Mathematics Mb (formerly Mathematics Xb). Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Robin Gottlieb, Juliana Belding, Janet Chen, David Ayala, 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.
Peter M. Garfield, Rehana Patel, 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 2, 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.
Clifford Taubes, Janet Chen, Mboyo Esole and Wei Zhang (fall term); Janet Chen, Juliana Belding, and Paul Bourgade (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: 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 1, 8:30 am, Science Center C. Required first meeting in spring: Monday, January 24, 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
Oliver Knill
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.
Wilfried Schmid, Jameel Al-Aidroos, Vaibhav Gadre, Antti Knowles, and Jun Yin (fall term); Jameel Al-Aidroos, Nir Avni, Peter Garfield, and Joseph Rabinoff (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 2, 8:30 am, Science Center C. Required first meeting in spring: Tuesday, 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.
Oliver Knill and members of the Department (fall term); Rehana Patel, John Hall, 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, M., W., F., at 12 (with sufficient enrollment); Section IV, Tu., Th., 10-11:30; Section V, Tu., Th., 11:30-1, and a weekly problem section to be arranged. EXAM GROUP: 1
Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as vectors, Euclidean spaces, linear transformations, determinants, eigenvalues, and eigenvectors. Of applications given, a regular section considers dynamical systems and both ordinary and partial differential equations plus an introduction to Fourier series.
Note: Required first meeting in fall: Wednesday, September 1, 8:30 am, Science Center B. Required first meeting in spring: Monday, January 24, 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
Benedict H. Gross
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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
Benedict H. Gross
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
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
Noam D. Elkies
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
Noam D. Elkies
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 - (New Course)
Catalog Number: 97995
Paul G. Bamberg
Half course (spring term). M., W., 1–2:30. EXAM GROUP: 6, 7
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 100level.

Mathematics 112. Introductory Real Analysis
Catalog Number: 1123
Yum Tong Siu
Half course (fall 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
Antti Knowles
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
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 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 118r. Dynamical Systems
Catalog Number: 6402
Vaibhav Suresh Gadre
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.
Prerequisite: Mathematics 21a,b.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Joseph David Rabinoff
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
Dennis Gaitsgory
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.
Prerequisite: Mathematics 122 or 55a.

Mathematics 124. Number Theory
Catalog Number: 2398
Joseph D. Harris
Half course (spring term). M., W., F. at 10. EXAM GROUP: 3
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
Sarah Colleen Koch
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
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
Jacob Lurie
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
Michael J. Hopkins
Half course (spring term). M., W., F., at 3. EXAM GROUP: 8
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
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Peter B. Kronheimer
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
Rachel Louise Epstein
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
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 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.
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.
Note: Expected to be given in 2011–12.
Prerequisite: Mathematics 21a,b.

Mathematics 154 (formerly Mathematics 191). Probability Theory
Catalog Number: 4306
Curtis T. McMullen
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Paul Bourgade
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 162. Introduction to Quantum Computing - (New Course)
Catalog Number: 93376
Clifford Taubes
Half course (spring term). M., W., F., at 2:30. EXAM GROUP: 7, 8
This course is meant to give an introduction to the fundamental mathematics of quantum computing. Notions from linear algebra, elementary number theory and probability theory are introduced along the way as needed.

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
Curtis T. McMullen
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
A second course in complex analysis: elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure and capacity.
Prerequisite: Mathematics 55b or 113.

Mathematics 213br. Advanced Complex Analysis
Catalog Number: 2641
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
Jacob Lurie
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
Xinwen Zhu
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
Mark Kisin
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.
Prerequisite: Mathematics 129.

Mathematics 223b (formerly Mathematics 251b). Algebraic Number Theory
Catalog Number: 2783
Wei Zhang
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.
Prerequisite: Mathematics 223a.

[Mathematics 224. Representations of Reductive Lie Groups]
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.
Note: Expected to be given in 2011–12.

[Mathematics 229x. Introduction to Analytic Number Theory]
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.
Note: Expected to be given in 2011–12.
Prerequisite: Mathematics 113, 123

Mathematics 230a. Differential Geometry
Catalog Number: 0372
Andrew W. Cotton-Clay
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
Andrew W. Cotton-Clay
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
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
Michael J. Hopkins
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
Nir David Avni
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
Hao Xu
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
Instructor to be determined
Half course (fall term). Hours to be arranged.
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.
Note: Expected to be given in 2011–12.
Prerequisite: Mathematics 221 and 232a or permission of instructor.

[Mathematics 233br (formerly Mathematics 261b). Theory of Schemes II]
Catalog Number: 3316
Instructor to be determined
Half course (spring term). Hours to be arranged.
A continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.
Note: Expected to be given in 2011–12.
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 253. Introduction to Computability and Randomness - (New Course)
Catalog Number: 67781
Rachel Louise Epstein
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
An introduction to computability theory and algorithmic randomness. Topics: Turing reducibility, computably enumerable sets, complexity, notions of randomness, and martingales, as well as interactions between computability and randomness.

Mathematics 255. Topics in General Relativity - (New Course)
Catalog Number: 10058
Lars Ake Andersson (Max Planck Institute)
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Following an introduction to the Cauchy problem in general relativity from the point of view of nonlinear hyperbolic equations, we review the geometry of black hole spacetimes and discuss methods for analyzing the global behavior of waves on such backgrounds.

Mathematics 258x. Random Matrix - (New Course)
Catalog Number: 80974
Horng-Tzer Yau
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The goal of this course is to give a detailed account of the recent advances concerning the local statistics of eigenvalue distributions of random matrices. Basic knowledge of probability theory and measure theory are required.

