Mathematics

Faculty of the Department of Mathematics

Benedict H. Gross, George Vasmer Leverett Professor of Mathematics (Chair)
Daniel Allcock, Benjamin Peirce Lecturer in Mathematics; Assistant Professor of Mathematics
Ilia A. Binder, Benjamin Peirce Lecturer of Mathematics; Assistant Professor of Mathematics
John David Boller, Preceptor in Mathematics
Tom C. Braden, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics (on leave 1999-00)
Lisa J. Carbone, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics
Brian Conrad, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics
Igor Dolgachev, Visiting Professor of Mathematics (University of Michigan) (fall term only)
Nathan Dunfield, Benjamin Pierce Lecturer in Mathematics, Assistant Professor of Mathematics
Noam D. Elkies, Professor of Mathematics
Andrew James Engelward, Preceptor in Mathematics
Kim Froyshov, Benjamin Peirce Lecturer of Mathematics; Assistant Professor of Mathematics
Daniel L. Goroff, Professor of the Practice of Mathematics
Robin Gottlieb, Senior Preceptor in Mathematics
Tom Graber, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics
Joseph D. Harris, Professor of Mathematics (Director of Graduate Study)
Arthur M. Jaffe, Landon T. Clay Professor of Mathematics and Theoretical Science
Kalle Karu, Benjamin Pierce Lecturer on Mathematics; Assistant Professor of Mathematics
David Kazhdan, Perkins Professor of Mathematics
Peter B. Kronheimer, Professor of Mathematics
Tamara R. Lefcourt, Preceptor in Mathematics
Yang Liu, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics
Barry C. Mazur, Gerhard Gade University Professor
Curtis T. McMullen, Professor of Mathematics
Grisha Mikhalkin, Benjamin Pierce Lecturer on Mathematics; Assistant Professor of Mathematics
Gerald E. Sacks, Professor of Mathematical Logic
Wilfried Schmid, Dwight Parker Robinson Professor of Mathematics
Richard M. Schoen, Visiting Professor of Mathematics (Stanford University) (fall term only)
Yum Tong Siu, William Elwood Byerly Professor of Mathematics (on leave 1999-00)
Eric Sommers, Benjamin Peirce Lecturer of Mathematics; Assistant Professor of Mathematics, Associate of Adams House
Shlomo Z. Sternberg, George Putnam Professor of Pure and Applied Mathematics
Dmitry Tamarkin, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics
Clifford Taubes, William Petschek Professor of Mathematics (Director of Undergraduate Studies)
Richard L. Taylor, Professor of Mathematics
Robert Winters, Preceptor in Mathematics
Shing-Tung Yau, William Caspar Graustein Professor of Mathematics, William Casper Graustein Professor of Mathematics
Yuhan Zha, Benjamin Peirce Lecturer on Mathematics; Assistant Professor of Mathematics

Other Faculty Offering Instruction in the Department of Mathematics

Cumrun Vafa, Professor of Physics

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.

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. 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 an honors version of Mathematics 21 which treats multivariable calculus and linear algebra in a rigorous, proof oriented way. Mathematics 25 and 55 are honors 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 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 mathematics background.

Placement in Mathematics Xa, 1a, 1b, 20, and 21a is based on the results of the Harvard Mathematics Placement Test 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 all the advice they can get. The Department has prepared a pamphlet with a detailed description of all its 100-level courses and their relationship to each other. This pamphlet also 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: Limited to 15 students per section.
Robin Gottlieb, Andrew James Engelward, Tamara R. Lefcourt 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; Section III: M., W., F., at 12, 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: Wednesday, September 22, 8:30 am, Science Center A. The sequence Xa, Xb gives solid preparation for Mathematics 1b.

Mathematics Xb. Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Limited to 15 students per section.
Robin Gottlieb, Andrew James Engelward, Tamara R. Lefcourt and members of the Department
Half course (spring term). Section I: M., W., F., at 10; Section II: M. W., F., at 11; Section III: M., W., F., at 12, 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
Robert Winters, Lisa J. Carbone, and Grisha Mikhalkin (fall term); Robert Winters (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 1, M., W., F., at 11, and a weekly problem session to be arranged. EXAM GROUP: 3, 4
Differential calculus of algebraic, logarithmic, and trigonometric functions with applications; an introduction to integration.
Note: Required first meeting in fall: Tuesday, September 21, 8:30 am, Science Center C.
Prerequisite: A solid background in precalculus.

