Harvard Mathematics Department : Mathematics Courses

Introduction

The Mathematics Department hopes that all students will 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 Harvards 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, 21a,b, 23a,b and 25a,b. The Harvard Mathematics Placement Test results recommend the appropriate starting level course, either Mathematics Ma, 1a, 1b, or 21. Recommendation for Mathematics 21 is sufficient qualification for Mathematics 19a,b, 20, 21a, 23a, and 25a.

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 Ma, Mb, 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 strong interest in mathematics. They assume a solid understanding of one-variable calculus, 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 departments 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 prior approval for taking a course that is listed as a prerequisite of one you have already taken. Our policy is that a student who takes and passes any calculus course is not normally permitted to then take a more elementary course for credit. A student who has passed Mathematics 21a, for example, will normally not be allowed to take Mathematics 1a, or 1b for credit. The Mathematics Department is prepared to make exceptions for sufficient academic reasons; in each case, however, a student must obtain written permission from the Mathematics Director of Undergraduate Studies in advance.

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

Faculty of the Department of Mathematics

Shing-Tung Yau	William Casper Graustein Professor of Mathematics (Chair)
Jameel Habeeb Al-Aidroos	Preceptor in Mathematics
Nir David Avni	Benjamin Peirce Lecturer in Mathematics
Paul G. Bamberg	Senior Lecturer on Mathematics
Clark Barwick	Benjamin Peirce Lecturer on Mathematics
Juliana Victoria Belding	Preceptor in Mathematics
Lydia Rosina Bieri	Benjamin Peirce Lecturer on Mathematics
Janet Chen	Preceptor in Mathematics
Andrew W. Cotton-Clay	Benjamin Peirce Lecturer on Mathematics
Noam D. Elkies	Professor of Mathematics
Dennis Gaitsgory	Professor of Mathematics
Peter McKee Garfield	Preceptor in Mathematics
Toby Gee	Benjamin Peirce Lecturer on Mathematics
Robin Gottlieb	Professor of the Practice of Mathematics
Benedict H. Gross	George Vasmer Leverett Professor of Mathematics
John T. Hall	Preceptor in Mathematics
Joseph D. Harris	Higgins Professor of Mathematics
Michael J. Hopkins	Professor of Mathematics (on leave spring term)
Arthur M. Jaffe	Landon T. Clay Professor of Mathematics and Theoretical Science
Mark Kisin	Professor of Mathematics
Oliver Knill	Preceptor in Mathematics
Antti Knowles	Lecturer on Mathematics
Sarah Colleen Koch	Benjamin Peirce Lecturer on Mathematics;
Peter B. Kronheimer	William Casper Graustein Professor of Mathematics (Director of Undergraduate Studies) (on leave spring term)
Jacob Lurie	Professor of Mathematics
Alina Marian	Visiting Assistant Professor of Mathematics (University of Illinois at Chicago) (fall term only)
Barry C. Mazur	Gerhard Gade University Professor
Curtis T. McMullen	Maria Moors Cabot Professor of the Natural Sciences
Sophie Marguerite Morel	Visiting Lecturer on Mathematics
Martin A. Nowak	Professor of Mathematics and of Biology
Rehana Patel	Preceptor in Mathematics
Joseph David Rabinoff	Benjamin Peirce Lecturer on 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
Shlomo Z. Sternberg	George Putnam Professor of Pure and Applied Mathematics (on leave fall term)
Clifford Taubes	William Petschek Professor of Mathematics (on leave 2009-10)
Christine Jiayou Taylor	Lecturer on Mathematics
Richard L. Taylor	Herchel Smith Professor of Mathematics (Director of Graduate Studies)
Kirsten Wickelgren	Lecturer on Mathematics (spring term only)
Lauren K. Williams	Benjamin Peirce Assistant Professor of Mathematics (on leave 2009-10)
Hao Xu	Lecturer on Mathematics (spring term only)
Horng-Tzer Yau	Professor of Mathematics
Jun Yin	Benjamin Peirce Lecturer in Mathematics
Xinwen Zhu	Benjamin Peirce Lecturer on Mathematics

Primarily for Undergraduates

Mathematics Ma (formerly Mathematics Xa). Introduction to Functions and Calculus I
Catalog Number: 1981 Enrollment: Normally limited to 15 students per section.
Juliana Belding, Jameel Al-Aidroos, Robin Gottlieb and members of the Department
Half course (fall term). Section meeting times: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M. W. F., at 12 (with sufficient enrollment); and a twice weekly lab session to be arranged.
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.
Note: Required first meeting: Wednesday, September 2, 8:30 am, Science Center D. Participation in a one and a half hour workshop is required each week. Participation in the weekly problem sessions is also encouraged. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. This course, when taken for a letter grade together with Mathematics Mb, meets the Core area requirement for Quantitative Reasoning.

