Mathematics Mb. Introduction to Functions and Calculus II
Catalog Number: 3857 Enrollment: Normally limited to 15 students per section.
Sarah Chisholm, Upendra Prasad, 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: 3
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 26, 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.
Prerequisite: Mathematics Ma.

Mathematics 1a. Introduction to Calculus
Catalog Number: 8434 Enrollment: Normally limited to 30 students per section.
Janet Chen, Jameel Al-Aidroos, Brendan Kelly, Sukhada Fadnavis, and members of the Department (fall term); Brendan Kelly (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: 3
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 3, 8:30 am, Science Center C . This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical 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.
Cliff Taubes, Rosalie Belanger-Rioux, Sarah Chisholm, Nina Zipser, and members of the Department (fall term) Jameel Al-Aidroos, Rosalie Belanger-Rioux, Yu-Wen Hsu, and members of the Dpartment (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: 3
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 2, 8:30 am, Science Center B. Required first meeting in spring: Monday, January 26, 8:30 am, Science Center D. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning.
Prerequisite: Mathematics 1a, or Ma and Mb, or equivalent.

Mathematics 18 (formerly Mathematics 20). Multivariable Calculus for Social Sciences
Catalog Number: 0906
Peter McKee Garfield
Half course (fall term). M., W., F., at 11. EXAM GROUP: 3
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
Upendra Prasad
Half course (fall term; repeated spring term). M., W., F., at 1, and a weekly discussion section to be arranged. EXAM GROUP: Fall: 1; Spring: 8
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.

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: 8
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.

Mathematics 21a. Multivariable Calculus
Catalog Number: 6760 Enrollment: Normally limited to 30 students per section.
Oliver Knill, Jameel Al-Aidroos, Rosalie Belanger-Rioux, Yu-Wen Hsu, Siu-Cheong Lau, and members of the Department (fall term); Peter Garfield, Rosalie Belanger -Rioux, Sarah Chisholm, Yu-Wen Hsu, 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: 3
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 3, 8:30 am, Science Center B . Required first meeting in spring: Monday, January 26, 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. 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.
Peter Garfield, and members of the Department (fall term); Oliver Knill, 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, Tu., Th., 10-11:30; Tu., Th., 11:30-1;, and a weekly problem section to be arranged. EXAM GROUP: 3
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: Wednesday, September 3, 8:30 am, Science Center D . Required first meeting in spring: Monday, January 26, 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.
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: 14
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.
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: 11
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.
Prerequisite: Mathematics 23a.

Mathematics 25a. Honors Linear Algebra and Real Analysis I
Catalog Number: 1525
Tasho Kaletha
Half course (fall term). M., W., F., at 10. EXAM GROUP: 5
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.
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
Daniel Anthony Cristofaro-Gardiner
Half course (spring term). M., W., F., at 10. EXAM GROUP: 5
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.
Prerequisite: Mathematics 25a or 55a.

*Mathematics 55a. Honors Abstract Algebra
Catalog Number: 4068
Dennis Gaitsgory
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 14
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. Instructor 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.

*Mathematics 55b. Honors Real and Complex Analysis
Catalog Number: 3312
Dennis Gaitsgory
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 11
A rigorous treatment of real and complex analysis.
Note: Mathematics 55b is an intensive course for students having significant experience with abstract mathematics. Instructor permission required. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning.

*Mathematics 60r. Reading Course for Senior Honors Candidates
Catalog Number: 8500
Jacob Lurie
Half course (fall term; repeated spring term). Hours to be arranged. EXAM GROUP: Fall: 11
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
Jacob Lurie
Half course (fall term; repeated spring term). Hours to be arranged. EXAM GROUP: Fall: 2
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
Jacob Lurie and members of the Department
Half course (fall term; repeated spring term). Hours to be arranged. EXAM GROUP: Fall: 16
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). M., W., 2–3:30. EXAM GROUP: 18
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
Tristan Collins
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 1
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
Adam Jacob
Half course (spring term). M., W., F., at 12. EXAM GROUP: 7
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
Curtis T. McMullen
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12
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: 14
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.
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.
Note: Expected to be given in 2015–16.
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: 11
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.
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.
Note: Expected to be given in 2015–16.
Prerequisite: Mathematics 21a,b.

Mathematics 121. Linear Algebra and Applications
Catalog Number: 7009
Yaim Cooper
Half course (fall term). M., W., F., at 1. EXAM GROUP: 1
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
Hiro Tanaka
Half course (fall term). M., W., F., at 12. EXAM GROUP: 11
Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.
Prerequisite: Mathematics 23a, 25a, 121; or 101 with the instructors permission. Should not be taken in addition to Mathematics 55a.