Mathematics 1b. Calculus, Series and Differential Equations
Catalog Number: 1804
Andrew James Engelward (fall term), Curtis T. McMullen (fall term), Tamara R. Lefcourt (fall term), Yuhan Zha (fall term), Tamara R. Lefcourt (spring term) and members of the Department.
Half course (fall term; repeated spring term). Section 1, 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
Integration and differential equations with applications. Approximations by polynomials and series.
Note: Required first meeting for fall: Thursday, September 23, 8:30 am, Science Center C. Required first meeting in spring: Tuesday, February 1, 8:30 am, Science Center D.
Prerequisite: Mathematics 1a, or Xa and Xb, or equivalent.

Mathematics 19. Mathematical Modeling
Catalog Number: 1256
Kalle Karu
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.
Prerequisite: Mathematics 1b, or permission of instructor.

Mathematics 20. Introduction to Linear Algebra and Multivariable Calculus
Catalog Number: 0906
Nathan Dunfield
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: Limited to 30 students per section.
Clifford Taubes (fall term), John David Boller (fall term), Yang Liu (fall term), Eric Sommers (fall term), Robert Winters (fall term), and Robert Winters (spring term), Daniel Allcock (spring term), Yang Liu (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
Functions of many variables; vectors and parametric curves. Partial differentiation, directional derivatives, and the gradient. Maxima, minima, and saddle-points. Multiple integration in cartesian and polar coordinates. Vector fields, line and surface integrals. The divergence and curl. Green’s, Stokes’ and Divergence Theorems.
Note: Required first meeting in fall: Tuesday, September 21, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, February 2, 8:00 am, Science Center C. A special section for students taking physics will be offered in both semesters. May not be taken for credit by students who have passed Applied Mathematics 21a.
Prerequisite: Mathematics 1b or equivalent.

Mathematics 21b. Linear Algebra and Differential Equations
Catalog Number: 1771
Noam D. Elkies and John David Boller (fall term); John David Boller and Dmitry Tamarkin (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 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; and a weekly problem session to be arranged. EXAM GROUP: 1
An introduction to linear algebra, including linear transformations and determinants, eigenvalues and eigenvectors. Ordinary differential equations and systems and their solution; applications.
Note: Required first meeting in fall: Thursday, September 23, 8:30 am, Science Center D. Required first meeting in spring: Thursday, February 3, 8:30 am, Science Center D. May not be taken for credit by students who have passed Applied Mathematics 21b.
Prerequisite: Mathematics 21a or equivalent experience with vectors.

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
Kalle Karu
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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
Kalle Karu
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
A continuation of Mathematics 25a. More advanced topics in differential and integral calculus: calculus of variations, Fourier series, and an introduction to the differential geometry of curves and surfaces in 3-space.
Prerequisite: Mathematics 25a or permission of instructor.

*Mathematics 55a. Honors Advanced Calculus and Linear Algebra
Catalog Number: 4068
Noam D. Elkies
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
A rigorous treatment of point-set and metric topology, linear and bilinear algebra, and differential and integral calculus.
Note: Intended for students with significant experience with abstract mathematics. Instructor’s permission required.

Mathematics 55b. Honors Advanced Multivariable Calculus and Linear Algebra
Catalog Number: 3312
Noam D. Elkies
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Continuation of Mathematics 55a. Calculus of functions in n variables. Additional topics may include normed linear spaces, differential equations, and Fourier analysis.
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 1999-00: (1) Logic and the Foundations of Mathematics (fall) Prerequisites: Math 25, 55, or 101. (2) Analytic Number Theory (fall) Prerequisites: Math 113 or 115. (3) Geometry and Gauge Theory (spring) Prerequisites: Point Set Topology as in Math 112 or 131, and basic Group Theory as in Math 122.
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
Daniel L. Goroff (fall term) and Curtis T. McMullen (spring term)
Half course (fall term; repeated spring term). Fall: Tu., Th., 10–11:30; Spring: M., W., F., at 10. EXAM GROUP: Fall: 12, 13; Spring: 3
An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, low-dimensional topology and the classification of knots.
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
Daniel Allcock
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral, sets of measure zero and conditions for integrability.
Prerequisite: Mathematics 21a,b, 22a,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
Ilia A. Binder
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 22a,b, 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
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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, 22a,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., 10–11:30. EXAM GROUP: 12, 13
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.
Prerequisite: Mathematics 22a,b, 23a,b, 25a,b, 55a,b, or 112.

