Paul G. Bamberg, Senior Lecturer on Mathematics

Bret J. Benesh, Preceptor in Mathematics, Contin Ed/Spec Prog Instructor

Lydia Rosina Bieri, Benjamin Peirce Lecturer on Mathematics

Sebastian B. Casalaina-Martin, Lecturer in Mathematics

Janet Chen, Preceptor in Mathematics

Danijela Damjanovic, Benjamin Peirce Lecturer on Mathematics

Samit Dasgupta, Benjamin Peirce Assistant Professor of Mathematics

Alberto De Sole, Benjamin Peirce Assistant Professor of Mathematics

John F. Duncan, Benjamin Peirce Lecturer on Mathematics

Noam D. Elkies, Professor of Mathematics, Associate of Lowell House

Dennis Gaitsgory, Professor of Mathematics

Véronique Godin, Benjamin Peirce Assistant Professor of Mathematics

Thomas Goodwillie, Visiting Professor of Mathematics, Visiting Scholar in Mathematics

Robin Gottlieb, Professor of the Practice in the Teaching of Mathematics

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

John T. Hall, Preceptor in Mathematics

Joseph D. Harris, Higgins Professor of Mathematics

Michael J. Hopkins, Professor of Mathematics

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

Thomas W. Judson, Preceptor in Mathematics, Contin Ed/Spec Prog Instructor

Oliver Knill, Preceptor in Mathematics

Toshiyuki Kobayashi, Visiting Professor of Mathematics

Peter B. Kronheimer, William Caspar Graustein Professor of Mathematics

Thomas Lam, Benjamin Peirce Assistant Professor of Mathematics

Matthew P. Leingang, Preceptor in Mathematics

Barry C. Mazur, Gerhard Gade University Professor

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

Brian Munson, Lecturer in Mathematics

Andreea C. Nicoara, Benjamin Peirce Assistant Professor of Mathematics, Fellow in the Department of Mathematics

Martin A. Nowak, Professor of Mathematics and of Biology

Rehana Patel, Preceptor in Mathematics

Gerald E. Sacks, Professor of Mathematical Logic

Pedram Safari, Lecturer in Mathematics

Wilfried Schmid, Dwight Parker Robinson Professor of Mathematics

Lior Silberman, Benjamin Peirce Assistant Professor of Mathematics

Yum Tong Siu, William Elwood Byerly Professor of Mathematics

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

Robert M. Strain, Benjamin Peirce Assistant Professor of Mathematics

Richard L. Taylor, Herchel Smith Professor of Mathematics

Benjamin Weinkove, Benjamin Peirce Assistant Professor of Mathematics

Lauren K. Williams, Benjamin Peirce Assistant Professor of Mathematics

Horng-Tzer Yau, Professor of Mathematics

Shing-Tung Yau, William Caspar Graustein Professor of Mathematics

The Mathematics Department recommends that all students take mathematics courses. This said, be careful to take only those courses that are appropriate for your level of experience. 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, 19a,b, 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 19a,b, 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 19a,b are courses that are designed for students concentrating in the life sciences, chemistry, and the environmental sciences. (These course are recommended over Math 21a,b by the various life science, environmental science, and chemistry concentrations). In any event, Math 19a can be taken either before or after Math 21a,b. Math 19b requires some multivariable calculus background, and should not be taken with Math 21b. Math 19a teaches differential equations, related techniques and modeling with applications to the life sciences. Math 19b focuses teaches linear algebra, probability and statistics with a focus on life science examples and 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. Mathematics 55, covers the material from Mathematics 25 plus much material from Mathematics 122 and Mathematics 113. Entrance into Mathematics 55 requires the consent of the instructor.

Students who have had substantial preparation beyond the level of the Advanced Placement Examinations are urged to consult the Director of Undergraduate Studies 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 supply 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.

Mathematics 113, 114, 122, 123, 131, and 132 form the core of the department’s more advanced courses. Mathematics concentrators are encouraged to consider taking these courses, particularly Mathematics 113, 122 and 131. (Those taking 55a,b will have covered the material of Mathematics 113 and 122, and are encouraged to take Mathematics 114, 123, and 132.)