Mathematics Mb (formerly Mathematics Xb). Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Juliana Belding, Jameel Al-Aidroos, Rehana Patel, 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: Participation in a one and a half hour workshop is required each week. Participation in the weekly problem sessions is also encouraged. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. This course, when taken for a letter grade together with Mathematics Ma, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics Ma.

Mathematics 1a. Introduction to Calculus
Catalog Number: 8434 Enrollment: Normally limited to 30 students per section.
Benedict H. Gross, Dennis Gaitsgory, Peter M. Garfield, and Joseph Rabinoff (fall term); Peter M. Garfield (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, 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 problems from many other disciplines.
Note: Required first meeting in fall: Thursday, September 3, 8:30 am, Science Center B. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or 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.
Janet Chen, Peter M. Garfield, and Christine Taylor (fall term); Robin Gottlieb, and John Hall (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. Spring: Section I, M., W., F., at 10; Section II, M., W., F., 11; Section III, M., W., F., 12; Section IV, Tu., Th., 10-11:30; Section V, Tu., Th., 11:30-1(with sufficient enrollment), 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: Wednesday, September 2, 8:30 am, Science Center C. Required first meeting in spring: Monday, January 25, 8:30 am, Science Center D. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitataive Reasoning.
Prerequisite: Mathematics 1a, or Ma and Mb, or equivalent.

Mathematics 19a. Modeling and Differential Equations for the Life Sciences
Catalog Number: 1256
John T. Hall (fall term) and 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 General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.

Mathematics 19b. Linear Algebra, Probability, and Statistics for the Life Sciences
Catalog Number: 6144
Peter M. Garfield
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 General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.

Mathematics 20. Algebra and Multivariable Mathematics for Social Sciences
Catalog Number: 0906
Rehana Patel (fall term); Rehana Patel (spring term)
Half course (fall term; repeated spring term). Fall: M., W., F., at 9; Spring: M., W., F., at 10. EXAM GROUP: Fall: 2; Spring: 3
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 General Education requirement for Empirical and Mathematical Reasoning or 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, Jameel Al-Aidroos, Andrew Cotton-Clay, Toby Gee, Horng-Tzer Yau, and Xinwen Zhu (fall term); Jameel Al-Aidroos, Janet Chen, and Jun Yin (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. Spring: 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., 12 (with sufficient enrollment); Section V, Tu., Th., 10-11:30; Section VI, Tu., Th., 11:30-1 (with sufficient enrollment), 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 Greens, Stokes, and Divergence Theorems.
Note: Required first meeting in fall: Thursday, September 3, 8:30 am, Science Center C. Required first meeting in spring: Monday, January 25, 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 General Education requirement for Empirical and Mathematical Reasoning or 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.
Rehana Patel, and Sophie Morel (fall term); Oliver Knill, and Juliana Belding (spring term)
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 10 (with sufficient enrollment); Section II, M., W., F., at 11; Section III, M., W., F., at 12; 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: MRequired first meeting in fall: Wednesday, September 2, 8:00 am, Science Center A. Required first meeting in spring: Monday, January 25, 8:30 am, Science Center B. May not be taken for credit by students who have passed Applied Mathematics 21b. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or 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). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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 General Education requirement for Empirical and Mathematical Reasoning or 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). Tu., Th., 2:30-4, and a weekly conference section to be arranged. EXAM GROUP: 16, 17
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. Stokess 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 General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 23a.

Mathematics 25a. Honors Linear Algebra and Real Analysis I
Catalog Number: 1525
Nir David Avni
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 General Education requirement for Empirical and Mathematical Reasoning or 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
Nir David Avni
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 General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 23a or 25a or 55a.

*Mathematics 55a. Honors Abstract Algebra
Catalog Number: 4068
Curtis T. McMullen
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
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. Instructors 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 General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.

