Mathematics Mb. Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Meredith Hegg, Meghan Anderson, Sarah Chisolm, Peter M. Garfield, 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 27, 8:30 am, Science Center A . Participation in two, one 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, Janet Chen, Sarah Chisolm, Sukhada Fadnavis, 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: Wednesday, September 4, 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.
Jameel Al-Aidroos, John Cain, Janet Chen, Sarah Chisolm, Keerthi Madapusi, and members of the Department (fall term); Robin Gottlieb, Sarah Chisolm, John Hall, and members of the Department (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: Fall: 2; Spring: 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: Tuesday, September 3, 8:30 am, Science Center B . Required first meeting in spring: Monday, January 27, 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 18 (formerly Mathematics 20). Multivariable Calculus for Social Sciences
Catalog Number: 0906
Meredith Hegg
Half course (fall term). M., W., F., at 9. EXAM GROUP: 2
Focus on concepts and techniques of multivariable calculus most useful to those studying the social sciences, particularly economics: functions of several variables; partial derivatives; directional derivatives and the gradient; constrained and unconstrained optimization, including the method of Lagrange multipliers. Covers linear and polynomial approximation and integrals for single variable and multivariable functions; modeling with derivatives. Covers topics from Math 21a most useful to social sciences.
Note: Should not ordinarily be taken in addition to Mathematics 21a or Applied Mathematics 21a. Mathematics 21b can be taken before or after Mathematics 18. Examples draw primarily from economics and the social sciences, though Mathematics 18 may be useful to students in certain natural sciences. Students whose main interests lie in the physical sciences, mathematics, or engineering should consider Math or Applied Mathematics 21a. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning.
Prerequisite: Mathematics 1b or equivalent, or a 5 on the BC Advanced Placement Examination in Mathematics.

Mathematics 19a. Modeling and Differential Equations for the Life Sciences
Catalog Number: 1256
John Hall (fall term) and John Wes Cain (spring term)
Half course (fall term; repeated spring term). M., W., F., at 1, and a weekly discussion 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 and ESPP. Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 18. This course can be taken before or after Mathematics 18. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.

Mathematics 19b. Linear Algebra, Probability, and Statistics for the Life Sciences
Catalog Number: 6144
Peter M. Garfield
Half course (spring term). M., W., F., at 1, and a weekly problem section to be arranged. EXAM GROUP: 6
Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental 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 and 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 21a. Multivariable Calculus
Catalog Number: 6760 Enrollment: Normally limited to 30 students per section.
Oliver Knill, Meghan Anderson, Michael Hopkins, and members of the Department (fall term); John Hall, John Cain, Meredith Hegg, 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: Fall: 2; Spring: 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, Stokess, and Divergence Theorems.
Note: Required first meeting in fall: Wednesday, September 4, 8:30 am, Science Center B . Required first meeting in spring: Monday, January 27, 8:30 am, Science Center D. 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.
Prerequisite: Mathematics 1b or equivalent.

Mathematics 21b. Linear Algebra and Differential Equations
Catalog Number: 1771 Enrollment: Normally limited to 30 students per section.
John Hall, Junecue Suh, and members of the Department (fall term); Janet Chen, Jameel Al-Aidroos, Meghan Anderson, Keerthi Madapusi, 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 (with sufficient enrollment); 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; Section V, M., W., F., 2 (with sufficient enrollment), 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 linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.
Note: Required first meeting in fall: Tuesday, September 3, 8:30 am, Science Center C . Required first meeting in spring: Monday, January 27, 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 1b 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.
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. 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.
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 10. EXAM GROUP: 3
A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.
Note: Only for students with a strong interest and background in mathematics. There will be a heavy workload. May not be taken for credit after Mathematics 23. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning.
Prerequisite: 5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.

Mathematics 25b. Honors Linear Algebra and Real Analysis II
Catalog Number: 1590
Noam D. Elkies
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
Dennis Gaitsgory
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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
Dennis Gaitsgory
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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
Yaiza Canzani
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
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
Clifford Taubes
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
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
Jacob Lurie
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
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
Siu Cheong Lau
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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. Real Analysis, Convexity, and Optimization
Catalog Number: 5253
Paul G. Bamberg
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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
Catalog Number: 45584
Sukhada Fadnavis
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
A self-contained treatment of the theory of probability and random processes with specific application to the theory of option pricing. Topics: axioms for probability, calculation of expectation by means of Lebesgue integration, conditional probability and conditional expectation, martingales, random walks and Wiener processes, and the Black-Scholes formula for option pricing. Students will work in small groups to investigate applications of the theory and to prove key results.
Note: A problem-solving section is required MW 2-3 or Th 7:30-9:30 PM
Prerequisite: A thorough knowledge of single-variable calculus and infinite series, plus at least one more advanced course such as MATH E-23a that provides experience with proofs and elementary real analysis. Acquaintance with elementary probability is desirable.