Courses numbered 200-249 are introductory graduate courses. They will include substantial homework and are likely to have a final exam, either in class or take home. Most are taught every year. They may be suitable for very advanced undergraduates. Mathematics 212a, 230a, 231a and 232a will help prepare graduate students for the qualifying examination in Mathematics. Courses numbered 250-299 are graduate topic courses, intended for advanced graduate students.

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.

*
Bret J. Benesh, John T. Hall, Brian Munson, and members of the Department
*

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: *Participation in a one and a half hour workshop is required each week, as well as required participation in a one hour problem session each week. 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, John Duncan, and Rehana Patel (fall term); Matthew P. Leingang (spring term)
*

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 18, 8:30 am, Science Center B. 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.

*
Thomas W. Judson, Danijela Damjanovic, John T. Hall, Brian Munson, and Robert Strain (fall term); Robin Gottlieb, Bret Benesh, and Lydia Bieri (spring term)
*

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 17, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center A. 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 19a. Modeling and Differential Equations for the Life Sciences**

Catalog Number: 1256

*
Thomas W. Judson (fall term); John T. Hall (spring term)
*

*
Half course (fall term; repeated spring 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: *This course is recommended over Math 21a for those planning to concentrate in the life sciences, chemistry, or environmental sciences. 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 19b. Linear Algebra, Probability, and Statistics for the Life Sciences**

Catalog Number: 6144

*
Clifford Taubes
*

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

Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.

*Note: *This course is recommended over Math 21b for those planning to concentrate in the life sciences, chemistry, or environmental sciences. Can be taken with Mathematics 21a. Students who have seen some multivariable calculus can take Math 19b before Math 19a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

**
Mathematics 20. Algebra and Multivariable Mathematics for Social Sciences**

Catalog Number: 0906

*
Matthew P. Leingang (fall term); Rehana Patel (spring term)
*

*
Half course (fall term; repeated spring term). 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.

*
Oliver Knill, Véronique Godin, and Rehana Patel (fall term); Thomas Judson, and 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 VI, Tu., Th., 11:30–1; and a weekly problem section 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 18, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 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 are offered each term.

*Prerequisite: *Mathematics 1b or equivalent.

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

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

*
Yum Tong Siu (fall term); Oliver Knill, Samit Dasgupta, and Rehana Patel (spring term)
*

*
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 (with sufficient enrollment); 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 plus an introduction to Fourier series.

*Note: *Required first meeting in fall: Monday, September 17, 8:30 am, Science Center A. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center D. 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. Linear Algebra and Real Analysis I**

Catalog Number: 2486

*
Paul G. Bamberg
*

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

A rigorous, integrated treatment of linear algebra and multivariable differential calculus, emphasizing topics that are relevant to fields such as physics and economics. Topics: fields, vector spaces and linear transformations, scalar and vector products, elementary topology of Euclidean space, limits, continuity, and differentiation in n dimensions, eigenvectors and eigenvalues, inverse and implicit functions, manifolds, and Lagrange multipliers. Students are expected to master twenty important proofs.

*Note: *Course content overlaps substantially with Mathematics 21a,b, 25a,b, so students should plan to continue in Mathematics 23b. See the description in the introductory paragraphs in the Mathematics section of the catalog about the differences between Mathematics 23 and Mathematics 25. 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, plus an interest both in proving mathematical results and in using them.

**
Mathematics 23b. Linear Algebra and Real Analysis II**

Catalog Number: 8571

*
Paul G. Bamberg
*

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

A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Applications of linear algebra to differential equations and Fourier analysis. Introduction to infinite-dimensional vector spaces. Stokes’s theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms. Students are expected to master twenty important proofs.

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

*Prerequisite: *Mathematics 23a.

**
Mathematics 25a. Honors Linear Algebra and Real Analysis I**

Catalog Number: 1525

*
Benjamin Weinkove
*

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

A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.

*Note: *Only for students with a strong interest and background in mathematics. There will be a heavy workload. 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: *5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.

