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 Kronheimer William Caspar Graustein Professor of Mathematics (Chair)
Jameel Al-Aidroos Senior Preceptor in Mathematics
Paul Bamberg Senior Lecturer on Mathematics
Rosalie Belanger-Rioux Preceptor in Mathematics
William Boney Benjamin Peirce Fellow
John Cain Senior Lecturer on Mathematics
Janet Chen Senior Preceptor in Mathematics
Yu-Ting Chen Lecturer on Mathematics
Man-Wai Cheung Benjamin Peirce Fellow (on leave 2016-2017)
Sarah Chisholm Preceptor in Mathematics
Tristan Collins Benjamin Peirce Fellow
Daniel Cristoforo-Gardiner Benjamin Peirce Fellow
Matthew Demers Lecturer on Mathematics
Noam Elkies Professor of Mathematics
Karen Edwards Lecturer in Mathematics
Philip Engel Lecture in Mathematics
Dennis Gaitsgory Professor of Mathematics
Robin Gottlieb Professor of the Practice of Mathematics (on leave Fall 2016)
Benedict Gross George Vasmer Leverett Professor of Mathematics (on leave 2016-2017)
Dusty Grundmeier Preceptor in Mathematics
Neha Gupta Preceptor in Mathematics
Fabian Haiden Benjamin Peirce Fellow
Joseph Harris Higgins Professor of Mathematic
Stephen Hermes Lecturer in Mathematics
Michael Hopkins Professor of Mathematics
Yu-Wen Hsu Preceptor in Mathematics
Aukosh Jagannath Benjamin Peirce Fellow (on leave 2016-2017)
Brendan Kelly Preceptor in Mathematics
Mark Kempton Lecturer in Mathematics
Aruna Kesavan Lecturer in Mathematics
Mark Kisin Professor of Mathematics (Director of Graduate Studies)
Oliver Knill Preceptor in Mathematics
Alexander Kupers Benjamin Peirce Fellow (on leave 2016-2017)
Jacob Lurie Professor of Mathematics (Director of Undergraduate Studie
Barry Mazur Gerhard Gade University Professor
Brendan McLellan Lecturer in Mathematics
Curtis McMullen Maria Moors Cabot Professor of the Natural Sciences
George Melvin Lecturer on Mathematics
Alison Miller Benjamin Peirce Fellow
Martin Nowak Professor of Mathematics and of Biology
Hector Pasten Benjamin Peirce Fellow
Anand Patel Visiting Lecturer
Katherine Penner Preceptor in Mathematics
Eric Peterson Benjamin Peirce fellow
Wilfried Schmid Dwight Parker Robinson Professor of Mathematics
Arul Shankar Benjamin Peirce Fellow
Yum Tong Siu William Elwood Byerly Professor of Mathematics (on leave fall term)
Philippe Sosoe Lecturer in Mathematics
Shlomo Sternberg George Putnam Professor of Pure and and Applied Mathematics
Hiro Tanaka Benjamin Peirce Fellow
Clifford Taubes William Petschek Professor of Mathematics
Arnav Tripathy Benjamin Peirce Fellow
Bena Tshishiku Benjamin Peirce Fellow
Brooke Ullery Benjamin Peirce Fellow
Hugh Woodin Professor of Philosophy and of Mathematics
Horng-Tzer Yau Professor of Mathematics
Shing-Tung Yau William Caspar Graustein Professor of Mathematics
Nina Zipser Lecturer in Mathematics and Dean of Faculty Affairs

Courses Fall 2016

MATH MAIntroduction to Functions and Calculus IKellySectionedThe 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.
MATH 1AIntroduction to CalculusCainSectionedThe 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.
MATH 1BCalculus, Series, and Differential EquationsPennerSectionedSpeaking 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.
MATH 18Multivariable Calculus for Social SciencesHsuMWF 11, SC 216Focus 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.
MATH 19AModeling and Differential Equations for the Life SciencesCainMWF 1, SC Hall EConsiders 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).
MATH 21AMultivariable CalculusKnillSectionedTo 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.