Mathematics 118r. Dynamical Systems
Catalog Number: 6402
Yaiza Canzani
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Paul G. Bamberg
Half course (fall term). M., W., at 1. EXAM GROUP: 6
Real and complex vector spaces, linear transformations, determinants, inner products, dual spaces, and eigenvalue problems. Applications to geometry, systems of linear differential equations, electric circuits, optimization, and Markov processes. Emphasizes learning to understand and write proofs. Students will work in small groups to solve problems and develop proofs.
Note: A problem-solving section is required M, W 2-3
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
Andrew W. Cotton-Clay
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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 10. EXAM GROUP: 3
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
Arul Shankar
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
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. Classical Geometry
Catalog Number: 5811
Andrew W. Cotton-Clay
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Presents several classical geometries, these being the affine, projective, Euclidean, spherical and hyperbolic geometries. They are viewed from many different perspectives, some historical and some very topical. 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
Curtis T. McMullen
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Andrew W. Cotton-Clay
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Siu Cheong Lau
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The exterior differential calculus and its applications to curves and surfaces in 3-space and to various notions of curvature. Introduction to Riemannian geometry in higher dimensions.
Prerequisite: Advanced calculus and linear algebra.

Mathematics 137. Algebraic Geometry
Catalog Number: 0556
Melody Tung Chan
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
Instructor to be determined
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.
Note: Expected to be given in 2014–15.
Prerequisite: Any mathematics course at the level of Mathematics 21a,b or higher, or permission of instructor.

Mathematics 143. Set Theory
Catalog Number: 6005
Peter Koellner
Half course (fall term). W., 1–3. EXAM GROUP: 6, 7
An introduction to set theory covering the fundamentals of ZFC (cardinal arithmetic, combinatorics, descriptive set theory) and the independence techniques (the constructible universe, forcing, the Solovay model). We will demonstrate the independence of CH (the Continuum Hypothesis), SH (Suslins Hypothesis), and some of the central statements of classical descriptive set theory.
Note: An additional hour of lecture will be scheduled independently.
Prerequisite: Any mathematics course at the level of Mathematics 21a or higher, or permission of instructor.

Mathematics 144. Model Theory
Catalog Number: 0690
Nathanael Ackerman
Half course (fall term). M., W., F., at 11. EXAM GROUP: 4
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 145. Set Theory II - (New Course)
Catalog Number: 19964
Peter Koellner
Half course (spring term). W., 1–3, and an additional hour of lecture will be scheduled independently. EXAM GROUP: 6, 7
An introduction to the hierarchy of axioms of infinity in set theory, their applications and their inner models.
Note: An additional hour of lecture will be scheduled independently.

[Mathematics 152. Discrete Mathematics]
Catalog Number: 8389
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Half course (spring 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: Expected to be given in 2014–15. Students who have taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit.

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. Probability Theory
Catalog Number: 4306
Clifford Taubes
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
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. Combinatorics
Catalog Number: 6612
Sukhada Fadnavis
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
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 168. Computability Theory]
Catalog Number: 31297
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Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
An introduction to computability theory (also known as recursion theory). A discussion of the problem of determining what it means for a set or function to be computable, including primitive recursion, Turing machines, and the Church-Turing Thesis. The theory of Turing degrees and the computably enumerable sets. Topics: the halting set, Turing reducibility and other reducibilities, Posts problem, the Recursion Theorem, priority arguments, and more.
Note: Expected to be given in 2014–15.
Prerequisite: The student must have the ability to read and write mathematical proofs.

Mathematics 212br. Advanced Real Analysis
Catalog Number: 7294
Yum Tong Siu
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
This class will be an introduction to harmonic analysis and singular integral. The textbook is Classical and Multilinear Harmonic Analysis, Volume 1, by Muscalu and Schlag. The topics covered in the course include maximum functions, interpolation of operators, Calderon-Zygmund theory and Littlewood-Paley theory. Some elementary probability theory will also be included. Good references of this course are Steins book on singular integrals and Fourier analysis.
Prerequisite: Mathematics 212ar and 213a.

Mathematics 213a. Complex Analysis
Catalog Number: 1621
Wilfried Schmid
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
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
Curtis T. McMullen
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
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
Junecue Suh
Half course (fall term). M., W., F., at 10. EXAM GROUP: 3
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., 2:30–4. EXAM GROUP: 16, 17
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. Algebraic Number Theory]
Catalog Number: 8652
----------
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 2014–15.
Prerequisite: Mathematics 129.