Mathematics 267x. Semiclassical Analysis - (New Course)
Catalog Number: 29452
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Typical subjects are high frequency approximations to solutions of hyperbolic equations and the asymptotics of Schrodingers equation when Plancks constant is thought of as a small parameter. Background in symplectic geometry and functional analysis will be provided.

Mathematics 276x. General Relativity Seminar - (New Course)
Catalog Number: 76687
Lars Ake Andersson (Max Planck Institute)
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
A literature course in mathematical general relativity, based on research papers. The presentation will involve significant participation from the students.

Mathematics 277x. Mapping Class Groups and Teichmuller Theory - (New Course)
Catalog Number: 96081
Vaibhav Suresh Gadre
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
Topics in the study of mapping class groups and Teichmuller Theory. These may include: random walks on mapping class groups and properties of Teichmuller flow.

Mathematics 278x. Analytic Methods in Complex and Algebraic Geometry - (New Course)
Catalog Number: 54129
Yum Tong Siu
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
A discussion of methods of d-bar estimates and multiplier ideal sheaves. Topics: effective results in algebraic geometry, invariance of plurigenera, finite generation of canonical rings, the abundance conjecture, and algebraic-geometric techniques in PDE estimates.

Mathematics 281x. Degeneration Methods in Enumerative Geometry - (New Course)
Catalog Number: 44559
Yu-jong Tzeng
Half course (fall term). Th., 1–2:30, W., 4:30–6.
Topics may include Hilbert schemes, degeneration methods, algebraic cobordism, theory of counting nodal curves and their applications.

Mathematics 282x. Algebraic Differential Equations - (New Course)
Catalog Number: 59666
Xinwen Zhu
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
An introduction to algebraic differential equations both in characteristic zero and characteristic p, and some applications.

Mathematics 283x. Some Aspects of Trace Formulae - (New Course)
Catalog Number: 60789
Wei Zhang
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
An introduction to the Arthur-Selberg trace formula and the relative trace formula. Subtopics: basic harmonic analysis on p-adic groups (orbital integrals and Shalika germs), counting points on Shimura varieties, etc.

Mathematics 286x. Finite Linear Groups and Their Representations - (New Course)
Catalog Number: 74773
Nir David Avni
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Structure of finite subgroups of GL(n,F) Kirillovs Orbit Method, structure of reductive groups over finite fields, Deligne-Lusztig theory.

Mathematics 287x. Algebraic L-Theory and Surgery - (New Course)
Catalog Number: 23251
Jacob Lurie
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
We will develop the algebraic L-theory of quadratic forms and its applications to the classification of manifolds of large dimension.

Mathematics 294x. Complex Manifolds, Its Complex Structure and the Metrics Supported by Them - (New Course)
Catalog Number: 38358
Shing-Tung Yau
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
A lecture on the space of complex structures and the intrinsic metrics associated to the complex structure and its bundles. A discussion of nonlinear equations. A possible discussion of some connections with string theory.

Mathematics 296. Complex Dynamics and Fractal Groups - (New Course)
Catalog Number: 53465
Sarah Colleen Koch
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Introduction to complex dynamics and self-similar groups. Topics: dynamics of rational maps, Thurstons topological characterization of rational maps, iterated monodromy groups, Teichmuller spaces, mapping class groups, and the classification of twisted rabbits.

Mathematics 297. Stochastic Analysis - (New Course)
Catalog Number: 68572
Paul Bourgade
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
This course consists of both the classical properties of stochastic processes (Brownian motion, Itô calculus) and a differential analysis of these random paths. This requires notions of probability theory and functional analysis.

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

*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
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

*Mathematics 318. Topics in Number Theory
Catalog Number: 7393
Barry C. Mazur 1975 (on leave spring term)

*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 - (New Course)
Catalog Number: 83679
Yu-jong Tzeng 6722 (fall term only)

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

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

*Mathematics 336. Topics in Mathematical Logic - (New Course)
Catalog Number: 32157
Rachel Louise Epstein 6721

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

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

*Mathematics 346y. Topics in Analysis: Quantum Dynamics
Catalog Number: 1053
Horng-Tzer Yau 5260 (on leave fall term)

*Mathematics 350. Topics in Mathematical Logic
Catalog Number: 5151
Gerald E. Sacks 3862 (on leave spring term)

*Mathematics 351. Topics in Algebraic Number Theory
Catalog Number: 3492
Richard L. Taylor 1453 (on leave 2010-11)

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

*Mathematics 353. Topics in Teichmüller Theory - (New Course)
Catalog Number: 98786
Vaibhav Suresh Gadre 6623

*Mathematics 356. Topics in Harmonic Analysis
Catalog Number: 6534
Wilfried Schmid 5097

*Mathematics 358. Topics in Arithmetic Geometry - (New Course)
Catalog Number: 30858
Junecue Suh 6835 (on leave 2010-11)

*Mathematics 365. Topics in Differential Geometry
Catalog Number: 4647
Shing-Tung Yau 1734

*Mathematics 366. Topics in Probability and Analytic Number Theory - (New Course)
Catalog Number: 64285
Paul Bourgade 6720

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

*Mathematics 377. Topics in Number Theory - (New Course)
Catalog Number: 90085
Sophie Marguerite Morel 6309 (on leave 2010-11)

*Mathematics 381. Introduction to Geometric Representation Theory
Catalog Number: 0800
Dennis Gaitsgory 5259 (on leave fall term)

*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 (on leave fall term)

*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
Catalog Number: 10029
Andrew Cotton-Clay

*Mathematics 397. Some Aspects of Trace Formula - (New Course)
Catalog Number: 15751
Wei Zhang 6723

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