Daniel Allcock, Benjamin Peirce Assistant Professor of Mathematics

Matthew Baker, Assistant Professor of Mathematics

Ilia A. Binder, Benjamin Peirce Assistant Professor of Mathematics

John Boller, Preceptor in Mathematics

Tom C. Braden, Benjamin Peirce Assistant Professor of Mathematics

Alexander Braverman, Benjamin Peirce Assistant Professor of Mathematics

Danny Calegari, Benjamin Peirce Assistant Professor of Mathematics

Lisa J. Carbone, Benjamin Peirce Assistant Professor of Mathematics

Nathan Dunfield, Benjamin Peirce Assistant Professor of Mathematics, Associate of Adams House

Noam D. Elkies, Professor of Mathematics

Andrew Engelward, Preceptor in Mathematics

Kim Anders Froyshov, Benjamin Peirce Assistant Professor of Mathematics

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

Robin Gottlieb, Senior Preceptor in Mathematics

Tom Graber, Benjamin Peirce Assistant Professor of Mathematics

Joseph D. Harris, Higgins Professor of Mathematics, Higgings Professor of Mathematics

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

Kalle Karu, Benjamin Peirce Assistant Professor of Mathematics

David Kazhdan, Perkins Professor of Mathematics

Sean M. Keel, Visiting Associate Professor of Mathematics, Visiting Scholar in Mathematics

Oliver Knill, Preceptor in Mathematics

Toshiyuki Kobayashi, Visiting Associate Professor of Mathematics

Peter B. Kronheimer, Professor of Mathematics

Yang Liu, Benjamin Peirce Assistant Professor of Mathematics

John F. Mackey, Preceptor in Mathematics

Barry C. Mazur, Gerhard Gade University Professor

Dusa Margaret McDuff, Visiting Professor of Mathematics

Curtis T. McMullen, 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

Eric Sommers, Benjamin Peirce Assistant Professor of Mathematics

Richard P. Stanley, Visiting Professor of Mathematics, Visiting Scholar in 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

Dale Winter, Preceptor in Mathematics

Robert Winters, Preceptor in Mathematics

Shing-Tung Yau, William Casper Graustein Professor of Mathematics

Yuhan Zha, Benjamin Peirce Assistant Professor of Mathematics

The Mathematics Department would like to place students in 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. In addition, some 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 wil lbegin in their mathematics education here with a deeper study of calculus and related topics.

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 21a. For example, Mathematics 19 can be taken either before or after Mathematics 21a,b (or Mathematics 20). Mathematics 19 covers modeling and differential equation topics for students interested in biological and other natural science applications. Mathematics 20 covers selected topics from Mathematics 21a and 21b for students particularly interested in economic 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 Xa, 1a, 1b, 20, and 21a is based on the results of the Harvard Mathematics Placement Test, and/or the Advanced Placement Examinations.

Placement in Mathematics 21b, 23a, 25a, and more advanced courses is based on material not covered in these 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 alot of advice. 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, 112, 121, 141.

The Department does not grant formal degree credit for courses in calculus taken in reverse order 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.

*
Robin Gottlieb, 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: T., Th., 10-11:30; Section IV: T., Th., 11:30-1:00; Section V: T., Th., 1-2:30 and a twice weekly lab session to be arranged. EXAM GROUP: 1*

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.

*Prerequisite: *Mathematics Xa.

**
Mathematics 1a. Introduction to Calculus**

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

*
Yum Tong Siu, Kim Froyshov, Yang Liu, Yuhan Zha (fall term); Tom Graber (spring term) and members of the Department
*

*
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 10; Section II, M., W., F., at 11; Section III, M., W., F., at 12; Section IV, Tu., Th., 10–11:30; Section V, Tu., Th., 11:30–1. Spring: Section I, T., Th., 10-11:30, and a weekly problem session to be arranged. EXAM GROUP: Fall: 3, 4; Spring: 12, 13*

The development of calculus by Newton and Leibniz ranks among the greatest acheivements of the millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how intregral 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 19, 8:00 am, Science Center C.

*Prerequisite: *A solid background in precalculus.

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

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

*
John Boller (fall term), John F. Mackey (spring term), Matthew Baker (spring term), Alexander Braverman (spring term), Peter B. Kronheimer (spring term), John W. Mackey (fall term), and Dmitry Tamarkin (fall term)
*

*
Half course (fall term; repeated spring term). Section I, M., W., F., at 10; Section II, M., W., F., at 11; Section III, M., W., F., at 12; Section IV, Tu., Th., 10–11:30; Section V, Tu., Th., 11:30–1, and a weekly problem session to be arranged. Required exams: Hours to be arranged. EXAM GROUP: 1*

Galileo said 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 differrential 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 interpretation, infinite series, power series, and Taylor series.

