Paul G. Bamberg, Senior Lecturer on Mathematics

Bret J. Benesh, Preceptor in Mathematics

Kevin Buzzard, Visiting Professor of Mathematics

Francesco Calegari, Benjamin Peirce Assistant Professor of Mathematics

Tom Coates, Benjamin Peirce Assistant Professor of Mathematics

Samit Dasgupta, Benjamin Peirce Assistant Professor of Mathematics

Alberto De Sole, Benjamin Peirce Assistant Professor of Mathematics

Elizabeth Denne, Benjamin Peirce Assistant Professor of Mathematics

Eaman Eftekhary, Benjamin Peirce Assistant Professor of Mathematics

Noam D. Elkies, Professor of Mathematics

Dennis Gaitsgory, Professor of Mathematics

Véronique Godin, Benjamin Peirce Assistant Professor of Mathematics

Daniel L. Goroff, Professor of the Practice of Mathematics

Robin Gottlieb, Senior Preceptor in Mathematics

Benedict H. Gross, George Vasmer Leverett Professor of Mathematics and Dean of Harvard College

Joseph D. Harris, Higgins Professor of Mathematics

David Helm, Benjamin Peirce Assistant Professor of Mathematics

Michael J. Hopkins, Professor of Mathematics

Arthur M. Jaffe, Landon T. Clay Professor of Mathematics and Theoretical Science

Thomas W. Judson, Preceptor in Mathematics

Oliver Knill, Preceptor in Mathematics

Joachim Krieger, Benjamin Peirce Assistant Professor of Mathematics

Peter B. Kronheimer, William Caspar Graustein Professor of Mathematics

Thomas Lam, Benjamin Peirce Assistant Professor of Mathematics

Matthew P. Leingang, Preceptor in Mathematics

Jacob Lurie, Lecturer on Mathematics

Barry C. Mazur, Gerhard Gade University Professor

Curtis T. McMullen, Maria Moors Cabot Professor of the Natural Sciences

Andreea C. Nicoara, Benjamin Peirce Assistant Professor of Mathematics

Martin A. Nowak, Professor of Mathematics and of Biology

David J. Pollack, Visiting Assistant Professor of Mathematics

Gerald E. Sacks, Professor of Mathematical Logic

Wilfried Schmid, Dwight Parker Robinson Professor of Mathematics

Yum Tong Siu, William Elwood Byerly Professor of Mathematics

Shlomo Z. Sternberg, George Putnam Professor of Pure and Applied Mathematics

Robert M. Strain, Benjamin Peirce Assistant Professor of Mathematics

Richard L. Taylor, Herchel Smith Professor of Mathematics

Angela G. Vierling-Claassen, Preceptor in Mathematics

Benjamin Weinkove, Benjamin Peirce Assistant Professor of Mathematics

Horng-Tzer Yau, Professor of Mathematics

Shing-Tung Yau, William Caspar Graustein Professor of Mathematics

Ilia Zharkov, Benjamin Peirce Assistant Professor of Mathematics

The Mathematics Department would like to welcome students into that course for which they are best qualified. Incoming students should take advantage of Harvard’s Mathematics Placement Test and of the science advising available in the Science Center the week before classes begin. Members of the Mathematics Department will be available during this period to consult with students. Generally, students with a strong precalculus background and some calculus experience will begin their mathematics education here with a deeper study of calculus and related topics in courses such as Mathematics 1a, 1b, 19, 20, and 21a, b. The Harvard Mathematics Placement Test results recommend the appropriate starting level course, either Mathematics Xa, 1a, 1b, or 21. Recommendation for Mathematics 21 is sufficient qualification for Mathematics 19, 20, 21a, 23a, and 25a.

In any event, what follows briefly describes these courses: Mathematics 1a introduces the basic ideas and techniques of calculus while Mathematics 1b covers integration techniques, differential equations, sequences and series. Mathematics 21a covers multi-variable calculus while Mathematics 21b covers basic linear algebra with applications to differential equations. Students who do not place into (or beyond) Mathematics 1a can take Mathematics Xa, Xb, a two-term sequence which integrates calculus and precalculus material and prepares students to enter Mathematics 1b.

There are a number of options available for students whose placement is to Mathematics 21. For example, Mathematics 19 can be taken either before or after Mathematics 21 (or Mathematics 20). Mathematics 19 covers modeling and differential equation topics for students interested in biological and other life science applications. Mathematics 20 covers selected topics from Mathematics 21a and 21b for students particularly interested in economic and social science applications.

Mathematics 23 is a theoretical version of Mathematics 21 which treats multivariable calculus and linear algebra in a rigorous, proof oriented way. Mathematics 25 and 55 are theory courses that should be elected only by those students who have a particular interest in, and commitment to, mathematics. They assume a solid understanding of one-variable calculus and a willingness to think rigorously and abstractly about mathematics, and to work extremely hard. Both courses study multivariable calculus and linear algebra plus many very deep related topics. Mathematics 25 differs from Mathematics 23 in that the work load in Mathematics 25 is significantly more than in Mathematics 23, but then Mathematics 25 covers more material. Mathematics 55 differs from Mathematics 25 in that the former assumes a very strong proof oriented mathematics background.

