Daniel Allcock, Benjamin Peirce Assistant Professor of Mathematics

Matthew Baker, Assistant Professor of Mathematics

Paul G. Bamberg, Senior Lecturer on Mathematics

John Boller, Preceptor in Mathematics

Alexander Braverman, Benjamin Peirce Assistant Professor of Mathematics

Danny Calegari, Benjamin Peirce Assistant Professor of Mathematics

Albert Chau, Benjamin Peirce Assistant Professor of Mathematics

Christophe Cornut, Benjamin Peirce Assistant Professor of Mathematics

Stephen DeBacker, Benjamin Peirce Assistant Professor of Mathematics

Nathan Dunfield, Benjamin Peirce Assistant Professor of Mathematics

Noam D. Elkies, Professor of Mathematics

Andrew Engelward, Preceptor in Mathematics

Daniel L. Goroff, Professor of the Practice of Mathematics, Associate Director of the Derek Bok Center for Teaching and Learning

Robin Gottlieb, Senior Preceptor in Mathematics

Thomas Benjamin Graber, Benjamin Peirce Assistant Professor of Mathematics

Joseph D. Harris, Higgins Professor of Mathematics

Kalle Karu, Benjamin Peirce Assistant Professor of Mathematics

David Kazhdan, Perkins Professor of Mathematics

Oliver Knill, Preceptor in Mathematics

Peter B. Kronheimer, Professor of Mathematics

John F. Mackey, Preceptor in Mathematics

Barry C. Mazur, Gerhard Gade University Professor

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

Mihnea Popa, Benjamin Peirce Assistant Professor of Mathematics

James G. Propp, Visiting Associate Professor of Mathematics

Fernando Rodriguez-Villegas, Visiting Associate Professor of Mathematics, Visiting Scholar in 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

Gigliola Staffilani, Visiting Associate Professor of Mathematics, Visiting Scholar in Mathematics

William A. Stein, Benjamin Peirce Assistant Professor of Mathematics

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

Dmitry Tamarkin, Benjamin Peirce Assistant Professor of Mathematics

Clifford Taubes, William Petschek Professor of Mathematics

Richard L. Taylor, Professor of Mathematics

Tatiana Toro, Visiting Associate Professor of Mathematics, Visiting Scholar in Mathematics

Dale Winter, Preceptor in Mathematics

Trevor Wooley, Visiting Professor of Mathematics

Shing-Tung Yau, William Casper Graustein Professor of Mathematics

Joy Ann Young

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 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.

Placement in Mathematics Xa, 1a, 1b, 20 and 21 is based on the results of the Harvard Mathematics Placement Test, and/or the Advanced Placement Examinations.

One calculus sequence is Mathematics 1a, 1b, 21a, 21b. 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-semester 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.

Placement in Mathematics 21b, 23a, 25a, and more advanced courses is based on material not covered in the placement examinations. Students who have had substantial preparation beyond the level of the Advanced Placement Examinations are urged to consult the Department concerning their proper placement in mathematics courses. Students should take this matter very seriously and solicit advice from faculty members in the Mathematics Department. The 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 Department Office. Many 100-level courses assume some familiarity with proofs. Courses that fulfill this prerequisite include Mathematics 23, 25, 55, 101, 102, 112, 121, 141. Of these, note that Mathematics 101 and 102 may be taken concurrently with Mathematics 1, 19, 20, or 21.

The 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 yearlong 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.

*
Dale Winter 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: *Both Math Xa and Xb must be taken together to fufill the Quantitative Reasoning requirement.

*Prerequisite: *Mathematics Xa.

**
Mathematics 1a. Introduction to Calculus**

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

*
John F. Mackey and members of the Department
*

*
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 session to be arranged. EXAM GROUP: 2*

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: Wednesday, September 12, 8:00 am, Science Center D.

*Prerequisite: *A solid background in precalculus.

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

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

*
Robin Gottlieb, Daniel Allcock, Albert Chau, Stephen DeBacker, and Dmitry Tamarkin (fall term); Robin Gottlieb, Stephen DeBacker, James Propp, Dmitry Tamarkin (spring term) and members of the Department.
*

*
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 in spring); Section V: Tu., Th., 10–11:30; Section Vl, Tu., Th., 11:30–1, and a weekly problem session to be arranged. Required exams: Hours to be arranged. EXAM GROUP: Fall: 2; Spring: 1*

Galileo wrote that, “The book of the universe is written in the language of mathematics.” Great problems in the physical, biological, and social sciences all find their expression as differential equations. This course builds on basic calculus to study differential equations of the first and second order. We develop both qualitative methods for visualizing solutions as well as analytical methods for writing out solutions, including techniques that evolve from our study of integration, infinite series, power series, and Taylor series.