Mathematics 55b. Honors Real and Complex Analysis
Catalog Number: 3312
Curtis T. McMullen
Half course (spring term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
A rigorous treatment of real and complex analysis.
Note: Mathematics 55b is an intensive course for students having significant experience with abstract mathematics. Instructors permission required. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or 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.
Supervised small group tutorial. Topics to be arranged.
Note: May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

For Undergraduates and Graduates

See also Applied Mathematics and Statistics.

Mathematics 101. Sets, Groups and Topology
Catalog Number: 8066
Benedict H. Gross
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.
Note: Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: An interest in mathematical reasoning.

Mathematics 112. Introductory Real Analysis
Catalog Number: 1123
Gerald E. Sacks
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
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
Wilfried Schmid
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Analytic functions of one complex variable: power series expansions, contour integrals, Cauchys 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
Catalog Number: 9111
Yum Tong Siu
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Lebesgue measure and integration; general topology; introduction to L ^p spaces, Banach and Hilbert spaces, and duality.
Prerequisite: Mathematics 23, 25, 55, or 112.

Mathematics 115. Methods of Analysis
Catalog Number: 1871
Jun Yin
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Complex functions; Fourier analysis; Hilbert spaces and operators; Laplaces equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.
Note: 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., 4–5:30. EXAM GROUP: 18
Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex funtionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students will be expected to understand and invent proofs of theorems in real and functional analysis.
Prerequisite: Mathematics 23ab, 25ab, or 55ab, or Mathematics 21ab plus at least one other more advanced course in mathematics.

Mathematics 118r. Dynamical Systems
Catalog Number: 6402
Antti Knowles
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory. Computer programs will be developed and used for visualization, approximation, and experimentation.
Prerequisite: Mathematics 21a,b.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Sarah Colleen Koch
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
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
Barry C. Mazur
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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 instructors permission. Should not be taken in addition to Mathematics 55a.

Mathematics 123. Algebra II: Theory of Rings and Fields
Catalog Number: 5613
Barry C. Mazur
Half course (spring term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
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
Richard L. Taylor
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pells 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
Sophie Marguerite Morel
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
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
Paul G. Bamberg
Half course (spring term). M., W., 4-5:30. EXAM GROUP: 9
Euclidean, spherical and hyperbolic geometry. No prior experience with proofs required.
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
Clark Barwick
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
Catalog Number: 7725
Lydia Rosina Bieri
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes theorem, introduction to cohomology.
Prerequisite: Mathematics 23a,b, 25a,b, 55a,b or 112.

Mathematics 136. Differential Geometry
Catalog Number: 1949
Lydia Rosina Bieri
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
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
Kirsten Wickelgren
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Affine and projective spaces, plane curves, Bezouts 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). M., W., F., at 1. EXAM GROUP: 6
An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.
Prerequisite: Any mathematics course at the level of Mathematics 21a,b or higher, or permission of instructor.

[Mathematics 143. Set Theory]
Catalog Number: 6005
Gerald E. Sacks
Half course (spring term). Hours to be arranged.
Axioms of set theory. Gödels constructible universe. Consistency of the axiom of choice and of the generalized continuum hypothesis. Cohens forcing method. Independence of the AC and GCH.
Note: Expected to be given in 2010–11.
Prerequisite: Any mathematics course at the level of Mathematics 21a or higher, or permission of instructor.

[Mathematics 144. Model Theory and Algebra]
Catalog Number: 0690
Gerald E. Sacks
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
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.
Note: Expected to be given in 2010–11.
Prerequisite: Mathematics 123 or the equivalent is suggested as a prerequisite, but not required.

Mathematics 152. Discrete Mathematics
Catalog Number: 8389
John T. Hall
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
Martin A. Nowak
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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,b.

Mathematics 154 (formerly Mathematics 191). Probability Theory
Catalog Number: 4306
Horng-Tzer Yau
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
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.
Note: This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning and the Core area requirement for Quantitative Reasoning.
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
Noam D. Elkies
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
In combinatorics and elsewhere one often encounters a "design", or a collection of subsets of some finite set S whose elements are evenly distributed in a suitable sense; for instance the collection of edges of a regular graph (each of whose vertices is contained in the same number of edges) or the collection of lines of a finite projective plane (any two of whose points are contained in a unique line). Of particular interest are designs symmetric under a large group of permutations of S. The consideration of specific classical designs and their symmetries will lead us to the general study of designs and permutation groups. We conclude with the construction and detailed analysis of the remarkable designs associated with Mathieus sporadic groups of permutations of 12- and 24- element sets.
Prerequisite: The ability to write proofs and some knowledge of linear algebra will be needed.

