading: see the course website https://math.berkeley.edu/~libland/teaching/math-55/ for details.

Homework: https://bcourses.berkeley.edu/courses/1347499/assignments

Course Webpage: https://math.berkeley.edu/~libland/teaching/math-55/

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Syllabus: 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

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Syllabus: 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

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Syllabus: Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required. Also listed as Economics C103.

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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:  This course provides an introduction to real analysis, and will emphasize on rigorous mathematical proof. Topics include: the real number system, sequences and limits, continuous functions, power series, uniform convergence, differentiation and integration, mean value theorem and metric spaces.

Office: 751 Evans

Office Hours: Tu 11am-12pm, W 11am-12pm, Th 1pm-2pm or by appointment.

Required Text: Ross, Elementary Analysis: The Theory of Calculus (2nd edition)

Recommended Reading: Pugh, Real Mathematical Analysis; Rudin, Principles of Mathematical Analysis

Grading: Homework 20%+Midterm 1 20%+Midterm 2 20%+Final 40%

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Course Webpage: https://math.berkeley.edu/~hongbins/m104f15.html

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: 54 or a course with equivalent linear algebra content.

Syllabus: 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.

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: 1055 Evans

Office Hours: Friday 9:30-12:30

Required Text: A First Course in Abstract Algebra, John Fraleigh (7th Edition)

Recommended Reading: An introduction to Abstract Algebra, D. Robinson (de Gruyter textbook)

Grading: %20 Homework, %20 Midterm  I, %20 Midterm II, %40 Final

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Course Webpage: tba

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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Required Text: Dummit and Foote, "Abstract Algebra" 

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Course Webpage: https://math.berkeley.edu/~kpmann/113algebra2015.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.

Office: 811 Evans

Office Hours: M 2-3; W 9-10; F 10-11

Required Text: Algebra, Abstract and Concrete, edition 2.6., F. M. Goodman, available on-line, FREE, at:

http://homepage.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html . 

Recommended Reading: Symmetry, Hermann Weyl, available on-line free at:

https://archive.org/details/Symmetry_482

Grading: The <b>final examination</b> will take place on<b>WEDNESDAY, DECEMBER  16, 2015, 7-10 PM</b>. 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, but will be due once per week. 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. Homework wil count for 10% of the course grade.

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.

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Prerequisites: 110 and 113, or consent of instructor.

Syllabus: Galois Theory and the surrounding requirements will be the primary focus of this course.

See the course web-page for a list of topics, and for more information about the grading/homework policy for this course.

Office: 889 Evans

Office Hours: TBA

Required Text: "Galois Theory" by J.-P. Escofier. (Available on SpringerLink)

Recommended Reading: (other materials will be discussed in class as needed)

Course Webpage: Course Webpage

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: 783 Evans

Office Hours:  M 12:30 - 1:45, Tu 11:15 - 11:50

Required Text: Mary L. Boas "Mathematical Methods in the Physical Sciences" 3rd edition.

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Homework:   homework assigned during a week  due on Thursday of following week at the beginning of  class

Prerequisites: 53 and 54.

Syllabus: Partial differential equations (PDE) are ubiquitous in science, engineering and physics. Applications include traffic flow, heat diffusion, wave propagation, quantum mechanics, computer vision, image processing, and optimal control theory. This course is a basic introduction to partial differential equations. Topics to be covered include waves, diffusion, Laplace's equation, Fourier series and Fourier transform techniques, numerical methods for PDE, Green's functions, distributions, and some topics in nonlinear PDE, such as shock waves, conservation laws, and the calculus of variations. 

Office: 1053 Evans

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Required Text: Partial Differential Equations: An Introduction (1st or 2nd Edition), Walter Strauss.

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Course Webpage: https://math.berkeley.edu/~jcalder/126F15

Prerequisites: 110 and 113.

Syllabus: 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, and/or other topics.  

Office: 793 Evans

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Required Text: Stillwell, "The four pillars of geometry" + additional readings supplied by instructor

Recommended Reading: Hartshorne, "Geometry: Euclid and Beyond Euclid" 

Grading: See syllabus on course website.  Note that class participation will be a portion of the grade.    

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Course Webpage: math.berkeley.edu/~kpmann/130geometry2015.html

Prerequisites: 104.

Syllabus: 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.

Office: 897 Evans

Office Hours: MWF 1:10-2:00

Required Text: "Elements of Differential Geometry" by Millman and Parker

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Grading: Homework 2/3,  Final 1/3

Homework: There will be weekly homework assignments.

