Paul G. Bamberg, Senior Lecturer on Mathematics

John D. Boller, Preceptor in Mathematics

Derek Bruff, Preceptor in Mathematics

Francesco Calegari, Benjamin Peirce Assistant Professor of Mathematics

Albert Chau, Benjamin Peirce Assistant Professor of Mathematics

Tom Coates, Benjamin Peirce Assistant Professor of Mathematics

Samit Dasgupta, Benjamin Peirce Assistant Professor of Mathematics

Alberto De Sole, Benjamin Peirce Assistant Professor of Mathematics

Elizabeth Denne, Benjamin Peirce Assistant Professor of Mathematics

Eaman Eftekhary, Benjamin Peirce Assistant Professor of Mathematics

Noam D. Elkies, Professor of Mathematics

Andrew Engelward, Preceptor in Mathematics

Daniel L. Goroff, Professor of the Practice of Mathematics

Robin Gottlieb, Senior Preceptor in Mathematics

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

David Helm, Benjamin Peirce Assistant Professor of Mathematics

Michael J. Hopkins, Visiting Professor of Mathematics

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

David S. Jerison, Visiting Professor of Mathematics

Thomas W. Judson, Preceptor in Mathematics

Oliver Knill, Preceptor in Mathematics

Joachim Krieger, Benjamin Peirce Assistant Professor of Mathematics

Peter B. Kronheimer, William Casper Graustein Professor of Mathematics

Joseph M. Landsberg, Visiting Associate Professor of Mathematics

Matthew P. Leingang, Preceptor in Mathematics

Laura F. Matusevich, Benjamin Peirce Assistant Professor of Mathematics

Barry C. Mazur, Gerhard Gade University Professor

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

Andreea C. Nicoara, Benjamin Peirce Assistant Professor of Mathematics

Martin A. Nowak, Professor of Mathematics and of Biology, Director of the Program for Evolutionary Dynamics

Mihnea Popa, Benjamin Peirce Assistant Professor of Mathematics

Weiyang Qiu, Benjamin Peirce Assistant Professor of Mathematics

Gerald E. Sacks, Professor of Mathematical Logic

Wilfried Schmid, Dwight Parker Robinson Professor of Mathematics

Yum Tong Siu, William Elwood Byerly Professor of Mathematics

William A. Stein, Benjamin Peirce Assistant Professor of Mathematics

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

Clifford Taubes, William Petschek Professor of Mathematics

Richard L. Taylor, Herchel Smith Professor of Mathematics

Dylan P. Thurston, Benjamin Peirce Assistant Professor of Mathematics

Angela G. Vierling-Claassen, Preceptor in Mathematics

Benjamin Weinkove, Benjamin Peirce Assistant Professor of Mathematics

Shing-Tung Yau, William Casper Graustein Professor of Mathematics

Ilia Zharkov, Benjamin Peirce Assistant Professor of Mathematics

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

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

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

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

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

The Mathematics Department does not grant formal degree credit without

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

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

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

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

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

*
Angela Vierling, Benjamin Weinkove, Thomas W. Judson, and members of the Department.
*

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

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

*Prerequisite: *Mathematics Xa.

**
Mathematics 1a. Introduction to Calculus**

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

*
Matthew P. Leingang and Derek Bruff (fall term), Matthew P. Leingang (spring term)
*

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

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

*Note: *Required first meeting in fall: Tuesday, September 21, 8:30 am, Science Center D. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Prerequisite: *A solid background in precalculus.

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

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

*
Robin Gottlieb, Albert Chau, and Weiyang Qiu (fall term), Robin Gottlieb, Matthew P. Leingang, and Andreea C. Nicoara (spring term)
*

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

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

*Note: *Required first meeting in fall: Monday, September 20, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, February 2, 8:30 am, Science Center D. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

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

**
Mathematics 19. Mathematical Modeling**

Catalog Number: 1256

*
Thomas W. Judson
*

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

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

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

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

Catalog Number: 0906

*
Matthew P. Leingang (fall term) and Derek Bruff (spring term)
*

*
Half course (fall term; repeated spring term). Fall: M., W., F., at 9, Spring: M., W., F., at 9, and a weekly problem section 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. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.

