### Courses Fall 2015

MA | Intro: Functions and Calculus I | Brendan Kelly | The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students. &This is a lecture course taught in small sections. In addition, participation in two one-hour workshops is 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 together with Mathematics Mb, can be followed by Mathematics 1b. | ||

1A | Introduction to Calculus | Janet Chen | 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: Participation in a weekly 90-minute workshop is required. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. Janet Chen, Wes Cain, Yaiza Canzani, and members of the Department (fall term); Brendan Kelly, and members of the Department (spring term) This is a lecture Course taught in small sections. Fall: section times: M., W., F., at 9 (with sufficient enrollment); M., W., F., at 10; | ||

1B | Calculus, Series and Differential equations | Robin Gottlieb | 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. This is a lecture taught in small sections. Note: This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning.Robin Gottlieb, Rosalie Belanger-Rioux, Sarah Chisholm, Yu-Wen Hsu, Eric Peterson, Nina Zipser, and members of the Department (fall term); Peter Garfield, Jameel Al-Aidroos, Sarah Chisholm, Dusty Grundmeier, and members of the Department (spring term). | ||

18 | Multivariable Calculus | Peter Garfield | MWF | 11:00 AM | 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. 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 Math 21a. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. |

19A | Modeling/Differential Equation | John Cain | MWF | 01:00 PM | 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). 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. |

21A | Multivariable Calculus | Oliver Knill | 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 Green's, Stokes's, and Divergence Theorems.Note: 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.Oliver Knill, William Boney, Matthew Demers, Dusty Grundmeier, Yu-Wen Hsu, and members of the Department (fall term); Janet Chen, Jameel Al-Aidroos, Matthew Demers, Yu-Wen Hsu, and members of the Department (spring term). This is a lecture taught in small sections. Fall: section times: M., W., F., at 9; M., W., F., at 10; M., W., F., at 11; M., W., F., at 12; Tu., Th., 10-11:30; Tu., Th., 11:30-1. | ||

21B | Linear Algebra and Differential Equ | Peter Garfield | 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: 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. Peter Garfield, Rosalie Belanger-Rioux, and members of the Department (fall term); Oliver Knill, Rosalie Belanger-Rioux, Peter Garfield, Fabian Halden, Yu-Wen Hsu, and members of the Department (spring term) This is a lecture taught in small sections. Fall: section times: M., W., F., at 10 (with sufficient enrollment); M., W., F., at 11; M., W., F., at 12 (with sufficient enrollment); | ||

23A | Linear Algebra and Real Analysis | Paul Bamberg | TTh | 02:30 PM | Linear algebra: vectors, linear transformations and matrices, scalar and vector products, basis and dimension, eigenvectors and eigenvalues, including an introduction to the R scripting language. Single-variable real analysis: sequences and series, limits and continuity, derivatives, inverse functions, power series and Taylor series.& Multivariable real analysis and calculus: topology of Euclidean space, limits, continuity, and differentiation in n dimensions, inverse and implicit functions, manifolds, Lagrange multipliers, path integrals, div, grad, and curl.& Emphasis on topics that are applicable to fields such as physics, economics, and computer science, but students are also expected to learn how to prove key results. Course content overlaps substantially with Mathematics 21a, 25a,b, so students should plan to continue in Mathematics 23b. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. |

25A | Honors Linear Algebra | Tasho Kaletha | MWF | 10:00 AM | 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. For students with a strong interest and background in mathematics. Expect to spend a lot of time doing mathematics. 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. |

55A | Honors Abstract Algebra | Yum-Tong Siu | TTh | 02:30 PM | A rigorous treatment of abstract algebra including linear algebra and group theory. 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. |

60R | Reading Course for Snr Honors Cand | Jacob Lurie | Advanced reading in topics not covered in courses. 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. | ||

91R | Supervised Reading and Research | Jacob Lurie | Programs of directed study supervised by a person approved by the Department. May not ordinarily count for concentration in Mathematics. | ||