**
Mathematics 25b. Honors Linear Algebra and Real Analysis II**

Catalog Number: 1590

*
Benjamin Weinkove
*

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

A rigorous treatment of basic analysis. Topics include: convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.

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

*Prerequisite: *Mathematics 23a or 25a or 55a.

**
*Mathematics 55a. Honors Abstract Algebra**

Catalog Number: 4068

*
Dennis Gaitsgory
*

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

A rigorous treatment of abstract algebra including linear algebra and group theory.

*Note: *Mathematics 55a is an intensive 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 Real and Complex Analysis**

Catalog Number: 3312

*
Samit Dasgupta
*

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

A rigorous treatment of real and complex analysis.

*Note: *Mathematics 55b is an intensive course for students having significant experience with abstract mathematics. Instructor’s permission required. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

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

Catalog Number: 8500

*
Peter B. Kronheimer
*

*
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

*
Peter B. Kronheimer
*

*
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

*
Peter B. Kronheimer and members of the Department
*

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

Topics for 2007-08: (1) Random graphs (fall), prerequisite: Math 122 or familiarity with abstract linear algebra and group theory. Knowledge of elementary probability theory helpful, but not required. (2) Clifford algebras and spinors (spring), prerequisite: Math 122 or familiarity with abstract linear algebra, groups and group actions.

*Note: *May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

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 12. EXAM GROUP: 5*

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 19a,b, 20 or 21a.

**
Mathematics 112. Introductory Real Analysis**

Catalog Number: 1123

*
Danijela Damjanovic
*

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

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.

*Prerequisite: *Mathematics 21a,b and either an ability to write proofs or concurrent enrollment in Mathematics 101. Should not ordinarily be taken in addition to Mathematics 23a,b, 25a,b or 55a,b.

**
Mathematics 113. Analysis I: Complex Function Theory**

Catalog Number: 0405

*
Robert M. Strain
*

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

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 112. Not to be taken after Mathematics 55b.

**
Mathematics 114. Analysis II: Measure, Integration and Banach Spaces - (New Course) **

Catalog Number: 9111

Lebesgue measure and integration; general topology; introduction to

**
[Mathematics 115. Methods of Analysis]**

Catalog Number: 1871

*
Wilfried Schmid
*

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

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

*Note: *Expected to be given in 2008–09. Mathematics 115 is especially for students interested in physics.

*Prerequisite: *Mathematics 23a,b, 25a,b, 55a,b, or 112.

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

Catalog Number: 5253

*
Paul G. Bamberg
*

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

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 121. Linear Algebra and Applications**

Catalog Number: 7009

*
Lydia R. Bieri
*

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

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

Catalog Number: 7855

*
Alberto DeSole
*

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

Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.

*Prerequisite: *Mathematics 23a, 25a, 121; or 101 with the instructor’s permission. Should not be taken in addition to Mathematics 55a.

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

Catalog Number: 5613

*
Alberto De Sole
*

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

Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.

*Prerequisite: *Mathematics 122 or 55a.

**
Mathematics 124. Number Theory**

Catalog Number: 2398

*
Joseph D. Harris
*

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

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 129. Number Fields**

Catalog Number: 2345

*
Richard L. Taylor
*

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

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 ideles.

*Prerequisite: *Mathematics 123.

**
Mathematics 130 (formerly Mathematics 138). Classical Geometry**

Catalog Number: 5811

*
Peter B. Kronheimer
*

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

Euclidean, spherical and hyperbolic geometry. No prior experience with proofs required.

*Note: *Not expected to be given 2008-09.

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

**
Mathematics 131. Topology I: Topological Spaces and the Fundamental Group**

Catalog Number: 2381

*
Véronique Godin
*

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

Abstract topological spaces; compactness, connectedness, continuity. Homeomorphism and homotopy, fundamental groups, covering spaces. Introduction to combinatorial topology.

*Prerequisite: *Some acquaintance with metric space topology (Mathematics 23a,b, 25a,b, 55a,b, 101, or 112) and with groups (Mathematics 101, 122 or 55a).