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

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

**
Mathematics 21a. Multivariable Calculus**

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

*
Andrew Engelward, Elizabeth Denne, and Eaman Eftekhary (fall term), Andrew Engelward, Eaman Eftekhary, and Joachim Krieger (spring term)
*

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

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

*Note: *Required first meeting in fall: Tuesday, September 21, 8:30 am, Science Center C. Required first meeting in the spring: Wednesday, February 2, 8:30 am, Science Center C. May not be taken for credit by students who have passed Applied Mathematics 21a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
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 term. The biochemistry/social sciences sections treat topics in probability and statistics in lieu of Green’s, Stokes’ and the Divergence Theorems.

*Prerequisite: *Mathematics 1b or equivalent.

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

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

*
Oliver Knill (fall term), Clifford Taubes, Elizabeth Denne, and Ilia Zharkov (spring term)
*

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

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

*Note: *Required first meeting in fall: Monday, September 20, 8:30 am, Science Center A. Required first meeting in spring: Wednesday, February 2, 8:30 am, Science Center A. May not be taken for credit by students who have passed Applied Mathematics 21b. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

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

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

Catalog Number: 2486

*
John D. Boller
*

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

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

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

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

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

Catalog Number: 8571

*
John D. Boller
*

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

Continuation of the subject matter of Mathematics 23a.

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

*Prerequisite: *Mathematics 23a.

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

Catalog Number: 1525

*
Tom Coates
*

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

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

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

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

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

Catalog Number: 1590

*
Tom Coates
*

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

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

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

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

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

Catalog Number: 4068

*
Wilfried Schmid
*

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

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

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

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

Catalog Number: 3312

*
Wilfried Schmid
*

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

Continuation of Mathematics 55a. Calculus of functions in n variables. Additional topics may include normed linear spaces, differential equations, and Fourier analysis.

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

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

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

Catalog Number: 8500

*
Clifford Taubes
*

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

Advanced reading in topics not covered in courses.

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

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

Catalog Number: 2165

*
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 Department
*

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

Topics for 2004-2005: (1) Additive Number Theory (fall), prerequisite: algebra (Math 122), some real analysis as in Math 112 or Math 23, 25, or 55) would be useful as well. (2) Algebraic Surfaces and Complex Manifolds of Higher Dimension (fall), prerequisites: complex analysis (Math 113) and knowledge of manifolds (such as Math 134 or Math 135). (3) Sheaves in Logic and Geometry (spring), prerequisites: topology (Math 131) and knowledge of manifolds (such as in Math 134 or Math 135). (4) Complex Multiplication (spring), prerequisites: complex analysis (Math 113) and algebraic number theory (such as in Math 123).

*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 term in which the tutorial is offered.

Catalog Number: 8066

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

**
Mathematics 106. Ordinary Differential Equations**

Catalog Number: 3377

*
Thomas W. Judson
*

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

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

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

**
Mathematics 112. Real Analysis**

Catalog Number: 1123

*
Weiyang Qiu
*

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

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

*
Mihnea Popa
*

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

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

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

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

Catalog Number: 1871

*
Joachim Krieger
*

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

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

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

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

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

Catalog Number: 5253

*
Daniel L. Goroff
*

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

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

*Prerequisite: *At least one course beyond Mathematics 21

**
Mathematics 118r. Dynamical Systems**

Catalog Number: 6402

*
Oliver Knill
*

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

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

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

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

Catalog Number: 7326

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

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

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

*Note: *Expected to be given in 2005–06.

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

**
Mathematics 121. Linear Algebra and Applications**

Catalog Number: 7009

*
Mihnea Popa
*

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

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

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

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

Catalog Number: 7855

*
Benedict H. Gross
*

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

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

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

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

Catalog Number: 5613

*
Benedict H. Gross
*

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

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

*Prerequisite: *Mathematics 122.

**
Mathematics 124. Number Theory**

Catalog Number: 2398

*
Samit Dasgupta
*

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

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

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

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

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

*Note: *Expected to be given in 2005–06.

