Fall 2014

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: 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.

Office: 1089 Evans

Office Hours: Wed 1:30-3:30pm

Required Text: Calculus for Life Sciences 1E for UC Berkeley by Sebastian J. Schreiber, Karl Smith and Wayne Getz. Available at a discounted rate of $55.95 online at http://www.wiley.com/WileyCDA/Section/id-822096.html.

Recommended Reading: Online lecture notes will be provided on the course web page.

Grading: 2 midterms (20% each), discussion section (20%), final exam (40%).

Homework: Provided on the course webpage.

Course Webpage: https://bcourses.berkeley.edu/courses/1251852

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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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: 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.

Course Webpage: math.berkeley.edu/~zworski/syllabus53.htm

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; second-order differential equations with constant coefficients. Fourier series and partial differential equations.

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

Prerequisites: 1B.

Syllabus: 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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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.

Office: 1053 Evans

Office Hours: Mon: 1pm–2pm, Wed: 1pm–2pm and 3pm–4pm in 1053 Evans

Required Text: Ross, Kenneth A. Elementary Analysis, 2nd Edition. Springer, 2013.

Recommended Reading:  Pugh, Charles C. Real mathematical analysis. Springer, 2002.
Rudin, Walter. Principles of Mathematical Analysis. , McGraw-Hill, 1976. 
Lebel, Jiří. Basic Analysis. Available online

Course Webpage: http://math.berkeley.edu/~jcalder/104F14/

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 Hall

Office Hours: M 1-2pm, 4-5pm, W 1-2pm

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

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

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Course Webpage: http://math.berkeley.edu/~hongbins/m104f14.html

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

Course Website: http://math.berkeley.edu/~giventh/11014.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.

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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.

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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.

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Prerequisites: 53 and 54.

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

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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.

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Prerequisites: 53 and 54.

Syllabus: The official course description lists the following topics: waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform. We will cover these topics and more.

Office: Evans 857

Office Hours: Tuesday 2-3:30pm and Friday 1:30-3pm.

Recommended Reading: Partial Differential Equations, An Introduction, Walter Strauss

Grading: Grades will be based off of weekly homework assignments, two midterms, and the final exam.

Homework: Homework will be posted on the course webpage.

Course Webpage: http://www.math.berkeley.edu/~murphy/126.html

Prerequisites: 53 and 54.

Syllabus: Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.

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Prerequisites: 113 and 104.

Syllabus: 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.

Syllabus: This course provides an introduction to point-set topology and to elementary algebraic topology. Topics include: topological spaces and continuous maps, connectedness and compactness, identification spaces, fundamental group and covering spaces, triangulation, classification of surfaces, Euler characteristic and simplicial homology.

Office:  751 Evans Hall

Office Hours: MWF 9-10am

Required Text: Armstrong, Basic Topology

Recommended Reading: Munkres, Topology (2nd edition)

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Course Webpage: http://math.berkeley.edu/~hongbins/m142f14.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.

Required Text: Introduction to Optimization.  Pablo Pedregal.  Springer, 2004, 978-1-4419-233
4-9 (Print) 978-0-387-21680-5 (Online). Available online from  http://www.springerlink.com/
(from university IP addresses).

Course Webpage: http://math.berkeley.edu/~strain/170.F14/index.html

Prerequisites: Math 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: Evans 841

Office Hours: Tue 10:00-11:30

Required Text: Complex Analysis by Elias M. Stein and Rami Shakarchi

Recommended Reading: Sarason, Complex Function Theory

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

Homework: Will be assigned weekly

Course Webpage: https://bcourses.berkeley.edu/courses/1267754

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: 837 Evans Hall

Office Hours: Mondays from 9:15am  to 11:15am, Wednesdays from 10:00am to 11:00am

Required Text: Complex Analysis by Elias M. Stein and Rami Shakarchi

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Grading: Homework 25%, Midterm 25%, Final 50%

Homework: weekly

Course Webpage: available soon

Prerequisites: 104, 110, 2 semesters lower division Physics.

