DEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS Bachelor of Science in Mathematics (Course 18)
The Department of Mathematics (http://math.mit.edu) oers General Mathematics Option training at the undergraduate, graduate, and postgraduate levels. In addition to the General Institute Requirements, the requirements Its expertise covers a broad spectrum of elds ranging from the consist of Dierential Equations, plus eight additional 12-unit traditional areas of "pure" mathematics, such as analysis, algebra, subjects in Course 18 of essentially dierent content, including at geometry, and topology, to applied mathematics areas such as least six advanced subjects (rst decimal digit one or higher) that are combinatorics, computational biology, fluid dynamics, theoretical distributed over at least three distinct areas (at least three distinct computer science, and theoretical physics. rst decimal digits). One of these eight subjects must be Linear Course 18 includes two undergraduate degrees: a Bachelor of Algebra. This leaves available 84 units of unrestricted electives. The Science in Mathematics and a Bachelor of Science in Mathematics requirements are flexible in order to accommodate students who with Computer Science. Undergraduate students may choose one pursue programs that combine mathematics with a related eld of three options leading to the Bachelor of Science in Mathematics: (such as physics, economics, or management) as well as students applied mathematics, pure mathematics, or general mathematics. who are interested in both pure and applied mathematics. More The general mathematics option provides a great deal of flexibility details can be found on the degree chart (http://catalog.mit.edu/ and allows students to design their own programs in conjunction degree-charts/mathematics-course-18/#generalmathematicstext). with their advisors. The Mathematics with Computer Science degree is oered for students who want to pursue interests in mathematics Applied Mathematics Option and theoretical computer science within a single undergraduate Applied mathematics focuses on the mathematical concepts and program. techniques applied in science, engineering, and computer science. Particular attention is given to the following principles and their At the graduate level, the Mathematics Department oers the PhD in mathematical formulations: propagation, equilibrium, stability, Mathematics, which culminates in the exposition of original research optimization, computation, statistics, and random processes. in a dissertation. Graduate students also receive training and gain experience in the teaching of mathematics. Sophomores interested in applied mathematics typically enroll in 18.200 Principles of Discrete Applied Mathematics and 18.300 The CLE Moore instructorships and Applied Mathematics Principles of Continuum Applied Mathematics. Subject 18.200 is instructorships bring mathematicians at the postdoctoral level to MIT devoted to the discrete aspects of applied mathematics and may be and provide them with training in research and teaching. taken concurrently with 18.03 Dierential Equations. Subject 18.300, oered in the spring term, is devoted to continuous aspects and makes considerable use of dierential equations. Undergraduate Study The subjects in Group I of the program correspond roughly to those An undergraduate degree in mathematics provides an excellent areas of applied mathematics that make heavy use of discrete basis for graduate work in mathematics or computer science, or mathematics, while Group II emphasizes those subjects that for employment in such elds as nance, business, or consulting. deal mainly with continuous processes. Some subjects, such as Students' programs are arranged through consultation with their probability or numerical analysis, have both discrete and continuous faculty advisors. aspects.
Undergraduates in mathematics are encouraged to elect an Students planning to go on to graduate work in applied mathematics undergraduate seminar during their junior or senior year. The should also take some basic subjects in analysis and algebra. experience gained from active participation in a seminar conducted by a research mathematician has proven to be valuable for students More detail on the Applied Mathematics option can be found on the planning to pursue graduate work as well as for those going on to degree chart (http://catalog.mit.edu/degree-charts/mathematics- other careers. These seminars also provide training in the verbal and course-18/#appliedmathematicstext). written communication of mathematics and may be used to fulll the Communication Requirement. Pure Mathematics Option Pure (or "theoretical") mathematics is the study of the basic concepts Many mathematics majors take 18.821 Project Laboratory in and structure of mathematics. Its goal is to arrive at a deeper Mathematics, which fullls the Institute's Laboratory Requirement understanding and an expanded knowledge of mathematics itself. and counts toward the Communication Requirement. Traditionally, pure mathematics has been classied into three general elds: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and
Department of Mathematics | 3 DEPARTMENT OF MATHEMATICS
geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students also may wish to Inquiries explore other topics such as logic, number theory, complex analysis, For further information, see the department's website (http:// and subjects within applied mathematics. math.mit.edu/academics/undergrad) or contact Math Academic Services, 617-253-2416. The subjects 18.701 Algebra I and 18.901 Introduction to Topology are more advanced and should not be elected until a student has had experience with proofs, as in Real Analysis (18.100A, 18.100B, Graduate Study 18.100P or 18.100Q) or 18.700 Linear Algebra. The Mathematics Department oers programs covering a broad For more details, see the degree chart (http:// range of topics leading to the Doctor of Philosophy or Doctor of catalog.mit.edu/degree-charts/mathematics-course-18/ Science degree. Candidates are admitted to either the Pure or #theoreticalmathematicstext). Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 115 doctoral students, about two thirds Bachelor of Science in Mathematics with Computer Science are in Pure Mathematics, one third in Applied Mathematics. (Course 18-C) Mathematics and computer science are closely related elds. The programs in Pure and Applied Mathematics oer basic and Problems in computer science are oen formalized and solved with advanced classes in analysis, algebra, geometry, Lie theory, logic, mathematical methods. It is likely that many important problems number theory, probability, statistics, topology, astrophysics, currently facing computer scientists will be solved by researchers combinatorics, fluid dynamics, numerical analysis, theoretical skilled in algebra, analysis, combinatorics, logic and/or probability physics, and the theory of computation. In addition, many theory, as well as computer science. mathematically oriented subjects are oered by other departments. Students in Applied Mathematics are especially encouraged to The purpose of this program is to allow students to study a take subjects in engineering and scientic subjects related to their combination of these mathematical areas and potential areas of research. application in computer science. Required subjects include linear algebra (18.06, 18.061, or 18.700) because it is so broadly used, All students pursue research under the supervision of the faculty and discrete mathematics (18.062[J] or 18.200) to give experience and are encouraged to take advantage of the many seminars and with proofs and the necessary tools for analyzing algorithms. colloquia at MIT and in the Boston area. The required subjects covering complexity (18.404 Theory of Computation or 18.400[J] Computability and Complexity Theory) and Doctor of Philosophy or Doctor of Science algorithms (18.410[J] Design and Analysis of Algorithms) provide an The requirements for these degrees are described on the introduction to the most theoretical aspects of computer science. department's website (http://math.mit.edu/academics/grad/ We also require exposure to other areas of computer science timeline). In outline, they consist of an oral qualifying examination, (6.031, 6.033, 6.034, or 6.036) where mathematical issues may a thesis proposal, completion of a minimum of 96 units (8 graduate also arise. More details can be found on the degree chart (http:// subjects), experience in classroom teaching, and a thesis containing catalog.mit.edu/degree-charts/mathematics-computer-science- original research in mathematics. course-18-c). Interdisciplinary Programs Some flexibility is allowed in this program. In particular, students may substitute the more advanced subject 18.701 Algebra I for 18.06 Computational Science and Engineering Linear Algebra, and, if they already have strong theorem-proving Students with primary interest in computational science may also skills, may substitute 18.211 Combinatorial Analysis or 18.212 consider applying to the interdisciplinary Computational Science and Algebraic Combinatorics for 18.062[J] Mathematics for Computer Engineering (CSE) program, with which the Mathematics Department Science or 18.200 Principles of Discrete Applied Mathematics. is aliated. For more information, see the CSE website (http:// cse.mit.edu/programs). Minor in Mathematics The requirements for a Minor in Mathematics are as follows: six 12- Mathematics and Statistics unit subjects in mathematics, beyond the Institute's Mathematics The Interdisciplinary Doctoral Program in Statistics provides training Requirement, of essentially dierent content, including at least three in statistics, including classical statistics and probability as well as advanced subjects (rst decimal digit one or higher). computation and data analysis, to students who wish to integrate these valuable skills into their primary academic program. The See the Undergraduate Section for a general description of the minor program is administered jointly by the departments of Aeronautics program (http://catalog.mit.edu/mit/undergraduate-education/ and Astronautics, Economics, Mathematics, Mechanical Engineering, academic-programs/minors). Physics, and Political Science, and the Statistics and Data Science
4 | Department of Mathematics DEPARTMENT OF MATHEMATICS
Center within the Institute for Data, Systems, and Society. It is open Laurent Demanet, PhD to current doctoral students in participating departments. For more Professor of Mathematics information, including department-specic requirements, see the Professor of Earth, Atmospheric and Planetary Sciences full program description (http://catalog.mit.edu/interdisciplinary/ graduate-programs/phd-statistics) under Interdisciplinary Graduate Alan Edelman, PhD Programs. Professor of Mathematics Pavel I. Etingof, PhD Financial Support Professor of Mathematics Financial support is guaranteed for up to ve years to students making satisfactory academic progress. Financial aid aer the rst Victor W. Guillemin, PhD year is usually in the form of a teaching or research assistantship. Professor of Mathematics