**
Mathematics 128. Lie Algebras**

Catalog Number: 6519

*
Shing-Tung Yau
*

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

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

**
Mathematics 129. Topics in Number Theory**

Catalog Number: 2345

*
William A. Stein
*

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

Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and an integrated discussion of how to compute.

*Prerequisite: *Mathematics 122 and 123.

**
Mathematics 131. Topology**

Catalog Number: 2381

*
Andreea C. Nicoara
*

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

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

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

**
Mathematics 134. Calculus on Manifolds**

Catalog Number: 7150

*
Elizabeth Denne
*

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

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

*
Eaman Eftekhary
*

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

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

*
Weiyang Qiu
*

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

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

*
David Helm
*

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

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

*Prerequisite: *Mathematics 122, 123.

**
Mathematics 138. Classical Geometry**

Catalog Number: 0162

*
Paul G. Bamberg
*

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

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

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

**
[Mathematics 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 2005–06.

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

**
[Mathematics 143. Set Theory]**

Catalog Number: 6005

*
Gerald E. Sacks
*

*
Half course (spring term). 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 2005–06.

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

**
Mathematics 144. Model Theory and Algebra**

Catalog Number: 0690

*
Gerald E. Sacks
*

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

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

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

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

Catalog Number: 8389

*
Paul G. Bamberg
*

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

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

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

*Prerequisite: *Mathematics 21b or equivalent.

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

Catalog Number: 3004 Enrollment: Limited to 30. Limited to seniors and graduate students.

*
Martin A. Nowak
*

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

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

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

**
Mathematics 191. Mathematical Probability**

Catalog Number: 4306

*
Paul G. Bamberg
*

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

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

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

**
[Mathematics 192r. Algebraic Combinatorics]**

Catalog Number: 6612

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

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

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

*Note: *Expected to be given in 2005–06. No prior knowledge of combinatorics is assumed, but familiarity with linear algebra will be helpful.

Catalog Number: 8330

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

*
Shlomo Z. Sternberg
*

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

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

*Prerequisite: *Experience with courses involving rigorous proofs: e.g. Mathematics 25a, b, 121, 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

*
Yum Tong Siu
*

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

Fundamentals of complex analysis, and further topics such as elliptic functions, theta functions, Riemann surfaces, uniformization theorem, the theorem of Riemann-Roch, and Abel’s theorem.

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

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

Catalog Number: 2641

*
Yum Tong Siu
*

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

Introduction to several complex variables, pseudoconvexity, domains of holomorphy, the d bar problem, sheaves and cohomology, Kaehler manifolds, Hodge decomposition, Kodaira’s vanishing and embedding theorems, abelian varieties, theta functions of several variables.

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

**
Mathematics 215. Transcendental Methods in Algebraic Geometry **

Catalog Number: 2363

*
Yum Tong Siu
*

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

Applications of L2 d-bar estimates and multiplier ideal sheaf techniques to problems in algebraic geometry such as the effective Nullstellensatz, the Fujita conjecture on the effective global generation, and very ampleness of line bundles, the effective Matsusaka big theorem, the invariance of plurigenera.

**
Mathematics 230ar. Differential Geometry**

Catalog Number: 0372

*
Benjamin Weinkove
*

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

Riemannian manifolds, geodesics, and curvature. Kähler geometry. A discussion of the heat equation and the Kahler-Ricci flow.

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

**
Mathematics 230br. Differential Geometry**

Catalog Number: 0504

*
Peter B. Kronheimer
*

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

A continuation of Mathematics 230ar. Topics in Riemannian geometry, Kähler geometry, Hodge theory, and Yang-Mills theory.

*Prerequisite: *Differential Topology.

**
Mathematics 232. Harnack Inequalities in Analysis and Geometry**

Catalog Number: 0620

*
David S. Jerison (Massachusetts Institute of Technology)
*

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

Moser’s version of the Harnack inequality of De Giorgi and Nash, leading to regularity of minimal surfaces. Inequalities of Li-Yau, Hamilton, and Perelman leading to a proof of the Poincaré hypothesis, insofar as it has been checked.

**
Mathematics 234. Evolutionary Dynamics **

Catalog Number: 8136

*
Martin A. Nowak
*

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

Advanced topics of evolutionary dynamics. Seminars and research projects.