Mathematics 119. Partial Differential Equations and Applications
Catalog Number: 7326
Peter B. Kronheimer
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
Partial differential equations with constant coefficients, hyperbolic elliptic, and parabolic equations, Fourier analysis, Green’s function.
Prerequisite: Familiarity with functions of a complex variable.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Yang Liu
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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 22a,b, 23a,b, 25a,b, or 55a,b.

Mathematics 122. Abstract Algebra I: Theory of Groups and Vector Spaces
Catalog Number: 7855
Eric Sommers
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
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
Eric Sommers
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
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
Nathan Dunfield
Half course (spring 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
Peter B. Kronheimer
Half course (spring 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
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 8:30–10. EXAM GROUP: 10, 11
General structures of Lie algebras, Poincaré-Birkhoff-Witt theorem, Campbell Hausdorf formula, Levi decomposition classification of simple Lie algebras and their representations.

Mathematics 129. Topics in Number Theory
Catalog Number: 2345
Barry C. Mazur
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Number fields, ideal class groups, Dirichlet unit theorem, zeta functions, with specific attention to cyclotomic fields.
Prerequisite: Mathematics 122 and 123.

Mathematics 131. Topology
Catalog Number: 2381
Lisa J. Carbone
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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 132x. Riemannian and Lorentzian Geometry
Catalog Number: 7149
Yuhan Zha
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Levi-Civita connection, curvature, Jacobi fields with applications to general relativity and cosmology.
Prerequisite: A good grounding in multivariable calculus, such as Mathematics 22, 23, 25, or 55.

Mathematics 134. Calculus on Manifolds
Catalog Number: 7150
Tom Graber
Half course (fall term). Hours to be arranged.
Generalization of multivariable calculus to the setting of manifolds in real n-space, as used in the study of global analysis and geometry. Differentiable mappings of linear spaces, the inverse and implicit function theorems, differential forms, integration on manifolds, the general version of Stokes’s theorem, integral geometry, applications.
Prerequisite: Mathematics 21a,b and familiarity with proofs as in Mathematics 101, 112, 121, or the equivalent.

[Mathematics 135. Differential Topology]
Catalog Number: 2107
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Half course (spring term). Hours to be arranged.
Smooth manifolds, intersection theory, vector fields, Hopf degree theorem, Euler characteristic, De Rham theory.
Note: Expected to be given in 2000–01.
Prerequisite: Mathematics 22a,b, 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
Brian Conrad
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
Grisha Mikhalkin
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Euclidean and non-Euclidean geometries in two and three dimensions.
Prerequisite: Mathematics 21a,b.

[Mathematics 141. Introduction to Mathematical Logic]
Catalog Number: 0600
Gerald E. Sacks
Half course (fall term). Hours to be arranged.
An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.
Note: Expected to be given in 2000–01. Expected to be given in 2000?01.
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). Hours to be arranged.
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.
Note: Expected to be given in 2000–01.
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). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Axioms of set theory. Gödel’s constructible universe. Consistency of the axiom of choice and of the generalized continuum hypothesis. Cohen’s forcing method. Independence of the AC and GCH.
Prerequisite: Any mathematics couse at the level of 21a or higher, or permission of instructor.

Mathematics 191. Mathematical Probability
Catalog Number: 4306
Clifford Taubes
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Yuhan Zha
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Generating functions and enumerative combinatorics. Combinatorial algorithms. Representations of the symmetric groups. Symmetric functions and Schur functions.

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 206. Lie Groups and Lie Algebras
Catalog Number: 8330
Wilfried Schmid
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Review of the basic results on Lie groups and Lie algebras, structure of compact Lie groups, finite dimensional representations, Borel-Weil-Bott theorem.

Mathematics 212a. Functions of a Real Variable
Catalog Number: 5446
Kim Froyshov
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
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
Kim Froyshov
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
Continuation of Mathematics 212a. Banach and Hilbert spaces. Self adjoint, normal operators and their functional calculus. Spectral theory. Integral and compact operators.
Prerequisite: Mathematics 212a.

Mathematics 213a. Functions of One Complex Variable
Catalog Number: 1621
Ilia A. Binder
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Review of basic complex analysis and further topics, including series and product developments, elliptic functions, Rieman surfaces and the uniformization theorem.

