Mathematics Xb. Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Bret J. Benesh, John T. Hall, Brian Munson, and members of the Department
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, as well as required participation in a one hour problem session each week. This course, when taken for a letter grade together with Mathematics Xa, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics Xa.

Mathematics 1a. Introduction to Calculus
Catalog Number: 8434 Enrollment: Normally limited to 30 students per section.
Matthew P. Leingang, John Duncan, and Rehana Patel (fall term); Matthew P. Leingang (spring term)
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10-11:30; Section Vl, Tu., Th., 11:30-1. Spring: Section I, M., W., F., at 10; Section II, Tu.Th. 10-11:30 (with sufficient enrollment) and a weekly problem section to be arranged. EXAM GROUP: 1
The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to optimization, graphing, mechanisms, and problems from many other disciplines.
Note: Required first meeting in fall: Tuesday, September 18, 8:30 am, Science Center B. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: A solid background in precalculus.

Mathematics 1b. Calculus, Series, and Differential Equations
Catalog Number: 1804 Enrollment: Normally limited to 30 students per section.
Thomas W. Judson, Danijela Damjanovic, John T. Hall, Brian Munson, and Robert Strain (fall term); Robin Gottlieb, Bret Benesh, and Lydia Bieri (spring term)
Half course (fall term; repeated spring term). Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12 (with sufficient enrollment); Section V: Tu., Th., 10-11:30; Section Vl, Tu., Th., 11:30-1, and a weekly problem section to be arranged. Required exams: Hours to be arranged. EXAM GROUP: 1
Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.
Note: Required first meeting in fall: Monday, September 17, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center A. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 1a, or Xa and Xb, or equivalent.

Mathematics 19a. Modeling and Differential Equations for the Life Sciences
Catalog Number: 1256
Thomas W. Judson (fall term); John T. Hall (spring term)
Half course (fall term; repeated spring term). M., W., F., at 1, and a weekly problem section to be arranged. EXAM GROUP: 6
Considers the construction and analysis of mathematical models that arise in the environmental sciences, biology, the ecological sciences, and in earth and atmospheric sciences. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).
Note: This course is recommended over Math 21a for those planning to concentrate in the life sciences, chemistry, or environmental sciences. Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 20. This course can be taken before or after Mathematics 20. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

Mathematics 19b. Linear Algebra, Probability, and Statistics for the Life Sciences
Catalog Number: 6144
Clifford Taubes
Half course (spring term). M., W., F., at 1 and a weekly problem section to be arranged. EXAM GROUP: 6
Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.
Note: This course is recommended over Math 21b for those planning to concentrate in the life sciences, chemistry, or environmental sciences. Can be taken with Mathematics 21a. Students who have seen some multivariable calculus can take Math 19b before Math 19a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

Mathematics 20. Algebra and Multivariable Mathematics for Social Sciences
Catalog Number: 0906
Matthew P. Leingang (fall term); Rehana Patel (spring term)
Half course (fall term; repeated spring term). M., W., F., at 9, and a weekly problem section to be arranged. EXAM GROUP: 2
Introduction to linear algebra, including vectors, matrices, and applications. Calculus of functions of several variables, including partial derivatives, constrained and unconstrained optimization, and applications. Covers the topics from Mathematics 21a,b which are most important in applications to economics, the social sciences, and some other fields.
Note: Should not ordinarily be taken in addition to Mathematics 21a,b. Examples drawn primarily from economics and the social sciences though Mathematics 20 may be useful to students in certain natural sciences. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 1b or equivalent, or an A or A- in Mathematics 1a, or a 5 on the AB or a 3 or higher on the BC Advanced Placement Examinations in Mathematics.

Mathematics 21a. Multivariable Calculus
Catalog Number: 6760 Enrollment: Normally limited to 30 students per section.
Oliver Knill, Véronique Godin, and Rehana Patel (fall term); Thomas Judson, and Matthew P. Leingang (spring term)
Half course (fall term; repeated spring term). Fall: Section I, M., W., F., at 9 (with sufficient enrollment); Section II, M., W., F., at 10; Section III, M., W., F., at 11; Section IV, M., W., F., at 12; Section V, Tu., Th., 10-11:30; Section VI, Tu., Th., 11:30-1; and a weekly problem section to be arranged. . EXAM GROUP: 1
To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives, and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces, and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Greens, Stokes, and Divergence Theorems.
Note: Required first meeting in fall: Tuesday, September 18, 8:30 am, Science Center C. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center C. May not be taken for credit by students who have passed Applied Mathematics 21a. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning. Activities using computers to calculate and visualize applications of these ideas will not require previous programming experience. Special sections for students interested in physics are offered each term.
Prerequisite: Mathematics 1b or equivalent.

