Mathematics Mb (formerly Mathematics Xb). Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Janet Chen, Meghan Anderson, Rachel Epstein, 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: Required first Meeting in spring: Monday, January 23, 8:30 am, Science Center D. Participation in two, one and a half hour workshops are required each week. 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.
Peter M. Garfield, Jameel Al-Aidroos, Juliana Belding, and members of the Department (fall term); Oliver Knill (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 1, 8:30 am, Science Center C. 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.
Clifford Taubes, John Hall, Meghan Anderson, David Ayala, and Keerthi Madapusi (fall term); Robin Gottlieb, Meghan Anderson, Juliana Belding and Janet Chen (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 (with sufficient enrollment); Section V, Tu., Th., 11:30-1(with sufficient enrollment), and a weekly problem section to be arranged. Required exams. 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, August 31, 8:30 am, Science Center C. Required first meeting in spring: Monday, January 23, 8:30 am, Science Center C. 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 1a, or Ma and Mb, or equivalent.

Mathematics 19a. Modeling and Differential Equations for the Life Sciences
Catalog Number: 1256
John T. Hall
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 life sciences, ecology and environmental life science. 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 ESPP. 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 McKee 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 life 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 ESPP. 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
Rachel Louise Epstein
Half course (fall term). M., W., F., at 9. 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 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, Paul Bourgade, Emily Riehl, Junecue Su, Yu-Jong Tzeng, and Members of the Department (fall term); Jameel Al-Aidroos, Peter Garfield, Sophie Morel, and Members of the Department (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 1, 8:30 am, Science Center B. Required first meeting in spring: Tuesday, January 24, 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.
Janet Chen, Sophie Morel, and members of the Department (fall term); John Hall, Juliana Belding, and members of the Department (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, Tu., Th., 10-11:30; Section IV, 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: Wednesday, August 31, 8:30 am, Science Center B. Required first meeting in spring: Monday, January 23, 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. 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
Sarah Colleen Koch
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Sarah Colleen Koch
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Yum Tong Siu
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
Yum Tong Siu
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
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.

Mathematics 110. Vector Space Methods for Differential Equations
Catalog Number: 97995
Paul G. Bamberg
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Develops the theory of inner product spaces, both finite-dimensional and infinite-dimensional, and applies it to a variety of ordinary and partial differential equations. Topics: existence and uniqueness theorems, Sturm-Liouville systems, orthogonal polynomials, Fourier series, Fourier and Laplace transforms, eigenvalue problems, and solutions of Laplaces equation and the wave equation in the various coordinate systems.
Prerequisite: Mathematics 23ab or 25 ab, or Mathematics 21ab plus any Mathematics course at the 100 level.

Mathematics 112. Introductory Real Analysis
Catalog Number: 1123
Jacob Lurie
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral.
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
Horng-Tzer Yau
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Peter B. Kronheimer
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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
Horng-Tzer Yau
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). M., W., 1–2:30. EXAM GROUP: 7
Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals 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 117. Probability and Random Processes with Economic Applications - (New Course)
Catalog Number: 45584
Paul G. Bamberg
Half course (spring term). M., W., 1–2:30. EXAM GROUP: 6, 7
A self-contained treatment of the theory of probability and random processes. Topics: axioms for probability, conditional probability, Poisson and normal distributions, random walks, laws of large numbers and the central limit theorem, Markov processes, martingales, and Poisson and diffusion processes. Applications to games of chance, the design of minimum-variance portfolios, and the Black-Scholes theory of option pricing. While emphasis will be on careful analysis of models, occasional guest lectures will explore applicability to the real world.
Prerequisite: Familiarity with multivariable calculus and linear algebra, e.g. Mathematics 21ab or 23ab. Prior experience with elementary probability (e.g. AP Statistics or Statistics 104) is desirable.