*Note: *Required first meeting for fall: Wednesday, September 12, 8:00 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:00 am, Science Center B.

*Prerequisite: *Mathematics 1a, or Xa and Xb, or equivalent.

**
Mathematics 19. Mathematical Modeling**

Catalog Number: 1256

*
Albert Chau
*

*
Half course (fall term). M., W., F., at 1, and a weekly problem session 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 includes 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.

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

Catalog Number: 0906

*
Daniel Allcock and John F. Mackey (spring term)
*

*
Half course (fall term; repeated spring term). Fall: M., W., F., at 9, Spring: M., W., F., at 9, and a weekly problem session to be arranged. EXAM GROUP: 2*

Introduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications.

*Note: *Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields. 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.

*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.

*
Clifford Taubes, Paul Bamberg, Christophe Cornut, Kalle Karu, Oliver Knill (fall term); Andrew Engelward, Christophe Cornut (spring term) and members of the Department.
*

*
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 session to be arranged. Required exams: Hours to be arranged. EXAM GROUP: 1*

To see how calculus applies in situations described by more than one variable, we study: Vectors, lines, planes, parametrization 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; Vector fields, line and surface integrals for work and flux; Divergence and curl of vector fields; the Green’s, Stokes’, and Divergence Theorems. Finally, there is an introduction to partial differential equations.

*Note: *Required first meeting in fall: Thursday, September 13, 8:00 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:00 am, Science Center C. May not be taken for credit by students who have passed Applied Mathematics 21a.
Activities using computers to calculate and visualize applications of these ideas will not require previous programming experience. Special sections for students interested in physics or biochemistry and social sciences are offered each semester. The biochemistry/social sciences sections treat topics in probability and statistics in lieu of Green’s, Stokes’ and the Divergence Theorems.

*Prerequisite: *Mathematics 1b or equivalent.

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

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

*
Richard L. Taylor (spring term), Dale Winter (fall term), Oliver Knill (spring term), Yum Tong Siu (spring term), and William A. Stein
*

*
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section ll, M., W., F., at 10; Section IIl, M., W., F., at 11. Spring: Section I, M., W., F., at 9 (with sufficient enrollment); Section ll, M., W., F., at 10; Section IIl, 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 session to be arranged. EXAM GROUP: 2*

By adding and multiplying arrays of numbers called vectors and matrices, linear algebra provides the structure for solving problems that arise in practical applications ranging from Markov processes to optimization and from Fourier series to statistics. To understand how, we develop thorough treatments of: euclidean spaces, including their bases, dimensions and geometry; and linear transformation of such spaces, including their determinants, eigenvalues, and eigenvectors. These concepts will be applied to solve dynamical systems, including both ordinary and partial differential equations.

*Note: *Required first meeting in fall: Wednesday, September 12, 8:00 am, Science Center A. Required first meeting in spring: Wednesday, January 30, 8:00 am, Science Center D. May not be taken for credit by students who have passed Applied Mathematics 21b.

*Prerequisite: *Mathematics lb or equivalent. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.

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

Catalog Number: 2486

*
John Boller
*

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

Vectors and matrices; eigenvalues and eigenvectors; systems of linear differential equations; differentiation and integration of functions of several variables; line integrals.

*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.

*Prerequisite: *Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination. This course does not correlate with the Physics 15 sequence. Mathematics 23 goes well beyond the concepts strictly necessary for Physics 15, which are more closely followed in Mathematics 21.

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

Catalog Number: 8571

*
John Boller
*

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

Continuation of the subject matter of Mathematics 23a. A rigorous treatment of linear algebra and the calculus of functions in n-dimensional space.

*Prerequisite: *Mathematics 23a.

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

Catalog Number: 1525

*
Kalle Karu
*

*
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: *This course should only be elected by students with a strong interest and background in mathematics. May not be taken for credit after Mathematics 23.

*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

*
Kalle Karu
*

*
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.

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

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

Catalog Number: 4068

*
Wilfried Schmid
*

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

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.

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

Catalog Number: 3312

*
Wilfried Schmid
*

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

Continuation of Mathematics 55a. Calculus on manifolds, de Rham cohomology. Additional topics may include differential equations.