**
Mathematics 132. Topology II: Smooth Manifolds - (New Course) **

Catalog Number: 7725

Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes’ theorem, introduction to cohomology.

**
Mathematics 136. Differential Geometry**

Catalog Number: 1949

*
Shlomo Z. Sternberg
*

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

The exterior differential calculus and its application to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions and to symplectic geometry.

*Prerequisite: *Advanced calculus and linear algebra.

**
Mathematics 137. Algebraic Geometry**

Catalog Number: 0556

*
John F. Duncan
*

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

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

*Prerequisite: *Mathematics 123.

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

Catalog Number: 0600

*
Gerald E. Sacks
*

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

An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.

*Note: *Expected to be given in 2008–09.

*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). 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.

*Note: *Not expected to be given 2008-09.

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

**
Mathematics 152. Discrete Mathematics**

Catalog Number: 8389

*
Bret J. Benesh
*

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

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 23a,b, 25a,b or 55a,b 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.

*
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.

*Note: *Expected to be given in 2008–09.

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

**
Mathematics 154 (formerly Mathematics 191). Probability Theory**

Catalog Number: 4306

*
Paul G. Bamberg
*

*
Half course (spring 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, random walks, and Markov processes.

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

**
Mathematics 155r (formerly Mathematics 192r). Combinatorics**

Catalog Number: 6612

*
Lauren K. Williams
*

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

Topics include enumerative and algebraic combinatorics related to representations of the symmetric group, symmetric functions, and Young tableaux.

*Prerequisite: *Mathematics 122 (or equivalent). Knowledge of representation theory of finite groups will be helpful.

Catalog Number: 5446

Banach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform.

**
Mathematics 212br. Advanced Real Analysis**

Catalog Number: 7294

*
Shlomo Z. Sternberg
*

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

Continuation of Mathematics 212ar. The spectral theorem for self-adjoint operators in Hilbert space. Applications to partial differential equations.

*Prerequisite: *Mathematics 212ar and 213a.

**
Mathematics 213a. Complex Analysis**

Catalog Number: 1621

*
Andreea C. Nicoara
*

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

A second course in complex analysis: elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure and capacity.

*Prerequisite: *Mathematics 55b or 113.

**
Mathematics 213br. Advanced Complex Analysis**

Catalog Number: 2641

*
Shing-Tung Yau
*

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

Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.

*Prerequisite: *Mathematics 213a.

**
Mathematics 221. Commutative Algebra - (New Course) **

Catalog Number: 8320

A first course in commutative algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, discrete valuation rings, filtrations, completions and dimension theory.

**
[Mathematics 222. Lie Groups and Lie Algebras]**

Catalog Number: 6738

*
Wilfried Schmid
*

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

Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.

*Note: *Expected to be given in 2008–09.

*Prerequisite: *Mathematics 114, 123 and 132.

**
[Mathematics 223a. (formerly Mathematics 251a.) Algebraic Number Theory]**

Catalog Number: 8652

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

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

A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.

*Note: *Expected to be given in 2008–09.

*Prerequisite: *Mathematics 129.

**
[Mathematics 223b. (formerly Mathematics 251b.) Algebraic Number Theory]**

Catalog Number: 2783

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

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

Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate’s thesis or Euler systems.

*Note: *Expected to be given in 2008–09.

*Prerequisite: *Mathematics 223a.

**
Mathematics 230a. Differential Geometry**

Catalog Number: 0372

*
Shing-Tung Yau
*

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

Elements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, geodesics, Riemann curvature, Darboux’s theorem, moment maps and symplectic quotients, complex and Kaehler manifolds, Dolbeault and de Rham cohomology.

*Prerequisite: *Mathematics 132 or equivalent.

**
Mathematics 230br. Advanced Differential Geometry**

Catalog Number: 0504

*
Shing-Tung Yau
*

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

A continuation of Mathematics 230a. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogov’s theorem and applications; sphere theorems; Gromov’s betti number bounds; Gromov-Hausdorff convergence; Cheeger’s finiteness theorem, and convergence theorems.

*Prerequisite: *Mathematics 230a.