The suitability of Mathematics 55 and higher numbered courses is not addressed by the placement examinations. Students who have had substantial preparation beyond the level of the Advanced Placement Examinations are urged to consult the Department Head Tutor in Mathematics concerning their initial Harvard mathematics courses. Students should take this matter very seriously. The Mathematics Department has also prepared a pamphlet with a detailed description of all its 100-level courses and their relationship to each other. This pamphlet gives sample lists of courses suitable for students with various interests. It is available at the Mathematics Department Office. Many 100-level courses assume some familiarity with proofs. Courses that fulfill this prerequisite include Mathematics 23, 25, 55, 101, 112, 121, and 141. Of these, note that Mathematics 101 may be taken concurrently with Mathematics 1, 19, 20, or 21.

The Mathematics Department does not grant formal degree credit without

In the case of students accepting admission as sophomores, this policy is administered as follows: students counting one half course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a, and students counting a full course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a and 1b.

Catalog Number: 1981 Enrollment: Normally limited to 15 students per section.

The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students.

**
Mathematics Xb. Introduction to Functions and Calculus II**

Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.

*
Angela G. Vierling-Claassen, Bret J. Benesh, and members of the Department
*

*
Half course (spring term). Section I: M., W., F., at 10; Section II: M. W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); and a twice weekly lab session to be arranged. EXAM GROUP: 3*

Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.

*Note: *This course, when taken for a letter grade together with Mathematics Xa, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics Xa.

**
Mathematics 1a. Introduction to Calculus**

Catalog Number: 8434 Enrollment: Normally limited to 30 students per section.

*
Thomas W. Judson, Bret J. Benesh, Robin Gottlieb (fall term), and David J. Pollack (spring term)
*

*
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10–11:30; Section Vl, Tu., Th., 11:30–1. Spring: Section I, M., W., F., at 10; Section II, Tu.Th. 10-11:30 (with sufficient enrollment) and a weekly problem section to be arranged. EXAM GROUP: 1*

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to optimization, graphing, mechanisms, and problems from many other disciplines.

*Note: *Required first meeting in fall: Tuesday, September 20, 8:30 am, Science Center D. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *A solid background in precalculus.

**
Mathematics 1b. Calculus, Series, and Differential Equations**

Catalog Number: 1804 Enrollment: Normally limited to 30 students per section.

*
Matthew P. Leingang and Angela G. Vierling-Claassen (fall term); Thomas W. Judson, Robin Gottlieb, and Robert M. Strain (spring term)
*

*
Half course (fall term; repeated spring term). Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12 (with sufficient enrollment); Section V: Tu., Th., 10–11:30; Section Vl, Tu., Th., 11:30–1, and a weekly problem section to be arranged. First meeting, spring:* W., Feb 1, 8:30 am, Science Center A. EXAM GROUP: Fall: 1; Spring: 2

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

**
Mathematics 19. Mathematical Modeling**

Catalog Number: 1256

*
Thomas W. Judson
*

*
Half course (fall term). M., W., F., at 1, and a weekly problem section to be arranged. EXAM GROUP: 6*

Considers the construction and analysis of mathematical models that arise in the environmental sciences, biology, the ecological sciences, and in earth and atmospheric sciences. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).

*Note: *Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 20. This course can be taken before or after Mathematics 20. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

**
Mathematics 20. Introduction to Linear Algebra and Multivariable Calculus**

Catalog Number: 0906

*
Matthew P. Leingang
*

*
Half course (fall term; repeated spring term). M., W., F., at 10, and a weekly problem section to be arranged. EXAM GROUP: 3*

Introduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.

*Note: *Should not ordinarily be taken in addition to Mathematics 21a,b. Examples drawn primarily from economics and the social sciences though Mathematics 20 may be useful to students in certain natural sciences. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics 1b or equivalent, or an A or A- in Mathematics 1a, or a 5 on the AB or a 3 or higher on the BC Advanced Placement Examinations in Mathematics.

**
Mathematics 21a. Multivariable Calculus**

Catalog Number: 6760 Enrollment: Normally limited to 30 students per section.

*
Oliver Knill, Paul G. Bamberg, and Thomas Lam (fall term); Oliver Knill, Matthew P. Leingang, Veronique Godin, Joachim Krieger, and Thomas Lam (spring term)
*

*
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10–11:30; Section VI, Tu., Th., 11:30–1; and a weekly problem section to be arranged. First meeting, spring:* W., Feb 1, 8:30 am, Science Center C . EXAM GROUP: Fall: 1; Spring: 2

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives, and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces, and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’, and Divergence Theorems.

**
Mathematics 21b. Linear Algebra and Differential Equations**

Catalog Number: 1771 Enrollment: Normally limited to 30 students per section.