MATH 21BLinear Algebra and Differential EquationsChenSectionedMatrices 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.
MATH 23ALinear Algebra and Real Analysis IPatelTTh 1:30-3, SC Hall ALinear 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
MATH 25AHonors Linear Algebra and Real Analysis IPetersonMWF 10, SC Hall EA 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.
MATH 55AHonors Abstract AlgebraElkiesMWF 11, SC 507A rigorous treatment of abstract algebra including linear algebra and group theory.
MATH 60RReading Course for Senior Honors CandidatesLurieTBAAdvanced reading in topics not covered in courses.
MATH 99RTutorialLurieTBASupervised small group tutorial. In the fall semester, the title is "Arithmetic of elliptic curves" taught by Zijian Yao More information.
MATH 101Sets, Groups and KnotsMcMullenTuTh 10-11:30, SC 216An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and low-dimensional topology.
MATH 114Analysis II: Measure, Integration and Banach SpacesHaidenTTh 1-2:30, SC 310Lebesgue measure and integration; general topology; introduction to Lp spaces, Banach and Hilbert spaces, and duality.
MATH 116Real Analysis, Convexity, and OptimizationBambergMW 12:30-2, SC 222Develops 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.
MATH 117Probability and Random Processes with Economic ApplicationsBambergTuTh 2:30-4, SC 104A self-contained treatment of the theory of probability and random processes with specific application to the theory of option pricing. Topics: axioms for probability, calculation of expectation by means of Lebesgue integration, conditional probability and conditional expectation, martingales, random walks and Wiener processes, and the Black-Scholes formula for option pricing. Students will work in small groups to investigate applications of the theory and to prove key results.
MATH 121Linear Algebra and ApplicationsUlleryMWF 11, Severs 206Real and complex vector spaces, linear transformations, determinants, inner products, dual spaces, and eigenvalue problems. Applications to geometry, systems of linear differential equations, optimization, and Markov processes.
MATH 122Algebra I: Theory of Groups and Vector SpacesHopkinsMWF 2, SC Hall EGroups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.
MATH 124Number TheoryTripathy TTh 10-11:30, SC 411Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms.
MATH 131Topology I: Topological Spaces and the Fundamental GroupHarrisMWF 10, SC 507First, 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.
MATH 136Differential GeometryTaubesMWF 12, SC 507The course is an introduction to Riemannian geometry with the focus (for the most part) being the Riemannian geometry of curves and surfaces in space where the fundamental notions can be visualized.
MATH 145ASet Theory IBoneyTTh 1-2:30, SC 411An introduction to set theory covering the fundamentals of ZFC (cardinal arithmetic, combinatorics, descriptive set theory) and the independence techniques (the constructible universe, forcing, the Solovay model). We will demonstrate the independence of CH (the Continuum Hypothesis), SH (Suslin's Hypothesis), and some of the central statements of classical descriptive set theory.
MATH 152Discrete MathematicsBambergTTh 11:30-1, NW Build B-104An 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.
MATH 153Mathematical Biology-Evolutionary DynamicsNowakTTh 2:30-4, 1 Brattle Sq 616Introduces 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.
MATH 212AReal AnalysisHT YauMWF 10, SC 310 Measure theory, functional analysis, Sobolev spaces and introduction to harmonic analysis.
MATH 213AComplex AnalysisSchmidTuTh 2:30-4, SC 216A 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.
MATH 221AlgebraPasten VasquezMWF 11, SC 310A 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.
MATH 230ADifferential GeometryCollinsMWF 1, SC 507Smooth 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.
MATH 231AAlgebraic TopologyKronheimerMWF 2, SC 507Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincare duality.
MATH 232AIntroduction to Algebraic Geometry IEngelMWF 12, SC 310Introduction to complex algebraic curves, surfaces, and varieties.
MATH 256XHeisenberg Calculus in Quantum TopologyMcLellanMWF 12, SC 411The main goal of this course is to provide a hands-on introduction to the Heisenberg calculus of Beals-Greiner and Taylor. This pseudodifferential calculus provides a natural method for studying non-elliptic problems where the classical abelian symbol calculus is not applicable. This theory is of indenpendent interest and also makes connections with diversive topics such as harmonic analysis, noncommutative geometry, number theory, partial differential equations, representation theory.