**
Mathematics 231a. (formerly Mathematics 272a.) Algebraic Topology**

Catalog Number: 7275

*
Danijela Damjanovic
*

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

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 and 132.

**
Mathematics 231br. (formerly Mathematics 272b.) Advanced Algebraic Topology**

Catalog Number: 9127

*
Michael J. Hopkins
*

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

Continuation of Mathematics 231a. 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 231a.

**
Mathematics 232a. (formerly Mathematics 260a.) Introduction to Algebraic Geometry I**

Catalog Number: 6168

*
Peter B. Kronheimer
*

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

Introduction to complex algebraic curves, surfaces, and varieties.

*Prerequisite: *Mathematics 123 and 132.

**
Mathematics 232br. (formerly Mathematics 260b.) Introduction to Algebraic Geometry II**

Catalog Number: 9205

*
Sebastian B. Casalaina-Martin
*

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

The course will cover the classification of complex algebraic surfaces.

*Prerequisite: *Mathematics 232a.

**
Mathematics 233a. (formerly Mathematics 261a.) Theory of Schemes I**

Catalog Number: 6246

*
Barry C. Mazur
*

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

An introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.

*Prerequisite: *Mathematics 221 and 232a or permission of instructor.

**
Mathematics 233br. (formerly Mathematics 261b.) Theory of Schemes II**

Catalog Number: 3316

*
Sebastian B. Casalaina-Martin
*

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

A continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.

*Prerequisite: *Mathematics 233a.

**
Mathematics 243 (formerly Mathematics 234). Evolutionary Dynamics **

Catalog Number: 8136

*
Martin A. Nowak
*

*
Half course (spring term). Tu., 1–4. EXAM GROUP: 15, 16, 17*

Advanced topics of evolutionary dynamics. Seminars and research projects.

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

**
Mathematics 244. Advanced Set Theory - (New Course) **

Catalog Number: 3138

Inner models of large cardinal axioms, focusing on recent work on inner models for large cardinals at the level of supercompact and beyond. Topics include: Continuum Hypothesis and Omega Conjecture.

**
Mathematics 262. Manifolds and Homotopy Theory**

Catalog Number: 5564

*
Thomas Goodwillie (Brown University)
*

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

Possible topics: Whitney’s embedding theorem and generalizations, the h-cobordism theorem and generalizations, surgery theory, and calculus of functors. Pace and emphasis of course may depend on the background and interest of the participants.

**
Mathematics 265. Infinite Dimensional Lie Algebras**

Catalog Number: 3191

*
Alberto De Sole
*

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

A detailed introduction to the structure and the representation theory of some important infinite-dimensional Lie algebras: the Heisenberg algebra, the Virasoro algebra and the affine Kac-Moody algebras.

**
Mathematics 271. Introduction to the Mathematics of General Relativity**

Catalog Number: 2400

*
Lydia Bieri
*

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

A focus on the Einstein field equations within GR. Brief review of SR and fundamentals from differential geometry. Discussion of Schwarzschild solution, black holes, energy-momentum tensor, non-localizability of gravitational energy, isolated gravitating systems.

**
Mathematics 273. Topics in Analysis and Mathematical Physics**

Catalog Number: 7810

*
Shlomo Z. Sternberg
*

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

Possible choices: exact models in statistical mechanics, asymptotic analysis, or supersymmetry.

**
Mathematics 275. Multiplicity-Free Representations: Complex Geometric Methods in Representation Theory**

Catalog Number: 0818

*
Toshiyuki Kobayashi (Research Institute for Mathematical Sciences, Kyoto)
*

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

Explanation of complex geometric methods such as reproducing kernels and "visible actions" for the study of infinite dimensional representations. From this viewpoint, various examples of multiplicity-free representations of Lie groups will be discussed.

**
Mathematics 282. Introduction to Seiberg-Witten Theory**

Catalog Number: 8399

*
Pedram Safari
*

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

Rudiments of Seiberg-Witten moduli spaces and invariants, including a review of gauge theory techniques. Selected applications to the geometry and topology of 4-manifolds.