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

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

Catalog Number: 8500

*
Clifford Taubes
*

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

Advanced reading in topics not covered in courses.

*Note: *Open only 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/UNS only.

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

Catalog Number: 2165

*
Clifford Taubes
*

*
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

*
Clifford Taubes and members of the Faculty
*

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

Topics for 2001-02: (1) Modular Forms (fall) Prerequisites: one semester of complex analysis (such as Math 113), also one semester of algebra or concurrent enrollment in algebra is strongly recommended. (2) Classical Mechanics and Geometry (fall) Prerequisites: multivariable calculus, linear algebra and comfort with reading and writing rigorous mathematical arguments are necessary prerequisites. For example, completion of Math 21a,b and Math 101, or Math 23a,b, or Math 25a,b or Math 55a,b satisfies this prerequisite. In addition knowledge of point-set topology at the level of Math 101, 112 or 131 will be assumed. (3) Statistics and its Applications to Biology (spring) Prerequisites: probability theory (at the level of Stat 110) and linear algebra. (4) Milnor’s Seven-Spheres (algebraic and differential topology) (spring) Prerequisites: multivariable calculus, linear algebra and comfort with reading and writing rigorous mathematical arguments are necessary prerequisites. For example, completion of Math 21a,b and Math 101, or Math 23a,b, or Math 25a,b or Math 55a,b satisfies this prerequisite. In addition knowledge of smooth manifolds and homology theory at the level of Math 135 will be assumed.

*Note: *May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit. Students must register their interest in taking a tutorial with the Assistant Director of Undergraduate Studies by the second day of the semester in which the tutorial is offered.

Catalog Number: 8066

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

**
Mathematics 102. Methods of Discrete Mathematics**

Catalog Number: 8389

*
Paul G. Bamberg and John Boller
*

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

An introduction to rigorous mathematics and proof drawing on topics from group theory, graph theory, combinatorics, and number theory. We explore how proofs are discovered and refined through observation, conjecture, and verification. As one component of this investigation, we will use Mathematica to explore interesting and otherwise intractable examples to develop computer-aided proofs.

*Note: *Acquaintance with Calculus is desirable. Students who have taken Mathematics 25ab or Mathematics 55ab should not take this course for credit. This course is designed to fulfill the goals of Mathematics 101.

*Prerequisite: *Mathematics 21b or equivalent. The student should also have some familiarity with some computer language though no prior familiarity with Mathematica will be assumed.

**
Mathematics 112. Real Analysis**

Catalog Number: 1123

*
Curtis T. McMullen
*

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

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

*
Matthew Baker
*

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

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 and the study of harmonic functions. An introduction to conformal geometry and conformal mappings.

*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

*
Dmitry Tamarkin
*

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

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 118r. Dynamical Systems]**

Catalog Number: 6402

*
Shlomo Z. Sternberg
*

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

Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory. Computer programs will be developed and used for visualization, approximation, and experimentation.

*Note: *Expected to be given in 2002–03.

*Prerequisite: *Mathematics 21a,b.

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

Catalog Number: 7326

*
Albert Chau
*

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

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

*
Peter B. Kronheimer
*

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

Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to linear programming, game theory 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

*
Stephen DeBacker
*

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

Introduction to the theory of groups and group actions, with emphasis on finite groups and matrix groups. Sylow theorems. A short introduction to rings and fields. Vector spaces and linear transformations. Bilinear forms: symmetric, Hermitian, and skew-symmetric forms.

*Prerequisite: *Mathematics 21b and the ability to write proofs as in Mathematics 101, 121, or the equivalent.

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

Catalog Number: 5613

*
Curtis T. McMullen
*

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

Rings, ideals, and modules; unique factorization domains, principal ideal domains and Euclidean domains and factorization of ideals in each; structure theorems for modules; fields, field extensions. Automorphism groups of fields are studied through the fundamental theorems of Galois theory.

*Prerequisite: *Mathematics 122.

**
Mathematics 124. Number Theory**

Catalog Number: 2398

*
William A. Stein
*

*
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

*
David Kazhdan
*

*
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

*
Shlomo Z. Sternberg
*

*
Half course (spring term). Tu., Th., 8:30–10. EXAM GROUP: 10, 11*

General structures of Lie algebras, Poincaré-Birkhoff-Witt theorem, Campbell Hausdorf formula, Levi decomposition classification of simple Lie algebras and their representations.