Mathematics 21b. Linear Algebra and Differential Equations
Catalog Number: 1771 Enrollment: Normally limited to 30 students per section.
Yum Tong Siu (fall term); Oliver Knill, Samit Dasgupta, and Rehana Patel (spring term)
Half course (fall term; repeated spring term). Fall: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); Spring: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M., W., F., at 12 (with sufficient enrollment); Section IV: Tu., Th., 10-11:30; Section V: Tu., Th., 11:30-1 and a weekly problem section to be arranged. EXAM GROUP: 1
Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as vectors, Euclidean spaces, linear transformations, determinants, eigenvalues, and eigenvectors. Of applications given, a regular section considers dynamical systems and both ordinary and partial differential equations plus an introduction to Fourier series.
Note: Required first meeting in fall: Monday, September 17, 8:30 am, Science Center A. Required first meeting in spring: Wednesday, January 30, 8:30 am, Science Center D. May not be taken for credit by students who have passed Applied Mathematics 21b. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics lb or equivalent. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.

Mathematics 23a. Linear Algebra and Real Analysis I
Catalog Number: 2486
Paul G. Bamberg
Half course (fall term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4
A rigorous, integrated treatment of linear algebra and multivariable differential calculus, emphasizing topics that are relevant to fields such as physics and economics. Topics: fields, vector spaces and linear transformations, scalar and vector products, elementary topology of Euclidean space, limits, continuity, and differentiation in n dimensions, eigenvectors and eigenvalues, inverse and implicit functions, manifolds, and Lagrange multipliers. Students are expected to master twenty important proofs.
Note: Course content overlaps substantially with Mathematics 21a,b, 25a,b, so students should plan to continue in Mathematics 23b. See the description in the introductory paragraphs in the Mathematics section of the catalog about the differences between Mathematics 23 and Mathematics 25. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 1b or a grade of 4 or 5 on the Calculus BC Advanced Placement Examination, plus an interest both in proving mathematical results and in using them.

Mathematics 23b. Linear Algebra and Real Analysis II
Catalog Number: 8571
Paul G. Bamberg
Half course (spring term). M., W., F., at 11, and a weekly conference section to be arranged. EXAM GROUP: 4
A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Applications of linear algebra to differential equations and Fourier analysis. Introduction to infinite-dimensional vector spaces. 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 Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 23a.

Mathematics 25a. Honors Linear Algebra and Real Analysis I
Catalog Number: 1525
Benjamin Weinkove
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.
Note: Only for students with a strong interest and background in mathematics. There will be a heavy workload. May not be taken for credit after Mathematics 23. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: 5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.

Mathematics 25b. Honors Linear Algebra and Real Analysis II
Catalog Number: 1590
Benjamin Weinkove
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
A rigorous treatment of basic analysis. Topics include: convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.
Note: There will be a heavy workload. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.
Prerequisite: Mathematics 23a or 25a or 55a.

*Mathematics 55a. Honors Abstract Algebra
Catalog Number: 4068
Dennis Gaitsgory
Half course (fall term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
A rigorous treatment of abstract algebra including linear algebra and group theory.
Note: Mathematics 55a is an intensive course for students having significant experience with abstract mathematics. 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 Core area requirement for Quantitative Reasoning.

Mathematics 55b. Honors Real and Complex Analysis
Catalog Number: 3312
Samit Dasgupta
Half course (spring term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
A rigorous treatment of real and complex analysis.
Note: Mathematics 55b is an intensive course for students having significant experience with abstract mathematics. Instructors permission required. This course, when taken for a letter grade, meets the Core area requirement for Quantitative Reasoning.

*Mathematics 60r. Reading Course for Senior Honors Candidates
Catalog Number: 8500
Peter B. Kronheimer
Half course (fall term; repeated spring term). Hours to be arranged.
Advanced reading in topics not covered in courses.
Note: Limited to candidates for honors in Mathematics who obtain the permission of both the faculty member under whom they want to work and the Director of Undergraduate Studies. May not count for concentration in Mathematics without special permission from the Director of Undergraduate Studies. Graded Sat/Unsat only.