**
Mathematics 283. Topics in Knot Theory**

Catalog Number: 7877

*
Brian Munson
*

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

The Conway polynomial, the Jones polynomial, Khovanov homology, Vassiliyev and finite type invariants, and configuration space methods. A discussion of links and their generalizations to higher dimensions.

**
Mathematics 287. Algebraic Curves**

Catalog Number: 7465

*
Joseph D. Harris
*

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

A development of the theory of algebraic curves/Riemann surfaces, touching on many of the classical aspects of their geometry. A focus on developing current research topics.

**
Mathematics 288. Algebraic K-Theory**

Catalog Number: 5052

*
Michael J. Hopkins
*

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

Begins with some of the classical invariants of algebraic topology (like the finiteness obstruction, simple homotopy, Reidemeister torsion) and locate them in "algebraic K-theory." More advanced topics will include Quillen K-groups, computations, Waldhausen K-theory.

**
Mathematics 299r. Graduate Tutorial in Geometry - (New Course) **

Catalog Number: 8799

Tutorial 1: Applications of Seiberg-Witten Theory. Applications of Seiberg-Witten theory, Floer homology and refinements to geometry and topology of 3-and 4-manifolds, including algebraic and symplectic manifolds and contact manifolds.Tutorial 2: Elliptic Surfaces. Self-contained introduction to the modern theory of elliptic surfaces for ground fields of characteristic zero and of positive characteristic. An in-depth analysis will be devoted to rational elliptic surfaces and elliptic K3 surfaces.

Catalog Number: 4344

**
*Mathematics 302. Topics in Dynamics of Group Actions**

Catalog Number: 5763

*
Danijela Damjanovic 5583
*

**
*Mathematics 304. Topics in Algebraic Topology**

Catalog Number: 0689

*
Michael J. Hopkins 4376
*

**
*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 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 spring term)
*

**
*Mathematics 319. Topics in Representation Theory**

Catalog Number: 9591

*
John F. Duncan 5505
*

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

Catalog Number: 2297

*
Arthur M. Jaffe 2095
*

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

Catalog Number: 0409

*
Yum Tong Siu 7550 (on leave fall term)
*

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

Catalog Number: 7003

*
Alberto De Sole 4627
*

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

Catalog Number: 9401

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

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

Catalog Number: 5498

*
Clifford Taubes 1243
*

**
*Mathematics 342. Topics in Combinatorics**

Catalog Number: 0751

*
Thomas Lam 5322 (on leave spring term)
*

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

Catalog Number: 4108

*
Peter B. Kronheimer 1759
*

**
*Mathematics 346y. Topics in Analysis: Quantum Dynamics**

Catalog Number: 1053

*
Horng-Tzer Yau 5260 (on leave spring 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 (on leave fall term)
*

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

Catalog Number: 3492

*
Richard L. Taylor 1453
*

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

Catalog Number: 6534

*
Wilfried Schmid 5097 (on leave 2007-08)
*

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

Catalog Number: 4647

*
Shing-Tung Yau 1734
*

**
*Mathematics 371. Topics in Partial Differential Equations and Mathematical Physics**

Catalog Number: 0777

*
Robert M. Strain 5323
*

**
*Mathematics 372. Topics in Mathematical Relativity**

Catalog Number: 1150

*
Lydia Rosina Bieri 5794
*

**
*Mathematics 379. Topics in Combinatorics**

Catalog Number: 3390

*
Lauren K. Williams 5499 (on leave spring term)
*

**
*Mathematics 381. Introduction to Geometric Representation Theory**

Catalog Number: 0800

*
Dennis Gaitsgory 5259 (on leave spring term)
*

**
*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
*

**
*Mathematics 384. Topics in Automorphic Forms**

Catalog Number: 8009

*
Lior Silberman 5506 (on leave 2007-08)
*

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

Catalog Number: 4687

*
Martin A. Nowak 4568 (on leave fall term)
*

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

Catalog Number: 6851

*
Noam D. Elkies 2604 (on leave 2007-08)
*

**
*Mathematics 398. Topics in Algebraic and Geometric Topology**

Catalog Number: 0863

*
Véronique Godin 5311
*