Mathematics

Faculty of the Department of Mathematics

Benedict H. Gross, George Vasmer Leverett Professor of Mathematics (Chair)
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 (on leave 2000-01)
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 (on leave fall term)
Tom Graber, Benjamin Peirce Assistant Professor of Mathematics (on leave 2001-02)
Joseph D. Harris, Higgins Professor of Mathematics, Higgings Professor of Mathematics (Director of Graduate Studies)
Arthur M. Jaffe, Landon T. Clay Professor of Mathematics and Theoretical Science
Kalle Karu, Benjamin Peirce Assistant Professor of Mathematics (on leave 2000-01)
David Kazhdan, Perkins Professor of Mathematics
Sean M. Keel, Visiting Associate Professor of Mathematics, Visiting Scholar in Mathematics (University of Texas, Austin) (fall term only)
Oliver Knill, Preceptor in Mathematics
Toshiyuki Kobayashi, Visiting Associate Professor of Mathematics (University of Tokyo) (spring term only)
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 ((SUNY))
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 (Massachusetts Institute of Technology) (fall term only)
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 (Director of Undergraduate Studies) (on leave spring term)
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

Other Faculty Offering Instruction in the Department of Mathematics

Warren Goldfarb, Walter Beverly Pearson Professor of Modern Mathematics and Mathematical Logic

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 prior approval. Our policy is that a student who takes and passes any calculus course is not normally permitted to then take a more elementary course for credit. A student who has passed Mathematics 21a, for example, will normally not be allowed to take Mathematics 1a, or 1b for credit. The Department is prepared to make exception for sufficient academic reasons; in each case, however, a student must obtain written permission from the Head Tutor in advance.

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.

Primarily for Undergraduates

Mathematics Xa. Introduction to Functions and Calculus I
Catalog Number: 1981 Enrollment: Normally limited to 15 students per section.
Andrew Engelward, Dale Winter, and members of the Department
Half course (fall term). Section meeting times: Section I: M., W., F., at 10; Section II: M., W., F., at 11; and a twice weekly lab session to be arranged. EXAM GROUP: 1
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.
Note: Required first meeting: Monday, September 18, 8:00 am, Science Center E. The sequence Xa, Xb gives solid preparation for Mathematics 1b.

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.

Cross-listed Courses

For Undergraduates and Graduates

See also Applied Mathematics and Statistics.
Mathematics 101. Sets, Groups and Topology
Catalog Number: 8066
Ilia A. Binder (fall term) and Lisa J. Carbone (spring term)
Half course (fall term; repeated spring term). Fall: Tu., Th., 10–11:30; Spring: Tu., Th., 11:30–1. EXAM GROUP: Fall: 12, 13; Spring: 13, 14
An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.
Note: Acquaintance with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 25a,b or 55a,b should not take this course for credit.
Prerequisite: An interest in mathematical reasoning.

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.

Cross-listed Courses

Applied Mathematics 105a. Complex and Fourier Analysis
Applied Mathematics 105b. Ordinary and Partial Differential Equations
Applied Mathematics 107. Graph Theory and Combinatorics
[Philosophy 144. Logic and Philosophy]

Primarily for Graduates

Mathematics 212a. Functions of a Real Variable
Catalog Number: 5446
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 8:30–10. EXAM GROUP: 10, 11
A review of measure and integration. Banach spaces, L^p spaces, and the Riesz representation theorem.
Prerequisite: Experience with courses involving rigorous proofs: e.g., Mathematics 25a,b, 112, 122.

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.

Graduate Courses of Reading and Research

*Mathematics 301. Theory and Practice of Teaching in the Mathematical Sciences
Catalog Number: 4344
Daniel L. Goroff 7683 and Wilfried Schmid 5097
Half course (spring term). Hours to be assigned.

*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)