Inquiries Lawrence Guth, PhD For further information, see the department's website (http:// Claude E. Shannon (1940) Professor of Mathematics math.mit.edu/academics/grad) or contact Math Academic Services, Anette E. Hosoi, PhD 617-253-2416. Neil and Jane Pappalardo Professor Professor of Mechanical Engineering Professor of Mathematics Faculty and Teaching Sta Member, Institute for Data, Systems, and Society Michel X. Goemans, PhD David S. Jerison, PhD RSA Professor of Mathematics Professor of Mathematics Head, Department of Mathematics Steven G. Johnson, PhD William Minicozzi, PhD Professor of Mathematics Singer Professor of Mathematics Professor of Physics Associate Head, Department of Mathematics Victor Kac, PhD Professors Professor of Mathematics Martin Z. Bazant, PhD E. G. Roos Professor Jonathan Adam Kelner, PhD Professor of Chemical Engineering Professor of Mathematics Professor of Mathematics Ju-Lee Kim, PhD Bonnie Berger, PhD Professor of Mathematics Simons Professor of Mathematics Frank Thomson Leighton, PhD Member, Health Sciences and Technology Faculty Professor of Mathematics Roman Bezrukavnikov, PhD George Lusztig, PhD Professor of Mathematics Edward A. Abdun-Nur (1924) Professor of Mathematics (On leave, fall) (On leave) Alexei Borodin, PhD Davesh Maulik, PhD Professor of Mathematics Professor of Mathematics John W. M. Bush, PhD Richard B. Melrose, PhD Professor of Mathematics Professor of Mathematics Hung Cheng, PhD Haynes R. Miller, PhD Professor of Mathematics Professor Post-Tenure of Mathematics Tobias Colding, PhD Ankur Moitra, PhD Cecil and Ida Green Distinguished Professor Norbert Wiener Professor of Mathematics Professor of Mathematics Associate Director, Institute for Data, Systems, and Society
Department of Mathematics | 5 DEPARTMENT OF MATHEMATICS
Elchanan Mossel, PhD Professor of Mathematics Associate Professors Member, Institute for Data, Systems, and Society Joern Dunkel, PhD Associate Professor of Mathematics Tomasz S. Mrowka, PhD Professor of Mathematics Semyon Dyatlov, PhD Associate Professor of Mathematics Pablo A. Parrilo, PhD Joseph F. and Nancy P. Keithley Professor Peter Hintz, PhD Professor of Electrical Engineering and Computer Science Associate Professor of Mathematics Professor of Mathematics (On leave) Member, Institute for Data, Systems, and Society Andrei Negut, PhD Bjorn Poonen, PhD Class of 1947 Career Development Chair Distinguished Professor in Science Associate Professor of Mathematics Professor of Mathematics Nike Sun, PhD Alexander Postnikov, PhD Associate Professor of Mathematics Professor of Mathematics Assistant Professors Philippe Rigollet, PhD Tristan Collins, PhD Professor of Mathematics Assistant Professor of Mathematics Member, Institute for Data, Systems, and Society (On leave) Jeremy Hahn, PhD Assistant Professor of Mathematics Rodolfo R. Rosales, PhD Professor of Mathematics Andrew Lawrie, PhD Assistant Professor of Mathematics Paul Seidel, PhD Levinson Professor of Mathematics Dor Minzer, PhD Assistant Professor of Mathematics Scott Roger Sheeld, PhD Leighton Family Professor of Mathematics Lisa Piccirillo, PhD Assistant Professor of Mathematics Peter W. Shor, PhD Henry Adams Morss and Henry Adams Morss, Jr. (1934) Professor of Lisa Sauermann, PhD Mathematics Assistant Professor of Mathematics
Michael Sipser, PhD Yufei Zhao, PhD Donner Professor of Mathematics Assistant Professor of Mathematics (On leave, spring) Visiting Professors Gigliola Stalani, PhD Sean Paul, PhD Abby Rockefeller Mauzé Professor of Mathematics Visiting Professor of Mathematics Gilbert Strang, PhD (Fall) MathWorks Professor of Mathematics Member, Institute for Data, Systems, and Society Visiting Associate Professors Bianca Viray, PhD Daniel W. Stroock, PhD Visiting Associate Professor of Mathematics Professor Post-Tenure of Mathematics Visiting Assistant Professors Zhiwei Yun, PhD Stephen Kleene, PhD Professor of Mathematics Visiting Assistant Professor of Mathematics
Wei Zhang, PhD Daniel Pomerleano, PhD Professor of Mathematics Visiting Assistant Professor of Mathematics (Fall)
6 | Department of Mathematics DEPARTMENT OF MATHEMATICS
Adjunct Professors Instructors Henry Cohn, PhD Daniel Alvarez-Gavela, PhD Adjunct Professor of Mathematics Instructor of Pure Mathematics
Lecturers Peter Baddoo, PhD Jennifer French, PhD Instructor of Applied Mathematics Lecturer in Digital Learning Keaton Burns, PhD Slava Gerovitch, PhD Instructor of Applied Mathematics Lecturer in Mathematics Zongchen Chen, PhD Peter J. Kempthorne, PhD Instructor of Mathematics Lecturer in Mathematics Gary Choi, PhD Tanya Khovanova, PhD Instructor of Applied Mathematics Lecturer in Mathematics Anthony Conway, PhD Instructor of Pure Mathematics CLE Moore Instructors Qing Deng, PhD Souvik Dhara, PhD CLE Moore Instructor of Mathematics Schramm Fellow Instructor
Tony Feng, PhD Matthew Durey, PhD CLE Moore Instructor of Mathematics Instructor of Applied Mathematics
Promit Ghosal, PhD William Cole Franks, PhD CLE Moore Instructor of Mathematics Instructor of Applied Mathematics
Jimmy He, PhD Felix Gotti, PhD CLE Moore Instructor of Mathematics Instructor of Applied Mathematics
Malo Pierig Jezequel, PhD Vili Heinonen, PhD CLE Moore Instructor of Mathematics Instructor of Applied Mathematics (Fall) Yang Li, PhD CLE Moore Instructor of Mathematics Andrew James Horning, PhD Instructor of Mathematics Tristan Ozuch-Meersseman, PhD CLE Moore Instructor of Mathematics Ousmane Kodio, PhD Instructor of Applied Mathematics Matthew Rosenzweig, PhD CLE Moore Instructor of Mathematics David Milton Kouskoulas, PhD Instructor of Mathematics Yair Shenfeld, PhD CLE Moore Instructor of Mathematics Dan Mikulincer, PhD Instructor of Mathematics Minh-Tam Trinh, PhD CLE Moore Instructor of Mathematics Hyunki Min, PhD Instructor of Mathematics Yilin Wang, PhD CLE Moore Instructor of Mathematics Christopher Rackauckas, PhD (Fall) Instructor of Applied Mathematics
Abigail Ward, PhD Anna Weigandt CLE Moore Instructor of Mathematics Instructor of Mathematics
Ziquan Zhuang, PhD Nicholas Wilkins, PhD CLE Moore Instructor of Mathematics Instructor of Pure Mathematics
Department of Mathematics | 7 DEPARTMENT OF MATHEMATICS
David A. Vogan, PhD Research Sta Professor Emeritus of Mathematics
Principal Research Scientists Andrew Victor Sutherland II, PhD General Mathematics Principal Research Scientist of Mathematics 18.0002[J] Introduction to Computational Science and Research Scientists Engineering (New) Shiva Chidambaram, PhD Same subject as 16.0002[J] Research Scientist of Mathematics Prereq: 6.0001; Coreq: 8.01 and 18.01 U (Fall, Spring; second half of term) Edgar Costa, PhD 3-0-3 units Research Scientist of Mathematics Credit cannot also be received for 6.0002
David Roe, PhD See description under subject 16.0002[J]. Research Scientist of Mathematics D. L. Darmofal, L. Demanet
Samuel Schiavone, PhD 18.01 Calculus Research Scientist of Mathematics Prereq: None Raymond van Bommel, PhD U (Fall, Spring) Research Scientist of Mathematics 5-0-7 units. CALC I Credit cannot also be received for 18.01A, ES.1801, ES.181A
Professors Emeriti Dierentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Michael Artin, PhD Dierentiation: denition, rules, application to graphing, rates, Professor Emeritus of Mathematics approximations, and extremum problems. Indenite integration; Daniel Z. Freedman, PhD separable rst-order dierential equations. Denite integral; Professor Emeritus of Mathematics fundamental theorem of calculus. Applications of integration Professor Emeritus of Physics to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Harvey P. Greenspan, PhD Innite series: geometric, p-harmonic, simple comparison tests, Professor Emeritus of Mathematics power series for some elementary functions. Fall: L. Guth. Spring: Information: W. Minicozzi Sigurdur Helgason, PhD Professor Emeritus of Mathematics 18.01A Calculus Steven L. Kleiman, PhD Prereq: Knowledge of dierentiation and elementary integration Professor Emeritus of Mathematics U (Fall; rst half of term) 5-0-7 units. CALC I Daniel J. Kleitman, PhD Credit cannot also be received for 18.01, ES.1801, ES.181A Professor Emeritus of Mathematics Six-week review of one-variable calculus, emphasizing material Arthur P. Mattuck, PhD not on the high-school AB syllabus: integration techniques and Professor Emeritus of Mathematics applications, improper integrals, innite series, applications to other topics, such as probability and statistics, as time permits. James R. Munkres, PhD Prerequisites: one year of high-school calculus or the equivalent, Professor Emeritus of Mathematics with a score of 5 on the AB Calculus test (or the AB portion of the BC Richard P. Stanley, PhD test, or an equivalent score on a standard international exam), or Professor Emeritus of Mathematics equivalent college transfer credit, or a passing grade on the rst half of the 18.01 advanced standing exam. Harold Stark, PhD D. Jerison Professor Emeritus of Mathematics
Alar Toomre, PhD Professor Emeritus of Mathematics
8 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.02 Calculus 18.03 Dierential Equations Prereq: Calculus I (GIR) Prereq: None. Coreq: Calculus II (GIR) U (Fall, Spring) U (Fall, Spring) 5-0-7 units. CALC II 5-0-7 units. REST Credit cannot also be received for 18.022, 18.02A, CC.1802, ES.1802, Credit cannot also be received for 18.032, CC.1803, ES.1803 ES.182A Study of dierential equations, including modeling physical Calculus of several variables. Vector algebra in 3-space, systems. Solution of rst-order ODEs by analytical, graphical, determinants, matrices. Vector-valued functions of one variable, and numerical methods. Linear ODEs with constant coecients. space motion. Scalar functions of several variables: partial Complex numbers and exponentials. Inhomogeneous equations: dierentiation, gradient, optimization techniques. Double integrals polynomial, sinusoidal, and exponential inputs. Oscillations, and line integrals in the plane; exact dierentials and conservative damping, resonance. Fourier series. Matrices, eigenvalues, elds; Green's theorem and applications, triple integrals, line and eigenvectors, diagonalization. First order linear systems: normal surface integrals in space, Divergence theorem, Stokes' theorem; modes, matrix exponentials, variation of parameters. Heat equation, applications. wave equation. Nonlinear autonomous systems: critical point Fall: B. Poonen. Spring: G. Stalani analysis, phase plane diagrams. Fall: J. Dunkel. Spring: T. Collins 18.02A Calculus Prereq: Calculus I (GIR) 18.031 System Functions and the Laplace Transform U (Fall, IAP, Spring; second half of term) Prereq: None. Coreq: 18.03 5-0-7 units. CALC II U (IAP) Credit cannot