99R | Tutorial | Jacob Lurie | Supervised small group tutorial. Topics to be arranged. May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit. | ||

101 | Sets and Groups and Topology | Jonathan Esole | MWF | 12:00 PM | An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology. Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. This course given fall term and repeated spring term. |

114 | Analysis II | Daniel Cristofaro-Gardiner | TTh | 10:00 AM | Lebesgue measure and integration; general topology; introduction to L spaces, Banach and Hilbert spaces, and duality. ^{p} |

116 | Real Analysis and Convexity | Paul Bamberg | MWF | 01:00 PM | 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. |

121 | Linear Algebra and Application | Michael Hopkins | MWF | 02:00 PM | 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. |

122 | Algebra I | Benedict Gross | MWF | 11:00 AM | Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups. |

124 | Number Theory | Barry Mazur | TTh | 10:00 AM | Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms. |

131 | Topology I | Clifford Taubes | MWF | 12:00 PM | First, an introduction to abstract topological spaces and their properties; and then, an introduction to algebraic topology and in particular homotopy theory, fundamental groups and covering spaces. |

136 | Differential Geometry | Tristan Collins | MWF | 10:00 AM | 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. |

142 | Descriptive Set Theory | Peter Koellner | TTh | 11:30 AM | An introduction to the study of definable subsets of reals and their regularity properties (such as Lebesque measurability and the property of Baire). A discussion of the unresolvability of the classical questions in ZFC and their resolution through the introduction of axioms of definable determinacy and strong axioms of infinity. |

152 | Discrete Mathematics | Paul Bamberg | TTh | 11:30 AM | An introduction to sets, logic, finite groups, finite fields, finite geometry, finite topology, combinatorics and graph theory.& A recurring theme of the course is the symmetry group of the regular icosahedron.& Elementary category theory will be introduced as a unifying principle. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard. Covers material used in Computer Science 121 and Computer Science 124. |

153 | Evolutionary Dynamics | Martin Nowak | TTh | 02:30 PM | 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. |

212A | Real Analysis | Shlomo Sternberg | TTh | 11:30 AM | Banach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform. |

213A | Complex Analysis | Wilfried Schmid | TTh | 02:30 PM | A second course in complex analysis: series, product and partial fraction expansions of holomorphic functions; Hadamard's theorem; conformal mapping and the Riemann mapping theorem; elliptic functions; Picard's theorem and Nevanlinna Theory. |

221 | Algebra | Alison Miller | MWF | 10:00 AM | A first course in Algebra: Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Galois theory, Noether normalization, the Nullstellensatz, localization, primary decomposition.Representation theory of finite groups. Introduction to Lie groups and Lie algebras:definitions, the exponential maps, semi-simple Lie algebras, examples. |

223A | Algebraic Number Theory | Arul Shankar | MWF | 12:00 PM | 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. |

229X | Intro to Analytic Number Theory | Noam Elkies | MWF | 11:00 AM | Fundamental methods, results, and problems of analytic number theory. Riemann zeta function and the Prime Number Theorem; Dirichlet's theorem on primes in arithmetic progressions; lower bounds on discriminants from functional equations; sieve methods, analytic estimates on exponential sums, and their applications. |

230A | Differential Geometry | Hiro Tanaka | TTh | 10:00 AM | Smooth manifolds (vector fields, differential forms, and their algebraic structures; Frobenius theorem), Riemannian geometry (metrics, connections, curvatures, geodesics, flatness, and manifolds of constant curvature), symplectic geometry, Lie groups, principal bundles. |

231A | Algebraic Topology | Peter Kronheimer | MWF | 02:00 PM | Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality. |

232A | Intro to Algebraic Geometry I | Jonathan Esole | MWF | 11:00 AM | Introduction to complex algebraic curves, surfaces, and varieties. |