Syllabus: Topics in mechanics presented from a mathematical viewpoint: e.g., Lagrangian mechanics, Hamiltonian mechanics and symplectic geometry, differential equations for fluids, spectral theory in quantum mechanics, probability theory and statistical mechanics. See department bulletins for specific topics each semester course is offered.

Office: 1071 Evans Hall

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Required Text: Mathematical Methods of Classical Mechanics (Arnol'd)

Recommended Reading: Lectures on Quantum Mechanics for Mathematics Students (Fadeev, Yakubovskiī)

Grading: Please consult the course website.

Homework: http://math.berkeley.edu/~libland/teaching/math-189/homework.html

Course Webpage: math.berkeley.edu/~libland/teaching/math-189/

Prerequisites: 104 or equivalent preparation in analysis. Further experience with upper-division mathematics courses such as Math 142 or 185, including writing proofs, is strongly recommended.

Syllabus: Metric spaces and general topological spaces. Compactness and connectedness. Quotient topologies. 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.

In my lectures I will try to give well-motivated careful presentations of the material.

Office: 811 Evans

Office Hours: TBA

Required Text: None

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The two texts whose treatment is closest to my approach to the topics of the course are:

Topology, 2nd Ed., J. R. Munkres, Prentice Hall, 2000

Real and Functional Analysis, 3rd Ed., S. Lang, Springer-Verlag, 1993

Students may also find useful the classic text General Topology by J. L. Kelley.

Grading: I plan to assign roughly-weekly problem sets. Collectively they will count for 50% of the course grade. Students are strongly encouraged to discuss the problem sets and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. 
There will be a final examination on WEDNESDAY, DECEMBER 17, 2014 7-10PM, which will count for 35% of the course grade. There will be a midterm exam. It will count for 15% of the course grade. There will be no early or make-up final examination. Nor will a make-up midterm exam be given; instead, if you tell me ahead of time that you must miss the midterm exam, then the final exam will count for 50% of your course grade. If you miss the 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 count for 50% of your course grade.

Course Webpage: http://math.berkeley.edu/~rieffel/202AannF14.html

Prerequisites: 206.

Syllabus: Unbounded self-adjoint operators. Stone's Theorem, Friedrichs extensions. Examples and applications, including differential operators. Perturbation theory. Further topics may include: unbounded operators in quantum mechanics, Stone-Von Neumann Theorem. Operator semigroups and evolution equations, some non-linear operators. Weyl theory of defect indices for ordinary differential operators.

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

Syllabus: Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. 

Office: 921 Evans

Office Hours: TBA

Required Text: Algebraic Topology, Allen Hatcher

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Course Webpage: https://bcourses.berkeley.edu/courses/1247157

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: 225B and 235A.

Syllabus: Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity.

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Prerequisites: 214 and 215A.

Syllabus: 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.

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Prerequisites: 250A (or my consent). The student is expected to have some familiarity with the vocabulary of Category Theory and basic Algebra (rings, modules)

Syllabus: The course will offer an intensive introduction into modern Homological Algebra. The latter since its birth has been seen more and more as permeating all of the modern Mathematics. Today it is present on the fundamental level in nearly every area of Mathematics, and especially so in Algebra, Geometry, Topology, Representation Theory, Algebraic Geometry, Mathematical Physics, and Number Theory.

I will cover adjointness and exactness of functors. I will discuss additive, exact, abelian and triangulated categories, as well as the fundmental operations on categories: forming a quotient and localization. Using quotients I will discuss homotopy categories associated with an exact category. Localizations of categories will be used to describe a passage from an exact category to its derived category. Time permeating I will also give a rapid overview of model categories in the sense of Quillen. For each of these contexts I will describe the correspoding theory of derived functors.

I will also cover the "mechanics" of Homological Algebra: complexes, double complexes, spectral sequences, developing as illustrations a number of classical spectral sequences (the First and the Second spectral sequences of a double complex, the Hochschild-Serre spectral sequence, Künneth spectral sequence).