also be received for 18.02, 18.022, CC.1802, ES.1802, 1-0-2 units ES.182A Studies basic continuous control theory as well as representation First half is taught during the last six weeks of the Fall term; covers of functions in the complex frequency domain. Covers generalized material in the rst half of 18.02 (through double integrals). Second functions, unit impulse response, and convolution; and Laplace half of 18.02A can be taken either during IAP (daily lectures) or transform, system (or transfer) function, and the pole diagram. during the second half of the Spring term; it covers the remaining Includes examples from mechanical and electrical engineering. material in 18.02. Information: H. R. Miller Fall, IAP: J. W. M. Bush. Spring: G. Stalani 18.032 Dierential Equations 18.022 Calculus Prereq: None. Coreq: Calculus II (GIR) Prereq: Calculus I (GIR) U (Spring) U (Fall) 5-0-7 units. REST 5-0-7 units. CALC II Credit cannot also be received for 18.03, CC.1803, ES.1803 Credit cannot also be received for 18.02, 18.02A, CC.1802, ES.1802, ES.182A Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local Calculus of several variables. Topics as in 18.02 but with more existence and uniqueness of solutions. focus on mathematical concepts. Vector algebra, dot product, T. Ozuch-Meersseman matrices, determinant. Functions of several variables, continuity, dierentiability, derivative. Parametrized curves, arc length, 18.04 Complex Variables with Applications curvature, torsion. Vector elds, gradient, curl, divergence. Multiple Prereq: Calculus II (GIR) and (18.03 or 18.032) integrals, change of variables, line integrals, surface integrals. U (Spring) Stokes' theorem in one, two, and three dimensions. 4-0-8 units A. Borodin Credit cannot also be received for 18.075, 18.0751
Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial dierential equations. P. Baddoo
Department of Mathematics | 9 DEPARTMENT OF MATHEMATICS
18.05 Introduction to Probability and Statistics 18.065 Matrix Methods in Data Analysis, Signal Processing, and Prereq: Calculus II (GIR) Machine Learning U (Spring) Subject meets with 18.0651 4-0-8 units. REST Prereq: 18.06 U (Spring) Elementary introduction with applications. Basic probability 3-0-9 units models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Reviews linear algebra with applications to life sciences, nance, Condence intervals. Introduction to linear regression. engineering, and big data. Covers singular value decomposition, J. Orlo weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed 18.06 Linear Algebra and undirected graphs, matrix factorizations, neural nets, machine Prereq: Calculus II (GIR) learning, and computations with large matrices. U (Fall, Spring) G. Strang 4-0-8 units. REST Credit cannot also be received for 18.061, 18.700 18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning Basic subject on matrix theory and linear algebra, emphasizing Subject meets with 18.065 topics useful in other disciplines, including systems of equations, Prereq: 18.06 vector spaces, determinants, eigenvalues, singular value G (Spring) decomposition, and positive denite matrices. Applications to 3-0-9 units least-squares approximations, stability of dierential equations, networks, Fourier transforms, and Markov processes. Uses linear Reviews linear algebra with applications to life sciences, nance, algebra soware. Compared with 18.700, more emphasis on matrix engineering, and big data. Covers singular value decomposition, algorithms and many applications. weighted least squares, signal and image processing, principal Fall: N. Sun. Spring: S. G. Johnson component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine 18.061 Linear Algebra and Optimization (New) learning, and computations with large matrices. Students in Course Prereq: Calculus II (GIR) 18 must register for the undergraduate version, 18.065. U (Fall) G. Strang 5-0-7 units. REST Credit cannot also be received for 18.06, 18.700 18.075 Methods for Scientists and Engineers Subject meets with 18.0751 Introductory course in linear algebra and optimization, assuming Prereq: Calculus II (GIR) and 18.03 no prior exposure to linear algebra and starting from the basics, U (Spring) including vectors, matrices, eigenvalues, singular values, and 3-0-9 units least squares. Covers the basics in optimization including convex Credit cannot also be received for 18.04 optimization, linear/quadratic programming, gradient descent, and regularization, building on insights from linear algebra. Explores a Covers functions of a complex variable; calculus of residues. variety of applications in science and engineering, where the tools Includes ordinary dierential equations; Bessel and Legendre developed give powerful ways to understand complex systems and functions; Sturm-Liouville theory; partial dierential equations; heat also extract structure from data. equation; and wave equations. A. Moitra, P. Parrilo H. Cheng
18.062[J] Mathematics for Computer Science Same subject as 6.042[J] Prereq: Calculus I (GIR) U (Fall, Spring) 5-0-7 units. REST
See description under subject 6.042[J]. Z. R. Abel, F. T. Leighton, A. Moitra
10 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.0751 Methods for Scientists and Engineers 18.086 Computational Science and Engineering II Subject meets with 18.075 Subject meets with 18.0861 Prereq: Calculus II (GIR) and 18.03 Prereq: Calculus II (GIR) and (18.03 or 18.032) G (Spring) Acad Year 2021-2022: Not oered 3-0-9 units Acad Year 2022-2023: U (Spring) Credit cannot also be received for 18.04 3-0-9 units
Covers functions of a complex variable; calculus of residues. Initial value problems: nite dierence methods, accuracy and Includes ordinary dierential equations; Bessel and Legendre stability, heat equation, wave equations, conservation laws and functions; Sturm-Liouville theory; partial dierential equations; heat shocks, level sets, Navier-Stokes. Solving large systems: elimination equation; and wave equations. Students in Courses 6, 8, 12, 18, and with reordering, iterative methods, preconditioning, multigrid, 22 must register for undergraduate version, 18.075. Krylov subspaces, conjugate gradients. Optimization and minimum H. Cheng principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, 18.085 Computational Science and Engineering I duality, adjoint methods. Subject meets with 18.0851 Information: W. G. Strang Prereq: Calculus II (GIR) and (18.03 or 18.032) U (Fall, Spring, Summer) 18.0861 Computational Science and Engineering II 3-0-9 units Subject meets with 18.086 Prereq: Calculus II (GIR) and (18.03 or 18.032) Review of linear algebra, applications to networks, structures, and Acad Year 2021-2022: Not oered estimation, nite dierence and nite element solution of dierential Acad Year 2022-2023: G (Spring) equations, Laplace's equation and potential flow, boundary-value 3-0-9 units problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientic and engineering Initial value problems: nite dierence methods, accuracy and applications. stability, heat equation, wave equations, conservation laws and M. Durey shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, 18.0851 Computational Science and Engineering I Krylov subspaces, conjugate gradients. Optimization and minimum Subject meets with 18.085 principles: weighted least squares, constraints, inverse problems, Prereq: Calculus II (GIR) and (18.03 or 18.032) calculus of variations, saddle point problems, linear programming, G (Fall, Spring, Summer) duality, adjoint methods. Students in Course 18 must register for the 3-0-9 units undergraduate version, 18.086. Information: W. G. Strang Review of linear algebra, applications to networks, structures, and estimation, nite dierence and nite element solution of 18.089 Review of Mathematics dierential equations, Laplace's equation and potential flow, Prereq: Permission of instructor boundary-value problems, Fourier series, discrete Fourier transform, G (Summer) convolution. Frequent use of MATLAB in a wide range of scientic and 5-0-7 units engineering applications. Students in Course 18 must register for the undergraduate version, 18.085. One-week review of one-variable calculus (18.01), followed by M. Durey concentrated study covering multivariable calculus (18.02), two hours per day for ve weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances. Information: W. Minicozzi
Department of Mathematics | 11 DEPARTMENT OF MATHEMATICS
18.094[J] Teaching College-Level Science and Engineering Analysis Same subject as 1.95[J], 5.95[J], 7.59[J], 8.395[J] Subject meets with 2.978 18.1001 Real Analysis Prereq: None Subject meets with 18.100A G (Fall) Prereq: Calculus II (GIR) 2-0-2 units G (Fall, Spring) See description under subject 5.95[J]. 3-0-9 units J. Rankin Credit cannot also be received for 18.1002, 18.100A, 18.100B, 18.100P, 18.100Q
18.095 Mathematics Lecture Series Covers fundamentals of mathematical analysis: convergence of Prereq: Calculus I (GIR) sequences and series, continuity, dierentiability, Riemann integral, U (IAP) sequences and series of functions, uniformity, interchange of limit 2-0-4 units operations. Shows the utility of abstract concepts and teaches Can be repeated for credit. understanding and construction of proofs. Proofs and denitions are Ten lectures by mathematics faculty members on interesting less abstract than in 18.100B. Gives applications where possible. topics from both classical and modern mathematics. All lectures Concerned primarily with the real line. Students in Course 18 must accessible to students with calculus background and an interest in register for undergraduate version 18.100A. mathematics. At each lecture, reading and exercises are assigned. Fall: T. Colding. Spring: M. Rosenzweig Students prepare these for discussion in a weekly problem session. Information: W. Minicozzi 18.1002 Real Analysis Subject meets with 18.100B 18.098 Internship in Mathematics Prereq: Calculus II (GIR) Prereq: Permission of instructor G (Fall, Spring) U (Fall, IAP, Spring, Summer) 3-0-9 units Units arranged [P/D/F] Credit cannot also be received for 18.1001, 18.100A, 18.100B, Can be repeated for credit. 18.100P, 18.100Q
Provides academic credit for students pursuing internships to gain Covers fundamentals of mathematical analysis: convergence of practical experience in the applications of mathematical concepts sequences and series, continuity, dierentiability, Riemann integral, and methods. sequences and series of functions, uniformity, interchange of limit Information: W. Minicozzi operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.099 Independent Study 18.100A, for students with more mathematical maturity. Places more Prereq: Permission of instructor emphasis on point-set topology and n-space. Students in Course 18 U (Fall, IAP, Spring, Summer) must register for undergraduate version 18.100B. Units arranged Fall: P. Seidel. Spring: W. Minicozzi Can be repeated for credit.
Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval. May not be used to satisfy Mathematics major requirements. Information: W. Minicozzi
12 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.100A Real Analysis 18.100Q Real Analysis Subject meets with 18.1001 Prereq: Calculus II (GIR) Prereq: Calculus II (GIR) U (Fall) U (Fall, Spring) 4-0-11 units 3-0-9 units Credit cannot also be received for 18.1001, 18.1002, 18.100A, Credit cannot also be received for 18.1001, 18.1002, 18.100B, 18.100B, 18.100P 18.100P, 18.100Q Covers fundamentals of mathematical analysis: convergence of Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, dierentiability, Riemann integral, sequences and series, continuity, dierentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than understanding and construction of proofs. Proofs and denitions are 18.100A, for students with more mathematical maturity. Places more less abstract than in 18.100B. Gives applications where possible. emphasis on point-set topology and n-space. Includes instruction Concerned primarily with the real line. and practice in written communication. Enrollment limited. Fall: T. Colding. Spring: M. Rosenzweig S. T. Paul
18.100B Real Analysis 18.101 Analysis and Manifolds Subject meets with 18.1002 Subject meets with 18.1011 Prereq: Calculus II (GIR) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or U (Fall, Spring) 18.100Q) 3-0-9 units U (Fall) Credit cannot also be received for 18.1001, 18.1002, 18.100A, 3-0-9 units 18.100P, 18.100Q Introduction to the theory of manifolds: vector elds and densities Covers fundamentals of mathematical analysis: convergence of on manifolds, integral calculus in the manifold setting and the sequences and series, continuity, dierentiability, Riemann integral, manifold version of the divergence theorem. 18.901 helpful but not sequences and series of functions, uniformity, interchange of limit required. operations. Shows the utility of abstract concepts and teaches R. B. Melrose understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more 18.1011 Analysis and Manifolds emphasis on point-set topology and n-space. Subject meets with 18.101 Fall: P. Seidel. Spring: W. Minicozzi Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q) 18.100P Real Analysis G (Fall) Prereq: Calculus II (GIR) 3-0-9 units U (Spring) 4-0-11 units Introduction to the theory of manifolds: vector elds and densities Credit cannot also be received for 18.1001, 18.1002, 18.100A, on manifolds, integral calculus in the manifold setting and the 18.100B, 18.100Q manifold version of the divergence theorem. 18.9011 helpful but not required. Students in Course 18 must register for the undergraduate Covers fundamentals of mathematical analysis: convergence of version, 18.101. sequences and series, continuity, dierentiability, Riemann integral, R. B. Melrose sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and denitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line. Includes instruction and practice in written communication. Enrollment limited. Q. Deng
Department of Mathematics | 13 DEPARTMENT OF MATHEMATICS
18.102 Introduction to Functional Analysis 18.104 Seminar in Analysis Subject meets with 18.1021 Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or U (Fall) 18.100Q) 3-0-9 units U (Spring) 3-0-9 units Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral Normed spaces, completeness, functionals, Hahn-Banach theorem, communication provided. Enrollment limited. duality, operators. Lebesgue measure, measurable functions, T. Ozuch-Meersseman integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem. 18.112 Functions of a Complex Variable P. I. Etingof Subject meets with 18.1121 Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.1021 Introduction to Functional Analysis 18.100Q) Subject meets with 18.102 U (Fall) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 3-0-9 units 18.100Q) G (Spring) Studies the basic properties of analytic functions of one complex 3-0-9 units variable. Conformal mappings and the Poincare model of non- Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral Normed spaces, completeness, functionals, Hahn-Banach theorem, formula. Taylor and Laurent decompositions. Singularities, duality, operators. Lebesgue measure, measurable functions, residues and computation of integrals. Harmonic functions and integrability, completeness of L-p spaces. Hilbert space. Compact, Dirichlet's problem for the Laplace equation. The partial fractions Hilbert-Schmidt and trace class operators. Spectral theorem. decomposition. Innite series and innite product expansions. The Students in Course 18 must register for the undergraduate version, Gamma function. The Riemann mapping theorem. Elliptic functions. 18.102. A. Borodin P. I. Etingof 18.1121 Functions of a Complex Variable 18.103 Fourier Analysis: Theory and Applications Subject meets with 18.112 Subject meets with 18.1031 Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q) 18.100Q) G (Fall) U (Fall) 3-0-9 units 3-0-9 units Studies the basic properties of analytic functions of one complex Roughly half the subject devoted to the theory of the Lebesgue variable. Conformal mappings and the Poincare model of non- integral with applications to probability, and half to Fourier series Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral and Fourier integrals. formula. Taylor and Laurent decompositions. Singularities, A. Lawrie residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions 18.1031 Fourier Analysis: Theory and Applications decomposition. Innite series and innite product expansions. The Subject meets with 18.103 Gamma function. The Riemann mapping theorem. Elliptic functions. Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or Students in Course 18 must register for the undergraduate version, 18.100Q) 18.112. G (Fall) A. Borodin 3-0-9 units
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, 18.103. A. Lawrie
14 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.116 Riemann Surfaces 18.152 Introduction to Partial Dierential Equations Prereq: 18.112 Subject meets with 18.1521 Acad Year 2021-2022: Not oered Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or Acad Year 2022-2023: G (Spring) 18.100Q) 3-0-9 units U (Spring) 3-0-9 units Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to Introduces three main types of partial dierential equations: number theory. diusion, elliptic, and hyperbolic. Includes mathematical tools, T. S. Mrowka real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar 18.117 Topics in Several Complex Variables conservation laws, rst order equations and trac problems. Prereq: 18.112 and 18.965 D. Jerison Acad Year 2021-2022: G (Spring) Acad Year 2022-2023: Not oered 18.1521 Introduction to Partial Dierential Equations 3-0-9 units Subject meets with 18.152 Can be repeated for credit. Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q) Harmonic theory on complex manifolds, Hodge decomposition G (Spring) theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of 3-0-9 units Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds. Introduces three main types of partial dierential equations: B. Poonen diusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes 18.118 Topics in Analysis equation, the European options problem, water waves, scalar Prereq: Permission of instructor conservation laws, rst order equations and trac problems. Acad Year 2021-2022: G (Spring) Students in Course 18 must register for the undergraduate version, Acad Year 2022-2023: Not oered 18.152. 3-0-9 units D. Jerison Can be repeated for credit. 18.155 Dierential Analysis I Topics vary from year to year. Prereq: 18.102 or 18.103 S. Dyatlov G (Fall) 3-0-9 units 18.125 Measure Theory and Analysis Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q First part of a two-subject sequence. Review of Lebesgue integration. G (Spring) Lp spaces. Distributions. Fourier transform. Sobolev spaces. 3-0-9 units Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and Provides a rigorous introduction to Lebesgue's theory of measure parabolic dierential operators. Recommended prerequisite: 18.112. and integration. Covers material that is essential in analysis, S. Dyatlov probability theory, and dierential geometry. D. W. Stroock 18.156 Dierential Analysis II Prereq: 18.155 18.137 Topics in Geometric Partial Dierential Equations G (Spring) Prereq: Permission of instructor 3-0-9 units Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Fall) Second part of a two-subject sequence. Covers variable coecient 3-0-9 units elliptic, parabolic and hyperbolic partial dierential equations. Can be repeated for credit. T. S. Mrowka
Topics vary from year to year. T. Colding
Department of Mathematics | 15 DEPARTMENT OF MATHEMATICS
18.157 Introduction to Microlocal Analysis 18.200A Principles of Discrete Applied Mathematics Prereq: 18.155 Prereq: None. Coreq: 18.06 Acad Year 2021-2022: G (Spring) U (Fall) Acad Year 2022-2023: Not oered 3-0-9 units 3-0-9 units Credit cannot also be received for 18.200
The semi-classical theory of partial dierential equations. Discussion Study of illustrative topics in discrete applied mathematics, of Pseudodierential operators, Fourier integral operators, including probability theory, information theory, coding theory, asymptotic solutions of partial dierential equations, and the secret codes, generating functions, and linear programming. spectral theory of Schroedinger operators from the semi-classical S. Dhara perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject. 18.204 Undergraduate Seminar in Discrete Mathematics R. B. Melrose Prereq: ((6.042[J] or 18.200) and (18.06, 18.700, or 18.701)) or permission of instructor 18.158 Topics in Dierential Equations U (Fall, Spring) Prereq: 18.157 3-0-9 units Acad Year 2021-2022: G (Spring) Acad Year 2022-2023: Not oered Seminar in combinatorics, graph theory, and discrete mathematics 3-0-9 units in general. Participants read and present papers from recent Can be repeated for credit. mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited. Topics vary from year to year. S. Dhara, J. He, A. Weigandt L. Guth 18.211 Combinatorial Analysis 18.199 Graduate Analysis Seminar Prereq: Calculus II (GIR) and (18.06, 18.700, or 18.701) Prereq: Permission of instructor U (Fall) G (Fall) 3-0-9 units Not oered regularly; consult department 3-0-9 units Combinatorial problems and methods for their solution. Can be repeated for credit. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior Studies original papers in dierential analysis and dierential experience with abstraction and proofs is helpful. equations. Intended for rst- and second-year graduate students. F. Gotti Permission must be secured in advance. V. W. Guillemin 18.212 Algebraic Combinatorics Prereq: 18.701 or 18.703 Discrete Applied Mathematics U (Spring) 3-0-9 units
18.200 Principles of Discrete Applied Mathematics Applications of algebra to combinatorics. Topics include walks Prereq: None. Coreq: 18.06 in graphs, the Radon transform, groups acting on posets, Young U (Spring) tableaux, electrical networks. 4-0-11 units A. Postnikov Credit cannot also be received for 18.200A
Study of illustrative topics in discrete applied mathematics, 18.217 Combinatorial Theory including probability theory, information theory, coding theory, Prereq: Permission of instructor secret codes, generating functions, and linear programming. G (Fall) Instruction and practice in written communication provided. 3-0-9 units Enrollment limited. Can be repeated for credit. M. X. Goemans, A. Moitra Content varies from year to year. A. Postnikov
16 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.218 Topics in Combinatorics 18.226 Probabilistic Methods in Combinatorics Prereq: Permission of instructor Prereq: (18.211, 18.600, and (18.100A, 18.100B, 18.100P, or 18.100Q)) G (Spring) or permission of instructor 3-0-9 units Acad Year 2021-2022: Not oered Can be repeated for credit. Acad Year 2022-2023: G (Fall) 3-0-9 units Topics vary from year to year. L. Sauermann Introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer 18.219 Seminar in Combinatorics science. Focuses on methodology as well as combinatorial Prereq: Permission of instructor applications. Suitable for students with strong interest and G (Fall) background in mathematical problem solving. Topics include Not oered regularly; consult department linearity of expectations, alteration, second moment, Lovasz local 3-0-9 units lemma, correlation inequalities, Janson inequalities, concentration Can be repeated for credit. inequalities, entropy method. Y. Zhao Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class. Continuous Applied Mathematics Information: Y. Zhao 18.300 Principles of Continuum Applied Mathematics 18.225 Graph Theory and Additive Combinatorics Prereq: Calculus II (GIR) and (18.03 or 18.032) Prereq: ((18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or U (Spring) 18.100Q)) or permission of instructor 3-0-9 units Acad Year 2021-2022: G (Fall) Covers fundamental concepts in continuous applied mathematics. Acad Year 2022-2023: Not oered Applications from trac flow, fluids, elasticity, granular flows, etc. 3-0-9 units Also covers continuum limit; conservation laws, quasi-equilibrium; Introduction to extremal graph theory and additive combinatorics. kinematic waves; characteristics, simple waves, shocks; diusion Highlights common themes, such as the dichotomy between (linear and nonlinear); numerical solution of wave equations; structure versus pseudorandomness. Topics include Turan- nite dierences, consistency, stability; discrete and fast Fourier type problems, Szemeredi's regularity lemma and applications, transforms; spectral methods; transforms and series (Fourier, pseudorandom graphs, spectral graph theory, graph limits, Laplace). Additional topics may include sonic booms, Mach cone, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete caustics, lattices, dispersion and group velocity. Uses MATLAB Fourier analysis, Freiman's theorem on sumsets and structure. computing environment. Discusses current research topics and open problems. R. R. Rosales Y. Zhao
Department of Mathematics | 17 DEPARTMENT OF MATHEMATICS
18.303 Linear Partial Dierential Equations: Analysis and 18.327 Topics in Applied Mathematics Numerics Prereq: Permission of instructor Prereq: 18.06 or 18.700 Acad Year 2021-2022: Not oered Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Spring) Acad Year 2022-2023: U (Fall) 3-0-9 units 3-0-9 units Can be repeated for credit.