*
Andreea C. Nicoara (fall term); Clifford Taubes, Angela G. Vierling-Claassen, and Ilia Zharkov (spring term)
*

*
Half course (fall term; repeated spring term). Fall: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Spring: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); Section IV: Tu., Th., 10–11:30; Section V: Tu., Th., 11:30–1 and a weekly problem section to be arranged. First Meeting, spring:* W., Feb 1, 8:30 am, Science Center D. EXAM GROUP: Fall: 1; Spring: 3

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as vectors, Euclidean spaces, linear transformations, determinants, eigenvalues, and eigenvectors. Of applications given, a regular section considers dynamical systems and both ordinary and partial differential equations. Accompanying an introduction to statistical techniques, applications from biology and other data-rich sciences are presented in a biology and statistics section.

**
Mathematics 23a. Theoretical Linear Algebra and Multivariable Calculus I**

Catalog Number: 2486

*
Alberto De Sole
*

*
Half course (fall term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4*

A rigorous treatment of linear algebra and the calculus of functions of n real variables. Topics include: Construction of number systems, fields; vector spaces and linear transformations, eigenvalues and eigenvectors, multilinear forms, and determinants; elementary topology of Euclidean space, inner products, and norms; differentiation and integration of functions of several real variables, the classical theorems of vector analysis.

*Note: *Mathematics 23a, b are honors courses, specifically designed for students with strong mathematics backgrounds who are seriously interested in continuing in the theoretical sciences. See the description in the introductory paragraphs in the Mathematics section of the catalog. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination. Mathematics 23 goes well beyond the concepts strictly necessary for Physics 15, which are more closely correlated with Mathematics 21.

**
Mathematics 23b. Theoretical Linear Algebra and Multivariable Calculus II**

Catalog Number: 8571

*
Alberto De Sole
*

*
Half course (spring term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4*

Continuation of the subject matter of Mathematics 23a.

*Note: *This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics 23a.

**
Mathematics 25a. Honors Multivariable Calculus and Linear Algebra**

Catalog Number: 1525

*
Elizabeth Denne
*

*
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3*

A rigorous treatment of linear algebra, point-set and metric topology, and the calculus of functions in n variables. Emphasis placed on careful reasoning, and on learning to understand and construct proofs.

*Note: *Only for students with a strong interest and background in mathematics. May not be taken for credit after Mathematics 23. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *A 5 on the Advanced Placement BC-Calculus Examination, or the equivalent as determined by the instructor.

**
Mathematics 25b. Honors Multivariable Calculus and Linear Algebra**

Catalog Number: 1590

*
Elizabeth Denne
*

*
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3*

A continuation of Mathematics 25a. More advanced topics, such as Fourier analysis, differential forms, and differential geometry, will be introduced as time permits.

*Note: *This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics 25a or permission of instructor.

**
*Mathematics 55a. Honors Advanced Calculus and Linear Algebra**

Catalog Number: 4068

*
Noam D. Elkies
*

*
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4*

A rigorous treatment of metric and general topology, linear and multi-linear algebra, differential and integral calculus.

*Note: *Mathematics 55a is an intense course for students having significant experience with abstract mathematics. Instructor’s permission required. Every effort will be made to accommodate students uncertain of whether the course is appropriate for them; in particular, Mathematics 55a and 25a will be closely coordinated for the first three weeks of instruction. Students can switch between the two courses during the first three weeks without penalty. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

**
Mathematics 55b. Honors Advanced Calculus and Linear Algebra**

Catalog Number: 3312

*
Yum Tong Siu
*

*
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3*

Continuation of Mathematics 55a. Calculus of functions of several variables. More advanced topics selected from functional analysis, Fourier analysis, differential equations, and differential geometry.

*Note: *This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *Mathematics 55a or permission of instructor.

**
*Mathematics 60r. Reading Course for Senior Honors Candidates**

Catalog Number: 8500

*
Peter B. Kronheimer
*

*
Half course (fall term; repeated spring term). Hours to be arranged.*

Advanced reading in topics not covered in courses.

*Note: *Limited to candidates for honors in Mathematics who obtain the permission of both the faculty member under whom they want to work and the Director of Undergraduate Studies. May not count for concentration in Mathematics without special permission from the Director of Undergraduate Studies. Graded Sat/Unsat only.

**
*Mathematics 91r. Supervised Reading and Research**

Catalog Number: 2165

*
Peter B. Kronheimer
*

*
Half course (fall term; repeated spring term). Hours to be arranged.*

Programs of directed study supervised by a person approved by the Department.

*Note: *May not ordinarily count for concentration in Mathematics.

**
*Mathematics 99r. Tutorial**

Catalog Number: 6024

*
Peter B. Kronheimer and members of the Department
*

*
Half course (fall term; repeated spring term). Hours to be arranged.*

Topics for 2005-06: Holomorphic Vector Bundles on Algebraic Curves (fall), prerequisite: basic complex analysis and elementary ideas from differential geometry. (2) Binary Quadratic Forms (fall), prerequisite: complex analysis (Math 113) and algebra (Math 122, 123). (3) Height Functions in Arithmetic Geometry (spring), prerequisite: Math 122, 123, and basic algebraic number theory. (4) Sphere Packings (spring), prerequisite: Math 112.