Mathematics 213b. Further Topics in Classical Complex Analysis
Catalog Number: 2641
Ilia A. Binder
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Topics in conformal geometry and theory of conformal mappings. Introduction to functions of several complex variables and complex manifolds.

Mathematics 217. Representations of Reductive Lie Groups
Catalog Number: 2006
Wilfried Schmid
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Harish Chandra modules, Characters, the discrete series, classification of irreducible representations, Plancherl theorem, arithmetic subgroups.

Mathematics 230ar. Differential Geometry
Catalog Number: 0372
Kim Froyshov
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 11. EXAM GROUP: 4
Topics in Riemannian geometry, Kähler geometry, Hodge theory, and Yang-Mills theory.
Note: Continuation of Mathematics 230ar.
Prerequisite: Differential Topology.

Mathematics 250a. Higher Algebra
Catalog Number: 4384
Dmitry Tamarkin
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
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
Dmitry Tamarkin
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Commutative algebra, infinite Galois theory, and fields and valuations.

Mathematics 254. Arithmetic of Elliptic Modular Curves
Catalog Number: 8391
Barry C. Mazur
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
A focus on an exposition of Kolyvagin’s technique of Euler systems and some of its arithmetic applications.
Prerequisite: Some acquaintance with Class Field Theory.

Mathematics 255a. Automorphic Representations and Galois Representations
Catalog Number: 4723
Richard L. Taylor
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
A study of the recent work on the local Langlands conjecture for GL(n) and on the 1-adic cohomology of certain simple Shimura varieties. An introduction to various background topics such as: class field theory, formal groups, Drinfeld’s theory of 1 dimensional formal groups, Shimura varieties, classification of abelian varieties over finite fields, automorphic forms, the trace formula, stabilization.

Mathematics 255b. Automorphic Representations and Galois Representations
Catalog Number: 1382
Richard L. Taylor
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Continuation of Mathematics 255a.

Mathematics 258. Deligne’s Work on the Ramanujan Conjecture
Catalog Number: 7655
Brian Conrad
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
An explanation of Deligne’s construction of 2-dimensional 1-adic representations associated to classical cuspidal modular forms, and his use of this to reduce the Ramanujan-Petersson Conjecture to the Weil Conjectures. The techniques involve a massive amount of etale cohomology and the arithmetic theory of modular curves, so we will try to carefully explain (if not prove) what we need from these areas.
Prerequisite: A solid understanding of schemes and classical theory of modular forms.

Mathematics 260a. Introduction to Algebraic Geometry
Catalog Number: 7004
Andreas Gathmann
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
Fundamentals of scheme theory, cohomology of quasi-coherent and coherent sheaves, algebraic curves, moduli schemes.
Prerequisite: Basic commutative algebra. Mathematics 137 desirable.

Mathematics 260b. Introduction to Algebraic Geometry
Catalog Number: 2745
Joseph D. Harris
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Continuation of Mathematics 260a.
Prerequisite: Mathematics 260a.

[Mathematics 261y. Geometry of Semisimple Groups and Their Representations]
Catalog Number: 2809
David Kazhdan
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
A study of finite-dimensional representations semisimple groups and special base in such representations. A study of the relations between this special base and the geometry of the corresponding semisimple group.
Note: Expected to be given in 2000–01.
Prerequisite: A basic knowledge of the structure theory of semisimple Lie algebras.

Mathematics 262y. Introduction to Invariant Theory
Catalog Number: 4254
Igor Dolgachev (University of Michigan)
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
A discussion of both the algebraic invariant theory that studies polynomial functions which are invariant under a group of linear transformations of the variables and the geometric invariant theory that studies the problem of construction of the orbit space of an algebraic variety under acted on by an algebraic linear group. If time permits, a discussion of some applications of the theory of construction of various moduli spaces studied in algebraic geometry.
Prerequisite: A basic knowledge of algebraic geometry on the level of Mathematics 260a.

Mathematics 263r. Arakelov Geometry for Novices
Catalog Number: 8534
Yuhan Zha
Half course (fall term). M., W., F., at 10.
An introduction to the basics of Arakelov geometry through a thorough study of very simple examples along with some of the basic, fundamental historical examples which motivated the development.
Note: No prior knowledge of the subject will be assumed. However, knowledge of some basic algebraic geometry (at the level of Mathematics 137) will be helpful.