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

Prerequisites: 110, 113, familiarity with using mathematical software

Syllabus: Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on Grobner bases

and their applications.

Office: 925 Evans

Office Hours: Wednesday, 3-4:30pm, or by appointment

Required Text:  D. Cox D. O'Shea, J. Little: Ideals, Varieties and Algorithms, Springer Undergraduate Texts, 2007

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Grading: Homework 35 %, Midterm Exam 30 %, Final Paper 35 %

Homework: Weekly

Course Webpage: https://math.berkeley.edu/~bernd/math143.html

Prerequisites: 53 and 54.

Syllabus: Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry, polyhedral geometry, the calculus of variations, and control theory.

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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.

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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.

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Prerequisites: essential prerequisite linear algebra (Math 54 or Math 110 )

Syllabus:  the course will be a mathematical introduction at the undergraduate level to the Hilbert space approach to quantum mechanics.

Office:  783 Evans

Office Hours: M 12:30 - 1:45 , Tu 11:15 - 11:50

Required Text:  the course will be based on the notes students will take in class from the lectures.

Recommended Reading:    Many of the topics covered can be found in  the book  by K. Hannabuss "An introduction to Quantum Theory" Oxford Graduate Texts in Mathematics.  Also recommended : L. D. Faddeev, O.A. Yakubovskii " Lectures on Quantum Mechanics for Mathematics Students" , Student Mathematical Library vol. 47, American Mathematical Society.

Grading:   30% Homework and discussion of problems in class

                  30% Midterm

                  40% Take-home Final Exam

Homework: weekly homework, due on Thursday of following week.

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Syllabus: 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

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Syllabus: 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

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Prerequisites: 104.

Syllabus: Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and Rn.

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Prerequisites: 202A-202B or equivalent is more than enough. Students who have studied only part of the material of Math 202AB and wish to enroll in Math 206 should discuss this with me.

Syllabus: Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*-algebras of operators. Commutative C*-algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Hilbert-Schmidt operators. Fredholm operators. The Fredholm index. Selected additional topics.

Office: 811 Evans

Office Hours: TBA

Required Text: 

Recommended Reading: A Course in Functional Analysis, John B. Conway, 2nd ed., Springer-Verlag.

My understanding is that through an agreement between UC and Springer, chapters of the first edition of this book are available for free download by students. They should be satisfactory - I do not plan to assign exercises from the Conway book.

Course description: 

The theory of Banach algebras is a very elegant blend of algebra and topology which provides unifying principles for a number of different parts of mathematics and its applications, notably operator theory, commutative and non-commutative harmonic analysis and the theory of group representations, and the theory of functions of one and several complex variables. But at the present time probably its most extensive use is as a foundation for non-commutative measure theory (von Neumann algebras, Math 209) and non-commutative topology and geometry (C*-algebras, Math 208). These in turn provide a foundation for quantum physics, but they also have myriad applications in many other directions, including group representations and harmonic analysis, ordinary topology and geometry, and even number theory. (For a vast panorama of the applications see Alain Connes' 1994 book "Noncommutative Geometry", which is available on the web as a free download.) I will cover the standard topics as listed in the catalog. Beyond the basic general theory of Banach algebras this will include several forms of the spectral theorem for self-adjoint operators on Hilbert space, compact and Fredholm operators, and group algebras and the Fourier transform and their relation to representation theory. Further topics as time permits, probably including Toeplitz operators and their index. The problem sets will involve a number of important specific examples beyond those presented in class.

Homework: Many weeks I will give out a problem set, and the course grade will be based on the work done on these. There will be no final examination.

Course Webpage: 

Prerequisites: 113, 110 and point-set topology (e.g. 202A).

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.

Office: Professor Jenny Harrison, 829 Evans Hall

Office Hours: MWF, 1-2 pm

Required Text:

• M J Greenberg and J Harper, Algebraic Topology: a First Course (Benjamin/Cummings 1981)
• A Hatcher, Algebraic Topology (Cambridge University Press, paperback, 2002)
• G E Bredon, Topology and Geometry (Springer GTM 139, 1993)

Hatcher is excellent and available as a free pdf, as is Greenberg & Harper. The latter went out of print some years ago, but it uses a more categorical and concise approach than Hatcher. Bredon provides useful material on differentiable manifolds which will give us concrete examples to work with.

Recommended Reading: Steve Awodey: Category Theory, Oxford University Press, 2010. This is a simpler version of the classic book by Eilenberg and Mac Lane. We will only be using some simple concepts from category theory, so this is optional reading.