Provides students with the basic analytical and computational Topics vary from year to year. tools of linear partial dierential equations (PDEs) for practical L. Demanet applications in science and engineering, including heat/diusion, wave, and Poisson equations. Analytics emphasize the viewpoint of 18.330 Introduction to Numerical Analysis linear algebra and the analogy with nite matrix problems. Studies Prereq: Calculus II (GIR) and (18.03 or 18.032) operator adjoints and eigenproblems, series solutions, Green's U (Spring) functions, and separation of variables. Numerics focus on nite- 3-0-9 units dierence and nite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/ Basic techniques for the ecient numerical solution of problems in explicit timestepping. Some programming required for homework science and engineering. Root nding, interpolation, approximation and nal project. of functions, integration, dierential equations, direct and iterative V. Heinonen methods in linear algebra. Knowledge of programming in a language such as MATLAB, Python, or Julia is helpful. 18.305 Advanced Analytic Methods in Science and Engineering L. Demanet Prereq: 18.04, 18.075, or 18.112 G (Fall) 18.335[J] Introduction to Numerical Methods 3-0-9 units Same subject as 6.337[J] Prereq: 18.06, 18.700, or 18.701 Covers expansion around singular points: the WKB method on G (Spring) ordinary and partial dierential equations; the method of stationary 3-0-9 units phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and Advanced introduction to numerical analysis: accuracy and eciency boundary-layer techniques; WKB method on partial dierential of numerical algorithms. In-depth coverage of sparse-matrix/iterative equations. and dense-matrix algorithms in numerical linear algebra (for linear H. Cheng systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational 18.306 Advanced Partial Dierential Equations with Applications topics (e.g., numerical integration or nonlinear optimization) may Prereq: (18.03 or 18.032) and (18.04, 18.075, or 18.112) also be surveyed. Final project involves some programming. G (Spring) A. J. Horning 3-0-9 units
Concepts and techniques for partial dierential equations, especially nonlinear. Diusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, trac flow, etc. R. R. Rosales
18 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.336[J] Fast Methods for Partial Dierential and Integral 18.338 Eigenvalues of Random Matrices Equations Prereq: 18.701 or permission of instructor Same subject as 6.335[J] Acad Year 2021-2022: G (Fall) Prereq: 6.336[J], 16.920[J], 18.085, 18.335[J], or permission of Acad Year 2022-2023: Not oered instructor 3-0-9 units G (Fall) 3-0-9 units Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include Unied introduction to the theory and practice of modern, near matrix calculus for nite and innite matrices (e.g., Wigner's semi- linear-time, numerical methods for large-scale partial-dierential circle and Marcenko-Pastur laws), free probability, random graphs, and integral equations. Topics include preconditioned iterative combinatorial methods, matrix statistics, stochastic operators, methods; generalized Fast Fourier Transform and other butterfly- passage to the continuum limit, moment methods, and compressed based methods; multiresolution approaches, such as multigrid sensing. Knowledge of MATLAB hepful, but not required. algorithms and hierarchical low-rank matrix decompositions; A. Edelman and low and high frequency Fast Multipole Methods. Example applications include aircra design, cardiovascular system 18.352[J] Nonlinear Dynamics: The Natural Environment modeling, electronic structure computation, and tomographic Same subject as 12.009[J] imaging. Prereq: Calculus II (GIR) and Physics I (GIR); Coreq: 18.03 K. Burns Acad Year 2021-2022: U (Spring) Acad Year 2022-2023: Not oered 18.337[J] Parallel Computing and Scientic Machine Learning 3-0-9 units Same subject as 6.338[J] Prereq: 18.06, 18.700, or 18.701 See description under subject 12.009[J]. Acad Year 2021-2022: Not oered D. H. Rothman Acad Year 2022-2023: G (Fall) 3-0-9 units 18.353[J] Nonlinear Dynamics: Chaos Same subject as 2.050[J], 12.006[J] Introduction to scientic machine learning with an emphasis on Prereq: Physics II (GIR) and (18.03 or 18.032) developing scalable dierentiable programs. Covers scientic U (Fall) computing topics (numerical dierential equations, dense and 3-0-9 units sparse linear algebra, Fourier transformations, parallelization of large-scale scientic simulation) simultaneously with modern See description under subject 12.006[J]. data science (machine learning, deep neural networks, automatic R. R. Rosales dierentiation), focusing on the emerging techniques at the connection between these areas, such as neural dierential 18.354[J] Nonlinear Dynamics: Continuum Systems equations and physics-informed deep learning. Provides direct Same subject as 1.062[J], 12.207[J] experience with the modern realities of optimizing code performance Subject meets with 18.3541 for supercomputers, GPUs, and multicores in a high-level language. Prereq: Physics II (GIR) and (18.03 or 18.032) C. Rackauckas U (Spring) 3-0-9 units
General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) dierential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology. O. Kodio
Department of Mathematics | 19 DEPARTMENT OF MATHEMATICS
18.3541 Nonlinear Dynamics: Continuum Systems 18.358[J] Nonlinear Dynamics and Turbulence Subject meets with 1.062[J], 12.207[J], 18.354[J] Same subject as 1.686[J], 2.033[J] Prereq: Physics II (GIR) and (18.03 or 18.032) Subject meets with 1.068 G (Spring) Prereq: 1.060A 3-0-9 units Acad Year 2021-2022: G (Spring) Acad Year 2022-2023: Not oered General mathematical principles of continuum systems. From 3-2-7 units microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) dierential equations. Exact solutions, See description under subject 1.686[J]. dimensional analysis, calculus of variations and singular L. Bourouiba perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid 18.367 Waves and Imaging and solid systems found, for example, in geophysics and biology. Prereq: Permission of instructor Students in Courses 1, 12, and 18 must register for undergraduate Acad Year 2021-2022: Not oered version, 18.354[J]. Acad Year 2022-2023: G (Spring) O. Kodio 3-0-9 units
18.355 Fluid Mechanics The mathematics of inverse problems involving waves, with Prereq: 2.25, 12.800, or 18.354[J] examples taken from reflection seismology, synthetic aperture Acad Year 2021-2022: G (Fall) radar, and computerized tomography. Suitable for graduate students Acad Year 2022-2023: Not oered from all departments who have anities with applied mathematics. 3-0-9 units Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and Topics include the development of Navier-Stokes equations, inviscid backprojection; adjoint-state methods; Radon and curvilinear flows, boundary layers, lubrication theory, Stokes flows, and surface Radon transforms; microlocal analysis of imaging; optimization, tension. Fundamental concepts illustrated through problems drawn regularization, and sparse regression. from a variety of areas, including geophysics, biology, and the L. Demanet dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes 18.369[J] Mathematical Methods in Nanophotonics classroom and laboratory demonstrations. Same subject as 8.315[J] J. W. Bush Prereq: 8.07, 18.303, or permission of instructor Acad Year 2021-2022: G (Fall) 18.357 Interfacial Phenomena Acad Year 2022-2023: Not oered Prereq: 2.25, 12.800, 18.354[J], 18.355, or permission of instructor 3-0-9 units Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Spring) High-level approaches to understanding complex optical media, 3-0-9 units structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical Fluid systems dominated by the influence of interfacial tension. phenomena such as photonic crystals and band gaps, anomalous Elucidates the roles of curvature pressure and Marangoni stress in diraction, mechanisms for optical connement, optical bers (new a variety of hydrodynamic settings. Particular attention to drops and old), nonlinearities, and integrated optical devices. Methods and bubbles, soap lms and minimal surfaces, wetting phenomena, covered include linear algebra and eigensystems for Maxwell's water-repellency, surfactants, Marangoni flows, capillary origami equations, symmetry groups and representation theory, Bloch's and contact line dynamics. Theoretical developments are theorem, numerical eigensolver methods, time and frequency- accompanied by classroom demonstrations. Highlights the role of domain computation, perturbation theory, and coupled-mode surface tension in biology. theories. J. W. Bush S. G. Johnson
20 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.376[J] Wave Propagation 18.385[J] Nonlinear Dynamics and Chaos Same subject as 1.138[J], 2.062[J] Same subject as 2.036[J] Prereq: 2.003[J] and 18.075 Prereq: 18.03 or 18.032 Acad Year 2021-2022: Not oered Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Spring) Acad Year 2022-2023: G (Fall) 3-0-9 units 3-0-9 units
See description under subject 2.062[J]. Introduction to the theory of nonlinear dynamical systems with T. R. Akylas, R. R. Rosales applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. 18.377[J] Nonlinear Dynamics and Waves Elementary bifurcations, normal forms. Phase plane, limit cycles, Same subject as 1.685[J], 2.034[J] relaxation oscillations, Poincare-Bendixson theory. Floquet Prereq: Permission of instructor theory. Poincare maps. Averaging. Near-equilibrium dynamics. Acad Year 2021-2022: G (Spring) Synchronization. Introduction to chaos. Universality. Strange Acad Year 2022-2023: Not oered attractors. Lorenz and Rossler systems. Hamiltonian dynamics and 3-0-9 units KAM theory. Uses MATLAB computing environment. R. R. Rosales A unied treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, 18.397 Mathematical Methods in Physics electrical and flow-structure interaction problems. Nonlinear free and Prereq: 18.745 or some familiarity with Lie theory forced vibrations; nonlinear resonances; self-excited oscillations; G (Fall) lock-in phenomena. Nonlinear dispersive and nondispersive waves; Not oered regularly; consult department resonant wave interactions; propagation of wave pulses and 3-0-9 units nonlinear Schrodinger equation. Nonlinear long waves and breaking; Can be repeated for credit. theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics Content varies from year to year. Recent developments in quantum and applications may vary from year to year. eld theory require mathematical techniques not usually covered in R. R. Rosales standard graduate subjects. V. G. Kac 18.384 Undergraduate Seminar in Physical Mathematics Prereq: 12.006[J], 18.300, 18.354[J], or permission of instructor Theoretical Computer Science U (Fall) 3-0-9 units 18.400[J] Computability and Complexity Theory Covers the mathematical modeling of physical systems, with Same subject as 6.045[J] emphasis on the reading and presentation of papers. Addresses Prereq: 6.042[J] a broad range of topics, with particular focus on macroscopic U (Spring) physics and continuum systems: fluid dynamics, solid mechanics, 4-0-8 units and biophysics. Instruction and practice in written and oral See description under subject 6.045[J]. communication provided. Enrollment limited. R. Williams, R. Rubinfeld O. Kodio
Department of Mathematics | 21 DEPARTMENT OF MATHEMATICS
18.404 Theory of Computation 18.408 Topics in Theoretical Computer Science Subject meets with 6.840[J], 18.4041[J] Prereq: Permission of instructor Prereq: 6.042[J] or 18.200 G (Spring) U (Fall) 3-0-9 units 4-0-8 units Can be repeated for credit.