Mathematics 265. The Langlands Lifting
Catalog Number: 9242
David Kazhdan
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
Let G be a reductive group over a local field F, We denote by Irr (G(F)) the set or irreducible complex representations of the group G(F). Let G^ be the Langlands dual group. By the definition G^ is a reductive complex group. Langlands conjectured that to any algebraic n-dimensional representation p of G one can associate the map from Irr (G(F)) to Irr (GL(n,F)). Moreover he conjectured the existence of analogous global lifting for the set of automorphic representations. In this course, we will discuss the possibilities for writing explicit formulas for the lifting.

Mathematics 267x. Topics in Differential Geometry
Catalog Number: 2140
Richard M. Schoen (Stanford University)
Half course (fall term). Tu., Th., 8:30–10. EXAM GROUP: 10, 11
A focus on volume minimization for submanifolds. We will develop the set-up for solving the Plateau problem, and explain how this can be extended to the setting in which the competing submanifolds are langrangian submanifolds of a symplectic of Kaehler manifold. Topics will include rectifiability, first and second variation of volume, monotonicity, and some regularity theory. Also some discussion of special lagrangian submanifolds.

Mathematics 268r. Topics in Differential Geometry
Catalog Number: 7379
Shing-Tung Yau
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Subjects on minimal surfaces and Kähler Einstein metrics. A discussion of general techniques on quasilinear and fully nonlinear elliptic equations.

Mathematics 269x. Geometry of Algebraic Curves
Catalog Number: 3247
Joseph D. Harris
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
Linear systems and maps to projective space; automorphisms, Weierstrass points, inflectionary points; geometry of the varieties parametrizing special linear series on C; theta-characteristics; enumerative geometry of the symmetric products of a curve and of its Jacobian, and applications. Local deformation theory of smooth and singular curves; limits of line bundles and linear systems on families of curves. Various parameter spaces of curves in projective space (the Hurwitz scheme, the Severi varieties, and the Hilbert schemes) and also the moduli space of abstract curves of genus g.

Mathematics 271. Topology of Algebraic Varieties
Catalog Number: 8063
Grisha Mikhalkin
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Possible topics include: introduction to topology of complex varieties: Lefschetz pencils and vanishing cycles; topology of complex isolated singularities: Milnor fibers and their monodromy; bifurcations of A—n, D—n, E—6, E—7, and E—8 singularities (over C and over R); introduction to topology of real varieties; real curves and surfaces; Hilbert’s 16th problem: constructions and prohibitions; construction of maximal real varieties in higher dimension; patchworking of algebraic varieties.

Mathematics 272a. Introduction to Algebraic Topology
Catalog Number: 1666
Daniel Allcock
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
Peter B. Kronheimer
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 274. Mathematical Aspects of String Theory
Catalog Number: 6210
Cumrun Vafa
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
A review of certain mathematical aspects of string theory and topological field theories.

Mathematics 275x. Topics in Complex Dynamics and Hyperbolic Geometry
Catalog Number: 0683
Curtis T. McMullen
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
A discussion of the interplay of complex analysis, negative curvature, conformal dynamics and low-dimensional topology.

Graduate Courses of Reading and Research

Mathematics 301. Theory and Practice of Teaching in the Mathematical Sciences
Catalog Number: 4344
---------------
Half course (fall term; repeated 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 (on leave 1999-00)

*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 320. Topics in Topology and Real Algebraic Geometry
Catalog Number: 5460
Grisha Mikhalkin 3590

*Mathematics 321. Topics in Mathematical Physics
Catalog Number: 2297
Arthur M. Jaffe 2095

*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 (on leave 1999-00)

*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

*Mathematics 337. Topics in Algebraic Geometry
Catalog Number: 9000
Kalle Karu 2366

*Mathematics 338. Topics in Algebra
Catalog Number: 5996
Lisa J. Carbone 3587

*Mathematics 345. Topics in Geometry and Topology
Catalog Number: 4108
Peter B. Kronheimer 1759

*Mathematics 346. Topics in Deformation Theory
Catalog Number: 8245
Dmitry Tamarkin 2463

*Mathematics 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 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 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 387. Topics in Number Theory
Catalog Number: 2665
Brian Conrad 2383

*Mathematics 389. Topics in Number Theory
Catalog Number: 6851
Noam D. Elkies 2604