*Note: *May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

Catalog Number: 8066

An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.

**
Mathematics 106. Ordinary Differential Equations**

Catalog Number: 3377

*
Thomas W. Judson
*

*
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6*

Analytic, numerical, and qualitative analysis of ordinary differential equations. Linear equations, linear and non-linear systems. Applications to mechanics, biology, physics, and the social sciences. Existence and uniqueness of solutions and visual analysis using computer graphics. Topics selected from Laplace transforms, power series solutions, chaos, and numerical solutions.

*Prerequisite: *Mathematics 19, 20 or 21a.

**
Mathematics 112. Real Analysis**

Catalog Number: 1123

*
Robert M. Strain
*

*
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral, sets of measure zero and conditions for integrability.

*Prerequisite: *Mathematics 21a,b or 23a,b, and either an ability to write proofs or concurrent enrollment in Mathematics 101. Should not ordinarily be taken in addition to Mathematics 25a,b or 55a,b.

**
Mathematics 113. Complex Analysis**

Catalog Number: 0405

*
Andreea C. Nicoara
*

*
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4*

Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy’s theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.

*Prerequisite: *Mathematics 23a,b, 25a,b, or 101. Students with an A grade in Mathematics 21a,b may also consider taking this course, but must understand proofs.

**
Mathematics 115. Methods of Analysis and Applications**

Catalog Number: 1871

*
Joachim Krieger
*

*
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6*

Some complex function theory; Fourier analysis; Hilbert spaces and operators; Laplace’s equations; Bessel and Legendre functions; symmetries; and Sturm-Liouville theory.

*Note: *Mathematics 115 is especially for students interested in physics.

*Prerequisite: *Mathematics 21a,b, 23a,b, or 25a,b, and permission of instructor.

**
[Mathematics 116. Convexity and Optimization with Applications]**

Catalog Number: 5253

*
----------
*

*
Half course (spring term). Hours to be arranged.*

Introduction to real and functional analysis through topics such as convex programming, duality theory, linear and non-linear programming, calculus of variations, and the maximum principle of optimal control theory.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *At least one course beyond Mathematics 21.

**
Mathematics 117. Topics in Analysis**

Catalog Number: 0630

*
Horng-Tzer Yau
*

*
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Sobolev space and logarithmic Sobolev inequalities. Schrodinger equation and semiclassical limit. Introduction to Bose-Einstein condensation and nonlinear Schrodinger equation.

*Prerequisite: *Math 112 or permission of the instructor.

**
[Mathematics 118r. Dynamical Systems]**

Catalog Number: 6402

*
----------
*

*
Half course (spring term). Hours to be arranged.*

A mathematical introduction to nonlinear dynamical system theory and its applications. Topics include concepts on the iteration of maps and the integration of flows, bifurcation theory, the role of equilibrium points, invariant manifolds, and attractors. Applications include examples from celestial mechanics, geometry or statistical mechanics or number theory. Computer demonstrations in class are used to visualize and understand the concepts and will encourage experimentation.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *Multi-variable calculus as well as linear algebra.

**
Mathematics 119. Partial Differential Equations and Applications **

Catalog Number: 7326

*
Robert M. Strain
*

*
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

Partial differential equations with constant coefficients, hyperbolic elliptic, and parabolic equations, Fourier analysis, Green’s function.

*Prerequisite: *Familiarity with functions of a complex variable.

**
Mathematics 121. Linear Algebra and Applications**

Catalog Number: 7009

*
Joachim Krieger
*

*
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4*

Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to differential equations, classical mechanics, and optimization theory. Emphasizes learning to understand and write proofs.

*Prerequisite: *Mathematics 21a,b or equivalent. Should not ordinarily be taken in addition to Mathematics 23a,b, 25a,b, or 55a,b.

**
Mathematics 122. Abstract Algebra I: Theory of Groups and Vector Spaces**

Catalog Number: 7855

*
Benedict H. Gross
*

*
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3*

Algebra is the language of modern mathematics. Provides an introduction to this language, through the study of groups and group actions, vector spaces and their linear transformations, and some general theory of rings and fields.

*Prerequisite: *Mathematics 21b, or the equivalent training in matrices and linear algebra.

**
Mathematics 123. Abstract Algebra II: Theory of Rings and Fields**

Catalog Number: 5613

*
Véronique Godin
*

*
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3*

Bilinear forms and group representations. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.

*Prerequisite: *Mathematics 122.

**
Mathematics 124. Number Theory**

Catalog Number: 2398

*
Véronique Godin
*

*
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4*

Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell’s equation; selected Diophantine equations; theory of integral quadratic forms.

*Prerequisite: *Mathematics 122 (which may be taken concurrently) or equivalent.

**
Mathematics 126. Representation Theory and Applications **

Catalog Number: 0369

*
Wilfried Schmid
*

*
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

Representation theory of finite groups including character theory, induced representations, Frobenius reciprocity, and interesting applications.