Office: 995 Evans Hall

Office Hours: TBA

Required Text: Cartan, Eilenberg. Homological Algebra. Princeton 1956 (still by far the best text written on the subject)

Recommended Reading: Freyd. Abelian categories. New York, Harper & Row 1964

Mac Lane. Category Theory for a Working Mathematician. Graduate Courses in Mathematics 5, Springer Verlag (either edition)

I will supply a number of suppemental texts and recommendations througout the semester

Grading: homework and the take home exam

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Prerequisites: 250A-250B or consent of instructor.

Syllabus: Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis.

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Subject: $p$-adic geometry

Instructor: Peter Scholze

Office: 1063 Evans Hall

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Scheduling Notes: Note that the class will be held in 740 Evans (not 285 Cory, as shown earlier), that the first class meeting will be September 2, and that there will be auxiliary weekly lectures by Jared Weinstein, as shown above.

Prerequisites: We will assume some background in algebraic geometry and Dieudonné theory. Tutorials held by Jared Weinstein will recall some of the intervening notions.  (See Scheduling Notes above, and Background Lectures, after the description of the main course below.)

Description: Originally defined by Drinfel'd, and then used extensively by L. Lafforgue and V. Lafforgue, moduli spaces of shtukas have proved to be a powerful tool in the study of the Langlands correspondence over function fields. It is an important question whether something similar could work over number fields.

In this course, we want to sketch a strategy to define moduli spaces of local shtukas over mixed-characteristic fields such as $\mathbb{Q}_p$ (leaving open the problem of assembling these spaces for varying primes $p$). These spaces should generalize Rapoport-Zink spaces, as well as the conjectural theory of local Shimura varieties that has recently been suggested by Rapoport-Viehmann. Notably, the spaces of local shtukas should overcome the 'minuscule' condition inherent in the theory of Shimura varieties, so that they can be seen as very general $p$-adic period domains (which exist even in situations where Griffiths transversality is a nontrivial condition). We wll start by reviewing the theory of (local and global) shtukas over function fields. Next, we will define shtukas in mixed characteristic in an absolute setup; this is closely related to the notion of Breuil-Kisin modules, and has been the subject of recent investigations of Fargues, some of whose results we will recall.

To set things into perspective, we will spend some time detailing the case of $p$-divisible groups, explaining the equivalence between $p$-divisible groups and certain shtukas (due to Breuil, Kisin, and Fargues). We will then use the joint work with Weinstein to identify Rapoport-Zink spaces with moduli spaces of shtukas.

We will then be able to define general spaces of shtukas. They will turn out to be somewhat esoteric objects, called 'diamonds', so quite a bit of time will be spent on explaining the definition of diamonds. In particular, we will explain how they overcome the problem that a general space of shtukas should live over 'a product of several copies of $\mathrm{Spa}\ \mathbb{Q}_p$' (taken over some absolute base '$\mathbb{F}_1$'), by giving a highly nontrivial definition of 'a product of several copies of $\mathrm{Spa}\ \mathbb{Q}_p$' in a way that makes Drinfel'd's lemma true.

If time permits, we will try to (conjecturally) understand the étale cohomology of these spaces, and their relation to the local Langlands correspondence.

Background lectures by Jared Weinstein: These weekly meetings are for graduate students who wish to follow Peter Scholze's Math 274 course on p-adic geometry.  We will discuss  Drinfeld's "elliptic modules" -- this is a formalism which encompasses cyclotomic fields, elliptic curves, and Drinfeld modules, which appear in characteristic p.  In the last case, Drinfeld modules (and the related concept of "shtukas") led to a really striking achievement -- L. Lafforgues' proof of Langlands' conjectures for a function field. Algebraic geometry background will be provided as necessary.

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Course Webpage: http://math.berkeley.edu/~jared/Math274/index.html

Prerequisites: Consent of instructor.

Syllabus: A. Capacity and Hausdorff measure methods for PDE

      B. Regularity estimates for linear and nonlinear elliptic PDE

      C. Convexity methods

      D. Special techniques for semilinear elliptic equations

Office: 1033 Evans Hall

Office Hours: TBA

Required Text: None

Recommended Reading: Evans, Partial Differential Equations (2nd ed), AMS; Evans and Gariepy, Measure Theory and Fine Properties of Functions, CRC Press

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