Mathematics 118r. Dynamical Systems
Catalog Number: 6402
Paul Bourgade
Half course (fall 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.
Prerequisite: Mathematics 21a,b.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Vaibhav Suresh Gadre
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to 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
Nir David Avni
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
Joseph D. Harris
Half course (spring term). M., W., F., at 2:30. EXAM GROUP: 7, 8
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
Barry C. Mazur
Half course (fall term). Tu. .,Th., 10-11:30. EXAM GROUP: 12, 13
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
Mark Kisin
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
Michael J. Hopkins
Half course (spring term). M., W., F., at 1:30. EXAM GROUP: 6, 7
Presents axioms for several geometries (affine, projective, Euclidean, spherical, hyperbolic). Develops models for these geometries using three-dimensional vector spaces over the reals, or over finite fields. Emphasis on reading and writing proofs.
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
Kirsten Graham Wickelgren
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
Clifford Taubes
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
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
Xinwen Zhu
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
Dennis Gaitsgory
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
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 11. EXAM GROUP: 4
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). M., W., F., at 12. EXAM GROUP: 5
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.
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 (fall 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 2012–13.
Prerequisite: Mathematics 123 or the equivalent is suggested as a prerequisite, but not required.

Mathematics 152. Discrete Mathematics
Catalog Number: 8389
Juliana Victoria Belding
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
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
Paul Bourgade
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: A previous mathematics course at the level of Mathematics 19ab, 21ab, or higher. For students from 19ab or 21ab, previous or concurrent enrollment in Math 101 or 112 may be helpful. Freshmen who did well in Math 23, 25 or 55 last term are also welcome to take the course.

Mathematics 155r (formerly Mathematics 192r). Combinatorics
Catalog Number: 6612
Jacob Lurie
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
An introduction to counting techniques and other methods in finite mathematics. Possible topics include: the inclusion-exclusion principle and Mobius inversion, graph theory, generating functions, Ramseys theorem and its variants, probabilistic methods.
Prerequisite: The ability to read and write mathematical proofs. Some familiarity with group theory (Math 122 or equivalent.)

Mathematics 167. Introduction to Symplectic Geometry - (New Course)
Catalog Number: 52244
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The basics of symplectic geometry with applications to Hamiltonian mechanics and other areas of physics and geometry.
Prerequisite: Linear algebra and advanced calculus.

Mathematics 212br. Advanced Real Analysis
Catalog Number: 7294
Antti Knowles
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
Joseph D. Harris
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
A second course in complex analysis: series, product and partial fraction expansions of holomorphic functions; Hadamards theorem; conformal mapping and the Riemann mapping theorem; elliptic functions; Picards theorem and Nevanlinna Theory.
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
Instructor to be determined
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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.
Note: Expected to be given in 2012–13.
Prerequisite: Mathematics 123.

Mathematics 222. Lie Groups and Lie Algebras
Catalog Number: 6738
Wilfried Schmid
Half course (spring 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). M., W., F., at 12. EXAM GROUP: 5
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 2012–13.
Prerequisite: Mathematics 129.

[Mathematics 223b (formerly Mathematics 251b). Algebraic Number Theory]
Catalog Number: 2783
Instructor to be determined
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
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 2012–13.
Prerequisite: Mathematics 223a.

[Mathematics 224. Representations of Reductive Lie Groups]
Catalog Number: 25927
Instructor to be determined
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.
Note: Expected to be given in 2012–13.

Mathematics 229x. Introduction to Analytic Number Theory
Catalog Number: 41034
Barry C. Mazur
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Hao Xu
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
Elements of differential geometry: Lie groups, vector bundles, principle bundles, connections, curvature, Chern classes, geodesics, Riemannian curvature, introduction to complex and Kahler manifolds.
Prerequisite: Mathematics 132 or equivalent.

Mathematics 230br. Advanced Differential Geometry
Catalog Number: 0504
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications.
Prerequisite: Mathematics 230a.

Mathematics 231a (formerly Mathematics 272a). Algebraic Topology
Catalog Number: 7275
Michael J. Hopkins
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
Peter B. Kronheimer
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Continuation of Mathematics 231a. 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
Xinwen Zhu
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
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
Xinwen Zhu
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
Junecue Suh
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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
Junecue Suh
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
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 251x. Vanishing of Torsion in the Cohomology of Arithmetic Groups - (New Course)
Catalog Number: 79421
Keerthi Shyam Madapusi Sampath
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
The goal of this course is to understand the results of Lan and Suh on the vanishing torsion in the cohomology of certain Shimura varieties of PEL types.