Mathematics 123. Algebra II: Theory of Rings and Fields
Catalog Number: 5613
Barry C. Mazur
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12
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
Benedict H. Gross
Half course (fall term). M., W., F., at 10. EXAM GROUP: 5
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: 8
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
Clifford Taubes
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 15
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
Emily Elizabeth Riehl
Half course (fall term). Tu., Th., 1–2:30. EXAM GROUP: 8
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
Daniel Anthony Cristofaro-Gardiner
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 1
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
Tristan Clifford Collins
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12
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
Yaim Cooper
Half course (spring term). M., W., F., at 11. EXAM GROUP: 14
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
Emily Elizabeth Riehl
Half course (fall term). M., W., F., at 11. EXAM GROUP: 18
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]
Catalog Number: 0690
Nathanael Ackerman
Half course (fall term). M., W., F., at 11.
An introduction to model theory with applications to fields and groups. First order languages, structures, and definable sets. Compactness, completeness, and back-and-forth constructions. Quantifier elimination for algebraically closed, differentially closed, and real closed fields. Omitting types, prime extensions, existence and uniqueness of the differential closure, saturation, and homogeneity. Forking, independence, and rank.
Note: Expected to be given in 2015–16.
Prerequisite: Mathematics 123 or the equivalent is suggested as a prerequisite, but not required.

Mathematics 145a. Set Theory I - (New Course)
Catalog Number: 95052
Peter Koellner
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 15
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.
Prerequisite: Any mathematics course at the level of Mathematics 21a or higher, or permission of instructor.

Mathematics 145b. Set Theory II - (New Course)
Catalog Number: 27354
Peter Koellner
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 15
An introduction to large cardinals and their inner models, with special emphasis on Woodins recent advances toward finding an ultimate version of Godels L. Topics include: Weak extender models, the HOD Dichotomy Theorem, and the HOD Conjecture.
Prerequisite: Mathematics 145a or permission of instructor.

Mathematics 152. Discrete Mathematics
Catalog Number: 8389
Paul G. Bamberg
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 15
An introduction to sets, logic, finite groups, finite fields, finite geometry, combinatorics, 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: students will gain experience in presenting proofs at the blackboard.
Note: Covers material used in Computer Science 121 and Computer Science 124. Students who have taken Computer Science 20, Mathematics 55, or Mathematics 122 should not take this course for credit.
Prerequisite: Mathematics 21b or 23a (may be taken concurrently). Previous experience with proofs is not required.

[Mathematics 153. Mathematical Biology-Evolutionary Dynamics]
Catalog Number: 3004
Martin A. Nowak
Half course (fall term). Tu., Th., 2:30–4.
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 2015–16.
Prerequisite: Mathematics 21a,b.

Mathematics 154. Probability Theory
Catalog Number: 4306
Horng-Tzer Yau
Half course (spring term). M., W., F., at 1. EXAM GROUP: 8
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.
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., 1–2:30. EXAM GROUP: 1
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 156. Mathematical Foundations of Statistical Software - (New Course)
Catalog Number: 80717
Paul G. Bamberg
Half course (fall term). Monday 2-4:30, plus weekly sections to be arranged. EXAM GROUP: 7
Presents the probability theory and statistical principles which underly the tools that are built into the open-source programming language R. Each class presents the theory behind a statistical tool, then shows how the implementation of that tool in R can be used to analyze real-world data. The emphasis is on modern bootstrapping and resampling techniques, which rely on computational power. Topics include discrete and continuous probability distributions, permutation tests, the central limit theorem, chi-square and Student t tests, linear regression, and Bayesian methods.
Prerequisite: An excellent backgound in single-variable calculus and infinite series, plus basic knowledge of multiple integration. Mathematics 18 or 21a, taken concurrently, would be sufficient. Students should be skillful software users but need not be programmers. Background in elementary probability (e.g. AP Statistics or Computer Science 20) would be helpful but is not required.

Mathematics 157. Mathematics in the World - (New Course)
Catalog Number: 66394
Joseph D. Harris
Half course (spring term). Tu., Th., 4–5:30. EXAM GROUP: 6
An interactive introduction to problem solving with an emphasis on subjects with comprehensive applications. Each class will be focused around a group of questions with a common topic. Possible topics: logic, information, number theory, probability, and algorithms.
Prerequisite: Mathematics 21b or 23a. More importantly, students should have a broad mathematical curiosity and be eager to brainstorm during in-class problem solving sessions.