A more extensive and theoretical treatment of the material in Study of areas of current interest in theoretical computer science. 6.045[J]/18.400[J], emphasizing computability and computational Topics vary from term to term. complexity theory. Regular and context-free languages. Decidable A. Moitra, J. A. Kelner and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy 18.410[J] Design and Analysis of Algorithms theorems, inherently complex problems, oracles, probabilistic Same subject as 6.046[J] computation, and interactive proof systems. Prereq: 6.006 M. Sipser U (Fall, Spring) 4-0-8 units 18.4041[J] Theory of Computation Same subject as 6.840[J] See description under subject 6.046[J]. Subject meets with 18.404 E. Demaine, M. Goemans Prereq: 6.042[J] or 18.200 G (Fall) 18.415[J] Advanced Algorithms 4-0-8 units Same subject as 6.854[J] Prereq: 6.046[J] and (6.042[J], 18.600, or 6.041) A more extensive and theoretical treatment of the material in G (Fall) 6.045[J]/18.400[J], emphasizing computability and computational 5-0-7 units complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. See description under subject 6.854[J]. Time and space measures on computation, completeness, hierarchy A. Moitra, D. R. Karger theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Students in Course 18 18.416[J] Randomized Algorithms must register for the undergraduate version, 18.404. Same subject as 6.856[J] M. Sipser Prereq: (6.041 or 6.042[J]) and (6.046[J] or 6.854[J]) Acad Year 2021-2022: Not oered 18.405[J] Advanced Complexity Theory Acad Year 2022-2023: G (Spring) Same subject as 6.841[J] 5-0-7 units Prereq: 18.404 See description under subject 6.856[J]. Acad Year 2021-2022: G (Spring) D. R. Karger Acad Year 2022-2023: Not oered 3-0-9 units 18.417 Introduction to Computational Molecular Biology Current research topics in computational complexity theory. Prereq: 6.006, 6.01, or permission of instructor Nondeterministic, alternating, probabilistic, and parallel G (Fall) computation models. Boolean circuits. Complexity classes and Not oered regularly; consult department complete sets. The polynomial-time hierarchy. Interactive proof 3-0-9 units systems. Relativization. Denitions of randomness. Pseudo- Introduces the basic computational methods used to model and randomness and derandomizations. Interactive proof systems and predict the structure of biomolecules (proteins, DNA, RNA). Covers probabilistically checkable proofs. classical techniques in the eld (molecular dynamics, Monte Carlo, R. Williams dynamic programming) to more recent advances in analyzing and predicting RNA and protein structure, ranging from Hidden Markov Models and 3-D lattice models to attribute Grammars and tree Grammars. B. Berger
22 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.418[J] Topics in Computational Molecular Biology 18.435[J] Quantum Computation Same subject as HST.504[J] Same subject as 2.111[J], 8.370[J] Prereq: 6.047, 18.417, or permission of instructor Prereq: 8.05, 18.06, 18.061, 18.700, or 18.701 G (Fall, Spring) G (Fall) 3-0-9 units 3-0-9 units Can be repeated for credit. Provides an introduction to the theory and practice of quantum Covers current research topics in computational molecular biology. computation. Topics covered: physics of information processing; Recent research papers presented from leading conferences such as quantum algorithms including the factoring algorithm and the International Conference on Computational Molecular Biology Grover's search algorithm; quantum error correction; quantum (RECOMB) and the Conference on Intelligent Systems for Molecular communication and cryptography. Knowledge of quantum mechanics Biology (ISMB). Topics include original research (both theoretical helpful but not required. and experimental) in comparative genomics, sequence and structure I. Chuang, A. Harrow, S. Lloyd, P. Shor analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, 18.436[J] Quantum Information Science and privacy. Recent research by course participants also covered. Same subject as 6.443[J], 8.371[J] Participants will be expected to present individual projects to the Prereq: 18.435[J] class. G (Spring) B. Berger 3-0-9 units
18.424 Seminar in Information Theory See description under subject 8.371[J]. Prereq: (6.041, 18.05, or 18.600) and (18.06, 18.700, or 18.701) I. Chuang, A. Harrow U (Spring) 3-0-9 units 18.437[J] Distributed Algorithms Same subject as 6.852[J] Considers various topics in information theory, including data Prereq: 6.046[J] compression, Shannon's Theorems, and error-correcting codes. Acad Year 2021-2022: Not oered Students present and discuss the subject matter. Instruction and Acad Year 2022-2023: G (Fall) practice in written and oral communication provided. Enrollment 3-0-9 units limited. P. W. Shor See description under subject 6.852[J]. N. A. Lynch 18.425[J] Cryptography and Cryptanalysis Same subject as 6.875[J] 18.453 Combinatorial Optimization Prereq: 6.046[J] Subject meets with 18.4531 G (Fall) Prereq: 18.06, 18.700, or 18.701 3-0-9 units Acad Year 2021-2022: Not oered Acad Year 2022-2023: U (Spring) See description under subject 6.875[J]. 3-0-9 units S. Goldwasser, S. Micali, V. Vaikuntanathan Thorough treatment of linear programming and combinatorial 18.434 Seminar in Theoretical Computer Science optimization. Topics include matching theory, network flow, matroid Prereq: 6.046[J] optimization, and how to deal with NP-hard optimization problems. U (Fall, Spring) Prior exposure to discrete mathematics (such as 18.200) helpful. 3-0-9 units Information: M. X. Goemans
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited. Fall: D. Minzer. Spring: W. C. Franks
Department of Mathematics | 23 DEPARTMENT OF MATHEMATICS
18.4531 Combinatorial Optimization 18.510 Introduction to Mathematical Logic and Set Theory Subject meets with 18.453 Prereq: None Prereq: 18.06, 18.700, or 18.701 Acad Year 2021-2022: U (Fall) Acad Year 2021-2022: Not oered Acad Year 2022-2023: Not oered Acad Year 2022-2023: G (Spring) 3-0-9 units 3-0-9 units Propositional and predicate logic. Zermelo-Fraenkel set theory. Thorough treatment of linear programming and combinatorial Ordinals and cardinals. Axiom of choice and transnite induction. optimization. Topics include matching theory, network flow, matroid Elementary model theory: completeness, compactness, and optimization, and how to deal with NP-hard optimization problems. Lowenheim-Skolem theorems. Godel's incompleteness theorem. Prior exposure to discrete mathematics (such as 18.200) helpful. H. Cohn Students in Course 18 must register for the undergraduate version, 18.453. 18.515 Mathematical Logic Information: M. X. Goemans Prereq: Permission of instructor G (Spring) 18.455 Advanced Combinatorial Optimization Not oered regularly; consult department Prereq: 18.453 or permission of instructor 3-0-9 units Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Spring) More rigorous treatment of basic mathematical logic, Godel's 3-0-9 units theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and Advanced treatment of combinatorial optimization with an emphasis completeness. Compactness and its consequences. Quantier on combinatorial aspects. Non-bipartite matchings, submodular elimination. Recursive sets and functions. Incompleteness and functions, matroid intersection/union, matroid matching, undecidability. Ordinals and cardinals. Set-theoretic formalization of submodular flows, multicommodity flows, packing and connectivity mathematics. problems, and other recent developments. Information: B. Poonen M. X. Goemans Probability and Statistics 18.456[J] Algebraic Techniques and Semidenite Optimization Same subject as 6.256[J] 18.600 Probability and Random Variables Prereq: 6.251[J] or 15.093[J] Prereq: Calculus II (GIR) Acad Year 2021-2022: Not oered U (Fall, Spring) Acad Year 2022-2023: G (Spring) 4-0-8 units. REST 3-0-9 units Credit cannot also be received for 6.041, 6.431, 15.079, 15.0791
See description under subject 6.256[J]. Probability spaces, random variables, distribution functions. P. Parrilo Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Logic Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit 18.504 Seminar in Logic theorem. Credit cannot also be received for 6.041A or 6.041B. Prereq: (18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, J. A. Kelner, S. Sheeld 18.100P, or 18.100Q) Acad Year 2021-2022: Not oered 18.615 Introduction to Stochastic Processes Acad Year 2022-2023: U (Fall) Prereq: 6.041 or 18.600 3-0-9 units G (Spring) 3-0-9 units Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and Basics of stochastic processes. Markov chains, Poisson processes, practice in written and oral communication provided. Enrollment random walks, birth and death processes, Brownian motion. limited. J. He H. Cohn
24 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.642 Topics in Mathematics with Applications in Finance 18.655 Mathematical Statistics Prereq: 18.03, 18.06, and (18.05 or 18.600) Prereq: (18.650[J] and (18.100A, 18.100A, 18.100P, or 18.100Q)) or U (Fall) permission of instructor 3-0-9 units G (Spring) 3-0-9 units Introduction to mathematical concepts and techniques used in nance. Lectures focusing on linear algebra, probability, statistics, Decision theory, estimation, condence intervals, hypothesis stochastic processes, and numerical methods are interspersed testing. Introduces large sample theory. Asymptotic eciency of with lectures by nancial sector professionals illustrating the estimates. Exponential families. Sequential analysis. Prior exposure corresponding application in the industry. Prior knowledge of to both probability and statistics at the university level is assumed. economics or nance helpful but not required. P. Kempthorne P. Kempthorne, V. Strela, J. Xia 18.656[J] Mathematical Statistics: a Non-Asymptotic Approach 18.650[J] Fundamentals of Statistics Same subject as 9.521[J], IDS.160[J] Same subject as IDS.014[J] Prereq: (6.436[J], 18.06, and 18.6501) or permission of instructor Subject meets with 18.6501 G (Spring) Prereq: 6.041 or 18.600 3-0-9 units U (Fall, Spring) 4-0-8 units See description under subject 9.521[J]. Credit cannot also be received for 15.075[J], IDS.013[J] S. Rakhlin, P. Rigollet