*Mathematics 91r. Supervised Reading and Research
Catalog Number: 2165
Peter B. Kronheimer
Half course (fall term; repeated spring term). Hours to be arranged.
Programs of directed study supervised by a person approved by the Department.
Note: May not ordinarily count for concentration in Mathematics.

*Mathematics 99r. Tutorial
Catalog Number: 6024
Peter B. Kronheimer and members of the Department
Half course (fall term; repeated spring term). Hours to be arranged.
Topics for 2007-08: (1) Random graphs (fall), prerequisite: Math 122 or familiarity with abstract linear algebra and group theory. Knowledge of elementary probability theory helpful, but not required. (2) Clifford algebras and spinors (spring), prerequisite: Math 122 or familiarity with abstract linear algebra, groups and group actions.
Note: May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

Mathematics 106. Ordinary Differential Equations
Catalog Number: 3377
Thomas W. Judson
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
Analytic, numerical, and qualitative analysis of ordinary differential equations. Linear equations, linear and non-linear systems. Applications to mechanics, biology, physics, and the social sciences. Existence and uniqueness of solutions and visual analysis using computer graphics. Topics selected from Laplace transforms, power series solutions, chaos, and numerical solutions.
Prerequisite: Mathematics 19a,b, 20 or 21a.

Mathematics 112. Introductory Real Analysis
Catalog Number: 1123
Danijela Damjanovic
Half course (spring term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral.
Prerequisite: Mathematics 21a,b and either an ability to write proofs or concurrent enrollment in Mathematics 101. Should not ordinarily be taken in addition to Mathematics 23a,b, 25a,b or 55a,b.

Mathematics 113. Analysis I: Complex Function Theory
Catalog Number: 0405
Robert M. Strain
Half course (spring term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
Analytic functions of one complex variable: power series expansions, contour integrals, 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 - (New Course)
Catalog Number: 9111
Curtis T. McMullen
Half course (fall term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
Lebesgue measure and integration; general topology; introduction to L ^p spaces, Banach and Hilbert spaces, and duality.
Prerequisite: Mathematics 23, 25, 55, or 112.

[Mathematics 115. Methods of Analysis]
Catalog Number: 1871
Wilfried Schmid
Half course (fall term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Complex functions; Fourier analysis; Hilbert spaces and operators; Laplaces equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.
Note: Expected to be given in 2008-09. Mathematics 115 is especially for students interested in physics.
Prerequisite: Mathematics 23a,b, 25a,b, 55a,b, or 112.

Mathematics 116. Convexity and Optimization with Applications
Catalog Number: 5253
Paul G. Bamberg
Half course (fall term). Tu., Th., 2:30-4. EXAM GROUP: 16, 17
Introduction to real and functional analysis through topics such as convex programming, duality theory, linear and non-linear programming, calculus of variations, and the maximum principle of optimal control theory.
Prerequisite: At least one course beyond Mathematics 21.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Lydia Bieri
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Real and complex vector spaces, dual spaces, linear transformations and Jordan normal forms. Inner product spaces. Applications to differential equations, classical mechanics, and optimization theory. Emphasizes learning to understand and write proofs.
Prerequisite: Mathematics 21b or equivalent. Should not ordinarily be taken in addition to Mathematics 23a, 25a, or 55a.

Mathematics 122. Algebra I: Theory of Groups and Vector Spaces
Catalog Number: 7855
Alberto DeSole
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.
Prerequisite: Mathematics 23a, 25a, 121; or 101 with the instructors permission. Should not be taken in addition to Mathematics 55a.

Mathematics 123. Algebra II: Theory of Rings and Fields
Catalog Number: 5613
Alberto DeSole
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.
Prerequisite: Mathematics 122 or 55a.

Mathematics 124. Number Theory
Catalog Number: 2398
Joseph D. Harris
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; 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
Richard L. Taylor
Half course (spring term). M., W., F., at 9. EXAM GROUP: 2
Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.
Prerequisite: Mathematics 123.

Mathematics 130 (formerly Mathematics 138). Classical Geometry
Catalog Number: 5811
Peter B. Kronheimer
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Euclidean, spherical and hyperbolic geometry. No prior experience with proofs required.
Note: Not expected to be given 2008-09.
Prerequisite: Mathematics 21a,b, 23a, 25a or 55a (may be taken concurrently).