Mathematics 161. Category Theory in Context - (New Course)
Catalog Number: 33581
Emily Elizabeth Riehl
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 15
An introduction to categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics as time permits with the aim of revisiting a broad range of mathematical examples from the categorical perspective.
Prerequisite: Mathematics 123 (may be taken concurrently) and Mathematics 131, or permission of instructor.

Mathematics 212br. Advanced Real Analysis
Catalog Number: 7294
Shlomo Z. Sternberg
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 15
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
Yum Tong Siu
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12
A second course in complex analysis: series, product and partial fraction expansions of holomorphic functions; Hadamards theorem; conformal mapping and the Riemann mapping theorem; elliptic functions; Picards theorem and Nevanlinna Theory.
Prerequisite: Mathematics 55b or 113.

Mathematics 213br. Advanced Complex Analysis
Catalog Number: 2641
Yum Tong Siu
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12
Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli.
Prerequisite: Mathematics 213a.

Mathematics 221. Algebra
Catalog Number: 8320
Hector Hardy Pasten Vasquez
Half course (fall term). M., W., F., at 10. 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
Tasho Kaletha
Half course (spring term). M., W., F., at 1.
Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.
Note: Expected to be given in 2015–16.
Prerequisite: Mathematics 114, 123 and 132.

Mathematics 223a. Algebraic Number Theory
Catalog Number: 8652
Igor Andreevich Rapinchuk
Half course (fall term). M., W., F., at 12. EXAM GROUP: 11
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. Algebraic Number Theory
Catalog Number: 2783
Igor Andreevich Rapinchuk
Half course (spring term). M., W., F., at 12. EXAM GROUP: 7
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 (fall term). Tu., Th., 2:30–4. EXAM GROUP: 14
Structure theory of reductive Lie groups, unitary representations, Harish Chandra modules, characters, the discrete series, Plancherel theorem.

Mathematics 229x. Introduction to Analytic Number Theory
Catalog Number: 41034
Arul Shankar
Half course (fall term). M., W., F., at 1. EXAM GROUP: 1
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
Hiro Tanaka
Half course (fall term). M., W., F., at 10. EXAM GROUP: 5
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
Shing-Tung Yau
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 5
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
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. Advanced Algebraic Topology
Catalog Number: 9127
Michael J. Hopkins
Half course (spring term). M., W., F., at 2. EXAM GROUP: 18
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 11. EXAM GROUP: 18
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: 14
The course will cover the classification of complex algebraic surfaces.
Prerequisite: Mathematics 232a.

Mathematics 233a. Theory of Schemes I
Catalog Number: 6246
Alison Beth Miller
Half course (fall term). M., W., F., at 1. EXAM GROUP: 1
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
----------
Half course (spring term). M., W., F., at 11.
A continuation of Mathematics 233a. Will cover the theory of schemes, sheaves, and sheaf cohomology.
Note: Expected to be given in 2015–16.
Prerequisite: Mathematics 233a.

Mathematics 243. Evolutionary Dynamics
Catalog Number: 8136
Martin A. Nowak
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 11
Advanced topics of evolutionary dynamics. Seminars and research projects.
Prerequisite: Experience with mathematical biology at the level of Mathematics 153.

Mathematics 250. Algebraic Invariants of Knots - (New Course)
Catalog Number: 53245
Alison Beth Miller
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12
Topics in classical and high-dimensional knot theory, with a focus on invariants related to the Alexander module. Possible topics: Seifert surfaces and pairings, Tristram-Levine signatures, the Blanchfield pairing, classification of simple n-knots, singularities of algebraic hypersurfaces, connections to arithmetic invariant theory.

Mathematics 258x. Random Matrix
Catalog Number: 80974
Horng-Tzer Yau
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 15
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 259. Diophantine Definability - (New Course)
Catalog Number: 73673
Hector Hardy Pasten Vasquez
Half course (spring term). Tu., Th., 11:30–1. EXAM GROUP: 15
A study of (positive existential) definability problems in number theory. The main topics to be considered will be definability of multiplication, interpretations and undecidability.

Mathematics 261. Topics in Symplectic Geometry: Langrangian Intersection Theory and SYZ - (New Course)
Catalog Number: 50451
Siu Cheong Lau
Half course (spring term). Tu., Th., 2:30–4. EXAM GROUP: 11
An investigation of geometric aspects of mirror symmetry in the SYZ approach using Langrangian intersection theory.