Mathematics 254y. Geometry with Valuations - (New Course)
Catalog Number: 64314
Nir David Avni
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
Possible topics: elimination of quantifiers in Henselian valued fields, motivic integration, stably dominated types, Berkovich spaces, tropicalization.

Mathematics 259x. Analytic Theory of Modular Forms - (New Course)
Catalog Number: 33309
Noam D. Elkies and members of the department
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
An introduction to automorphic forms on GL(2) from a classical perspective and an explanation of their use in studying analytic problems such as Dukes theorem, Hilberts eleventh problem, and subconvexity.

Mathematics 261y. von Neumann Algebras - (New Course)
Catalog Number: 44588
Jacob Lurie
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
An introduction to the theory of von Neumann algebras, emphasizing their applications to the study of topological field theories.

Mathematics 262y. Perverse Sheaves in Representation Theory - (New Course)
Catalog Number: 84831
Carl Mautner
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
An introduction to perverse sheaves with a view towards modular representation theory. Possible topics: the geometric Satake theorem, Springer theory and (modular) Deligne-Lusztig theory.

Mathematics 265y. Topics in the Moduli Theory of Sheaves - (New Course)
Catalog Number: 94528
Baosen Wu
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
An introduction to the geometry of moduli spaces of stable sheaves on curves and surfaces. Topics may include Verlinde formula, Donaldson-Thomas invariants, etc.

Mathematics 266x. Categorical Homotopy Theory - (New Course)
Catalog Number: 29481
Emily Elizabeth Riehl
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
A survey of categorical tools for homotopy theory emphasizing the formal relationships among the following topics: weak factorization systems, model categories; enriched categories; Kan extensions, derived functors; homotopy colimits, the bar construction; infinity categories.

Mathematics 268x. Graph Limits - (New Course)
Catalog Number: 12792
Gabor P. Lippner
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Introduction to the emerging field of relating large graphs to analytical objects. Topics may include: ultra-limit method and Szemeredi regularity, constant-time algorithms, Borel graphs and measurable equivalence relations, Gromovs sofic groups.

Mathematics 270. Advanced Probability Theory - (New Course)
Catalog Number: 44129
Antti Knowles
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The axioms of Kolmogorov, convergence and limit theorems, random walks, martingales and Markov processes in discrete time, concentration of measure and large deviations.
Prerequisite: Basic measure theory; some elementary probability theory is recommended but not required.

Mathematics 271y. Probability Theory and Stochastic Process - (New Course)
Catalog Number: 82126
Horng-Tzer Yau
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
We will cover the construction of Brownian motions and develop the Ito calculus. We will review discrete martingale and stopping time. A knowledge of measure theory and basic probability is required.

Mathematics 273y. Contact Geometry in 3 Dimensions - (New Course)
Catalog Number: 27899
Steven Sivek
Half course (spring term). W., F., 1:30–3. EXAM GROUP: 6, 7, 8
An introduction to contact structures on 3-manifolds. Topics: the classification of overtwisted structures, symplectic fillings and tightness, convex surface theory and bypasses, Legendrian knots, open book decompositions and the Giroux correspondence.

Mathematics 285x. Representations of Reductive Groups over Local Non-Archimedian Fields - (New Course)
Catalog Number: 48416
David Kazhdan
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
A presentation of the theory of irreducible representations of split groups and reductive groups G over local non-archimedian fields. An explanation of parametrization of connected components of the space of irreducible representations of G and a description of the local behavior of characters of irreducible representations of G.

Mathematics 287y. Geometry of Algebraic Curves - (New Course)
Catalog Number: 27977
Joseph D. Harris
Half course (fall term). W., F., 3–4:30. EXAM GROUP: 8, 9
This course will survey the geometry of Riemann surfaces/algebraic curves, in the abstract and in projective space, with examples.

Mathematics 288x. The Kahler-Einstein Metrics - (New Course)
Catalog Number: 36805
Hao Xu
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
An introduction to the existence problem of Kahler-Einstein metric on algebraic manifolds.