[Mathematics 223b. Algebraic Number Theory]
Catalog Number: 2783
----------
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 2014–15.
Prerequisite: Mathematics 223a.

Mathematics 224. Representations of Reductive Lie Groups
Catalog Number: 25927
Igor Andreevich Rapinchuk
Half course (spring term). M., W., F., at 10. EXAM GROUP: 3
Lie groups, Lie algebras and their representation theory, focusing on the classical groups.

Mathematics 229x. Introduction to Analytic Number Theory
Catalog Number: 41034
Arul Shankar
Half course (fall term). M., W., F., at 1. EXAM GROUP: 6
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
Shlomo Z. Sternberg
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
Connections on the tangent bundle, Levi-Civitas theorem, Gausss lemma, curvature, distance and volume, general relativity, connections on principle bundles.
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 10. EXAM GROUP: 3
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. Algebraic Topology
Catalog Number: 7275
Hiro Tanaka
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. Advanced Algebraic Topology
Catalog Number: 9127
Gereon Quick
Half course (spring term). M., W., F., at 2:30. EXAM GROUP: 7, 8
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. Introduction to Algebraic Geometry I
Catalog Number: 6168
Jonathan Mboyo Esole
Half course (fall term). M., W., F., at 12. EXAM GROUP: 5
Introduction to complex algebraic curves, surfaces, and varieties.
Prerequisite: Mathematics 123 and 132.

Mathematics 232br. Algebraic Geometry II
Catalog Number: 9205
Jonathan Mboyo Esole
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
The course will cover the classification of complex algebraic surfaces.
Prerequisite: Mathematics 232a.

Mathematics 233a. Theory of Schemes I
Catalog Number: 6246
Igor Andreevich Rapinchuk
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. 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.
Note: Expected to be omitted in 2012–13.
Prerequisite: Mathematics 233a.

Mathematics 241. Fine Structure Theory - (New Course)
Catalog Number: 81434
W. Hugh Woodin
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
There is evidence now that there is an ultimate version of Godels L. This course will develop the fine structure theory of a "Penultimate L" which seems a necessary precursor to Ultimate L.

Mathematics 243. Evolutionary Dynamics
Catalog Number: 8136
Martin A. Nowak
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 16, 17
Advanced topics of evolutionary dynamics. Seminars and research projects.
Prerequisite: Experience with mathematical biology at the level of Mathematics 153.

Mathematics 253x. Analysis on Manifolds via the Laplace Operator - (New Course)
Catalog Number: 61468
Yaiza Canzani
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
The study of the Laplace operator on Riemannian manifolds and explanation of how it encodes geometric information of the manifold. We will cover basic examples, existence of L^2 basis of eigenfunctions, nodal domain theorems, heat and wave operators, comparison theorems and eigenvalue estimates.

Mathematics 253y. Probability and Brownian Motion - (New Course)
Catalog Number: 16994
Alexander Bloemendal
Half course (spring term). M., W., F., at 11. EXAM GROUP: 4
An introduction to rigorous probability theory. Standard topics include laws of large numbers, central limit theorems, random walks and martingales; the main goal is a thorough understanding of Brownian motion from several points of view.
Prerequisite: Some exposure to measure theory such as taught in Mathematics 114, and some familiarity with elementary probability such as taught in Mathematics 154 and 117, Statistics 110 or Engineering Science 150.

Mathematics 255x. Topics in Diophantine Geometry - (New Course)
Catalog Number: 79431
Arul Shankar
Half course (spring term). M., W., F., at 1. EXAM GROUP: 6
Siegels theorem on the finiteness of integral points on elliptic curves, the Thue-Siegel-Roth theorem on approximations of algebraic integers, and results on the representation of integers by binary forms.

Mathematics 255y. Spin Geometry and SuperSymmetry - (New Course)
Catalog Number: 14096
Jonathan Mboyo Esole
Half course (spring term). M., W., 1–2:30. EXAM GROUP: 6, 7
Introduction to Clifford algebra, spinors and the geometry of supersymmetry in various dimensions.

Mathematics 256x. The Theory of Error-Correcting Codes - (New Course)
Catalog Number: 13741
Noam D. Elkies
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Code and linear codes; Hamming weight and distance; Weight enumerators; the dual code and the MacWilliams identity; Gleasons theorems and consequences. Construction and properties of Reed-Muller, Reed-Solomon, BCH, Golay, and Goppa codes. Generalizations and connections with sphere packing and other topics as time and students backgrounds permit.