Mathematics 131. Topology I: Topological Spaces and the Fundamental Group
Catalog Number: 2381
Véronique Godin
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Abstract topological spaces; compactness, connectedness, continuity. Homeomorphism and homotopy, fundamental groups, covering spaces. Introduction to combinatorial topology.
Prerequisite: Some acquaintance with metric space topology (Mathematics 23a,b, 25a,b, 55a,b, 101, or 112) and with groups (Mathematics 101, 122 or 55a).

Mathematics 132. Topology II: Smooth Manifolds - (New Course)
Catalog Number: 7725
Véronique Godin
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
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
The exterior differential calculus and its application to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions and to symplectic geometry.
Prerequisite: Advanced calculus and linear algebra.

Mathematics 137. Algebraic Geometry
Catalog Number: 0556
John F. Duncan
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
Affine and projective spaces, plane curves, 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). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.
Note: Expected to be given in 2008-09.
Prerequisite: Any mathematics course at the level of Mathematics 21a,b or higher, or permission of instructor.

Mathematics 143. Set Theory
Catalog Number: 6005
Gerald E. Sacks
Half course (spring term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Axioms of set theory. Gödels constructible universe. Consistency of the axiom of choice and of the generalized continuum hypothesis. Cohens forcing method. Independence of the AC and GCH.
Note: Not expected to be given 2008-09.
Prerequisite: Any mathematics course at the level of 21a or higher, or permission of instructor.

Mathematics 152. Discrete Mathematics
Catalog Number: 8389
Bret J. Benesh
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
An introduction to finite groups, finite fields, finite geometry, discrete probability, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. Taught in a seminar format, and students will gain experience in presenting proofs at the blackboard.
Note: Students who have taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit.
Prerequisite: Mathematics 21b or equivalent.

[Mathematics 153. Mathematical Biology-Evolutionary Dynamics]
Catalog Number: 3004 Enrollment: Limited to 30.
Martin A. Nowak
Half course (fall term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
Introduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer.
Note: Expected to be given in 2008-09.
Prerequisite: Mathematics 21a,b, Biological Sciences 50 and 53 or equivalent.

Mathematics 154 (formerly Mathematics 191). Probability Theory
Catalog Number: 4306
Paul G. Bamberg
Half course (spring term). Tu., Th., 4-5:30. EXAM GROUP: 18
An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.
Prerequisite: Any mathematics course at the level of Mathematics 19a,b, or 21a,b or higher, or knowledge of multivariable calculus as demonstrated on the online placement test.

Mathematics 155r (formerly Mathematics 192r). Combinatorics
Catalog Number: 6612
Lauren K. Williams
Half course (fall term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Topics include enumerative and algebraic combinatorics related to representations of the symmetric group, symmetric functions, and Young tableaux.
Prerequisite: Mathematics 122 (or equivalent). Knowledge of representation theory of finite groups will be helpful.

Mathematics 212br. Advanced Real Analysis
Catalog Number: 7294
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
Continuation of Mathematics 212ar. The spectral theorem for self-adjoint operators in Hilbert space. Applications to partial differential equations.
Prerequisite: Mathematics 212ar and 213a.

Mathematics 213a. Complex Analysis
Catalog Number: 1621
Andreea C. Nicoara
Half course (fall term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
A second course in complex analysis: elliptic functions, canonical products, conformal mapping, extremal length, harmonic measure and capacity.
Prerequisite: Mathematics 55b or 113.

Mathematics 213br. Advanced Complex Analysis
Catalog Number: 2641
Shing-Tung Yau
Half course (spring term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.
Prerequisite: Mathematics 213a.

Mathematics 221. Commutative Algebra - (New Course)
Catalog Number: 8320
Thomas Lam
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
A first course in commutative algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, discrete valuation rings, filtrations, completions and dimension theory.
Prerequisite: Mathematics 123.

[Mathematics 222. Lie Groups and Lie Algebras]
Catalog Number: 6738
Wilfried Schmid
Half course (spring term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.
Note: Expected to be given in 2008-09.
Prerequisite: Mathematics 114, 123 and 132.

[Mathematics 223a. (formerly Mathematics 251a.) Algebraic Number Theory]
Catalog Number: 8652
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Half course (fall term). Hours to be arranged.
A graduate introduction to algebraic number theory. Topics: the structure of ideal class groups, groups of units, a study of zeta functions and L-functions, local fields, Galois cohomology, local class field theory, and local duality.
Note: Expected to be given in 2008-09.
Prerequisite: Mathematics 129.