**
Mathematics 129. Topics in Number Theory**

Catalog Number: 2345

*
Christophe Cornut
*

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

Number fields, ideal class groups, Dirichlet unit theorem, zeta functions, with specific attention to cyclotomic fields.

*Prerequisite: *Mathematics 122 and 123.

**
Mathematics 131. Topology**

Catalog Number: 2381

*
Barry C. Mazur
*

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

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

*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

*
Alexander Braverman
*

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

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.

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

**
Mathematics 135. Differential Topology**

Catalog Number: 2107

*
Shing-Tung Yau
*

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

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

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

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

Curves and surfaces in 3-space, Gaussian curvature and its intrinsic meaning, Gauss-Bonnet theorem, surfaces of constant curvature.

*Note: *Expected to be given in 2002–03.

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

**
Mathematics 137. Algebraic Geometry**

Catalog Number: 0556

*
Daniel Allcock
*

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

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 introduction to spherical, Euclidean and hyperbolic geometry in two and three dimensions, with an emphasis on the similarities and differences between these flavors of geometry. The most important tool in analyzing these geometries will be a study of their symmetries; we will see how this leads naturally to basic notions in group theory and topology. Topics to be covered include the connection between hyperbolic geometry and special relativity and applications of symmetry groups to physics.

*Prerequisite: *Mathematics 21a,b.

**
[Mathematics 141. Introduction to Mathematical Logic]**

Catalog Number: 0600

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

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

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.

*Note: *Expected to be given in 2002–03.

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

**
Mathematics 144. Model Theory and Algebra**

Catalog Number: 1258

*
Gerald E. Sacks
*

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

An introduction to model theory with applications to theories of fields. First order languages, elementary monomorphisms, and the fundamental existence theorem. Model completeness of algebraically closed fields and elimination quantifiers for real closed fields. The Morley derivative and categoricity in power. Saturated structures. Existence and uniqueness of differential closure.

*Note: *Mathematics 123 suggested, but not required.

**
Mathematics 191. Mathematical Probability**

Catalog Number: 4306

*
David Kazhdan
*

*
Half course (fall term). Tu., Th., 8:30–10. EXAM GROUP: 10, 11*

An introduction to probability theory. Discrete and continuous random variables; univariate and multivariate distributions; conditional probability. Weak and strong laws of large numbers and the central limit theorem. Elements of stochastic processes: the Poisson process, random walks, and Markov chains.

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

**
Mathematics 192r. Algebraic Combinatorics**

Catalog Number: 6612

*
James G. Propp (University of Wisconsin)
*

*
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17*

This course will enable students to be able to conduct original research in low-dimensional combinatorics. Methods taught will include recurrence relations (linear and non-linear), transfer matrices, and generating functions; topics include frieze patterns, number walls and tilings.There will be an emphasis on discovery and the use of computers.

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

Catalog Number: 5446

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

**
Mathematics 212b. Functions of a Real Variable**

Catalog Number: 7294

*
Shlomo Z. Sternberg
*

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

Continuation of Mathematics 212a. Banach and Hilbert spaces. Self adjoint, normal operators and their functional calculus. Spectral theory. Integral and compact operators. Wavelets and other applications.

*Prerequisite: *Mathematics 212a.

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

Catalog Number: 1621

*
Yum Tong Siu
*

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

Fundamentals of complex analysis, and further topics such as elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure, capacity, hyperbolic geometry, quasiconformal maps.

*Prerequisite: *Basic complex analysis, topology of covering spaces, differential forms.

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

Catalog Number: 2641

*
Yum Tong Siu
*

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

Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, uniformization, moduli, several complex variables.

**
Mathematics 216. Topics in Harmonic Analysis and Geometric Measure Theory**

Catalog Number: 1468

*
Tatiana Toro (University of Washington)
*

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

A study of how harmonic analysis, geometric measure theory and potential theory interact together to produce results concerning the regularity of free boundaries. The course will be self contained and therefore it will include a brief introduction to certain aspects of the 3 fields mentioned above.

**
Mathematics 219. The Analysis of the Kortweg-de Vries Equation**

Catalog Number: 0546

*
Gigliola Staffilani (Stanford University)
*

*
Half course (spring term). M., W., 2–3:30. EXAM GROUP: 7, 8*

The Kortweg-de Vries (KdV) equation is a typical dispersive equation. The guiding theme of the course will be the study of the following questions. How rough can an initial profile for the K-dV Cauchy problem be and still be able to claim well-posedness in an interval of time? What happens at time? Does the solution “blow-up” or can it be continued at all times?