Mathematics 160. John Wallis and Transcendence: Measuring the Circle - (New Course)
Catalog Number: 17177 Enrollment: Limited to 15. Undergraduates may enroll with permission of the instructor.
Barry C. Mazur
Half course (spring term). Tu., 2–4. EXAM GROUP: 16, 17
What does it mean to "know" a length, an area, a volume? We do it by relating lengths to lengths, areas to areas, volumes to volumes. Archimedes had his "mechanical" method and modern mathematics has the Calculus. John Wallis grappled with an approach that lies somewhere in between Archimedes and the Calculus. We will read John Walliss treatise "The arithmetic of infinitesimals" (and related works) to learn what this is.
Prerequisite: A working knowledge of the Calculus.

New Course

Topics in Additive Combinatorics
Ben Green
Entire Year 2009-10 M W F at 10 beginning Friday September 18 In Science Center 232
A large part of Additive Combinatorics is concerned with approximate algebraic objects and their relation to genuine ones. What is an approximate group? An approximate ring? An approximate homomorphism? An approximate polynomial? We will study these questions and more. To give an example, one notion of an approximate subgroup of Z is that it is a finite set A with the property that A+A = {x + y: x, y in A} is covered by a few translates of A. A remarkable theorem of Freiman and Ruzsa, which we shall prove in the course, describes the structure of sets like this in quite explicit terms. Additive Combinatorics is perhaps better known for its applications: Szemeredi's theorem about arithmetic progressions in dense sets of integers, or the theorem that the primes contain arbitrarily long progressions. We will discuss these and other applications according to the tastes of the audience (for example, we could look at the work of Tao and Vu on random matrices, or some of the work of Bourgain-Gamburd-Sarnak on nonlinear sieve problems). A special emphasis will be given to the role of nilpotent Lie groups, although this is not yet fully understood. These seem to form part of a "higher Fourier analysis", the proper development of which is something I am keen to get people interested in.

Primarily for Graduates

Mathematics 212a (formerly Mathematics 212ar). Real Analysis
Catalog Number: 5446
Horng-Tzer Yau
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Banach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform.
Prerequisite: Mathematics 114 or equivalent.

Mathematics 212br. Advanced Real Analysis
Catalog Number: 7294
Jun Yin
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
Yum Tong Siu
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Yum Tong Siu
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.
Prerequisite: Mathematics 213a.

Mathematics 221. Commutative Algebra
Catalog Number: 8320
Toby Gee
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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.
Prerequisite: Mathematics 123.

Mathematics 222. Lie Groups and Lie Algebras
Catalog Number: 6738
Wilfried Schmid
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.
Prerequisite: Mathematics 114, 123 and 132.

[Mathematics 223a (formerly Mathematics 251a). Algebraic Number Theory]
Catalog Number: 8652
Instructor to be determined
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 2010–11.
Prerequisite: Mathematics 129.

[Mathematics 223b (formerly Mathematics 251b). Algebraic Number Theory]
Catalog Number: 2783
Instructor to be determined
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: Tates thesis or Euler systems.
Note: Expected to be given in 2010–11.
Prerequisite: Mathematics 223a.

Mathematics 224. Representations of Reductive Lie Groups - (New Course)
Catalog Number: 25927
Wilfried Schmid
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Harish-Chandra modules, characters, the discrete series, classification of irreducible representations, Plancherel theorem.

Mathematics 229x. Introduction to Analytic Number Theory - (New Course)
Catalog Number: 41034
Noam D. Elkies
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlets theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications.
Prerequisite: Mathematics 113, 123

Mathematics 230a. Differential Geometry
Catalog Number: 0372
Lydia Rosina Bieri
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
Elements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, geodesics, Riemann curvature, Darbouxs 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
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
A continuation of Mathematics 230a. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogovs theorem and applications; sphere theorems; Gromovs betti number bounds; Gromov-Hausdorff convergence; Cheegers finiteness theorem, and convergence theorems.
Prerequisite: Mathematics 230a.