**
[Mathematics 128. Lie Algebras]**

Catalog Number: 6519

*
----------
*

*
Half course (spring term). Hours to be arranged.*

Definition of Lie algebras, examples, Poincare-Birkhoff-Witt theorem, Levy decomposition. Semi-simple algebras, their classification and finite-dimensional representations, Verma modules, and Weyl character formula.

*Note: *Expected to be given in 2006–07.

**
Mathematics 129. Topics in Number Theory**

Catalog Number: 2345

*
Benedict H. Gross
*

*
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3*

Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.

*Prerequisite: *Mathematics 122 and 123.

**
Mathematics 131. Topology**

Catalog Number: 2381

*
Tom Coates
*

*
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Basic notions of point set topology such as continuity, compactness, metrizability. Algebraic topology including fundamental groups, covering spaces, and higher homotopy groups.

*Prerequisite: *Some acquaintance with metric space topology (Mathematics 25a,b, 55a,b, 101, or 112) and with groups (Mathematics 101 or 122).

**
[Mathematics 134. Calculus on Manifolds]**

Catalog Number: 7150

*
----------
*

*
Half course (fall term). Hours to be arranged.*

Generalization of multivariable calculus to the setting of manifolds in real n-space, as used in the study of global analysis and geometry. Differentiable mappings of linear spaces, the inverse and implicit function theorems, differential forms, integration on manifolds, the general version of Stokes’s theorem, integral geometry, applications.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *Mathematics 21a,b and familiarity with proofs as in Mathematics 101, 112, 121, or the equivalent.

**
Mathematics 135. Differential Topology**

Catalog Number: 2107

*
Eaman Eftekhary
*

*
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

Smooth manifolds, intersection theory, vector fields, Hopf degree theorem, Euler characteristic, De Rham theory.

*Prerequisite: *Mathematics 23a,b, 25a,b, 55a,b, or 134.

**
Mathematics 136. Differential Geometry**

Catalog Number: 1949

*
Ilia Zharkov
*

*
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4*

Curves and surfaces in 3-space: curvature and its intrinsic meaning, Gauss-Bonnet theorem, surfaces of constant curvature, introduction to Riemannian geometry with applications in dimension two.

*Prerequisite: *Mathematics 21a,b and familiarity with proofs as in Mathematics 101, 112, 121, or equivalent.

**
Mathematics 137. Algebraic Geometry**

Catalog Number: 0556

*
Shing-Tung Yau
*

*
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7*

Affine and projective spaces, plane curves, Bezout’s theorem, singularities and genus of a plane curve, Riemann-Roch theorem.

*Prerequisite: *Mathematics 122, 123.

**
Mathematics 138. Classical Geometry**

Catalog Number: 0162

*
Paul G. Bamberg
*

*
Half course (spring term). M., W., 4–5:30. EXAM GROUP: 9*

An exploration of the many different flavors of plane geometry. The course begins with finite geometry, then surveys the nine possible Cayley-Klein plane geometries, focusing on Euclidean geometry, the Galilean geometry of uniform motion, spherical and elliptic geometry, and geometries related to relativistic physics such as Minkowskian geometry and hyperbolic geometry. An important tool in the study of these geometries is a study of their symmetry groups.

*Prerequisite: *Mathematics 21a and 21b (may be taken concurrently), or Mathematics 23a, 25a, or 55a.

**
Mathematics 139x. Topics in Symplectic Geometry**

Catalog Number: 0911

*
Shlomo Z. Sternberg
*

*
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

An introduction to symplectic geometry which lies at the heart of Hamiltonian mechanics and has recent applications to topology.

*Prerequisite: *Solid background in linear algebra and multivariable calculus.

**
Mathematics 141. Introduction to Mathematical Logic**

Catalog Number: 0600

*
Gerald E. Sacks
*

*
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.

*Prerequisite: *Any mathematics course at the level of Mathematics 21a,b or higher, or permission of instructor.

**
Mathematics 143. Set Theory**

Catalog Number: 6005

*
Gerald E. Sacks
*

*
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Axioms of set theory. Gödel’s constructible universe. Consistency of the axiom of choice and of the generalized continuum hypothesis. Cohen’s forcing method. Independence of the AC and GCH.

*Prerequisite: *Any mathematics course at the level of 21a or higher, or permission of instructor.

**
[Mathematics 144. Model Theory and Algebra]**

Catalog Number: 0690

*
----------
*

*
Half course (spring term). Hours to be arranged.*

An introduction to model theory with applications to fields and groups. First order languages, structures, and definable sets. Compactness, completeness, and back-and-forth constructions. Quantifier elimination for algebraically closed, differentially closed, and real closed fields. Omitting types, prime extensions, existence and uniqueness of the differential closure, saturation, and homogeneity. Forking, independence, and rank.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *Mathematics 123 or the equivalent is suggested as a prerequisite, but not required.

