Spring 2013

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: 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 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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Prerequisites: Some high school calculus and physics would be useful, but not essential, preparation for this seminar.

Syllabus: Our main focus will be a careful reading and discussion of the book "Geometry, Relativity, and the Fourth Dimension," by Rudolf von Bitter Rucker. The author may be better known to some students as Rudy Rucker, described in Wikipedia as "...an American philosopher mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement." Depending upon student interests, we may look at other work of the author, and/or other material on relativity and higher (or lower, as in "Flatland," by the repetitively named Edwin Abbott Abbott) dimensions. 

The seminar will meet ten times, beginning the first week of the semester. The planned meeting dates are January 24 and 31; February 7, 21 and 28; March 14 and 21; and April 4, 11 and 18, 2013.

Office: 825 Evans Hall

Office Hours: Tuesdays, 1-2 PM

Required Text: Rucker, Geometry, Relativity, and the Fourth Dimension

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Course Webpage: see http://math.berkeley.edu/~alanw

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. (9 weeks)

Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations. (5 weeks)

Office: 905 Evans Hall

Office Hours: TBA

Required Text: Lay, Linear Algebra;Nagel, Saff & Snyder, Differential Equations. Custom Berkeley edition in 1 volume 

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Grading: Three in-class tests and a final exam, 20% each. Lowest in-class test score is discarded. Homework+Section participation+weekly quizzes, 40%. 

Homework: Due twice a week, in section (Tu and Th)

Course Webpage: Will be on bSpace

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: Evans 805

Office Hours: MW 2-3P, F 12-1P

Required Text: Ross, Elementary Analysis: The Theory of Calculus

Recommended Reading: Rudin, Principles of Mathematical Analysis

Grading: 25% homework, 15% first written exam, 20% second written exam, 40% final written exam.

Homework: Homework will be assigned every week.
A selection of problems will be graded for each homework assignment. Late homework will not be accepted, but the two lowest homework scores will be dropped.

Course Webpage: http://math.berkeley.edu/~jdahl/104/index.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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Course Webpage: http://math.berkeley.edu/~dherin/Math110_2013.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: 887 Evans

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Required Text: Fraleigh, "A First Course in Abstract Algebra"

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Grading: Two midterms (20% each), final exam (30%), homework (18%), writing project (12%). 

Homework: Weekly assignments

Course Webpage: http://math.berkeley.edu/~oeding/math113_sp2013.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: 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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Course Webpage: http://math.berkeley.edu/~dherin/Math123_2013.html

Prerequisites: Math 128A or equivalent, basic MATLAB skills.

Syllabus: Second part of the course sequence on introductory numerical analysis: Iterative methods for linear systems; Approximation theory; Eigenvalue approximations; Solution of systems of equations; Numerical methods for ODEs and PDEs.

Office: 1089 Evans

Office Hours: TBD

Required Text: R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, Brooks-Cole, 2010. ISBN-13: 978-0-538-73351-9; ISBN-10: 0-538-73351-9 or R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition, Brooks-Cole, 2004. ISBN-13: 978-0-534-39200-0; ISBN-10: 0-534-39200-8.

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Grading: Homework assignments, programming assignments, midterm exam, and final exam.

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

Course Webpage: http://persson.berkeley.edu/128B

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: 1A-1B, 53, or equivalent.

Syllabus: Theory of rational numbers based on the number line, the Euclidean algorithm and fractions in lowest terms. The concepts of congruence and similarity, equation of a line, functions, and quadratic functions.

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Prerequisites: 53, 54, and 113.

Syllabus: History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.

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Prerequisites: 55.

Syllabus: Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional topics at the discretion of the instructor.

Office: 793 Evans Hall

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

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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.

Office: 749 Evans

Office Hours: TBA

Required Text: Brown and Churchill, Complex Variables and Applications, 8th edition. Note: you will probably want to get the paperback edition, which is about $70, versus the hardcover edition, which is around $170, and put the other $100 to better use. (Be careful not to accidentally get the solutions manual instead of the textbook, however! The ISBN is 978-0071263283.) There's also an e-book version.

Recommended Reading: Needham's Visual Complex Analysis can be very helpful. I'll ask the library to have a copy on reserve. An inexpensive supplementary text (with lots of exercises) is Fisher's Complex Variables; it's a $15 Dover book. Marsden and Hoffman's Basic Complex Analysis is a good resource for a slightly more advanced treatment.

Grading: TBA

Homework: TBA; expect weekly assignments.

Course Webpage: bspace.berkeley.edu

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:  Math 113

Syllabus: The topic for this Math 191 is an introduction to elliptic curves and modular forms 

Course Description:   Elliptic curves and modular forms are seemingly disparate objects which are (each on their own) wildly intricate, theoretically rich and of the utmost importance to modern number theory.  In fact, the two types of objects share an intimate and non-obvious relationship - a relationship deep enough to imply a result whose proof eluded the world's top mathematicians for more than 350 years.