Mathematics 262. The Geometry of the Complex Monge-Ampere Equation - (New Course)
Catalog Number: 16248
Tristan Collins
Half course (spring term). M., W., F., at 12. EXAM GROUP: 7
A discussion of the complex Monge-Ampere equation, and its applications in the geometry of Kahler manifolds. Topics: Yaus solution of the Calabi Conjecture, and the geometry of Gromov-Hausdorff limits of Ricci flat metrics. Further topics may include the degenerate Monge-Ampere equation and singular Calabi-Yau metrics, as well as Ricci flat metrics on non-compact manifolds, particularly conical Calabi-Yau metrics and their connection to the geometry of Fano varieties.

Mathematics 262x. Topics in Geometric Analysis - (New Course)
Catalog Number: 64721
Shing-Tung Yau
Half course (fall term). Tu., Th., 11:30–1. EXAM GROUP: 15
Basic analysis of Riemannian manifolds and their applications in geometry and theoretical physics including general relativity and string theory.

Mathematics 263y. Topics in Geometry and Physics: K-Theory - (New Course)
Catalog Number: 76548
Daniel Stuart Freed
Half course (fall term). Tu., Th., 2:30–4. EXAM GROUP: 14
An introduction to topological K-theory followed by recent applications. Specific topics may include: twisted K-theory and representations of loop groups, differential K-theory and the index theorem, Ramond-Ramond fields in superstring theory, topological insulators.

[Mathematics 265. Reductive Groups Over Local and Global Fields] - (New Course)
Catalog Number: 48986
Tasho Kaletha
Half course (spring term). M., W., F., at 11.
An introduction to the theory of reductive groups, beginning with their structure theory over algebraically closed fields, discussing rationality questions, and a study of special phenomena that occur when the field of definition is a local or global field.
Note: Expected to be given in 2015–16.

Mathematics 266. Intersection Theory in Algebraic Geometry - (New Course)
Catalog Number: 59157
Joseph D. Harris
Half course (spring term). M., W., F., at 10. EXAM GROUP: 5
A second course in algebraic geometry, centered around intersection theory but intended in addition to introduce the student to basic tools of algebraic geometry, such as deformation theory, characteristic classes, Hilbert schemes and specialization methods.

Mathematics 269. Topics in Kinetic Theory - (New Course)
Catalog Number: 43676
Laure Saint-Raymond
Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12
An introduction to PDEs for statistical physics out of thermodynamic equilibrium. 1. Mathematical tools for the study of kinetic transport equations. 2. Mean field approximation: the case of the Vlasov-Poisson system. 3. Collisional kinetic theory: an introduction to the Boltzmann equation.
Prerequisite: A knowledge of basic functional analysis and Fourier analysis.

Mathematics 271. Topics in Arithmetic Statistics - (New Course)
Catalog Number: 21406
Arul Shankar
Half course (spring term). Tu., Th., 1–2:30. EXAM GROUP: 1
Topics: Cohen Lenstra heuristics, prehomogeneous vector spaces, applications to statistics of number fields and class groups, Poonen-Rains heuristics, and ranks of elliptic curves. Tools will include Davenport and Bhargavas geometry-of-numbers methods.

Mathematics 275. Topics in Geometry and Dynamics - (New Course)
Catalog Number: 86549
Curtis T. McMullen
Half course (spring term). Tu., Th., 10–11:30. EXAM GROUP: 12
A survey of fundamental results and current research. Topics may include: ergodic theory, hyperbolic manifolds, Mostow rigidity, Kazhdans property T, Ratners theorem, and dynamics over moduli space.

Mathematics 281. Algebraic K-theory and Manifold Topology - (New Course)
Catalog Number: 77391
Jacob Lurie
Half course (fall term). M., W., F., at 12. EXAM GROUP: 11
An introduction to the algebraic K-theory of rings and ring spectra, emphasizing connections with simple homotopy theory and the topology of manifolds.

*Mathematics 303. Topics in Diophantine Problems - (New Course)
Catalog Number: 89215
Hector Hardy Pasten Vasquez 7765
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 11; Spring: 7

*Mathematics 304. Topics in Algebraic Topology
Catalog Number: 0689
Michael J. Hopkins 4376
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 16; Spring: 6

*Mathematics 308. Topics in Number Theory and Modular Forms
Catalog Number: 0464
Benedict H. Gross 1112 (on leave spring term)
Half course (fall term; repeated spring term). . EXAM GROUP: 3

*Mathematics 314. Topics in Differential Geometry and Mathematical Physics
Catalog Number: 2743
Shlomo Z. Sternberg 1965
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 1; Spring: 8