**
Mathematics 230ar. Differential Geometry**

Catalog Number: 0372

*
Shing-Tung Yau
*

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

A study of Riemannian manifolds, geodesics and curvature, and relations between curvature and topology. Also, a discussion of connections in principal bundles, spinors and Dirac operators, and the Bochner method.

*Prerequisite: *Math 131 and familiarity with smooth manifolds.

**
Mathematics 230br. Differential Geometry**

Catalog Number: 0504

*
Shing-Tung Yau
*

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

Topics in Riemannian geometry, Kähler geometry, Hodge theory, and Yang-Mills theory.

*Note: *Continuation of Mathematics 230ar.

*Prerequisite: *Differential Topology.

**
Mathematics 250a. Higher Algebra**

Catalog Number: 4384

*
Noam D. Elkies
*

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

Course introduces ubiquitous algebraic structures and their properties and discusses some of their diverse applications. Highlights of the first term: Galois theory; the Brauer theory of central simple algebras; and representation theory of finite groups.

*Prerequisite: *Mathematics 123 or equivalent.

**
Mathematics 250b. Higher Algebra**

Catalog Number: 8464

*
Noam D. Elkies
*

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

Continuation of Mathematics 250a. Main project will be the study of Lie groups, Lie algebras, and their finite-dimensional representations.

**
Mathematics 251x. Chiral Algebras**

Catalog Number: 1035

*
David Kazhdan
*

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

Invariant description of vertex algebras (Beilinson-Drinfeld). Chiral de Rham complex. (BD and Malikov, Schectman and Vaintrob.)

**
Mathematics 252x. Arithmetic of Elliptic Curves**

Catalog Number: 6770

*
Barry C. Mazur
*

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

Selected recent results concerning the arithmetic of elliptic curves and their L-functions, with relevant background material.

*Prerequisite: *Knowledge of basic algebraic number theory and the theory of algebraic curves, as well as some acquaintance with the cohomology of groups.

**
Mathematics 255r. Topics in Number Theory: Heights **

Catalog Number: 0622

*
Matthew Baker
*

*
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17*

An introduction to the theory of heights in arithmetic geometry, with an emphasis on problems related to lower bounds for heights. Discussions of heights of algebraic numbers, including Mahler measure, Lehmer’s problem and Bilu’s equidistribution theorem.

**
Mathematics 256. Theory of Error-Correcting Codes**

Catalog Number: 5333

*
Noam D. Elkies
*

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

Codes and linear codes, Hamming weight and distance, rates of transmission and reliability. Weight enumerators, dual code and the MacWilliams identity, Gleason’s theorems and consequences. Construction and properties of the Reed-Muller, Reed-Solomon, BCH, and Golay codes. Good Goppa codes from curves with many points over finite fields. Connections with sphere packing and other topics as time permits.

**
Mathematics 258x. Analytic Methods for Diophantine Problems**

Catalog Number: 8021

*
Trevor Wooley (University of Michigan)
*

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

An introduction to the Hardy-Littlewood (circle) method, concentrating on its applications to counting integral/rational solutions of diophantine problems.

*Prerequisite: *Elementary Fourier analysis and number theory.

**
Mathematics 259x. Topics in K-Theory and L-functions**

Catalog Number: 6955

*
Fernando Rodriguez-Villegas (University of Texas, Austin)
*

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

An overview of the relations between K-theory, special values of L-functions and polylogarithms. A discussion of Borel’s theorem for number fields and the conjectures of Bloch and Beilinson for elliptic curves, as well as the connection with the Mahler measure of polynomials.

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

Catalog Number: 7004

*
Mihnea Popa
*

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

Introduction to complex algebraic varieties. Hodge theory. Curves, surfaces, moduli problems.

*Prerequisite: *Some familiarity with manifolds, differential forms and singular homology.

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

Catalog Number: 2745

*
Mihnea Popa
*

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

Continuation of Mathematics 260a.

*Prerequisite: * Mathematics 260a.

**
Mathematics 262. Modular Forms**

Catalog Number: 2368

*
Wilfried Schmid
*

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

The analytic theory of modular forms and Maas forms, from both the classical and the representation-theoretic perspective. Topics include L-functions, functional equation, Euler products, converse theorems.

**
Mathematics 264. A Second Course in Algebraic Geometry**

Catalog Number: 7235

*
Joseph D. Harris
*

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

An introduction to some of the basic tools used in algebraic geometry. Topics include: Intersection theory and enumerative geometry; Schubert calculus; Chern and Segre classes; linearization; and parameter spaces.