*Note: *Required first meeting for fall: Monday, September 18, 8:00 am, Science Center B. Required first meeting in spring: Wednesday, January 31, 8:00 am, Science Center C.

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

**
Mathematics 19. Mathematical Modeling**

Catalog Number: 1256

*
John F. Mackey
*

*
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

*
Eric Sommers
*

*
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, Daniel Allcock, Oliver Knill, Dale Winter, Robert Winters, Danny Calegari (fall term); Robert Winters, Dale Winter, Yang Liu and Oliver Knill(spring term) and members of the Department.
*

*
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9; 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. Spring: Section I, M., W., F., at 10; Section II, M., W., F., at 11; Section III, M., W., F., at 12; Section IV, Tu., Th., 10–11:30; Section V, 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: Tuesday, September 19, 8:00 am, Science Center B. Required first meeting in spring: Wednesday, January 31, 8:00 am, Science Center B. 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 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.

*
Robert Winters and Dmitry Tamarkin (fall term); Richard Taylor, Daniel Allcock (spring term), and members of the department.
*

*
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 session to be arranged. EXAM GROUP: 1*

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: Monday, September 18, 8:00 am, Science Center A. Required first meeting in spring: Thursday, February 1, 8:00 am, Science Center C. May not be taken for credit by students who have passed Applied Mathematics 21b.

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

Catalog Number: 2486

*
David Kazhdan
*

*
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

*
David Kazhdan
*

*
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

*
Matthew Baker
*

*
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.

*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

*
Matthew Baker
*

*
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). M., W., F., at 2. EXAM GROUP: 7*

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., 2:30–4. EXAM GROUP: 16, 17*

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.*

Small group tutorials, ordinarily limited to Mathematics concentrators. Supervised individual projects and class presentations required. Topics for 2000-01: (1) Fourier Series and Applications (fall) Prerequisites: Math 25, 55, or 101, Math 115 would be helpful, but not necessary. (2) Computational Algebraic Geometry (fall) Prerequisites: Math 25, 55, or 101 and an interest in computational mathematics. (3) The Symmetric Group and its Representations (spring) Prerequisites: basics group theory as in Math 122, and linear algebra as in Math 121, previous exposure to representation theory would be helpful, but not necessary. A second tutorial topic will be announced later for the spring semester.

*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 112. Real Analysis**

Catalog Number: 1123

*
Yang Liu
*

*
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

*
Yum Tong Siu
*

*
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

*
Ilia A. Binder
*

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

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.

*Prerequisite: *Mathematics 21a,b.

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

Catalog Number: 7326

*
Peter B. Kronheimer
*

*
Half course (fall term). Hours to be arranged. EXAM GROUP: 4*

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

*Note: *Expected to be given in 2001–02.

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

**
Mathematics 121. Linear Algebra and Applications**

Catalog Number: 7009

*
Tom Graber
*

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

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

*
Barry C. Mazur
*

*
Half course (fall term). M., W., F., at 9. EXAM GROUP: 2*

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

*
Barry C. Mazur
*

*
Half course (spring term). M., W., F., at 9. EXAM GROUP: 2*

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

*
Daniel Allcock
*

*
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

*
Richard L. Taylor
*

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

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

**
Mathematics 128. Lie Algebras**

Catalog Number: 6519

*
Lisa J. Carbone
*

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

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

**
Mathematics 131. Topology**

Catalog Number: 2381

*
Curtis T. McMullen
*

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

Basic notions of point set topology such as continuity, compactness, separation theorems, 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

*
Tom C. Braden
*

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

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

*
Kim Anders Froyshov
*

*
Half course (fall 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

*
Shing-Tung Yau
*

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

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

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

**
Mathematics 137. Algebraic Geometry**

Catalog Number: 0556

*
Yuhan Zha
*

*
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

*
Danny Calegari
*

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

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 might include classical tessellations, the Gauss-Bonnet theorum, scissors congruence, orbifolds, and fibered geometries.

*Prerequisite: *Mathematics 21a,b.

**
Mathematics 139. Classical Geometry and Low-Dimensional Topology**

Catalog Number: 6979

*
Danny Calegari
*

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

A continuation of the study of spherical, Euclidean and especially hyperbolic geometry in two and three dimensions begun in Mathematics 138. The emphasis will be on the relationship with topology, and the existence of metrics of constance curvature on a vast class of two and three dimensional manifolds. We will concentrate mainly on a detailed study of examples, and we will try to be as explicit and as elementary as possible. Topics to be covered might include: uniformization for surfaces, shapes and volumes of hyperbolic polyhedra, circle packing and Andreev’s theorem, and hyperbolic structures on knot complements.

*Prerequisite: *Mathematics 21ab, 113, 138 would be vary useful, but not essential.