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

**
Mathematics 237. Analysis, Geometry, and Algebraic Geometry Related to Calabi-Yau Manifolds**

Catalog Number: 8335

*
Shing-Tung Yau
*

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

A study of the material on how to solve the equations to construct Ricciflat metrics and the basic properties of such manifolds that have interest for string theory and other topics.

**
Mathematics 242. Set Theory: Large Cardinals from Determinacy**

Catalog Number: 9033

*
Peter Koellner
*

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

A course on the strength of the axiom of determinacy. First we prove a classic result of Woodin: ‘ZF + AD’ is consistent, then ‘ZFC + there are (omega)-many Woodin cardinals’ is consistent. Second goal: to discuss recent work of Woodin in this area, in particular, the HOD-analysis, a key ingredient in his results on CH.

**
Mathematics 250. Higher Algebra**

Catalog Number: 9334

*
Noam D. Elkies
*

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

An introduction to Galois theory, Brauer groups (which describe the structure of central simple algebras over a given field) and linear representations of finite groups, and some applications of these structures in various mathematical disciplines.

*Prerequisite: *Mathematics 123 or equivalent.

**
Mathematics 251a. Algebraic Number Theory**

Catalog Number: 1703

*
Richard L. Taylor
*

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

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

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

**
Mathematics 251b. Algebraic Number Theory**

Catalog Number: 7441

*
Richard L. Taylor
*

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

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

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

**
Mathematics 255x. The Eisenstein Ideal**

Catalog Number: 7016

*
Francesco Calegari
*

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

The course is built around Mazur’s paper, “Modular curves and the Eisenstein ideal.” Topics: finite group schemes, and Mazur’s classification of torsion subgroups of elliptic curves over Q.

**
Mathematics 257. The Arithmetic of Abelian Varieties: Classical and Computational Results**

Catalog Number: 4304

*
William A. Stein
*

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

Quick review of abelian varieties, statement of the conjecture over global fields, Tate’s theorem about isogeny invariance, Milne’s theorem about restriction of scalars, Cassels-Tate pairing on Sha and squareness properties, computations with the conjecture.

**
Mathematics 258. Explicit Constructions of Rational Points on Eliptical Curves**

Catalog Number: 4297

*
Samit Dasgupta
*

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

A discussion of Hegner points, the Shimura Reciprocity Law, the Gross-Zagier Formula, and Kolyvagin’s Theorem. Additional topics may include mock Heegner points, complez multiplication points on Shimura curves, and Darmon’s construction of Stark-Heegner points.

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

Catalog Number: 7004

*
Joseph D. Harris
*

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

Classical theory of projective varieties, covering concepts like dimension, degree, smoothness and singularity, tangent spaces and tangent cones, parameter spaces and moduliar spaces. Emphasis will be on examples and problems.

*Prerequisite: *Mathematics 250.

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

Catalog Number: 2745

*
Joseph D. Harris
*

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

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

**
Mathematics 263y. Projective Geometry and Representation Theory**

Catalog Number: 2593

*
Joseph Landsberg (Georgia Institute of Technology)
*

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

Classical questions about the geometry of subvarieties of projective space, a study of the geometry of rational homogeneous varieties, and the conjectures of Deligne and Vogel regarding categorical generalizations of Lie algebras. Emphasis on how these perspectives can be useful to work on open questions in related areas.

*Prerequisite: *Mathematics 260a or the equivalent.

**
Mathematics 268y. Asymptotic Methods in Higher Dimensional Geometry**

Catalog Number: 3896

*
Mihnea Popa
*

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

Recent developments in higher dimensional algebraic geometry based on Mori theory, asymptotic multiplier ideals, volumes of divisors, and other asymptotic invariants; a study of cones of divisors and curves on projective varieties.

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

Catalog Number: 1666

*
Dylan P. Thurston
*

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

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

*Prerequisite: *Mathematics 131 or permission of instructor.

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

Catalog Number: 6502

*
Michael J. Hopkins (Massachusetts Institute of Technology)
*

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

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

*Prerequisite: *Mathematics 272a.