MATH 258YDegenerations in Algebraic GeometryPatelMWF 2, SC 411Many results in algebraic geometry can be proved by degenerating the varieties under consideration. The construction of useful degenerations is more an art rather than a science, highly dependent on context, but it is helpful to have an understanding of some "standard" examples. This course is intended to study examples which serve to illustrate common technical issues that arise under degeneration. We will attempt to cover instances of degeneration in three areas.
MATH 263Analytic Techniques in Algebraic GeometryCollinsTTh 10-11:30, SC 310This course will discuss applications of analysis to algebraic geometry. Topics: the d-bar equations, plurpotential theory and techniques of multiplier ideal sheaves with applications to Kodaira embedding, the Fujita conjecture, invariance of plurigenera (after Siu) and asymptotic invariants of line bundles. Possible further topics: applications of harmonic maps to the Frankel conjecture (after Siu-Yau), and applications to Yang-Mills theory to classification of class VII
MATH 274Symplectic DualityTripathyTu Th 2:30-4, SC 310An introduction to three-dimensional mirror symmetry, known as symplectic duality in the mathematical literature. Possible topics: the foundational papers of Braden-Licata-Proudfoot-Webster, the proposed construction of the Coulomb branch in the moduli space of vacua by Braverman-Finkelbert-Nakajima, and relations to geometric Langlands and two-dimensional mirror symmetry, especially quantum cohomology of symplectic resolutions and mirror symmetry for hypertoric varieties.
MATH 283Geometric Langlands Correspondence in Characteristic pGaitsgoryTTh 11:30-1, SC 310The genuine geometric Langlands correspondence is a difficult open problem. But it has a variant that is much more easily accessible (but still very interesting): when we consider D-modules in characteristic p. We will cover recent developments in this field, surveying the works of Braverman, Bezrukanikov, Chen, Travkin, and Zhu.
MATH 286Nonlinear Analysis in GeometryST. YauTTh 11:30-1, CMSA Building, 20 Garden Street G10We shall cover nonlinear equations appearing in complex geometry and their applications to algebraic geometry and physics.
MATH 300Teaching Undergraduate MathematicsAl-AidroosTue 1-2:30, SC 507Become 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.
MATH 303Topics in Diophantine ProblemsPasten VasquezTBANA
MATH 304Topics in Algebraic TopologyHopkinsTBANA
MATH 308Topics in Number Theory and Modular FormsGrossTBANA
MATH 314Topics in Differential Geometry and Mathematical PhysicsSternbergTBANA
MATH 316Topics in Algebraic GeometryTBATBANA
MATH 318Topics in Number TheoryMazurTBANA
MATH 321Topics in Mathematical PhysicsJaffeTBANA
MATH 327Topics in Several Complex VariablesSiuTBANA
MATH 333Topics in Complex Analysis, Dynamics and GeometryMcMullenTBANA
MATH 335Topics in Differential Geometry and AnalysisTaubesTBANA
MATH 343Topics in Complex GeometryCollinsTBANA
MATH 345Topics in Geometry and TopologyKronheimerTBANA
MATH 346YTopics in Analysis: Quantum DynamicsYauTBANA
MATH 348Topics in Representation TheoryHaidenTBANA
MATH 352Topics in Algebraic Number TheoryKisinTBANA
MATH 356Topics in Harmonic AnalysisSchmidTBANA
MATH 357Topics in Model TheoryBoneyTBANA
MATH 361Topics in Differential Geometry and AnalysisCanzaniTBANA
MATH 362Topics in Number TheoryMillerTBANA
MATH 364Topics in Algebraic GeometryUlleryTBANA
MATH 365Topics in Differential GeometryS.T. YauTBANA
MATH 368Topics in Algebraic TopologyPetersonTBANA
MATH 373Topics in Algebraic TopologyLurieTBANA
MATH 381Introduction to Geometric Representation TheoryGaitsgoryTBANA
MATH 382Topics in Algebraic GeometryHarrisTBANA
MATH 385Topics in Set TheoryWoodinTBANA
MATH 387Topics in Mathematical Physics: Bridgeland Stability ConditionsTanakaTBANA
MATH 388Topics in Mathematics and BiologyNowakTBANA
MATH 389Topics in Number TheoryElkiesTBANA

Courses Spring 2017

MATH MBIntroduction to Functions and Calculus IIKellySectionedContinued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.