**
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 142. Recursion Theory**

Catalog Number: 6531

*
Gerald E. Sacks
*

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

Normal forms. Universal Turing machines. Recursively enumerable sets. Turing degrees. Post’s problems. Finite injury arguments. Splitting and density. Infinite injury arguments. Minimal pairs. Minimal degrees and P not equal to NP.

*Prerequisite: *Mathematics 141, or Computer Science 121 or permission of instructor.

**
[Mathematics 143 (formerly Mathematics 143r). Set Theory]**

Catalog Number: 6005

*
Gerald E. Sacks
*

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

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.

*Note: *Expected to be given in 2001–02.

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

**
Mathematics 191. Mathematical Probability**

Catalog Number: 4306

*
Joseph D. Harris
*

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

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 192. Algebraic Combinatorics**

Catalog Number: 5806

*
Richard P. Stanley (Massachusetts Institute of Technology)
*

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

A basic introduction to enumerative and algebraic combinatorics, focusing on applications of linear algebra, group theory, and ring theory to combinatorics. Topics: generating functions, transfer matrices ,the Matrix-Tree Theorem, and Young tableaux.

*Note: *No prior knowledge of combinatorics is assumed.

*Prerequisite: *Math 122 or equivalent.

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

*
Curtis T. McMullen
*

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

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

*
Curtis T. McMullen
*

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

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

**
Mathematics 214. Harmonic Measure**

Catalog Number: 3329

*
Ilia A. Binder
*

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

Possible topics will include: Dimension properties of harmonic measure in the plane: relations with rational dynamics, thermodynanmical formalism: connections with probability theory, dimension properties of Browman motion.

**
Mathematics 215. Topics in Several Complex Variables**

Catalog Number: 6772

*
Yum Tong Siu
*

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

Kaehler geometry, applications of L2 estimates of d-bar to problems in algebraic geometry, relationship between diophantine approximation and the higher dimensional Nevanlinna theory of value distribution.

**
Mathematics 218. Restriction of Representations to Reductive Subgroups**

Catalog Number: 7861

*
Toshiyuki Kobayashi (University of Tokyo)
*

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

A study of the restriction of infinite dimensional representations of real reductive Lie groups to certain reductive Lie groups to certain reductive subgroups. Techniques for the geometric construction of representations, branching laws with emphasis on discrete decomposable cases such as K-type formulas, and multiplicities will be studied in detail.

**
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 245. Proof Theory**

Catalog Number: 0756

*
Warren Goldfarb
*

*
Half course (fall term). Tu., 2–4 and an hour to be arranged. EXAM GROUP: 16, 17*

Herbrand’s and Gentzen’s analysis of logical inference; Hilbert’s program for consistency proofs by metamathematical treatment of proof structures; consistency of number theory and subsystems of analysis; ordinal-theoretic measures of the strength of axiomatic theories; the logic of provability.

**
Mathematics 250a. Higher Algebra**

Catalog Number: 4384

*
Alexander Braverman
*

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

Rudiments of category theory, modules and rings, representation theory of finite groups, and some homological algebra.

*Prerequisite: *Mathematics 123 or equivalent.

**
Mathematics 250b. Higher Algebra**

Catalog Number: 8464

*
Alexander Braverman
*

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

Commutative algebra, infinite Galois theory, and fields and valuations.

**
Mathematics 251. Kac-Moody Algebras**

Catalog Number: 1993

*
Lisa J. Carbone
*

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

An introduction to Kac-Moody algebras, a particular class of infinite dimensional Lie algebras, their representations, Kac-Moody groups, and the buildings of Kac-Moody groups over finite fields.

*Prerequisite: *Some familiarity with finite dimensional Lie algebras and their representations, as in Mathematics 128.

**
Mathematics 253. Arithmetic Curves and Surfaces**

Catalog Number: 7783

*
Yuhan Zha
*

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

Arithmetic Riemann-Roch theorems for arithmetic curves and surfaces. Arithmetic adjuction formula, Noether formula, Hodge index theorem. Small sections of ample line bundles and other applications.

*Prerequisite: *An understanding of the first 3 chapters of Hartshorne’s book in algebraic geometry or its equivalent.

**
Mathematics 257. Topics in Deformation Theory**

Catalog Number: 8481

*
Dmitry Tamarkin
*

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

A discussion of how to associate a Lie algebra with a given deformation problem. We will consider several examples (complex structures, flat connections, Goldman-Millson theory, deformation quantization, etc.), and provide a proof of M. Kontsevich formality theorem.

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

Catalog Number: 7004

*
Peter B. Kronheimer
*

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

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

*
Peter B. Kronheimer
*

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

Continuation of Mathematics 260a.

*Prerequisite: * Mathematics 260a.