254 | Topics in Random Matrices | Horng-Tzer Yau | MWF | 12:00 PM | A discussion of a few advanced topics in random matrix theory. Topics include: analysis of Dyson Brownian motions via PDE method, eigenvector flows and beta ensembles. Depending on the progress of the class, other topics will be added. &We will assume stochastic calculus and a basic knowledge of random matrix theory. &A quick overview of random matrix theory will be given in the beginning of the course. |

256 | Dynamics, Stability, noncomm. Algebra | Haiden | TTh | 02:30 PM | An introduction leading up to current research, to notions and results in the theory of dynamical systems, in particular Teichmuller theory, and their analogs or generalizations in non-commutative (derived) algebraic geometry in the sense of Kontsevich. |

276 | Topics in Probability Theory | Chen | TTh | 10:00 AM | This course introduces stochastic calculus. Topics: Brownian motion, martingales, and stochastic integration and differential equations. &Measure theory and basic probability theory are prerequisites. |

277 | Fukaya Categories and Sheaves | Hiro Tanaka | MWF | 01:00 PM | After setting up the foundations for defining Fukaya categories, we will explore results showing that various Fukaya categories "glue". Little analytic background will be assumed, but we will attempt to cover the foundations. |

282 | Geometry of Algebraic Curves | Joseph Harris | MWF | 10:00 AM | Algebraic curves are some of the most fascinating objects in algebraic geometry: we know a good deal about them, but many major open questions remain.& At the heart of these questions is the relation between the classical notion of curves in projective space and the modern notion of abstract curves. The theory of linear systems on curves, which we'll study in this course, represents a bridge between these two notions. We'll cover the basic theorems governing linear systems on curves (Riemann-Roch, Clifford's theorem, etc.) and go on to discuss the Castelnuovo and Brill-Noether theorems, ending with a survey of open problems (and what we know about them so far). |

289 | Topics on Geometric Analysis | Shing-Tung Yau | TTh | 11:30 AM | We shall cover topics related to existence and regularity for nonlinear equations appearing in Kahler geometry and general relativity. |

300 | Teaching Undergraduate Math | Brendan Kelly | T | 01:00 PM | Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students. |

300 | Teaching Undergraduate Math | Robin Gottlieb | T | 01:00 PM | Become an effective instructor. This course focuses on observation, practice, feedback, and reflection providing insight into teaching and learning. Involves iterated videotaped micro-teaching sessions, accompanied by individual consultations. Required of all mathematics graduate students. |

303 | Topics in Diophantine Problems | Hector Pasten Vasquez | |||

304 | Topics in Algebraic Topology | Michael Hopkins | |||

308 | Number Theory and Modular Form | Benedict Gross | |||

314 | Topics in Differential Geometry | Shlomo Sternberg | |||

318 | Topics in Number Theory | Barry Mazur | |||

321 | Topics in Mathematical Physics | Arthur Jaffe | |||

327 | Several Complex Variables | Yum-Tong Siu | |||

333 | Complex Analysis, Dynamics, Geometry | Curtis McMullen | |||

335 | Topics in Differential Geometry | Clifford Taubes | |||

343 | Topics in Complex Geometry | Tristan Collins | |||

345 | Topics in Geometry and Topology | Peter Kronheimer | |||

346Y | Topics in Analysis | Horng-Tzer Yau | |||

348 | Topics in Representation Theory | ||||

352 | Topics in Algebraic Number | Mark Kisin | |||

356 | Topics in Harmonic Analysis | Wilfried Schmid | |||

357 | Topics in Model Theory | William Boney | |||

361 | Topics in Differential Geometry | Yaiza Canzani | |||

362 | Topics in Number Theory | Alison Miller | |||

363 | Topics in Elliptic Fibrations | Jonathan Esole | |||

365 | Topics in Differential Geometry | Shing-Tung Yau | |||

368 | Topics in Algebraic Topology | ||||

373 | Topics in Algebraic Topology | Jacob Lurie | |||

374 | Topics in Number Theory | Arul Shankar | |||

381 | Geometric Representation Theory | Dennis Gaitsgory | |||

382 | Topics in Algebraic Geometry | Joseph Harris | |||

385 | Topics in Set Theory | W. Hugh Woodin | |||

387 | Topics in Mathematical Physics | Hiro Tanaka | |||

388 | Topics in Mathematics and Biology | Martin Nowak |

### Courses Spring 2016

MB | Intro: Functions and Calculus II | Sarah Chisholm | 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. | ||