We will give an introduction to curves, coordinate rings, projective and affine space, elliptic curves, their group law, elliptic curves over various fields, maps between elliptic curves, differentials.  Divisors, Jacobians, elementary moduli of elliptic curves over the complex numbers, and the Mordell-Weil theorem.

We will also talk about lattices, congruence subgroups, modular forms, double coset and Hecke operators, Eisenstein series, Fourier expansions, the Peterson product, and the statement of a first version of the modularity theorem.

Office: 901 Evans

Office Hours: Friday 3-4

Required Text: Elliptic Curves by J.S. Milne (See http://www.jmilne.org/math/Books/ectext0.pdf) and Rational Points on Elliptic Curves by Joseph Silverman and John Tate

Recommended Reading:   Silverman - The Arithmetic of Elliptic Curves

Grading:  To Be Announced

Homework:  To Be Announced

Course Webpage:  math.berkeley.edu/~coleman/Courses/Sp13/ecmf.html

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: 206.

Syllabus: Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.

Office: 811 Evans

Office Hours: TBA

Required Text: none

Recommended Reading: "C*-algebras by Example", K. R. Davidson, Fields Institute Monographs, A. M. S.

Grading: see course webpage

Homework: see course webpage

Course Webpage: http://math.berkeley.edu/~rieffel

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.

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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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Prerequisites: Math 128A/B or equivalent knowledge of basic numerical analysis, MATLAB programming experience.

Syllabus: Theory and practical methods for numerical solution of differential equations. Finite volume methods for hyperbolic conservation laws, finite element methods for elliptic equations, discontinuous Galerkin methods for first and second order systems of conservation laws.

Office: 1089 Evans

Office Hours: TBD

Required Text: R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002. ISBN 978-0521009249 and C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009. ISBN 978-0486469003.

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Grading: Problem sets.

Homework: 7 Problem sets.

Course Webpage: persson.berkeley.edu/228B

Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor.

Syllabus: Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Also listed as Molecular and Cell Biology C244.

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Prerequisites: 214 and 215A.

Syllabus: Riemann surfaces, divisors and line bundles on Riemann surfaces, differentials and the Riemann-Roch theorem. Line bundles and Abel-Jacobi theorem. Complex manifolds, Kahler metrics, sheaf and Dolbeault cohomology. Basic Hodge theory, line bundles, Kodaira's vanishing and embedding theorems.

Office: 905 Evans

Office Hours: TBA

Required Text: 

Recommended Reading: Notes on Riemann surfaces, http://math.berkeley.edu/~teleman/math/Riemann.pdf.

Griffiths&Harris, Principles of Algebraic Geometry. Wells, Differential Analysis on complex Manifolds.

Grading: Homework and a final paper/presentation

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Prerequisites: 250A.

Syllabus: Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods.

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Prerequisites: 256A.

Syllabus: 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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Course Webpage: http://math.berkeley.edu/~nadler/256bspring2013

Prerequisites: For courses of this level, I used to write: A dense subset in Griffitth-Harris' "Principles of Algebraic Geometry", but then somebody added in handwriting: ... in discrete topology. Well, the truth is, that by now I myself have so firmly forgotten those Principles, that saying "a nowhere dense set" (in that topology) would be more appropriate.   

Syllabus: The ultimate goal of the course is to give an exposition of the recent joint work of the instructor with Valentin Tonita "The quantum Riemann-Roch-Hirzebruch theorem in true genus-0 Gromov-Witten theory". The theorem completely characterises genus-0 Gromov-Witten invariants of K-theoretic origin in terms of the usual, cohomological ones. This recent development opens many new, largely unexplored opportunities in enumerative algebraic geometry. On the other hand, the proof (and even formulation) of the theorem is "technology-dependent", that is, relies on a series of previous advances by many authors (e.g. - among many others - Chen-Ruan's Gromov-Witten invariants of orbifolds and Jarvis-Kimura's computation of such for the orbifolds Point/Finite Group).

We plan to begin with an overview of classical Riemann-Roch formulas in homotopic topology and algebraic geometry. Then we will give an introduction to Gromov-Witten invariants and study in detal general properties of such invariants in genus 0. At this point we will have to develop the so-called "symplectic loop space formalism" - a combinatorial framework (replacing the notion of Frobenius structures) suitable for  comparing  Gromov-Witten invariants of various sorts. Our intermediate goal will be to establish a number of general "twisting" formulas (they typically show that various attempts to invent new GW-invariants are unsuccessful, by expressing those new invariants in terms of the old ones), including the Quantum Hirzebruch-Riemann-Roch formula of Tom Coates, expressing certain GW-invariants with values in complex cobordisms in terms of cohomologica ones. We'll conclude with proving the title theorem, providing some applications (among them - the intrinsic role of finite-difference equations in quantum K-theory), and outlining a number of open problems and directions promising for further research.

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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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