**
Mathematics 266x. Algebraic D-modules**

Catalog Number: 3590

*
Alexander Braverman
*

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

D-modules on affine space, Bernstein’s inequality. Kashiwara’s theorem, Fourier D-modules on general algebraic varieties, inverse and direct image, holonomic D-modules with regular singularities, introduction to Riemann-Hilbert correspondence.

*Prerequisite: *The second half of the course will assume the knowledge of basic geometry and homological algebra.

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

Catalog Number: 1666

*
Peter B. Kronheimer
*

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

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

*
Peter B. Kronheimer
*

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

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

*Prerequisite: *Mathematics 272a.

**
Mathematics 275. Topics in Conformal Dynamics**

Catalog Number: 5906

*
Curtis T. McMullen
*

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

Advanced topics in complex dynamics, hyperbolic geometry, Kleinian groups and Teichmueller theory.

**
Mathematics 277x. Foliations and the Topology of 3-Manifolds**

Catalog Number: 1294

*
Danny Calegari
*

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

A study of 2-dimensional foliations and laminations, mostly in the context of 3-manifold topology. The aim of the course is to give an exposition of some of these results which are not widely available.

*Prerequisite: *Prior coursework in 3-manifold topology and/or geometrization helpful, but not required.

**
Mathematics 278x. Progress Towards Geometrization**

Catalog Number: 5916

*
Danny Calegari
*

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

A discussion of progress made towards the resolution of the geometrization conjecture in the last decade. Topics may include: resolution of the Seifert fiber space conjecture; homotopy hyperbolic 3 manifolds are hyperbolic; bounds on exceptional surgeries; and the orbifold theorem.

*Prerequisite: *A good basic background in 3-manifold topology. Some familiarity with the Thurston theory of 3-manifolds, such as might be obtained by reading “Thurston’s Notes.”

Catalog Number: 4344

**
*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
*

**
*Mathematics 311. Topics in Representation Theory**

Catalog Number: 0157

*
Stephen DeBacker 4110
*

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

Catalog Number: 2743

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

**
*Mathematics 315. Topics in Number Theory**

Catalog Number: 3335

*
Christophe Cornut 4014 (on leave 2002-03)
*

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

Catalog Number: 7393

*
Barry C. Mazur 1975
*

**
*Mathematics 322. Topics in Representation Theory**

Catalog Number: 2962

*
Alexander Braverman 3630 (on leave spring term)
*

**
*Mathematics 323. Topics in Algebraic Geometry**

Catalog Number: 4659

*
Mihnea Popa 4015
*

**
*Mathematics 325. Topics in Mathematics**

Catalog Number: 5928

*
David Kazhdan 4668 (on leave 2002-03)
*

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

Catalog Number: 0409

*
Yum Tong Siu 7550 (on leave 2002-03)
*

**
*Mathematics 331. Topics in Topology and Geometry**

Catalog Number: 7992

*
Nathan Dunfield 2311 (on leave 2001-02)
*

**
*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 337. Topics in Algebraic Geometry**

Catalog Number: 9000

*
Kalle Karu 2366
*

**
*Mathematics 341. Topics in Arithmetic Algebraic Geometry**

Catalog Number: 9365

*
Matthew Baker 3325
*

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

Catalog Number: 4108

*
Peter B. Kronheimer 1759
*

**
*Mathematics 346. Topics in Deformation Theory**

Catalog Number: 8245

*
Dmitry Tamarkin 2463
*

**
*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 (on leave 2002-03)
*

**
*Mathematics 353. Topics in Lattices and Arithmetic Groups in Algebraic Geometry**

Catalog Number: 0570

*
Daniel Allcock 2186
*

**
*Mathematics 354. Topics in Number Theory**

Catalog Number: 1217

*
William A. Stein 4016
*

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

Catalog Number: 6534

*
Wilfried Schmid 5097 (on leave fall term)
*

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

Catalog Number: 4647

*
Shing-Tung Yau 1734
*

**
*Mathematics 367. Topics in Geometry and Partial Differential Equations**

Catalog Number: 9037

*
Albert Chau 4017
*

**
*Mathematics 374. Topics in Geometric Topology**

Catalog Number: 3971

*
Danny Calegari 3332
*

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

Catalog Number: 2037

*
Joseph D. Harris 2055
*

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

Catalog Number: 6851

*
Noam D. Elkies 2604
*