1A | Introduction to Calculus | Brendan Kelly | 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: Participation in a weekly 90-minute workshop is required. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. Janet Chen, Wes Cain, Yaiza Canzani, and members of the Department (fall term); Brendan Kelly, and members of the Department (spring term) This is a lecture Course taught in small sections. Fall: section times: M., W., F., at 9 (with sufficient enrollment); M., W., F., at 10; | ||

1B | Calculus, Series and Differential equations | Peter Garfield | 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. This is a lecture taught in small sections. Note: This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning.Robin Gottlieb, Rosalie Belanger-Rioux, Sarah Chisholm, Yu-Wen Hsu, Eric Peterson, Nina Zipser, and members of the Department (fall term); Peter Garfield, Jameel Al-Aidroos, Sarah Chisholm, Dusty Grundmeier, and members of the Department (spring term). | ||

19A | Modeling/Differential Equation | John Cain | MWF | 01:00 PM | 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). 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. |

19B | Linear Algebra and Probability | Rosalie Belanger-Rioux | MWF | 01:00 PM | 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. 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. |

21A | Multivariable Calculus | Janet Chen | 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 Green's, Stokes's, and Divergence Theorems.Note: 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.Oliver Knill, William Boney, Matthew Demers, Dusty Grundmeier, Yu-Wen Hsu, and members of the Department (fall term); Janet Chen, Jameel Al-Aidroos, Matthew Demers, Yu-Wen Hsu, and members of the Department (spring term). This is a lecture taught in small sections. Fall: section times: M., W., F., at 9; M., W., F., at 10; M., W., F., at 11; M., W., F., at 12; Tu., Th., 10-11:30; Tu., Th., 11:30-1. | ||

21B | Linear Algebra and Differential Equ | Oliver Knill | 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: 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. Peter Garfield, Rosalie Belanger-Rioux, and members of the Department (fall term); Oliver Knill, Rosalie Belanger-Rioux, Peter Garfield, Fabian Halden, Yu-Wen Hsu, and members of the Department (spring term) This is a lecture taught in small sections. Fall: section times: M., W., F., at 10 (with sufficient enrollment); M., W., F., at 11; M., W., F., at 12 (with sufficient enrollment); | ||

23B | Linear Algebra and Real Analysis | Paul Bamberg | TTh | 02:30 PM | 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. Stokes's theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. |

25B | Honors Linear Algebra II | Tasho Kaletha | MWF | 10:00 AM | 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. Expect to spend a lot time doing mathematics. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. |

55B | Honors Real and Complex Analysis | Yum-Tong Siu | TTh | 02:30 PM | A rigorous treatment of real and complex analysis. 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. |

60R | Reading Course for Snr Honors Cand | Jacob Lurie | Advanced reading in topics not covered in courses. 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. | ||

91R | Supervised Reading and Research | Jacob Lurie | Programs of directed study supervised by a person approved by the Department. May not ordinarily count for concentration in Mathematics. | ||

99R | Tutorial | Jacob Lurie | Supervised small group tutorial. Topics to be arranged. May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit. | ||

101 | Sets and Groups and Topology | Clifford Taubes | TTh | 11:30 AM | An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology. Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. This course given fall term and repeated spring term. |

110 | Vector Space Methods for | Paul Bamberg | TTh | 11:30 AM | 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 Laplace's equation and the wave equation in the various coordinate systems. |

112 | Introductory Real Analysis | Tristan Collins | TTh | 01:00 PM | 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. |