In-depth introduction to the theoretical foundations of statistical 18.657 Topics in Statistics methods that are useful in many applications. Enables students to Prereq: Permission of instructor understand the role of mathematics in the research and development Acad Year 2021-2022: Not oered of ecient statistical methods. Topics include methods for Acad Year 2022-2023: G (Spring) estimation (maximum likelihood estimation, method of moments, 3-0-9 units M-estimation), hypothesis testing (Wald's test, likelihood ratio Can be repeated for credit. test, T tests, goodness of t), Bayesian statistics, linear regression, generalized linear models, and principal component analysis. Topics vary from term to term. E. Mossel P. Rigollet
18.6501 Fundamentals of Statistics 18.675 Theory of Probability Subject meets with 18.650[J], IDS.014[J] Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q Prereq: 6.041 or 18.600 G (Fall) G (Fall, Spring) 3-0-9 units 4-0-8 units Sums of independent random variables, central limit phenomena, Credit cannot also be received for 15.075[J], IDS.013[J] innitely divisible laws, Levy processes, Brownian motion, In-depth introduction to the theoretical foundations of statistical conditioning, and martingales. Prior exposure to probability (e.g., methods that are useful in many applications. Enables students to 18.600) recommended. understand the role of mathematics in the research and development Y. Shenfeld of ecient statistical methods. Topics include methods for estimation (maximum likelihood estimation, method of moments, M-estimation), hypothesis testing (Wald's test, likelihood ratio test, T tests, goodness of t), Bayesian statistics, linear regression, generalized linear models, and principal component analysis. E. Mossel
Department of Mathematics | 25 DEPARTMENT OF MATHEMATICS
18.676 Stochastic Calculus 18.702 Algebra II Prereq: 18.675 Prereq: 18.701 G (Spring) U (Spring) 3-0-9 units 3-0-9 units
Introduction to stochastic processes, building on the fundamental Continuation of 18.701. Focuses on group representations, rings, example of Brownian motion. Topics include Brownian motion, ideals, elds, polynomial rings, modules, factorization, integers in continuous parameter martingales, Ito's theory of stochastic quadratic number elds, eld extensions, and Galois theory. dierential equations, Markov processes and partial dierential R. Bezrukavnikov equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its 18.703 Modern Algebra application to probability. Prereq: Calculus II (GIR) N. Sun U (Spring) 3-0-9 units 18.677 Topics in Stochastic Processes Prereq: 18.675 Focuses on traditional algebra topics that have found greatest G (Fall, Spring) application in science and engineering as well as in mathematics: 3-0-9 units group theory, emphasizing nite groups; ring theory, including Can be repeated for credit. ideals and unique factorization in polynomial and Euclidean rings; eld theory, including properties and applications of nite elds. Topics vary from year to year. 18.700 and 18.703 together form a standard algebra sequence. Fall: S. Sheeld. Spring: D. W. Stroock V. G. Kac
Algebra and Number Theory 18.704 Seminar in Algebra Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703) 18.700 Linear Algebra U (Spring) Prereq: Calculus II (GIR) 3-0-9 units U (Fall) Topics vary from year to year. Students present and discuss 3-0-9 units. REST the subject matter. Instruction and practice in written and oral Credit cannot also be received for 18.06, 18.061 communication provided. Some experience with proofs required. Vector spaces, systems of linear equations, bases, linear Enrollment limited. independence, matrices, determinants, eigenvalues, inner products, M.-T. Trinh quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06. 18.705 Commutative Algebra J.-L. Kim Prereq: 18.702 G (Fall) 18.701 Algebra I 3-0-9 units Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of Exactness, direct limits, tensor products, Cayley-Hamilton theorem, instructor integral dependence, localization, Cohen-Seidenberg theory, U (Fall) Noether normalization, Nullstellensatz, chain conditions, primary 3-0-9 units decomposition, length, Hilbert functions, dimension theory, 18.701-18.702 is more extensive and theoretical than the completion, Dedekind domains. 18.700-18.703 sequence. Experience with proofs necessary. 18.701 W. Zhang focuses on group theory, geometry, and linear algebra. D. Maulik
26 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.706 Noncommutative Algebra 18.725 Algebraic Geometry I Prereq: 18.702 Prereq: None. Coreq: 18.705 Acad Year 2021-2022: Not oered G (Fall) Acad Year 2022-2023: G (Fall) 3-0-9 units 3-0-9 units Introduces the basic notions and techniques of modern algebraic Topics may include Wedderburn theory and structure of Artinian geometry. Covers fundamental notions and results about algebraic rings, Morita equivalence and elements of category theory, varieties over an algebraically closed eld; relations between localization and Goldie's theorem, central simple algebras and the complex algebraic varieties and complex analytic varieties; Brauer group, representations, polynomial identity rings, invariant and examples with emphasis on algebraic curves and surfaces. theory growth of algebras, Gelfand-Kirillov dimension. Introduction to the language of schemes and properties of Z. Yun morphisms. Knowledge of elementary algebraic topology, elementary dierential geometry recommended, but not required. 18.708 Topics in Algebra Z. Yun Prereq: 18.705 Acad Year 2021-2022: Not oered 18.726 Algebraic Geometry II Acad Year 2022-2023: G (Spring) Prereq: 18.725 3-0-9 units G (Spring) Can be repeated for credit. 3-0-9 units
Topics vary from year to year. Continuation of the introduction to algebraic geometry given in Z. Yun 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology. 18.715 Introduction to Representation Theory D. Maulik Prereq: 18.702 or 18.703 Acad Year 2021-2022: Not oered 18.727 Topics in Algebraic Geometry Acad Year 2022-2023: G (Fall) Prereq: 18.725 3-0-9 units Acad Year 2021-2022: Not oered Acad Year 2022-2023: G (Spring) Algebras, representations, Schur's lemma. Representations of SL(2). 3-0-9 units Representations of nite groups, Maschke's theorem, characters, Can be repeated for credit. applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers. Topics vary from year to year. G. Lusztig A. Negut
18.721 Introduction to Algebraic Geometry 18.737 Algebraic Groups Prereq: 18.702 and 18.901 Prereq: 18.705 Acad Year 2021-2022: U (Fall) Acad Year 2021-2022: Not oered Acad Year 2022-2023: Not oered Acad Year 2022-2023: G (Spring) 3-0-9 units 3-0-9 units
Presents basic examples of complex algebraic varieties, ane and Structure of linear algebraic groups over an algebraically closed projective algebraic geometry, sheaves, cohomology. eld, with emphasis on reductive groups. Representations of groups E. Costa, S. Schiavone, R. van Bommel over a nite eld using methods from etale cohomology. Some results from algebraic geometry are stated without proof. B. Poonen
Department of Mathematics | 27 DEPARTMENT OF MATHEMATICS
18.745 Lie Groups and Lie Algebras I 18.757 Representations of Lie Groups Prereq: (18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or Prereq: 18.745 or 18.755 18.100Q) Acad Year 2021-2022: G (Fall) G (Fall) Acad Year 2022-2023: Not oered 3-0-9 units 3-0-9 units
Covers fundamentals of the theory of Lie algebras and related Covers representations of locally compact groups, with emphasis on groups. Topics may include theorems of Engel and Lie; enveloping compact groups and abelian groups. Includes Peter-Weyl theorem algebra, Poincare-Birkho-Witt theorem; classication and and Cartan-Weyl highest weight theory for compact Lie groups. construction of semisimple Lie algebras; the center of their P. I. Etingof enveloping algebras; elements of representation theory; compact Lie groups and/or nite Chevalley groups. 18.781 Theory of Numbers V. G. Kac Prereq: None U (Spring) 18.747 Innite-dimensional Lie Algebras 3-0-9 units Prereq: 18.745 Acad Year 2021-2022: Not oered An elementary introduction to number theory with no algebraic Acad Year 2022-2023: G (Fall) prerequisites. Primes, congruences, quadratic reciprocity, 3-0-9 units diophantine equations, irrational numbers, continued fractions, partitions. Topics vary from year to year. J-L Kim P. I. Etingof 18.782 Introduction to Arithmetic Geometry 18.748 Topics in Lie Theory Prereq: 18.702 Prereq: Permission of instructor Acad Year 2021-2022: Not oered Acad Year 2021-2022: Not oered Acad Year 2022-2023: U (Fall) Acad Year 2022-2023: G (Fall) 3-0-9 units 3-0-9 units Can be repeated for credit. Exposes students to arithmetic geometry, motivated by the problem of nding rational points on curves. Includes an introduction to p- Topics vary from year to year. adic numbers and some fundamental results from number theory J-L. Kim and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include 18.755 Lie Groups and Lie Algebras II Mordell's theorem, the Weil conjectures, and Jacobian varieties. Prereq: 18.745 or permission of instructor D. Roe G (Spring) 3-0-9 units 18.783 Elliptic Curves Subject meets with 18.7831 A more in-depth treatment of Lie groups and Lie algebras. Topics Prereq: 18.702, 18.703, or permission of instructor may include homogeneous spaces and groups of automorphisms; Acad Year 2021-2022: U (Spring) representations of compact groups and their geometric realizations, Acad Year 2022-2023: Not oered Peter-Weyl theorem; invariant dierential forms and cohomology of 3-0-9 units Lie groups and homogeneous spaces; complex reductive Lie groups, classication of real reductive groups. Computationally focused introduction to elliptic curves, with Z. Yun applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. A. Sutherland
28 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.7831 Elliptic Curves 18.787 Topics in Number Theory Subject meets with 18.783 Prereq: Permission of instructor Prereq: 18.702, 18.703, or permission of instructor Acad Year 2021-2022: Not oered Acad Year 2021-2022: G (Spring) Acad Year 2022-2023: G (Fall) Acad Year 2022-2023: Not oered 3-0-9 units 3-0-9 units Can be repeated for credit.