*Mathematics 318. Topics in Number Theory
Catalog Number: 7393
Barry C. Mazur 1975
Half course (fall term; repeated spring term). . EXAM GROUP: 4

*Mathematics 321. Topics in Mathematical Physics
Catalog Number: 2297
Arthur M. Jaffe 2095
Half course (fall term; repeated spring term). . EXAM GROUP: 5

*Mathematics 327. Topics in Several Complex Variables
Catalog Number: 0409
Yum Tong Siu 7550
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 7; Spring: 18

*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry
Catalog Number: 9401
Curtis T. McMullen 3588
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 11; Spring: 7

*Mathematics 335. Topics in Differential Geometry and Analysis
Catalog Number: 5498
Clifford Taubes 1243
Half course (fall term; repeated spring term). . EXAM GROUP: 10

*Mathematics 339. Topics in Combinatorics
Catalog Number: 83942
Sukhada Fadnavis 7084
Half course (fall term; repeated spring term). . EXAM GROUP: 4

*Mathematics 341. Topics in Number Theory
Catalog Number: 28563
Keerthi Shyam Madapusi Sampath 2232
Half course (fall term; repeated spring term). . EXAM GROUP: 10

*Mathematics 343. Topics in Complex Geometry - (New Course)
Catalog Number: 70763
Tristan Clifford Collins 7773
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 18; Spring: 14

*Mathematics 345. Topics in Geometry and Topology
Catalog Number: 4108
Peter B. Kronheimer 1759
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 11; Spring: 7

*Mathematics 346y. Topics in Analysis: Quantum Dynamics
Catalog Number: 1053
Horng-Tzer Yau 5260
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 13; Spring: 9

*Mathematics 352. Topics in Algebraic Number Theory
Catalog Number: 86228
Mark Kisin 6281
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 2; Spring: 16

*Mathematics 355. Topics in Category Theory and Homotopy Theory
Catalog Number: 95192
Emily Elizabeth Riehl 1416
Half course (fall term; repeated spring term). . EXAM GROUP: 5

*Mathematics 356. Topics in Harmonic Analysis
Catalog Number: 6534
Wilfried Schmid 5097 (on leave spring term)
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 7; Spring: 18

*Mathematics 358. Topics in Arithmetic Geometry
Catalog Number: 30858
Junecue Suh 6835
Half course (fall term; repeated spring term). . EXAM GROUP: 12

*Mathematics 361. Topics in Differential Geometry and Analysis
Catalog Number: 61965
Yaiza Canzani 7325 (on leave 2014-15)
Half course (fall term; repeated spring term). . EXAM GROUP: 3

*Mathematics 362. Topics in Number Theory - (New Course)
Catalog Number: 57512
Alison Beth Miller 7777 (on leave spring term)
Half course (fall term; repeated spring term). . EXAM GROUP: 5

*Mathematics 363. Topics in Elliptic Fibrations and String Theory
Catalog Number: 89264
Jonathan Mboyo Esole 3362
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 2; Spring: 16

*Mathematics 365. Topics in Differential Geometry
Catalog Number: 4647
Shing-Tung Yau 1734
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 6; Spring: 17

*Mathematics 373. Topics in Algebraic Topology
Catalog Number: 49813
Jacob Lurie 5450
Half course (fall term; repeated spring term). . EXAM GROUP: 4

*Mathematics 374. Topics in Number Theory
Catalog Number: 83329
Arul Shankar 7303
Half course (fall term; repeated spring term). . EXAM GROUP: 4

*Mathematics 381. Introduction to Geometric Representation Theory
Catalog Number: 0800
Dennis Gaitsgory 5259
Half course (fall term; repeated spring term). . EXAM GROUP: 15

*Mathematics 382. Topics in Algebraic Geometry
Catalog Number: 2037
Joseph D. Harris 2055
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 17; Spring: 13

*Mathematics 385. Topics in Set Theory
Catalog Number: 33405
W. Hugh Woodin 7421 (on leave spring term)
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 13; Spring: 9

*Mathematics 387. Topics in Mathematical Physics: Bridgeland Stability Conditions
Catalog Number: 47551
Hiro Tanaka 7326 (on leave spring term)
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 16; Spring: 6

*Mathematics 388. Topics in Mathematics and Biology
Catalog Number: 4687
Martin A. Nowak 4568 (on leave fall term)
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 17; Spring: 13

*Mathematics 389. Topics in Number Theory
Catalog Number: 6851
Noam D. Elkies 2604
Half course (fall term; repeated spring term). . EXAM GROUP: Fall: 1; Spring: 8