113 | Analysis I | Horng-Tzer Yau | MWF | 12:00 PM | Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy's theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions. |

115 | Methods of Analysis | Babak Haghighat | TTh | 01:00 PM | Complex functions; Fourier analysis; Hilbert spaces and operators; Laplace's equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory. Mathematics 115 is especially for students interested in physics. |

118R | Dynamical Systems | Fabian Haiden | TTh | 10:00 AM | 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. |

123 | Algebra II | Arul Shankar | TTh | 11:30 AM | Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules. |

129 | Number Fields | Barry Mazur | TTh | 10:00 AM | 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. |

130 | Classical Geometry | Eric Peterson | MWF | 11:00 AM | 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. |

132 | Topology II: Smooth Manifolds | Michael Hopkins | MWF | 12:00 PM | Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes' theorem, introduction to cohomology. |

137 | Algebraic Geometry | Joseph Harris | MWF | 10:00 AM | Affine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem. |

144 | Model Theory | William Boney | MWF | 11:00 AM | 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. |

154 | Probability Theory | Jacob Lurie | MWF | 01:00 PM | 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. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. |

155R | Combinatorics | Noam Elkies | MWF | 11:00 AM | 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, Ramsey's theorem and its variants, probabilistic methods. |

157 | Mathematics in the World | Joseph Harris | TTh | 02:30 PM | 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: logic, information, number theory, probability, and algorithms. |

212BR | Advanced Real Analysis | Shlomo Sternberg | TTh | 10:00 AM | Functional analysis related to quantum mechanics. Topics include (but not limited to) The Stone-von Neumann theorem, Gruenwald-van Hove theorem, Ruelle's theorem on the continuous spectrum and scattering states, Agmon's theorem on the exponential decay of bound states, scattering theory. |

213BR | Advanced Complex Analysis | Shing-Tung Yau | TTh | 11:30 AM | Fundamentals of Riemann surfaces. Topics may include sheaves and cohomology, potential theory, uniformization, and moduli. |

222 | Lie Groups and Lie Algebras | Alison Miller | MWF | 10:00 AM | Lie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations. |

223B | Algebraic Number Theory | Arul Shankar | TTh | 01:00 PM | Continuation of Mathematics 223a. Topics: adeles, global class field theory, duality, cyclotomic fields. Other topics may include: Tate's thesis or Euler systems. |

230BR | Advanced Differential Geometry | Shing-Tung Yau | TTh | 11:30 AM | A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications. |

231BR | Advanced Algebraic Topology | Peter Kronheimer | MWF | 02:00 PM | Continuation of Mathematics 231a. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories. |

232BR | Algebraic Geometry II | Jonathan Esole | MWF | 12:00 PM | The course will cover the classification of complex algebraic surfaces. |

233A | Theory of Schemes I | Yaim Cooper | MWF | 11:00 AM | 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. |

243 | Evolutionary Dynamics | Martin Nowak | TTh | 02:30 PM | Advanced topics of evolutionary dynamics. Seminars and research projects. |

253X | Spin Systems | TTh | 10:00 AM | Classical spin systems generalize the Ising model to a general number of components and general spin distributions. They are fundamental models of phase transitions. Topics: the proof of existence of phase transitions, continuous symmetry, methods based on convexity, random walk representations, correlation inequalities, critical phenomena, and the relation of spin systems to the self-avoiding walk. | |

264 | Nonlinear Partial Differential | Tristan Collins | MWF | 11:00 AM | An introduction to techniques in nonlinear elliptic equations. Topics: the Schauder theory, the Cordes-Nirenberg estimates for nonlinear equations, viscosity techniques and the ABP estimate, the Krylov-Safonov Harnack inequality and the Evans-Krylov theorem. Applications include solvability of the Dirichlet problem for convex equations and the Monge-Ampere equation in a domain. Possible further topics: minimal surfaces, and the sigma_K equations. |