Mathematics 289x. Equivariant Stable Homotopy Theory - (New Course)
Catalog Number: 43338
Michael J. Hopkins
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
This course will cover the basics of equivariant stable homotopy theory and go through the solution of the Kervaire invariant problem.

Mathematics 291x. Seminar on Geometric Representation Theory - (New Course)
Catalog Number: 23026
David Kazhdan
Half course (fall term). Tu., 4:30–7 p.m. EXAM GROUP: 8, 9
A study of topics on geometric representation theory.

Mathematics 298. Random Matrices - (New Course)
Catalog Number: 38719
Alexander Bloemendal
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
An introduction to random matrix theory. Topics: Wigner matrices, Gaussian and circular ensembles, Dysons Brownian motion, determinantal processes, orthogonal polynomials, bulk and edge scaling limits, beta ensembles, continuum limits, and various recent applications.

*Mathematics 299. Graduate Tutorial in Number Theory
Catalog Number: 8799
Mark Kisin (fall term) and Dennis Gaitsgory (spring term)
Half course (fall term; repeated spring term). M., 1:30–3. EXAM GROUP: 6, 7
An advanced topics course in algebraic number theory. Material will be taken from research papers, both classical and contemporary, and the presentation will involve significant participation from the students.
Note: Enrollment limited, please contact the professor before registering.

*Mathematics 304. Topics in Algebraic Topology
Catalog Number: 0689
Michael J. Hopkins 4376

*Mathematics 308. Topics in Number Theory and Modular Forms
Catalog Number: 0464
Benedict H. Gross 1112 (on leave 2011-12)

*Mathematics 313. Topics in Geometrical Representation Theory
Catalog Number: 65047
Xinwen Zhu 6373

*Mathematics 314. Topics in Differential Geometry and Mathematical Physics
Catalog Number: 2743
Shlomo Z. Sternberg 1965

*Mathematics 316. Topics in Algebraic Topology / Arithmetic Geometry
Catalog Number: 97966
Kirsten Graham Wickelgren 6374

*Mathematics 317. Topics in Number Theory and Algebraic Geometry
Catalog Number: 46444
Joseph David Rabinoff 6356 (on leave 2011-12)

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

*Mathematics 320. Topics in Deformation Theory
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 332. Topics in Algebraic Geometry
Catalog Number: 83679
Yu-jong Tzeng 6722

*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry
Catalog Number: 9401
Curtis T. McMullen 3588 (on leave 2011-12)

*Mathematics 335. Topics in Differential Geometry and Analysis
Catalog Number: 5498
Clifford Taubes 1243

*Mathematics 336. Topics in Mathematical Logic
Catalog Number: 32157
Rachel Louise Epstein 6721

*Mathematics 338. Topics in Complex Dynamics
Catalog Number: 61551
Sarah Colleen Koch 6308

*Mathematics 341. Topics in Number Theory - (New Course)
Catalog Number: 28563
Keerthi Shyam Madapusi Sampath 2232

*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

*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 (on leave 2011-12)

*Mathematics 352. Topics in Algebraic Number Theory
Catalog Number: 86228
Mark Kisin 6281

*Mathematics 353. Topics in Teichmüller Theory
Catalog Number: 98786
Vaibhav Suresh Gadre 6623

*Mathematics 355. Topics in Category Theory and Homotopy Theory - (New Course)
Catalog Number: 95192
Emily Elizabeth Riehl 1416

*Mathematics 356. Topics in Harmonic Analysis
Catalog Number: 6534
Wilfried Schmid 5097 (on leave fall term)

*Mathematics 358. Topics in Arithmetic Geometry
Catalog Number: 30858
Junecue Suh 6835

*Mathematics 365. Topics in Differential Geometry
Catalog Number: 4647
Shing-Tung Yau 1734 (on leave 2011-12)

*Mathematics 366. Topics in Probability and Analytic Number Theory
Catalog Number: 64285
Paul Bourgade 6720

*Mathematics 373. Topics in Algebraic Topology
Catalog Number: 49813
Jacob Lurie 5450

*Mathematics 377. Topics in Number Theory
Catalog Number: 90085
Sophie Marguerite Morel 6309

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

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