Mathematics 265x. Reasoning via Models - (New Course)
Catalog Number: 73059 Enrollment: Limited to 20.
Eric S. Maskin, Barry C. Mazur, and Amartya Sen
Half course (fall term). Tu., 2–4. EXAM GROUP: 15, 16, 17
An examination of how formal models are used in different disciplines. Examples will be taken from economics, mathematics, physics and philosophy, among other fields.
Note: This course may not be counted towards the required eight letter-graded half-courses in mathematics for the concentration requirement 1a, but may be counted as one of the four half-courses in mathematics or related fields, requirement 1b. This is cross-listed in Economics, History of Science, and Philososphy.
Prerequisite: There are no specific course prerequisites, but ease and familiarity with formal reasoning is essential.

Mathematics 268. Pure Motives and Rigid Local Systems - (New Course)
Catalog Number: 72509
Stefan Patrikis
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 15, 16
An introduction to pure motives, balanced between the general theory and examples arising from rigid local systems. Topics include: unconditionally Tannakian variants of the category of pure homological motives; Katzs theory of middle convolution; the realization of G—2 as a motivic Galois group.

Mathematics 270x. Topics in Automorphic Forms - (New Course)
Catalog Number: 70229
Benedict H. Gross
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
We will give an introduction to the theory of modular and automorphic forms, with an emphasis on applications to algebraic number theory. Topics to be covered include the formalism of L-groups, functoriality, trace formulae, and the construction by Chenevier and Clozel of number fields with limited ramification.

Mathematics 273x. Topics in Algebraic Geometry - (New Course)
Catalog Number: 98825
Yaim Cooper
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 13, 14
Intersection theory with a view toward Gromov-Witten theory.

Mathematics 280x. Topics in Mathematical Physics: Bridgeland Stability Conditions - (New Course)
Catalog Number: 90433
Hiro Tanaka
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
The basics of Bridgeland stability conditions for stable oo-catagories will be covered. The courses ultimate goal is to represent Hall (co)algebra-like structures as a co/sheaf on a Ran space.

Mathematics 280y. Topics in Symplectic Geometry - (New Course)
Catalog Number: 42209
Siu Cheong Lau
Half course (spring term). M., W., 2:30–4. EXAM GROUP: 7, 8
Symplectic geometry has grown into an important branch of mathematics due to its intimate relationship with physics. A focus on symplectic enumerative invariants and Lagrangian Floer theory, which have great developments in recent years brought by string theory and mirror symmetry.

Mathematics 282y. Tamagawa Numbers of Algebraic Groups over Function Fields - (New Course)
Catalog Number: 31459
Jacob Lurie
Half course (spring term). M., W., F., at 12. EXAM GROUP: 5
A discussion of some recent work (jointly with Dennis Gaitsgory) on the Tamagawa numbers of algebraic groups defined over function fields, using a variety of techniques inspired by algebraic topology.

Mathematics 284x. Relations between Clifford Algebras and Lie Algebras - (New Course)
Catalog Number: 87316
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13
We will mostly follow the recent book by Meinrenken with applications to differential geometry.

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

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

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

*Mathematics 327. Topics in Several Complex Variables
Catalog Number: 0409
Yum Tong Siu 7550

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

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

*Mathematics 339. Topics in Combinatorics
Catalog Number: 83942
Sukhada Fadnavis 7084

*Mathematics 341. Topics in Number Theory
Catalog Number: 28563
Keerthi Shyam Madapusi Sampath 2232

*Mathematics 345. Topics in Geometry and Topology
Catalog Number: 4108
Peter B. Kronheimer 1759 (on leave 2013-14)

*Mathematics 346y. Topics in Analysis: Quantum Dynamics
Catalog Number: 1053
Horng-Tzer Yau 5260 (on leave 2013-14)

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

*Mathematics 355. Topics in Category Theory and Homotopy Theory
Catalog Number: 95192
Emily Elizabeth Riehl 1416 (on leave 2013-14)

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

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

*Mathematics 361. Topics in Differential Geometry and Analysis - (New Course)
Catalog Number: 61965
Yaiza Canzani 7325

*Mathematics 363. Topics in Elliptic Fibrations and String Theory - (New Course)
Catalog Number: 89264
Jonathan Mboyo Esole 3362

*Mathematics 365. Topics in Differential Geometry
Catalog Number: 4647
Shing-Tung Yau 1734 (on leave 2013-14)

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

*Mathematics 374. Topics in Number Theory - (New Course)
Catalog Number: 83329
Arul Shankar 7303

*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 (on leave fall term)

*Mathematics 385. Topics in Set Theory - (New Course)
Catalog Number: 33405
W. Hugh Woodin 7421

*Mathematics 387. Topics in Mathematical Physics: Bridgeland Stability Conditions - (New Course)
Catalog Number: 47551
Hiro Tanaka 7326 (on leave spring term)

*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