267 | Reductive Groups Over Local an | Tasho Kaletha | MWF | 01:00 PM | The study of the structure of connected reductive groups over p-adic fields (i.e. non-archimedean local fields of characteristic zero) and their smooth irreducible representations. In addition, we will touch upon the corresponding theory over the real numbers and then discuss automorphic representations of the adelic points of connected reductive groups defined over number fields. |

273 | Introduction to Semiclassical | Yaiza Canzani | TTh | 01:00 PM | Semiclassical analysis is a branch of the general theory of partial differential operators.& It is used, for example, to study qualitative properties of solutions of partial differential equations and spectral asymptotics like Weyl's law.& We shall learn how to microlocalize in phase space and to use Hamiltonian dynamical systems in this space to study the partial differential equations problems. The beauty of the field lies in this interaction between analysis and geometry.& We will develop the basic setup of the theory and then give a guided tour through some of the applications in spectral asymptotics and quantum ergodicity. |

278 | Formal Geom in Algebr Topology | Eric Peterson | MWF | 12:00 PM | Topics in algebraic topology and formal geometry, with a focus on complex-oriented phenomena and the construction of the homotopical sigma-orientation. |

287 | Contact Homology | Daniel Cristofaro-Gardiner | TTh | 02:30 PM | The purpose of this course is to introduce some invariants of contact manifolds that are defined by counting pseudolomorphic curves, and discuss some applications. We will also discuss some related invariants of symplectic manifolds. The particular invariants we will discuss will be chosen in part based on audience input; an emphasis of the course will be a recently developed invariant called embedded contact homology, and other topics may include cylindrical and linearized contact homology, Legendrian contact homology, the contact homology algebra, symplectic homology, and symplectic field theory. Some of these invariants have yet to be rigorously defined, and part of the course will involve explaining some of the issues that remain to be resolved, and some possible approaches.& Applications will include generalizations of the Weinstein conjecture, symplectic embedding problems, distinguishing contact structures, and calculating Gromov-Witten invariants by cutting along contact-type hypersurfaces. |

303 | Topics in Diophantine Problems | Hector Pasten Vasquez | |||

304 | Topics in Algebraic Topology | Michael Hopkins | |||

308 | Number Theory and Modular Form | Benedict Gross | |||

314 | Topics in Differential Geometry | Shlomo Sternberg | |||

318 | Topics in Number Theory | Barry Mazur | |||

321 | Topics in Mathematical Physics | Arthur Jaffe | |||

327 | Several Complex Variables | Yum-Tong Siu | |||

333 | Complex Analysis, Dynamics, Geometry | Curtis McMullen | |||

335 | Topics in Differential Geometry | Clifford Taubes | |||

343 | Topics in Complex Geometry | Tristan Collins | |||

345 | Topics in Geometry and Topology | Peter Kronheimer | |||

346Y | Topics in Analysis | Horng-Tzer Yau | |||

348 | Topics in Representation Theory | ||||

352 | Topics in Algebraic Number | Mark Kisin | |||

356 | Topics in Harmonic Analysis | Wilfried Schmid | |||

357 | Topics in Model Theory | William Boney | |||

361 | Topics in Differential Geometry | Yaiza Canzani | |||

362 | Topics in Number Theory | Alison Miller | |||

363 | Topics in Elliptic Fibrations | Jonathan Esole | |||

365 | Topics in Differential Geometry | Shing-Tung Yau | |||

368 | Topics in Algebraic Topology | ||||

373 | Topics in Algebraic Topology | Jacob Lurie | |||

374 | Topics in Number Theory | Arul Shankar | |||

381 | Geometric Representation Theory | Dennis Gaitsgory | |||

382 | Topics in Algebraic Geometry | Joseph Harris | |||

385 | Topics in Set Theory | W. Hugh Woodin | |||

387 | Topics in Mathematical Physics | Hiro Tanaka | |||

388 | Topics in Mathematics and Biology | Martin Nowak | |||

389 | Topics in Number Theory | Noam Elkies |

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