**
Mathematics 274r. Topics in Geometric Topology**

Catalog Number: 0163

*
Dylan P. Thurston
*

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

Geometry, topology and algebra in dimensions 2 and 3, including the Geometrization Conjecture (every 3-manifold can be split canonically into geometric pieces) and the loop and sphere theorems.

**
Mathematics 275y. Teichmüller Theory**

Catalog Number: 8541

*
Curtis T. McMullen
*

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

Topics in the complex analytic theory of the moduli space of Riemann surfaces, and its relations to topology, hyperbolic geometry, and dynamics.

**
Mathematics 276. Motivic Homotopy Theory**

Catalog Number: 2739

*
Michael J. Hopkins (Massachusetts Institute of Technology)
*

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

Study of the Morel-Voevodsky homotopy theory of smooth varieties. Topics: Morel’s relationship between motivic homotopygroups and the Grothendieck-Witt ring, and the work of Dugger-Isaksen on “sums of squares” formulae.

**
Mathematics 281. The Symplectic Category and the WKB Approximation**

Catalog Number: 4904

*
Shlomo Z. Sternberg
*

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

Applications of symplectic geometry to high frequency solutions of partial differential equations using the language of category theory.

*Prerequisite: *Some familiarity with differential geometry and with partial differential equations.

Catalog Number: 4344

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

Catalog Number: 5133

*
Benjamin Weinkove 4942
*

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

Catalog Number: 0464

*
Benedict H. Gross 1112
*

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

Catalog Number: 0552

*
Daniel L. Goroff 7683 (on leave fall term)
*

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

Catalog Number: 3874

*
Samit Dasgupta 5030
*

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

Catalog Number: 2743

*
Shlomo Z. Sternberg 1965
*

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

Catalog Number: 7393

*
Barry C. Mazur 1975 (on leave fall term)
*

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

Catalog Number: 2297

*
Arthur M. Jaffe 2095 (on leave spring term)
*

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

Catalog Number: 4659

*
Mihnea Popa 4015
*

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

Catalog Number: 2696

*
David Helm 4630
*

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

Catalog Number: 0409

*
Yum Tong Siu 7550
*

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

Catalog Number: 7003

*
Alberto De Sole 4627 (on leave 2004-2005)
*

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

Catalog Number: 2194

*
Elizabeth Denne 5031
*

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

Catalog Number: 9401

*
Curtis T. McMullen 3588 (on leave fall term)
*

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

Catalog Number: 5498

*
Clifford Taubes 1243
*

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

Catalog Number: 2526

*
Francesco Calegari 4435
*

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

Catalog Number: 4108

*
Peter B. Kronheimer 1759 (on leave fall term)
*

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

Catalog Number: 7227

*
Eaman Eftekhary 5045
*

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

Catalog Number: 5151

*
Gerald E. Sacks 3862
*

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

Catalog Number: 3492

*
Richard L. Taylor 1453
*

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

Catalog Number: 1217

*
William A. Stein 4016
*

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

Catalog Number: 6534

*
Wilfried Schmid 5097
*

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

Catalog Number: 4647

*
Shing-Tung Yau 1734
*

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

Catalog Number: 9037

*
Albert Chau 4017
*

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

Catalog Number: 1023

*
Joachim Krieger 4632
*

**
*Mathematics 378. Topics in Computational and Combinatorial Algebraic Geometry**

Catalog Number: 4436

*
Laura F. Matusevich 4357 (on leave 2004-05)
*

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

Catalog Number: 2037

*
Joseph D. Harris 2055
*

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

Catalog Number: 7736

*
Ilia Zharkov 4631 (on leave fall term)
*

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

Catalog Number: 3746

*
Andreea C. Nicoara 4374
*

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

Catalog Number: 4687

*
Martin A. Nowak 4568
*

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

Catalog Number: 6851

*
Noam D. Elkies 2604
*

**
*Mathematics 391. Topics in Differential Geometry and Partial Differential Equations**

Catalog Number: 2974

*
Weiyang Qiu 4359
*

**
*Mathematics 392. Topics in Geometry**

Catalog Number: 8778

*
Tom Coates 4633
*