Computationally focused introduction to elliptic curves, with Topics vary from year to year. applications to number theory and cryptography. Topics include W. Zhang point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality Mathematics Laboratory proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. Students 18.821 Project Laboratory in Mathematics in Course 18 must register for the undergraduate version, 18.783. Prereq: Two mathematics subjects numbered 18.10 or above A. Sutherland U (Fall, Spring) 3-6-3 units. Institute LAB 18.784 Seminar in Number Theory Prereq: 18.701 or (18.703 and (18.06 or 18.700)) Guided research in mathematics, employing the scientic U (Fall) method. Students confront puzzling and complex mathematical 3-0-9 units situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain Topics vary from year to year. Students present and discuss them mathematically. Students choose three projects from a large the subject matter. Instruction and practice in written and oral collection of options. Each project results in a laboratory report communication provided. Enrollment limited. subject to revision; oral presentation on one or two projects. Projects T. Feng drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, 18.785 Number Theory I and probability. Enrollment limited. Prereq: None. Coreq: 18.705 Fall: S. Kleene. Spring: A. Negut G (Fall) 3-0-9 units Topology and Geometry Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, niteness of the class group, Dirichlet's 18.900 Geometry and Topology in the Plane unit theorem. Local elds, ramication, discriminants. Zeta and Prereq: 18.03 or 18.06 L-functions, analytic class number formula. Adeles and ideles. U (Fall, Spring) Statements of class eld theory and the Chebotarev density 3-0-9 units theorem. A. Sutherland Covers selected topics in geometry and topology, which can be visualized in the two-dimensional plane. Polygons and polygonal 18.786 Number Theory II paths. Billiards. Closed curves and immersed curves. Algebraic Prereq: 18.785 curves. Triangulations and complexes. Hyperbolic geometry. G (Spring) Geodesics and curvature. Other topics may be included as time 3-0-9 units permits. J. Hahn Continuation of 18.785. More advanced topics in number theory, such as Galois cohomology, proofs of class eld theory, modular forms and automorphic forms, Galois representations, or quadratic forms. W. Zhang
Department of Mathematics | 29 DEPARTMENT OF MATHEMATICS
18.901 Introduction to Topology 18.906 Algebraic Topology II Subject meets with 18.9011 Prereq: 18.905 Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of G (Spring) instructor 3-0-9 units U (Fall, Spring) 3-0-9 units Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic Introduces topology, covering topics fundamental to modern analysis classes, and cohomology operations. and geometry. Topological spaces and continuous functions, P. Seidel connectedness, compactness, separation axioms, covering spaces, and the fundamental group. 18.917 Topics in Algebraic Topology Fall: A. Conway. Spring: D. Alvarez-Gavela Prereq: 18.906 Acad Year 2021-2022: Not oered 18.9011 Introduction to Topology Acad Year 2022-2023: G (Spring) Subject meets with 18.901 3-0-9 units Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of Can be repeated for credit. instructor G (Fall, Spring) Content varies from year to year. Introduces new and signicant 3-0-9 units developments in algebraic topology with the focus on homotopy theory and related areas. Introduces topology, covering topics fundamental to modern analysis Information: H. R. Miller and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, 18.919 Graduate Topology Seminar and the fundamental group. Students in Course 18 must register for Prereq: 18.906 the undergraduate version, 18.901. G (Fall) Fall: A. Conway. Spring: D. Alvarez-Gavela 3-0-9 units
18.904 Seminar in Topology Study and discussion of important original papers in the various Prereq: 18.901 parts of algebraic topology. Open to all students who have taken U (Spring) 18.906 or the equivalent, not only prospective topologists. 3-0-9 units H. R. Miller
Topics vary from year to year. Students present and discuss 18.937 Topics in Geometric Topology the subject matter. Instruction and practice in written and oral Prereq: Permission of instructor communication provided. Enrollment limited. Acad Year 2021-2022: Not oered A. Conway Acad Year 2022-2023: G (Spring) 3-0-9 units 18.905 Algebraic Topology I Can be repeated for credit. Prereq: 18.901 and (18.701 or 18.703) G (Fall) Content varies from year to year. Introduces new and signicant 3-0-9 units developments in geometric topology. T. S. Mrowka Singular homology, CW complexes, universal coecient and Künneth theorems, cohomology, cup products, Poincaré duality. J. Hahn
30 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.950 Dierential Geometry 18.965 Geometry of Manifolds I Subject meets with 18.9501 Prereq: 18.101, 18.950, or 18.952 Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or G (Fall) 18.100Q) 3-0-9 units U (Spring) 3-0-9 units Dierential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a Introduction to dierential geometry, centered on notions of continuation of 18.965 and focuses more deeply on various aspects curvature. Starts with curves in the plane, and proceeds to higher of the geometry of manifolds. Contents vary from year to year, and dimensional submanifolds. Computations in coordinate charts: rst can range from Riemannian geometry (curvature, holonomy) to and second fundamental form, Christoel symbols. Discusses the symplectic geometry, complex geometry and Hodge-Kahler theory, or distinction between extrinsic and intrinsic aspects, in particular smooth manifold topology. Prior exposure to calculus on manifolds, Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. as in 18.952, recommended. Examples such as hyperbolic space. W. Minicozzi S. Kleene 18.966 Geometry of Manifolds II 18.9501 Dierential Geometry Prereq: 18.965 Subject meets with 18.950 G (Spring) Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 3-0-9 units 18.100Q) G (Spring) Continuation of 18.965, focusing more deeply on various aspects 3-0-9 units of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to Introduction to dierential geometry, centered on notions of symplectic geometry, complex geometry and Hodge-Kahler theory, or curvature. Starts with curves in the plane, and proceeds to higher smooth manifold topology. dimensional submanifolds. Computations in coordinate charts: rst T. Colding and second fundamental form, Christoel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular 18.968 Topics in Geometry Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Prereq: 18.965 Examples such as hyperbolic space. Students in Course 18 must Acad Year 2021-2022: Not oered register for the undergraduate version, 18.950. Acad Year 2022-2023: G (Spring) S. Kleene 3-0-9 units Can be repeated for credit. 18.952 Theory of Dierential Forms Prereq: 18.101 and (18.700 or 18.701) Content varies from year to year. U (Spring) P. Seidel 3-0-9 units 18.979 Graduate Geometry Seminar Multilinear algebra: tensors and exterior forms. Dierential forms Prereq: Permission of instructor on Rn: exterior dierentiation, the pull-back operation and the G (Spring) Poincaré lemma. Applications to physics: Maxwell's equations from Not oered regularly; consult department the dierential form perspective. Integration of forms on open sets 3-0-9 units of Rn. The change of variables formula revisited. The degree of a Can be repeated for credit. dierentiable mapping. Dierential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The Content varies from year to year. Study of classical papers in push-forward operation for forms. Thom forms and intersection geometry and in applications of analysis to geometry and topology. theory. Applications to dierential topology. T. Mrowka V. W. Guillemin
Department of Mathematics | 31 DEPARTMENT OF MATHEMATICS
18.994 Seminar in Geometry 18.S096 Special Subject in Mathematics Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or Prereq: Permission of instructor 18.100Q) U (IAP, Spring) U (Fall) Units arranged 3-0-9 units Can be repeated for credit.
Students present and discuss subject matter taken from current Opportunity for group study of subjects in mathematics not journals or books. Topics vary from year to year. Instruction and otherwise included in the curriculum. Oerings are initiated by practice in written and oral communication provided. Enrollment members of the Mathematics faculty on an ad hoc basis, subject to limited. departmental approval. 18.S097 is graded P/D/F. Information: T. S. Mrowka Sta
18.999 Research in Mathematics 18.S097 Special Subject in Mathematics Prereq: Permission of instructor Prereq: Permission of instructor G (Fall, IAP, Spring, Summer) U (IAP) Units arranged Units arranged [P/D/F] Can be repeated for credit. Can be repeated for credit.
Opportunity for study of graduate-level topics in mathematics Opportunity for group study of subjects in mathematics not under the supervision of a member of the department. For graduate otherwise included in the curriculum. Oerings are initiated by students desiring advanced work not provided in regular subjects. members of the Mathematics faculty on an ad hoc basis, subject to Information: W. Minicozzi departmental approval. 18.S097 is graded P/D/F. Sta 18.UR Undergraduate Research Prereq: Permission of instructor 18.S190 Special Subject in Mathematics U (Fall, IAP, Spring, Summer) Prereq: Permission of instructor Units arranged [P/D/F] Acad Year 2021-2022: Not oered Can be repeated for credit. Acad Year 2022-2023: U (Fall, Spring) Units arranged Undergraduate research opportunities in mathematics. Permission Can be repeated for credit. required in advance to register for this subject. For further information, consult the departmental coordinator. Opportunity for group study of subjects in mathematics not Information: W. Minicozzi otherwise included in the curriculum. Oerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to 18.THG Graduate Thesis departmental approval. Prereq: Permission of instructor Sta G (Fall, IAP, Spring, Summer) Units arranged 18.S191 Special Subject in Mathematics Can be repeated for credit. Prereq: Permission of instructor U (Spring) Program of research leading to the writing of a Ph.D. thesis; to be Units arranged arranged by the student and an appropriate MIT faculty member. Can be repeated for credit. Information: W. Minicozzi Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Oerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. Sta
32 | Department of Mathematics DEPARTMENT OF MATHEMATICS
18.S995 Special Subject in Mathematics Prereq: Permission of instructor G (Spring) Units arranged Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Oerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval. Sta
18.S996 Special Subject in Mathematics Prereq: Permission of instructor G (Spring) Units arranged Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Oerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval. Sta
18.S997 Special Subject in Mathematics Prereq: Permission of instructor G (IAP) Units arranged Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Oerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval. Sta
18.S998 Special Subject in Mathematics Prereq: Permission of instructor G (Spring) Units arranged Can be repeated for credit.
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Oerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. Sta
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