Fall 2020

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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Required Text Custom Math 10 Bio Calculus 2018, by Stewart and Day

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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.

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Prerequisites 16A

Description Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization.

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Prerequisites Three years of high school mathematics

Description Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences.

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Prerequisites Mathematics 1B or N1B

Description Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.

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Prerequisites 1B, N1B, 10B, or N10B

Description Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product spaces. Eigenvalues and eigenvectors; orthogonality, symmetric matrices. Linear second-order differential equations; first-order systems with constant coefficients. Fourier series.

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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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Prerequisites 

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 Math 53 and 54

Description 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.

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Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

Description The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

Description The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

Description The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Recommended Reading 

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Course Webpage 

Prerequisites 53 and 54. 55 or an equivalent exposure to proofs

Description The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

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Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs is recommended

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. 55 or an equivalent exposure to proofs

Description Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs

Description Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs

Description Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites 54 or a course with equivalent linear algebra content. 55 or an equivalent exposure to proofs

Description Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.

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Prerequisites Math 55 is recommended

Description Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

Office 883 Evans (inaccessible)

Office Hours TTh 12:10–1:30

Required Text Niven, Zuckerman, Montgomery, An introduction to the theory of numbers, Wiley, 5th edition.

Recommended Reading 

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

Homework Due weekly, on Thursdays

Course Webpage https://math.berkeley.edu/~vojta/115.html

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 104 and 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 Math 53 and 54, basic programming skills.

Description Basic concepts and methods in numerical analysis: Solution of equations in one variable; Polynomial interpolation and approximation; Numerical differentiation and integration; Initial-value problems for ordinary differential equations; Direct methods for solving linear systems.

Office Remotely

Office Hours TBD

Required Text 

R. L. Burden and J. D. Faires, Numerical Analysis, 10th edition, Brooks-Cole, 2015. ISBN-13: 978-1305253667; ISBN-10: 1305253663.

Recommended Reading 

Grading Homework assignments, bi-weekly quizzes, programming assignments, midterm exam, and final exam.

Homework Weekly homework assignments and four programming assignments will be posted on the course web page.

Course Webpage http://persson.berkeley.edu/math128a (summer version until fall semester starts)

Prerequisites Math 104 and 113 or consent of instructor

Description Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. Proof by transfinite induction and definitions by transfinite recursion. Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences.

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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 53 and 54

Description 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

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 Strong command of Math 53 and Math 54, upper-division level of mathematical maturity; Math 110, Math 113 and 2 semesters of physics are recommended.   

Description In this reincarnation of the course, we will follow closely the instructor's own "Introduction to Quantum Mechanics" 

https://math.berkeley.edu/~giventh/qm_book.pdf 

As can be seen from the book's Preface, it will be an attempt to understand physics, yet from a point of view that might be appealing to a mathematician. The emphasis will be on quantum mechanics, although some brief (an certainly insuficient) excursions into Hamiltonian and Lagrangian classical mechanics will also occur. 

Formal mathematical rigor will be a second priority. Respectively, Math 53 and Math 54 will suffice as the starting prerequisites. The students will be expected, however, to digest quickly many new (and often abstract) ideas. So, by the required degree of general mathematical maturity, this course will perhaps be on a par with other senior level electives.   

Office 

Office Hours I will hold weekly office hours on Zoom

Required Text: https://math.berkeley.edu/~giventh/qm_book.pdf 

Recommended Reading: Reading the textbook is highly recommended  

Grading 1/5 take home final + 4/5 homework; I have, however, the following (ingenious, in my view) grading scheme. The score of any particular homework which is below your score on the final will be dropped - together with its weight in the total grade. E.g. if half of your hw is below the final score, then only the other half counts, but it weighs 2/3 rather than 4/5, while the final score weighs 1/3 rather than 1/5.    

Homework: weekly

Course Webpage: is found here  

Prerequisites Consent of instructor

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

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Prerequisites 

Description Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.

Office 

Office Hours 

Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 

Description Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.

Office 

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Required Text 

Recommended Reading 

Grading 

Homework 

Course Webpage 

Prerequisites 202A-202B

Description 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 783 Evans

Office Hours 

Required Text 

Recommended Reading R.G. Douglas Banach algebra techniques in operator theory. 2nd edition

Also recommended : M.Reed, B.Simon Methods of modern mathematical physics vol.1 Functional analysis

W,B. Arveson A short course on spectral theory

G.K. Pedersen Analysis now.

Grading Letter grade.

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Prerequisites Consent of instructor

Description 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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Grading Letter grade.

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Prerequisites Statistics C205A-C205B or consent of instructor

Description 

Brownian motion on the real line spends zero time at the origin --- but, despite this, it always returns there. The exceptional set of times at which Brownian motion is at the origin is a random fractal, which has a fractal (or Hausdorff) dimension of one-half. A measure on the real line may be constructed which is supported on the exceptional set in such a way that, when a clock ticks according to rises in the measure's distribution function, time passes at the rate at which Brownian motion visits the origin. This notion of time is called Brownian local time. It is an example of how a singular notion of the passage of time, attached to a natural random fractal, arises in the scaling limit of a discrete random system --- of random walk, in this case. In a second example, a particle advances randomly on the three-dimensional integer lattice with a bias in a preferred direction, such as downwards. It is obstructed, however, by encounters with a randomly scattered field of hard and immovable obstacles. Lengthy sojourns, during which the particle idles in traps forged from the obstacles, are interrupted by flickering bursts of advance, interludes between the escape of the particle from one trap and its descent in another. The particle's scaled progression is a random devil's staircase, akin to the graph of Brownian local time. Such staircases arise in the study of planar critical percolation when it is studied under a dynamical perturbation. Finally, percolation --- and random fractals --- emerge in the analysis of random turn games. The class will probe these directions, and will offer modes of entry for research into several of them.

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Grading Letter grade.

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Prerequisites 125A and (135 or 136)

Description Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.

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Grading Letter grade.

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Prerequisites 214

Description Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds.

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Grading Letter grade.

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Prerequisites 250A

Description Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.

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Grading Letter grade.

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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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Grading Letter grade.

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Prerequisites 261A or equivalent

Description Second semester of the 261A-B sequence.  I'll assume familiarity with basic Lie theory, as covered in 261A, although I will probably review some of it as needed.  This course will function essentially as a seminar on more advanced topics, from Algebraic groups to Quantum groups.  See course web page for more info.

Office No in person office meetings

Office Hours By appointment online

Required Text None

Homework Problems assigned from time to time on various topics covered in the class.

Course Webpage https://math.berkeley.edu/~mhaiman/math261B-fall20/

Prerequisites Consent of instructor

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

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Grading Letter grade.

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