MATH 1AIntroduction to CalculusGuptaSectionedThe 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.
MATH 1BCalculus, Series, and Differential EquationsGottliebSectionedSpeaking 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.
MATH 19BLinear Algebra, Probability, and Statistics for the Life SciencesBelanger-RiouxTBAProbability, 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.
MATH 21AMultivariable CalculusChenSectionedTo 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.
MATH 21BLinear Algebra and Differential EquationsKnillSectionedMatrices 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.
MATH 23BLinear Algebra and Real Analysis IIPatelTBAA 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.
MATH 25BHonors Linear Algebra and Real Analysis IIPetersonTBAA 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.
MATH 55BHonors Real and Complex AnalysisElkiesTBAA rigorous treatment of real and complex analysis.
MATH 60RReading Course for Senior Honors CandidatesLurieTBAAdvanced reading in topics not covered in courses.
MATH 91RSupervised Reading and ResearchLurieTBAPrograms of directed study supervised by a person approved by the Department.
MATH 99RTutorialLurieTBASupervised small group tutorial. Topics to be arranged.
MATH 101Sets, Groups and TopologyGrundmeierTBAAn introduction to rigorous mathematics, axioms, and proofs, via topics including set theory, symmetry groups, and low-dimensional topology.
MATH 110Vector Space Methods for Differential EquationsMcLellanTBADevelops 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.
MATH 112Introductory Real AnalysisHaiden310An 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.
MATH 113Analysis I: Complex Function TheorySiuTBAAnalytic 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.
MATH 115Methods of AnalysisTBATBAComplex functions; Fourier analysis; Hilbert spaces and operators; Laplace's equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.
MATH 118RDynamical SystemsCainTBAIntroduction 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.
MATH 123Algebra II: Theory of Rings and FieldsMazurTBARings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.
MATH 129Number FieldsKisinTBAAlgebraic 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.
MATH 130Classical GeometryHopkinsTBAPresents 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.
MATH 132Topology II: Smooth ManifoldsMelvinTBADifferential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes' theorem, introduction to cohomology.
MATH 137Algebraic GeometryKronheimerTBAAffine and projective spaces, plane curves, Bezout's theorem, singularities and genus of a plane curve, Riemann-Roch theorem.
MATH 154Probability TheoryTaubesTBAAn 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.
MATH 155RCombinatoricsHarrisTBAAn 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.
MATH 157Mathematics in the WorldHarrisTBAAn 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.
MATH 212BRAdvanced Real AnalysisTBATBAFunctional 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.
MATH 222Lie Groups and Lie AlgebrasHaidenTBALie theory, including the classification of semi-simple Lie algebras and/or compact Lie groups and their representations.
MATH 224Representations of Reductive Lie GroupsGaitsgoryTBAStructure theory of reductive Lie groups, unitary representations, Harish Chandra modules, characters, the discrete series, Plancherel theorem.
MATH 229XIntroduction to Analytic Number TheoryPasten VasquezTBAFundamental 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.
MATH 230BRAdvanced Differential GeometryST YauTue Thu 11:30-1A continuation of Mathematics 230a. Topics in differential geometry: Analysis on manifolds. Laplacians. Hodge theory. Spin structures. Clifford algebras. Dirac operators. Index theorems. Applications.
MATH 231BRAdvanced Algebraic TopologyPetersonTBAContinuation of Mathematics 231a. Vector bundles and characteristic classes. Bott periodicity. K-theory, cobordism and stable cohomotopy as examples of cohomology theories.
MATH 232BRAlgebraic Geometry IITBATBAThe course will cover the classification of complex algebraic surfaces.