**
Mathematics 266r. An Introduction to the Theory of Representations of p-adic Groups**

Catalog Number: 4183

*
David Kazhdan
*

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

The definition of admissible representations. A study of the basic properties of the category of admissible representations (including the description of the Bernstein’s center and the Paley-Wiener theorem)

**
Mathematics 267. The Minimal Model Program and Moduli Spaces of Curves**

Catalog Number: 6966

*
Sean M. Keel (University of Texas, Austin)
*

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

A description of the geometry of moduli spaces of curves, both local (the deformation theory of pointed curves) and global (the Picard groups of moduli spaces; their ample and effective cones). An overview of the minimal model program and how it is carried out in general. Finally, we will consider the application of the minimal model program to moduli spaces of curves, and what it says about their geometry.

**
Mathematics 269. Topics in Lie Theory**

Catalog Number: 0427

*
Eric Sommers
*

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

An overview of some topics in the represenation theory of algebraic groups over different fields. Will begin with a short treatment of algebraic groups which will be followed by a survey of some results in the representation theory of reductive groups over finite fields. Other topics: the Borel-Weil-Bott theorem, the Springer correspondence, and properties of nilpotent orbits.

*Note: *A familiarity with Lie algebras on the level of Mathematics 128 would be helpful. Also some previous exposure to algebraic geometry.

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

Catalog Number: 1666

*
Tom C. Braden
*

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

Homotopy theory. Covering spaces and fibrations. Simplicial and CW complexes. Manifolds. Homology theories. Universal coefficients and Künneth formulas. Hurewicz theorem. Applications to fixed point theory and other topics.

*Prerequisite: *Mathematics 131 or permission of instructor.

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

Catalog Number: 6502

*
Tom C. Braden
*

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

Cohomology theories. Duality theorems. Fibre bundles. Spectral sequences. Eilenberg-MacLane spaces.

*Prerequisite: *Mathematics 272a.

**
Mathematics 273. Topics in Symplectic Topology**

Catalog Number: 8608

*
Dusa Margaret McDuff ((SUNY))
*

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

A discussion of J-holomorphic curves and applications. A study of some of the technical results needed to set up the theory, and then a discussion of various applications, such as Polterovich’s proof that the Hofer diameter of the group of symplectomorphisms of S^2 is infinite and the Lalonde-McDuff result that the homotopy groups of HAM (M, \omega)-- the group of Hamiltonian symplectomorphisms of (M, \omega)-- act trivially on the rational homology of M.

*Note: *The course is intended for students without much knowledge of symplectic geometry.

**
Mathematics 277. Topology and Geometry of 3-Manifolds**

Catalog Number: 5131

*
Nathan Dunfield
*

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

A study of the foundations of the theory of 3-manifolds, as well as selected advanced topics.

*Note: *No prior experience with 3-manifolds will be assumed.

**
Mathematics 278. Floer Homology**

Catalog Number: 5093

*
Kim Anders Froyshov
*

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

Floer homology groups in Yang-Mills and Seiberg-Witten theory and numerical invariants derived from these.

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 312. Topics in Geometry and Representation Theory**

Catalog Number: 5174

*
Tom C. Braden 3586
*

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

Catalog Number: 2743

*
Shlomo Z. Sternberg 1965
*

**
*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 322. Topics in Representation Theory**

Catalog Number: 2962

*
Alexander Braverman 3630
*

**
*Mathematics 325. Topics in Mathematics**

Catalog Number: 5928

*
David Kazhdan 4668
*

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

Catalog Number: 0409

*
Yum Tong Siu 7550
*

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

Catalog Number: 7992

*
Nathan Dunfield 2311
*

**
*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 (on leave spring term)
*

**
*Mathematics 337. Topics in Algebraic Geometry**

Catalog Number: 9000

*
Kalle Karu 2366 (on leave 2000-01)
*

**
*Mathematics 338. Topics in Algebra**

Catalog Number: 5996

*
Lisa J. Carbone 3587
*

**
*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
*

*
Half course (fall term; repeated spring term). Tu., Th., 11–12:30.*

**
*Mathematics 347. Topics in Complex Analysis**

Catalog Number: 7343

*
Ilia A. Binder 3585
*

**
*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 352. Topics in Complex Manifolds**

Catalog Number: 7458

*
Yang Liu 2158
*

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

Catalog Number: 0570

*
Daniel Allcock 2186
*

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

Catalog Number: 6534

*
Wilfried Schmid 5097
*

**
*Mathematics 358. Topics in Gauge Theory**

Catalog Number: 8246

*
Kim Anders Froyshov 1104
*

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

Catalog Number: 4647

*
Shing-Tung Yau 1734
*

**
*Mathematics 372. Topics in Arakelov Geometry**

Catalog Number: 9471

*
Yuhan Zha 2282
*

**
*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 385. Topics in Representation Theory**

Catalog Number: 7037

*
Eric Sommers 2247
*

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

Catalog Number: 6851

*
Noam D. Elkies 2604 (on leave 2000-01)
*