**
Mathematics 152 (formerly Mathematics 102). Methods of Discrete Mathematics**

Catalog Number: 8389

*
Paul G. Bamberg
*

*
Half course (fall term). Section I: Tu., Th., 1–2:30; Section II: Tu., Th., 2:30–4. EXAM GROUP: 15, 16*

An introduction to finite groups, finite fields, finite geometry, discrete probability, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. Taught in a seminar format, and students will gain experience in presenting proofs at the blackboard.

*Note: *Students who have taken Mathematics 25ab or 55ab should not take this course for credit.

*Prerequisite: *Mathematics 21b or equivalent.

**
Mathematics 153. Mathematical Biology-Evolutionary Dynamics**

Catalog Number: 3004 Enrollment: Limited to 30.

*
Martin A. Nowak
*

*
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16*

Introduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer.

*Note: *Limited to seniors and graduate students.

*Prerequisite: *Mathematics 21a and b, Biological Sciences 50 and 53 or equivalent.

**
Mathematics 191. Mathematical Probability**

Catalog Number: 4306

*
Paul G. Bamberg
*

*
Half course (spring term). Tu., Th., 4–5:30. EXAM GROUP: 18*

An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability. Elements of random processes: the Poisson process, random walks, and Markov chains.

*Prerequisite: *Any mathematics course at the level of Mathematics 19, or 21a, b or higher, or knowledge of multivariable calculus as demonstrated on the online placement test.

**
Mathematics 192r. Algebraic Combinatorics**

Catalog Number: 6612

*
Thomas Lam
*

*
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5*

An introduction to the relation between algebra and combinatorics. Topics include generating functions, partially ordered sets and mobius functions, partitions and tableaux theory, and algebraic graph theory.

*Note: *No prior knowledge of combinatorics is assumed, but familiarity with linear algebra will be helpful.

**
Mathematics 194. Combinatorial Game Theory**

Catalog Number: 0658

*
Jacob Lurie
*

*
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6*

An introduction to the theory of combinatorial games (games of no chance and perfect information). Topics: the arithmetic of games, structure theory for impartial games, surreal numbers, temperature theory for finite games, and Norton’s analysis of “all small” games via the atomic weight calculus.

*Prerequisite: *Mathematics 123 or equivalent.

Catalog Number: 5446

A review of measure and integration. Hilbert and Banach spaces, L^ p spaces, and the Riesz representation theorem.

**
Mathematics 212b. Advanced Real Analysis**

Catalog Number: 7294

*
Curtis T. McMullen
*

*
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

Continuation of Mathematics 212a. Functional analysis and applications. Topics may include distributions, elliptic regularity, spectral theory, operator algebras, unitary representations, and ergodic theory.

*Prerequisite: *Mathematics 212a and 213a.

**
Mathematics 213a. Functions of One Complex Variable**

Catalog Number: 1621

*
Wilfried Schmid
*

*
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Review of basic complex analysis. Further topics will include series and product developments, uniformization, and special functions.

*Prerequisite: *Basic complex analysis or ability to learn quickly.

**
Mathematics 213b. Further Topics in Classical Complex Analysis**

Catalog Number: 2641

*
Wilfried Schmid
*

*
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Compact Riemann surfaces, Riemann-Roch theorem, and introduction to the theory of modular forms.

*Prerequisite: *Mathematics 213a and previous or concurrent enrollment in 212a and b preferred.

**
Mathematics 214x. Geometric Analysis**

Catalog Number: 0926

*
Yum Tong Siu
*

*
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4*

A study of major techniques of estimates in partial differential equations which have been developed for global problems in Riemannian geometry, Kaehler geometry, Yang-Mills theory, Cauchy-Riemann geometry, and the complex Neumann problem.

**
Mathematics 230ar. Differential Geometry**

Catalog Number: 0372

*
Peter B. Kronheimer
*

*
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6*

Elements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, Geodesics, Riemann curvature, Darboux’s theorem, moment maps and symplectic quotients, complex and Kaehler manifolds, Dolbeault and de Rham cohomology.

**
Mathematics 230br. Differential Geometry**

Catalog Number: 0504

*
Peter B. Kronheimer
*

*
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6*

A continuation of Mathematics 230ar. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogov’s theorem and applications; sphere theorems; Gromov’s betti number bounds; Gromov-Hausdorff convergence; Cheeger’s finiteness theorem, and convergence theorems.

*Prerequisite: *Mathematics 135.

**
Mathematics 233. General Relativity**

Catalog Number: 0676

*
Shing-Tung Yau
*

*
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5*

Descriptions of topology and geometry of space time within the frameworks of general relativity. A discussion of the problems of defining conversed quantities of general relativity that are important to both physical and geometric considerations. Discussion of evolutions of the Einstein equation.

**
Mathematics 234. Evolutionary Dynamics **

Catalog Number: 8136

*
Martin A. Nowak
*

*
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16*

Advanced topics of evolutionary dynamics. Seminars and research projects.

*Prerequisite: *Experience with mathematical biology at the level of Mathematics 153.

**
Mathematics 235. Minimal Surfaces**

Catalog Number: 0677

*
Shing-Tung Yau
*

*
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5*

Discussions of classical minimal surfaces in Euclidean space and their constructions through variational means and also through mean curvature flow. A discussion of special Lagrangian minimal submanifolds that are important for Calabi Yau manifolds.