Mathematics 231a (formerly Mathematics 272a). Algebraic Topology
Catalog Number: 7275
Clark Barwick
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
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
Clark Barwick
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
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
Joseph D. Harris
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
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
Joseph D. Harris
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
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
Dennis Gaitsgory
Half course (fall term). Tu., Th., 3–4:30. EXAM GROUP: 16, 17, 18
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
Dennis Gaitsgory
Half course (spring term). Tu., Th., 3–4:30. EXAM GROUP: 16, 17, 18
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 252x. Chromatic Homotopy Theory - (New Course)
Catalog Number: 95261
Jacob Lurie
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
A discussion of the chromatic picture of stable homotopy theory, beginning with Quillens work on cohomology theories and formal group laws and culminating with the resolution of the Ravenel conjectures via the work of Devinatz-Hopkins-Smith.

Mathematics 264. Introduction to the Dynamics of Large Quantum Systems - (New Course)
Catalog Number: 43739
Antti Knowles
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
Possible topics: the N-body Schrodinger equation, self adjointness of Schrodinger operators, path integrals, coherent states, limiting dynamics and nonlinear Schrodinger equations.
Prerequisite: Basic knowledge of functional analysis and measure theory.

Mathematics 269x. Integrable Systems and Algebraic Geometry - (New Course)
Catalog Number: 58846
Hao Xu
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Possible topics: Gelfand-Dickey formalism of integrable hierarchies, theta function solutions of KdV equations, Witten-Kontsevich theorem and intersection theory on moduli spaces of curves.

Mathematics 272x. p-adic Modular Forms and Analytic Continuation - (New Course)
Catalog Number: 92556
Joseph Rabinoff
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
Fundamentals of the theory of p-adic modular forms. Topics may include: the canonical subgroup, p-adic congruence properties, analytic continuation including the control theorem, and applications to Galois representations.

Mathematics 275z. Riemann Surfaces and Hyperbolic Geometry - (New Course)
Catalog Number: 52645
Curtis T. McMullen
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Fundamental results and topics in Teichmüller theory, hyperbolic 3-manifolds, complex dynamics and moduli spaces.

Mathematics 278y. Algebraic Topology Literature - (New Course)
Catalog Number: 25136
Michael J. Hopkins
Half course (fall term). W., F., 2–3:30. EXAM GROUP: 7, 8
A literature course in advanced topics in algebraic topology. Material will be taken from research papers, both classical and contemporary, and the presentation will involve significant participation from the students.
Note: Enrollment limited to 8 students, please contact the professor before registering.

Mathematics 279. Aspects of the p-adic Langlands Correspondence - (New Course)
Catalog Number: 86355
Toby Gee
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
An introduction to the p-adic Langlands correspondence. Precise details to be decided on nearer to the time.

Mathematics 284. Loop Groups and their Flag Varieties - (New Course)
Catalog Number: 73953
Xinwen Zhu
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Applications of loop groups and their flag varieties to representation theory and algebraic geometry. Possible topics: reductive groups over local fields, flag varieties of loop groups, uniformization of G-bundles on curves.

Mathematics 285. Topics in Low-Dimensional Topology - (New Course)
Catalog Number: 67752
Peter B. Kronheimer
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
An introduction to Floer homology theories and to some of the questions in topology to which they have been applied.

Mathematics 291. Symplectic and Contact Topology - (New Course)
Catalog Number: 37538
Andrew Cotton-Clay
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
A tour of symplectic and contact topology. Hamiltonian and Reeb dynamics. Symplectomorphism groups. Symplectic fibrations and Lefschetz pencils. Open books. Surgery fillings. Holomorphic curves, Floer homology, and symplectic field theory.

Mathematics 292. Galois Representations - (New Course)
Catalog Number: 78113
Mark Kisin
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
An introduction to p-adic Hodge theory and deformations of Galois representations as well as some applications.

Mathematics 293. Topics in the Moduli Theory of Sheaves - (New Course)
Catalog Number: 22431
Alina Marian (University of Illinois at Chicago)
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
After building a standard background, we will focus on sheaves over curves and surfaces. Topics may include: Lie algebra actions on the cohomology of moduli, geometry of rank 0 sheaves, wallcrossing, strange dualities.

Mathematics 299r. Graduate Tutorial in Number Theory
Catalog Number: 8799
Richard L. Taylor
Half course (spring term). M., 1:30–3. EXAM GROUP: 6, 7
Student seminars on topics from Galois representations and automorphic forms. Participation by invitation only.