[Mathematics 223b. (formerly Mathematics 251b.) Algebraic Number Theory]
Catalog Number: 2783
----------
Half course (spring term). Hours to be arranged.
Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tates thesis or Euler systems.
Note: Expected to be given in 2008-09.
Prerequisite: Mathematics 223a.

Mathematics 230a. Differential Geometry
Catalog Number: 0372
Shing-Tung Yau
Half course (fall term). M., W., F., at 2. EXAM GROUP: 7
Elements of differential geometry: Riemannian geometry, symplectic and Kaehler geometry, geodesics, Riemann curvature, Darbouxs theorem, moment maps and symplectic quotients, complex and Kaehler manifolds, Dolbeault and de Rham cohomology.
Prerequisite: Mathematics 132 or equivalent.

Mathematics 230br. Advanced Differential Geometry
Catalog Number: 0504
Shing-Tung Yau
Half course (spring term). M., W., F., at 2. EXAM GROUP: 7
A continuation of Mathematics 230a. Topics in global Riemannian geometry: Ricci curvature and volume comparison; sectional curvature and distance comparison; Toponogovs theorem and applications; sphere theorems; Gromovs betti number bounds; Gromov-Hausdorff convergence; Cheegers finiteness theorem, and convergence theorems.
Prerequisite: Mathematics 230a.

Mathematics 231a. (formerly Mathematics 272a.) Algebraic Topology
Catalog Number: 7275
Danijela Damjanovic
Half course (fall term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.
Prerequisite: Mathematics 131 and 132.

Mathematics 231br. (formerly Mathematics 272b.) Advanced Algebraic Topology
Catalog Number: 9127
Michael J. Hopkins
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Continuation of Mathematics 231a. Spectral sequences and techniques of computation. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.
Prerequisite: Mathematics 231a.

Mathematics 232a. (formerly Mathematics 260a.) Introduction to Algebraic Geometry I
Catalog Number: 6168
Peter B. Kronheimer
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
Introduction to complex algebraic curves, surfaces, and varieties.
Prerequisite: Mathematics 123 and 132.

Mathematics 232br. (formerly Mathematics 260b.) Introduction to Algebraic Geometry II
Catalog Number: 9205
Sebastian B. Casalaina-Martin
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
The course will cover the classification of complex algebraic surfaces.
Prerequisite: Mathematics 232a.

Mathematics 233a. (formerly Mathematics 261a.) Theory of Schemes I
Catalog Number: 6246
Barry C. Mazur
Half course (fall term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
An introduction to the theory and language of schemes. Textbooks: Algebraic Geometry by Robin Hartshorne and Geometry of Schemes by David Eisenbud and Joe Harris. Weekly homework will constitute an important part of the course.
Prerequisite: Mathematics 221 and 232a or permission of instructor.

Mathematics 233br. (formerly Mathematics 261b.) Theory of Schemes II
Catalog Number: 3316
Sebastian B. Casalaina-Martin
Half course (spring term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
A continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.
Prerequisite: Mathematics 233a.

Mathematics 243 (formerly Mathematics 234). Evolutionary Dynamics
Catalog Number: 8136
Martin A. Nowak
Half course (spring term). Tu., 1-4. EXAM GROUP: 15, 16, 17
Advanced topics of evolutionary dynamics. Seminars and research projects.
Prerequisite: Experience with mathematical biology at the level of Mathematics 153.

Mathematics 244. Advanced Set Theory - (New Course)
Catalog Number: 3138
Peter Koellner
Half course (spring term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
Inner models of large cardinal axioms, focusing on recent work on inner models for large cardinals at the level of supercompact and beyond. Topics include: Continuum Hypothesis and Omega Conjecture.
Prerequisite: Course in Set Theory or permission of the instructor.

Mathematics 262. Manifolds and Homotopy Theory
Catalog Number: 5564
Thomas Goodwillie (Brown University)
Half course (spring term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
Possible topics: Whitneys embedding theorem and generalizations, the h-cobordism theorem and generalizations, surgery theory, and calculus of functors. Pace and emphasis of course may depend on the background and interest of the participants.

Mathematics 265. Infinite Dimensional Lie Algebras
Catalog Number: 3191
Alberto De Sole
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
A detailed introduction to the structure and the representation theory of some important infinite-dimensional Lie algebras: the Heisenberg algebra, the Virasoro algebra and the affine Kac-Moody algebras.