MATH 233ATheory of Schemes ITBATBAAn 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.
MATH 243Evolutionary DynamicsNowakTBAAdvanced topics of evolutionary dynamics. Seminars and research projects.
MATH 252Linear Series and Positivity of Vector BundlesUlleryTBAIn this course, we will cover a variety of topics related to positivity in algebraic geometry. We will start with the basics of divisors, line bundles, vector bundles, and linear series and their cohomological and numerical properties. Other topics may include but are not limitied to: syzygies and Castelnuovo-Mumford regularity, vanishing theorems, asymptotic linear series, degeneracy loci, Fujita's conjecture and Reider's Theorem, and stability of vector bundles
MATH 260Low Dimensional Topology: Mapping Class GroupsTshishikuTBAAn introduction to mapping class groups and their cohomology. Possible topics: surface bundles, monodromy Earle-Eells theorem; Miller-Morita-Mumford classes, nontriviality, Atiyah-Kodaira constructions; curve complexes and homological stability; flat surface bundles and Nielsen realization problems; Morita's nonllifting theorem, Bott vanishing, Thurston stability; sections of surface bundles and Milnor-Wood inequalities.
MATH 268YDiophantine ApproximationPasten VasquezTBARoth's theorem. The subspace theorem. Arithmetic discriminants. The abc conjecture: partial results and applications.
MATH 275XTopics in Geometry and DynamicsMcMullenTBAA survey of fundamental results and current research. Topics may include: Riemann surfaces, hyperbolic 3-manifolds, moduli spaces, complex dynamics and rigidity.
MATH 284XCanonical Bases in Representation TheoryMelvinTBAAn introduction to (dual) canonical bases in representation theory and some of their applications/appearance in geometry. Topics: Kashiwara crystals and their realisations, total positivity, geometric crystals of Berenstein-Kazhdan, toric degenerations of Schubert varieties, mirror constructions.
MATH 288Probability Theory and Stochastic ProcessH.T. YauTBAWe will cover the construction of Brownian motions and develop the Ito calculus. We will review discrete martingale and stopping time.
MATH 303Topics in Diophantine ProblemsPasten VasquezTBANA
MATH 304Topics in Algebraic TopologyHopkinsTBANA
MATH 308Topics in Number Theory and Modular FormsGrossTBANA
MATH 314Topics in Differential Geometry and Mathematical PhysicsSternbergTBANA
MATH 316Topics in Algebraic GeometryTBATBANA
MATH 318Topics in Number TheoryMazurTBANA
MATH 321Topics in Mathematical PhysicsJaffeTBANA
MATH 327Topics in Several Complex VariablesSiuTBANA
MATH 333Topics in Complex Analysis, Dynamics and GeometryMcMullenTBANA
MATH 335Topics in Differential Geometry and AnalysisTaubesTBANA
MATH 343Topics in Complex GeometryCollinsTBANA
MATH 345Topics in Geometry and TopologyKronheimerTBANA
MATH 346YTopics in Analysis: Quantum DynamicsYauTBANA
MATH 348Topics in Representation TheoryHaidenTBANA
MATH 352Topics in Algebraic Number TheoryKisinTBANA
MATH 356Topics in Harmonic AnalysisSchmidTBANA
MATH 357Topics in Model TheoryBoneyTBANA
MATH 361Topics in Differential Geometry and AnalysisCanzaniTBANA
MATH 362Topics in Number TheoryMillerTBANA
MATH 364Topics in Algebraic GeometryUlleryTBANA
MATH 365Topics in Differential GeometryS.T. YauTBANA
MATH 368Topics in Algebraic TopologyPetersonTBANA
MATH 373Topics in Algebraic TopologyLurieTBANA
MATH 381Introduction to Geometric Representation TheoryGaitsgoryTBANA
MATH 382Topics in Algebraic GeometryHarrisTBANA
MATH 385Topics in Set TheoryWoodinTBANA
MATH 387Topics in Mathematical Physics: Bridgeland Stability ConditionsTanakaTBANA
MATH 388Topics in Mathematics and BiologyNowakTBANA
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