Grading: Homework will be graded, hopefully, but the scores will not count towards your course grade.

Homework: There will be weekly homework assignments and a final exam which will be based 75% on homework and 25% on material from the lectures.

Course Webpage: https://bcourses.berkeley.edu/courses/1325765

Prerequisites: Consent of instructor.

Syllabus: Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions.

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Prerequisites: Statistics C205A-C205B.

Syllabus: The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probabilty offered according to students demand and faculty availability. Also listed as Statistics C206A.

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Prerequisites: 125B and 135.

Syllabus: 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: 128A.

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This class focuses on two aspects for solving differential equations. 1) How to discretize a differential equation (temporal, spatial). 2) After obtaining a discretized system, how to solve it. Modern scientific applications increasingly require one to consider two aspects together to design efficient and accurate numerical schemes. We aim at discussing the following topics. 

Linear multistep methods; Runge-Kutta methods; Stability theory; Stiff equations; Discretization techniques; Sparse direct method; Krylov methods and preconditioning; Domain decomposition method; Matrix functions 

Office: 1083 Evans

Office Hours:  W 1:30-3:30PM or by appointment.

Required Text: 

[L] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM, 2007. ISBN 978-0-898716-29-0.

Course Webpage: https://math.berkeley.edu/~linlin/228A/

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Syllabus: Hamiltonian systems appear in conservative problems in mechanics as in celestial mechanics but also in the statistical mechanics governing the motion of atoms and molecules in matter. The discoveries of last century have opened up new perspectives for the old field of Hamiltonian systems and let to the formation of the new field of symplectic geometry. In the course, I will give a detailed acount of some basic methods and results in symplectic geometry and its application to physics and other fields of mathematics.
Here is an outline of the course:

1. Sympletic linear algebra. Quadratic Hamiltonians.
2. Symplectic and Contact manifolds, cotangent bundles. Darboux's theorem.
3. Variational problems and Minimax principle. Hofer-Zehnder Capacity and Hofer Geometry.
4. Weinstein's conjecture, Viterbo's theorem. Gromov-Eliashberg $C^0$-rigidity.
5. Holomorphic Curves and Gromov's Symplectic Width.

 

Office: 803 Evans

Office Hours: 2-3:30 pm

Required Text: None

Recommended Reading: F. Rezakhanlou: Lectures on Symplectic Geometry

Grading: based on homework assignments

Homework: 

Course Webpage: https://math.berkeley.edu/~rezakhan/math242_html/

Prerequisites: 114 or consent of instructor.

Syllabus: 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.

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

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.

Office: 927 Evans Hall

Office Hours: Tu Th 9:30-11:00

Required Text: Algebraic number theory by Neukirch

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Course Webpage: https://bcourses.berkeley.edu/courses/1316003

Prerequisites: Officially, Math 214, but not really necessary.

Syllabus: Lie groups, Lie algebras, and their representations. In the spring semester we will also go into other topics, such as algebraic groups and flag varieties, Hopf algebras and quantum groups, and canonical bases; perhaps also some combinatorial applications and open problems.

Office: 855 Evans

Office Hours: MWF 12-1 or by appointment

Required Text: V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. This book is available online from campus computers or through the library proxy server.

Recommended Reading: see course web page

Grading: Based on homework

Course Webpage: https://math.berkeley.edu/~mhaiman/math261-fall15-spring16/

Prerequisites:  Differential Geometry (manifolds, bundles, differential forms) and PDEs (Sobolev spaces, elliptic regularity)

Syllabus: This course aims to build a knowledge base in gauge theory, starting with classical analytic results (Uhlenbeck compactness, Yang-Mills moduli spaces), and going into some applications (Donaldson theory / Floer theory / Atiyah-Floer conjectures) according to participant's interests. We will use an adaptation of the 'dice reading seminar' format; see

Course Webpage: www.math.berkeley.edu/~katrin/teach/gauge

Prerequisites: Consent of instructor.

Syllabus: 

Title: On singularity formation in nonlinear PDE’s: a constructive approach.

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Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor.

Syllabus: Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty.

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Prerequisites: For qualified graduate students.

Syllabus: Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree.

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Office Hours: 

Required Text: 

Recommended Reading: 

Grading: 

Homework: 

Course Webpage: 

Prerequisites: 

Syllabus: 

Office: 

Office Hours: 

Required Text: 

Recommended Reading: 

Grading: 

Homework: 

Course Webpage: