Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132

Introduction

The Mathematics Department hopes that all students will take mathematics courses. This said, be careful to take only those courses that are appropriate for your level of experience. Incoming students should take advantage of Harvards Mathematics Placement Test and of the science advising available in the Science Center the week before classes begin. Members of the Mathematics Department will be available during this period to consult with students. Generally, students with a strong precalculus background and some calculus experience will begin their mathematics education here with a deeper study of calculus and related topics in courses such as Mathematics 1a, 1b, 18,19a,b, 21a,b, 23a,b and 25a,b. The Harvard Mathematics Placement Test results recommend the appropriate starting level course, either Mathematics Ma, 1a, 1b, or 21. Recommendation for Mathematics 21 is sufficient qualification for Mathematics 18, 19a,b, 21a, 23a, and 25a.

What follows briefly describes these courses: Mathematics 1a introduces the basic ideas and techniques of calculus while Mathematics 1b covers integration techniques, differential equations, and series. Mathematics 21a covers multi-variable calculus while Mathematics 21b covers basic linear algebra with applications to differential equations. Students who do not place into (or beyond) Mathematics 1a can take Mathematics Ma, Mb, a two-term sequence which integrates calculus and precalculus material and prepares students to enter Mathematics 1b.

There are a number of options available for students whose placement is to Mathematics 21. For example, Mathematics 19a,b are courses that are designed for students concentrating in the life sciences. (These course are recommended over Math 21a,b by the various life science concentrations). In any event, Math 19a can be taken either before or after Math 21a,b. Math 19b should not be taken with Math 21b. Math 19a teaches differential equations, related techniques and modeling with applications to the life sciences. Math 19b teaches linear algebra, probability and statistics with a focus on life science examples and applications. Mathematics 18 covers selected topics from Mathematics 1b and 21a for students particularly interested in economic and social science applications.

Mathematics 23 is a theoretical version of Mathematics 21 which treats multivariable calculus and linear algebra in a rigorous, proof oriented way. Mathematics 25 and 55 are theory courses that should be elected only by those students who have a strong interest in mathematics. They assume a solid understanding of one-variable calculus, a willingness to think rigorously and abstractly about mathematics, and to work extremely hard. Both courses study multivariable calculus and linear algebra plus many very deep related topics. Mathematics 25 differs from Mathematics 23 in that the work load in Mathematics 25 is significantly more than in Mathematics 23, but then Mathematics 25 covers more material. Mathematics 55 differs from Mathematics 25 in that the former assumes a very strong proof oriented mathematics background. Mathematics 55, covers the material from Mathematics 25 plus much material from Mathematics 122 and Mathematics 113. Entrance into Mathematics 55 requires the consent of the instructor.

Students who have had substantial preparation beyond the level of the Advanced Placement Examinations are urged to consult the Director of Undergraduate Studies in Mathematics concerning their initial Harvard mathematics courses. Students should take this matter very seriously. The Mathematics Department has also prepared a pamphlet with a detailed description of all its 100-level courses and their relationship to each other. This pamphlet gives sample lists of courses suitable for students with various interests. It is available at the Mathematics Department Office. Many 100-level courses assume some familiarity with proofs. Courses that supply this prerequisite include Mathematics 23, 25, 55, 101, 112, 121, and 141. Of these, note that Mathematics 101 may be taken concurrently with Mathematics 1, 18, 19, or 21.

Mathematics 113, 114, 122, 123, 131, and 132 form the core of the departments more advanced courses. Mathematics concentrators are encouraged to consider taking these courses, particularly Mathematics 113, 122 and 131. (Those taking 55a,b will have covered the material of Mathematics 113 and 122, and are encouraged to take Mathematics 114, 123, and 132.)

Courses numbered 200-249 are introductory graduate courses. They will include substantial homework and are likely to have a final exam, either in class or take home. Most are taught every year. They may be suitable for very advanced undergraduates. Mathematics 212a, 230a, 231a and 232a will help prepare graduate students for the qualifying examination in Mathematics. Courses numbered 250-299 are graduate topic courses, intended for advanced graduate students.

The Mathematics Department does not grant formal degree credit without prior approval for taking a course that is listed as a prerequisite of one you have already taken. Our policy is that a student who takes and passes any calculus course is not normally permitted to then take a more elementary course for credit. A student who has passed Mathematics 21a, for example, will normally not be allowed to take Mathematics 1a, or 1b for credit. The Mathematics Department is prepared to make exceptions for sufficient academic reasons; in each case, however, a student must obtain written permission from the Mathematics Director of Undergraduate Studies in advance.

In the case of students accepting admission as sophomores, this policy is administered as follows: students counting one half course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a, and students counting a full course of advanced standing credit in mathematics are deemed to have passed Mathematics 1a and 1b.

Faculty of the Department of Mathematics


Peter B. Kronheimer William Caspar Graustein Professor of Mathematics (Chair)
Jameel Habeeb Al-Aidroos Senior Preceptor in Mathematics
Paul G. Bamberg Senior Lecturer on Mathematics
Roland Bauerschmid Lecturer on Mathematics
Rosalie Belanger-Rioux Preceptor in Mathematics
William Boney Benjamin Peirce Fellow
John Cain Senior Lecturer on Mathematics
Yaiza Canzani Benjamin Peirce Fellow
Janet Chen Senior Preceptor in Mathematics
Yu-Ting Chen Lecturer on Mathematics
Sarah Chisholm Preceptor in Mathematics
Tristan Collins Benjamin Peirce Fellow
Yaim Cooper Lecturer on Mathematics
Daniel Cristoforo-Gardiner Benjamin Peirce Fellow
Matthew Demers Lecturer on Mathematics
Noam D. Elkies Professor of Mathematics
Mboyo Esole Benjamin Peirce Fellow
Dennis Gaitsgory Professor of Mathematics (on leave 2015-16)
Peter McKee Garfield Preceptor in Mathematics
Robin Gottlieb Professor of the Practice of Mathematics (on leave spring term)
Benedict H. Gross George Vasmer Leverett Professor of Mathematics (on leave spring term)
Dusty Grundmeier Lecturer on Mathematics
Babak Haghighat Lecturer on Mathematics
Fabian Haiden Benjamin Peirce Fellow
Joseph D. Harris Higgins Professor of Mathematic
Michael J. Hopkins Professor of Mathematics
Yu-Wen Hsu Preceptor in Mathematics
Tasho Kaletha Benjamin Peirce Fellow
Brendan Kelly Preceptor in Mathematics
Mark Kisin Professor of Mathematics (Director of Graduate Studies) (on leave spring term)
Oliver Knill Preceptor in Mathematics
Jacob Lurie Professor of Mathematics (Director of Undergraduate Studies)
Barry C. Mazur Gerhard Gade University Professor
Curtis T. McMullen Maria Moors Cabot Professor of the Natural Sciences (on leave 2015-16)
Alison Miller Benjamin Peirce Fellow
Martin A. Nowak Professor of Mathematics and of Biology
Hector Pasten Benjamin Peirce Fellow (on leave 2015-16)
Eric Peterson Benjamin Peirce fellow
Wilfried Schmid Dwight Parker Robinson Professor of Mathematics (on leave spring term)
Arul Shankar Benjamin Peirce Fellow
Yum Tong Siu William Elwood Byerly Professor of Mathematics
Shlomo Z. Sternberg George Putnam Professor of Pure and and Applied Mathematics
Hiro Tanaka Benjamin Peirce Fellow
Clifford Taubes William Petschek Professor of Mathematics
Bena Tshishiku Benjamin Peirce Fellow (on leave 2015-16)
Brooke Ullery Benjamin Peirce Fellow (on leave 2015-16)
Hugh Woodin Professor of Philosophy and of Mathematics (on leave fall term)
Horng-Tzer Yau Professor of Mathematics
Shing-Tung Yau William Caspar Graustein Professor of Mathematics
Nina Zipser Lecturer on Mathematics and Dean for Faculty Affairs (fall term only)
Other Faculty Offering Instruction in the Department of Mathematics
Arthur M. Jaffe Landon T. Clay Professor of Mathematics and Theoretical Science
Peter Koellner Professor of Philosophy

Courses Fall 2015

MA Intro: Functions and Calculus I Brendan Kelly MWF 10,11,12The 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 MWF 9,10,11,12 TTh 10,11:30 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 intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, at 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1. Weekly workshop times to be arranged. Spring: MWF, at 10. Weekly workshop times to be arranged.
1B Calculus, Series and Differential equations Robin Gottlieb MWF 9,10,11,12TTh 10,11:30 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). This is a lecture taught in small sections. Fall intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, at 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1. Spring: spring times: MWF, at 10; MWF, 11; MWF, 12; TuTh, 10-11:30 (with sufficient enrollment); TuTh, 11:30-1(with sufficient enrollment), and a weekly problem section to be arranged.
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 MWF 9,10,11,12 TTh 10,11:30 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. A project 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 intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9; MWF, at 10; MWF, at 11; MWF, at 12; TuTh, 10-11:30; TuTh, 11:30-1. Spring: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1 (with sufficient enrollment), and a weekly problem section to be arranged
21B Linear Algebra and Differential Equ Peter Garfield MWF 10,11,12 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 course taught in small sections. Fall intro meeting: September 2: 8:30 AM. Fall intro meeting: September 2: 8:30. Fall: section times: MWF, at 10 (with sufficient enrollment); MWF, at 11; MWF, at 12 (with sufficient enrollment); Spring: section times: MWF, at 9; MWF, at 10; MWF, at 11; MWF, 12; TuTh, 10-11:30; TuTh, 11:30-1;, and a weekly problem section to be arranged.
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 p spaces, Banach and Hilbert spaces, and duality.
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 SheavesHiro 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 MWF 10,11,12Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b. 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 Ma, can be followed by Mathematics 1b.
1A Introduction to Calculus Brendan Kelly MWF 10 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 intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, at 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1. Weekly workshop times to be arranged. Spring: MWF, at 10. Weekly workshop times to be arranged.
1B Calculus, Series and Differential equations Peter Garfield MWF 10,11,12TTh 10,11:30 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). This is a lecture taught in small sections. Fall intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, at 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1. Spring: spring times: MWF, at 10; MWF, 11; MWF, 12; TuTh, 10-11:30 (with sufficient enrollment); TuTh, 11:30-1(with sufficient enrollment), and a weekly problem section to be arranged.
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 MWF 9,10,11,12 TTh 10,11:30 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. A project 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 intro meeting: September 2: 8:30 AM. Fall: section times: MWF, at 9; MWF, at 10; MWF, at 11; MWF, at 12; TuTh, 10-11:30; TuTh, 11:30-1. Spring: section times: MWF, at 9 (with sufficient enrollment); MWF, at 10; MWF, at 11; MWF, 12 (with sufficient enrollment); TuTh, 10-11:30; TuTh, 11:30-1 (with sufficient enrollment), and a weekly problem section to be arranged
21B Linear Algebra and Differential Equ Oliver Knill MWF 9,10,11,12 TTh 10,11:30 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 course taught in small sections. Fall intro meeting: September 2: 8:30 AM. Fall intro meeting: September 2: 8:30. Fall: section times: MWF, at 10 (with sufficient enrollment); MWF, at 11; MWF, at 12 (with sufficient enrollment); Spring: section times: MWF, at 9; MWF, at 10; MWF, at 11; MWF, 12; TuTh, 10-11:30; TuTh, 11:30-1;, and a weekly problem section to be arranged.
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 HaidenTTh 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 PetersonMWF 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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