Mathematics 271. Introduction to the Mathematics of General Relativity
Catalog Number: 2400
Lydia Bieri
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
A focus on the Einstein field equations within GR. Brief review of SR and fundamentals from differential geometry. Discussion of Schwarzschild solution, black holes, energy-momentum tensor, non-localizability of gravitational energy, isolated gravitating systems.

Mathematics 273. Topics in Analysis and Mathematical Physics
Catalog Number: 7810
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 11:30-1. EXAM GROUP: 13, 14
Possible choices: exact models in statistical mechanics, asymptotic analysis, or supersymmetry.

Mathematics 275. Multiplicity-Free Representations: Complex Geometric Methods in Representation Theory
Catalog Number: 0818
Toshiyuki Kobayashi (Research Institute for Mathematical Sciences, Kyoto)
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
Explanation of complex geometric methods such as reproducing kernels and "visible actions" for the study of infinite dimensional representations. From this viewpoint, various examples of multiplicity-free representations of Lie groups will be discussed.

Mathematics 282. Introduction to Seiberg-Witten Theory
Catalog Number: 8399
Pedram Safari
Half course (fall term). Tu., Th., 1-2:30. EXAM GROUP: 15, 16
Rudiments of Seiberg-Witten moduli spaces and invariants, including a review of gauge theory techniques. Selected applications to the geometry and topology of 4-manifolds.

Mathematics 283. Topics in Knot Theory
Catalog Number: 7877
Brian Munson
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
The Conway polynomial, the Jones polynomial, Khovanov homology, Vassiliyev and finite type invariants, and configuration space methods. A discussion of links and their generalizations to higher dimensions.

Mathematics 287. Algebraic Curves
Catalog Number: 7465
Joseph D. Harris
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
A development of the theory of algebraic curves/Riemann surfaces, touching on many of the classical aspects of their geometry. A focus on developing current research topics.

Mathematics 288. Quasi-categories and topological field theories
Catalog Number: 5052
Michael J. Hopkins
Half course (fall term). Tu., Th., 10-11:30. EXAM GROUP: 12, 13
An introduction to the theory of quasi-categories with applications to string topology and topological conformal field theories. Prerequisites: model categories and simplicial sets as covered in 272b, spring 07

*Mathematics 302. Topics in Dynamics of Group Actions
Catalog Number: 5763
Danijela Damjanovic 5583

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

*Mathematics 307. Topics in Differential Geometry and Partial Differential Equations
Catalog Number: 5133
Benjamin Weinkove 4942

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

*Mathematics 310. Topics in Number Theory
Catalog Number: 3874
Samit Dasgupta 5030

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

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

*Mathematics 319. Topics in Representation Theory
Catalog Number: 9591
John F. Duncan 5505

*Mathematics 321. Topics in Mathematical Physics
Catalog Number: 2297
Arthur M. Jaffe 2095

*Mathematics 327. Topics in Several Complex Variables
Catalog Number: 0409
Yum Tong Siu 7550 (on leave spring term)

*Mathematics 328. Topics in Lie Algebra
Catalog Number: 7003
Alberto De Sole 4627

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

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

*Mathematics 342. Topics in Combinatorics
Catalog Number: 0751
Thomas Lam 5322 (on leave spring term)

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

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

*Mathematics 347. Topics in Floer Homology and Low Dimensional Topology
Catalog Number: 7227
Eaman Eftekhary 5045

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

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

*Mathematics 356. Topics in Harmonic Analysis
Catalog Number: 6534
Wilfried Schmid 5097 (on leave 2007-08)

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

*Mathematics 371. Topics in Partial Differential Equations and Mathematical Physics
Catalog Number: 0777
Robert M. Strain 5323

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

*Mathematics 379. Topics in Combinatorics
Catalog Number: 3390
Lauren K. Williams 5499 (on leave spring term)

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

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

*Mathematics 383. Topics in Algebraic Geometry
Catalog Number: 7736
Ilia Zharkov 4631

*Mathematics 384. Topics in Automorphic Forms
Catalog Number: 8009
Lior Silberman 5506 (on leave 2007-08)

*Mathematics 388. Topics in Mathematics and Biology
Catalog Number: 4687
Martin A. Nowak 4568 (on leave fall term)

*Mathematics 389. Topics in Number Theory
Catalog Number: 6851
Noam D. Elkies 2604 (on leave 2007-08)

*Mathematics 398. Topics in Algebraic and Geometric Topology
Catalog Number: 0863
Véronique Godin 5311

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