Cross-listed Courses

Applied Mathematics 105a. Complex and Fourier Analysis
Applied Mathematics 105b. Ordinary and Partial Differential Equations
Applied Mathematics 107. Graph Theory and Combinatorics
[Empirical and Mathematical Reasoning 14. Fat Chance] - (New Course)
*Freshman Seminar 24i. Mathematical Problem Solving
*Freshman Seminar 26s. Mathematical Structures and Gödel’s Completeness Theorem
*History of Science 206r. The Continuum of Motion, Space and Change in Aristotle and the Aristotelian Tradition: Seminar - (New Course)
Philosophy 144. Logic and Philosophy

Reading and Research

*Mathematics 300. Teaching Undergraduate Mathematics
Catalog Number: 3996
Robin Gottlieb and Jameel Al-Aidroos
Half course (fall term). Tu., 1–2:30.
This course is for all first-year graduate students in Mathematics.

[*Mathematics 301. Theory and Practice of Teaching in the Mathematical Sciences]
Catalog Number: 4344
Instructor to be determined
Note: Expected to be given in 2010–11.

*Mathematics 304. Topics in Algebraic Topology
Catalog Number: 0689
Michael J. Hopkins 4376 (on leave spring term)

*Mathematics 306. Topics in Representation Theory
Catalog Number: 9397
Nir Avni

*Mathematics 308. Topics in Number Theory and Modular Forms
Catalog Number: 0464
Benedict H. Gross 1112

*Mathematics 313. Topics in Geometrical Representation Theory - (New Course)
Catalog Number: 65047
Xinwen Zhu 6373

*Mathematics 314. Topics in Differential Geometry and Mathematical Physics
Catalog Number: 2743
Shlomo Z. Sternberg 1965 (on leave fall term)

*Mathematics 316. Topics in Algebraic Topology / Arithmetic Geometry - (New Course)
Catalog Number: 97966
Kirsten Wickelgren 6374 (spring term only)

*Mathematics 317. Topics in Number Theory and Algebraic Geometry - (New Course)
Catalog Number: 46444
Joseph David Rabinoff 6356

*Mathematics 318. Topics in Number Theory
Catalog Number: 7393
Barry C. Mazur 1975

*Mathematics 320. Topics in Deformation Theory - (New Course)
Catalog Number: 84773
Hao Xu 6520

*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

*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry
Catalog Number: 9401
Curtis T. McMullen 3588

*Mathematics 335. Topics in Differential Geometry and Analysis
Catalog Number: 5498
Clifford Taubes 1243 (on leave 2009-10)

*Mathematics 338. Topics in Complex Dynamics - (New Course)
Catalog Number: 61551
Sarah Colleen Koch 6308

*Mathematics 345. Topics in Geometry and Topology
Catalog Number: 4108
Peter B. Kronheimer 1759 (on leave spring term)

*Mathematics 346y. Topics in Analysis: Quantum Dynamics
Catalog Number: 1053
Horng-Tzer Yau 5260

*Mathematics 349. Topics in Algebraic Number Theory
Catalog Number: 0278
Toby Gee 6018

*Mathematics 350. Topics in Mathematical Logic
Catalog Number: 5151
Gerald E. Sacks 3862

*Mathematics 351. Topics in Algebraic Number Theory
Catalog Number: 3492
Richard L. Taylor 1453

*Mathematics 352. Topics in Algebraic Number Theory - (New Course)
Catalog Number: 86228
Mark Kisin 6281

*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 369. Topics in Derived Algebraic Geometry
Catalog Number: 3401
Clark Barwick

*Mathematics 372. Topics in Mathematical Relativity
Catalog Number: 1150
Lydia Rosina Bieri 5794

*Mathematics 373. Topics in Algebraic Topology - (New Course)
Catalog Number: 49813
Jacob Lurie 5450

*Mathematics 379. Topics in Combinatorics
Catalog Number: 3390
Lauren K. Williams 5499 (on leave 2009-10)

*Mathematics 381. Introduction to Geometric Representation Theory
Catalog Number: 0800
Dennis Gaitsgory 5259

*Mathematics 382. Topics in Algebraic Geometry
Catalog Number: 2037
Joseph D. Harris 2055

*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 394. Topics in Many-Body Quantum System
Catalog Number: 9143
Jun Yin

*Mathematics 395. Topics in Symplectic, Contact, and Low - Dimensional Topology - (New Course)
Catalog Number: 10029
Andrew Cotton-Clay

General education course

Fat Chance

Source

This page was last fetched from here and adapted to this page on Fri Jul 16 04:00:10 EDT 2010.