**
Mathematics 250. Higher Algebra**

Catalog Number: 9334

*
Barry C. Mazur
*

*
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7*

Focuses on Galois theory and a small amount of representation theory of finite groups. A study of the important applications of this material to number theory, and the study of algebraic curves.

*Prerequisite: *Mathematics 123 or equivalent.

**
[Mathematics 251a. Algebraic Number Theory]**

Catalog Number: 1703

*
----------
*

*
Half course (fall term). Hours to be arranged.*

A graduate introduction to algebraic number theory. Topics: local fields, Galois cohomology, local class field theory, and local duality.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *Mathematics 123 or 250 and permission of instructor.

**
[Mathematics 251b. Algebraic Number Theory]**

Catalog Number: 7441

*
----------
*

*
Half course (spring term). Hours to be arranged.*

Continuation of Mathematics 251a. Topics: global fields, adeles, class field theory, and duality. Other topics may include: Tate’s thesis, cyclotomic fields, or Euler systems.

*Note: *Expected to be given in 2006–07.

*Prerequisite: *Mathematics 251a or permission of instructor.

**
Mathematics 254z. Current Results in Modular Forms**

Catalog Number: 0941

*
Barry C. Mazur
*

*
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7*

A study of the background material necessary for the comprehension of some recent results in the theory of modular and automorphic forms, and specifically p-adic modular forms. The course will involve substantial student presentations.

**
Mathematics 255y. A Concrete Approach to p-adic Forms**

Catalog Number: 3603

*
Kevin Buzzard (Imperial College)
*

*
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16*

Brief introduction to classical and overconvergent modular forms. How parts of the Coleman-Mazur eigencurve can be “computed.” The Gouvea-Mazur conjectures. Eigencurves at the centre and boundaryof weight space. Concrete accessible open problems.

**
Mathematics 260a. Introduction to Algebraic Geometry**

Catalog Number: 7004

*
Yum Tong Siu
*

*
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Introduction to complex algebraic curves, surfaces, and varieties.

*Prerequisite: *Mathematics 250.

**
Mathematics 260b. Introduction to Algebraic Geometry**

Catalog Number: 2745

*
Joseph D. Harris
*

*
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3*

Continuation of 260a: Introduction to the theory of coherent sheaves, schemes, and sheaf cohomology, with examples and applications.

**
Mathematics 261a. Theory of Schemes**

Catalog Number: 0947

*
Francesco Calegari
*

*
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5*

An introduction to the theory and language of schemes. We will follow closely Chapters II and III of Hartshorne’s book *Algebraic Geometry.*

*Note: *Weekly homework will constitute an important part of the course.

*Prerequisite: *Mathematics 137 and 250 or permission of instructor.

**
Mathematics 261b. Theory of Schemes**

Catalog Number: 0956

*
Samit Dasgupta
*

*
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5*

Continuation of Mathematics 261a.

**
Mathematics 267y. Geometric Representation Theory**

Catalog Number: 0996

*
Dennis Gaitsgory
*

*
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6*

Category O, Beilinson-Bernstein localization, Kac-Moody algebras, affine Grassmannian and Langland’s duality.

*Prerequisite: *Algebraic geometry, basics of D-modules, basics of semi-simple groups and Lie algebras.

**
Mathematics 271x. Manifolds of Special Holonomy**

Catalog Number: 1007

*
Ilia Zharkov
*

*
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14*

Berger’s classification of holonomy: Kahler, hyper-Kahler, Calabi-Yau manifolds. Calabi conjecture and Yau’s proof. ALE metrics. Examples of G2 and Spin(7) manifolds. Calibrated submanifolds.

**
Mathematics 272a. Introduction to Algebraic Topology**

Catalog Number: 1666

*
Joseph D. Harris
*

*
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3*

Covering spaces and fibrations. Simplicial and CW complexes. Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

*Prerequisite: *Mathematics 131 or permission of instructor.

**
Mathematics 272b. Introduction to Algebraic Topology**

Catalog Number: 6502

*
Eaman Eftekhary
*

*
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4*

Spectral sequences and techniques of computation. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.

*Prerequisite: *Mathematics 272a.

**
Mathematics 274x. Heegaard Floer Homology**

Catalog Number: 1018

*
Eaman Eftekhary
*

*
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13*

Heegaard diagrams for three and four dimentional manifolds, Heegaard Floer homology as the Floer homology of the symmetric product and as an example of symplectic field theory, exact triangles, computations, applications to knot theory.

**
Mathematics 277y. Topics in Homotopy Theory**

Catalog Number: 1021

*
Michael J. Hopkins
*

*
Half course (fall term). Th., 3–5. EXAM GROUP: 17, 18*

An investigation of several topics in “string topology” including the recent work of Galatius-Madsen-Tillmann-Weiss on the stable mapping class group, recent work of Sullivan on BV structures and the master equation.

**
Mathematics 278y. Algebraic Topology Literature**

Catalog Number: 1038 Enrollment: Limited to 8.

*
Michael J. Hopkins
*

*
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16*

A literature course in advanced topics in algebraic topology. Material will be taken from research papers, both classical and contemporary, and the presentation will involve significant participation from the students.

*Note: *Enrollment limited, please contact the professor before registering.

Catalog Number: 4344

**
*Mathematics 304. Topics in Algebraic Topology**

Catalog Number: 0689

*
Michael J. Hopkins 4376
*

**
*Mathematics 307. Topics in Differential Geometry and Partial Differential Equations **

Catalog Number: 5133

*
Benjamin Weinkove 4942 (on leave 2005-06)
*

**
*Mathematics 308. Topics in Number Theory and Modular Forms**

Catalog Number: 0464

*
Benedict H. Gross 1112
*

**
*Mathematics 309. Topics in Dynamical Systems Theory**

Catalog Number: 0552

*
Daniel L. Goroff 7683 (on leave 2005-06)
*

**
*Mathematics 310. Topics in Number Theory**

Catalog Number: 3874

*
Samit Dasgupta 5030
*

**
*Mathematics 314. Topics in Differential Geometry and Mathematical Physics**

Catalog Number: 2743

*
Shlomo Z. Sternberg 1965 (on leave spring term)
*

**
*Mathematics 318. Topics in Number Theory**

Catalog Number: 7393

*
Barry C. Mazur 1975
*

**
*Mathematics 321. Topics in Mathematical Physics**

Catalog Number: 2297

*
Arthur M. Jaffe 2095
*

**
*Mathematics 326. Topics in Arithmetic Geometry of Modular Curves and Shimura Curves**

Catalog Number: 2696

*
David Helm 4630 (on leave 2005-06)
*

**
*Mathematics 327. Topics in Several Complex Variables**

Catalog Number: 0409

*
Yum Tong Siu 7550
*

**
*Mathematics 328. Topics in Lie Algebra**

Catalog Number: 7003

*
Alberto De Sole 4627
*

**
*Mathematics 329. Topics in Knot Theory**

Catalog Number: 2194

*
Elizabeth Denne 5031
*

**
*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry**

Catalog Number: 9401

*
Curtis T. McMullen 3588
*

**
*Mathematics 335. Topics in Differential Geometry and Analysis**

Catalog Number: 5498

*
Clifford Taubes 1243
*

**
*Mathematics 342. Topics in Combinatorics**

Catalog Number: 0751

*
Thomas Lam 5322
*

**
*Mathematics 344. Topics in Number Theory**

Catalog Number: 2526

*
Francesco Calegari 4435
*

**
*Mathematics 345. Topics in Geometry and Topology**

Catalog Number: 4108

*
Peter B. Kronheimer 1759
*

**
*Mathematics 346y. Topics in Analysis: Quantum Dynamics**

Catalog Number: 1053

*
Horng-Tzer Yau 5260
*

**
*Mathematics 347. Topics in Floer Homology and Low Dimensional Topology **

Catalog Number: 7227

*
Eaman Eftekhary 5045
*

**
*Mathematics 350. Topics in Mathematical Logic**

Catalog Number: 5151

*
Gerald E. Sacks 3862
*

**
*Mathematics 351. Topics in Algebraic Number Theory**

Catalog Number: 3492

*
Richard L. Taylor 1453
*

**
*Mathematics 356. Topics in Harmonic Analysis**

Catalog Number: 6534

*
Wilfried Schmid 5097
*

**
*Mathematics 365. Topics in Differential Geometry**

Catalog Number: 4647

*
Shing-Tung Yau 1734
*

**
*Mathematics 371. Topics in Partial Differential Equations and Mathematical Physics**

Catalog Number: 0777

*
Robert M. Strain 5323
*

**
*Mathematics 376. Topics in Analysis of Partial Differential Equations**

Catalog Number: 1023

*
Joachim Krieger 4632
*

**
*Mathematics 381. Introduction to Geometric Representation Theory**

Catalog Number: 0800

*
Dennis Gaitsgory 5259
*

**
*Mathematics 382. Topics in Algebraic Geometry**

Catalog Number: 2037

*
Joseph D. Harris 2055
*

**
*Mathematics 383. Topics in Algebraic Geometry**

Catalog Number: 7736

*
Ilia Zharkov 4631
*

**
*Mathematics 386. Topics in Several Complex Variables and CR Geometry**

Catalog Number: 3746

*
Andreea C. Nicoara 4374
*

**
*Mathematics 388. Topics in Mathematics and Biology**

Catalog Number: 4687

*
Martin A. Nowak 4568
*

**
*Mathematics 389. Topics in Number Theory**

Catalog Number: 6851

*
Noam D. Elkies 2604
*

**
*Mathematics 392. Topics in Geometry**

Catalog Number: 8778

*
Tom Coates 4633 (on leave spring term)
*

**
*Mathematics 398. Topics in Algebraic and Geometric Topology**

Catalog Number: 0863

*
Véronique Godin 5311
*