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Course 18: Mathematics 
  18.0118.499    18.5018.THG   
General Mathematics18.01 Calculus
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Prereq: None Units: 507 Credit cannot also be received for 18.014, 18.01A, CC.181A, ES.1801, ES.181A Lecture: TR1,F2 (54100) Recitation: MW10 (36112) or MW11 (36112) or MW12 (26322) or MW1 (26322) or MW2 (26322) +final Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable firstorder differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, pharmonic, simple comparison tests, power series for some elementary functions. Fall: J. Speck Spring: Information: G. Staffilani No textbook information available 18.01A Calculus
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Prereq: Knowledge of differentiation and elementary integration Units: 507 Credit cannot also be received for 18.01, 18.014, CC.181A, ES.1801, ES.181A Ends Oct 17. Lecture: TR1,F2 (10250) Recitation: MW10 (133101) or MW11 (4159) or MW12 (4159, 36112) or MW1 (36112, 36155) or MW2 (36155) Sixweek review of onevariable calculus, emphasizing material not on the highschool AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of highschool calculus or the equivalent, with a score of 4 or 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam. J. W. Bush No textbook information available 18.014 Calculus with Theory
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Prereq: None Units: 507 Credit cannot also be received for 18.01, 18.01A, CC.181A, ES.1801, ES.181A Lecture: TR1,F2 (E17133) Recitation: MW2 (E17133) +final Covers the same material as 18.01, but at a deeper and more rigorous level. Emphasizes careful reasoning and understanding of proofs. Assumes knowledge of elementary calculus. Topics: axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. C. Barwick No textbook information available 18.02 Calculus
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Prereq: Calculus I (GIR) Units: 507 Credit cannot also be received for 18.022, 18.023, 18.024, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A Lecture: TR1,F2 (26100) Recitation: MW9 (E17139) or MW10 (E17139, 24307, 8119) or MW11 (E17139, 8119, 24307, 26302) or MW12 (E17139, 26302, 24307, 133101) or MW1 (133101, 24307, 131143, 66154) or MW2 (66154, 131143, 36153, 4149) or MW3 (4149, 36153, 131143) +final Calculus of several variables. Vector algebra in 3space, determinants, matrices. Vectorvalued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. Fall: W. Minicozzi Spring: P. Etingof No textbook information available 18.02A Calculus
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Prereq: Calculus I (GIR) Units: 507 Credit cannot also be received for 18.02, 18.022, 18.023, 18.024, CC.1802, CC.182A, ES.1802, ES.182A Begins Oct 20. Lecture: TR1,F2 (10250) Recitation: MW10 (133101) or MW11 (4159) or MW12 (4159, 36112) or MW1 (36112, 36155) or MW2 (36155) +final First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the first half of the Spring term; it covers the remaining material in 18.02. Fall: J. W. Bush Spring: Information: G. Staffilani No textbook information available 18.022 Calculus
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Prereq: Calculus I (GIR) Units: 507 Credit cannot also be received for 18.02, 18.023, 18.024, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A Lecture: TR1,F2 (E25111) Recitation: MW11 (131143) or MW12 (E17128, 131143) or MW1 (E17128) or MW2 (26302) +final Calculus of several variables. Topics as in 18.02 but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions. A. Borodin No textbook information available 18.024 Calculus with Theory
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Prereq: Calculus I (GIR), permission of Instructor Units: 507 Credit cannot also be received for 18.02, 18.022, 18.023, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A Continues 18.014. Parallel to 18.02, but at a deeper level, emphasizing careful reasoning and understanding of proofs. Considerable emphasis on linear algebra and vector integral calculus. C. Barwick 18.03 Differential Equations
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Prereq: None. Coreq: Calculus II (GIR) Units: 507 Credit cannot also be received for 18.034, 18.036, CC.1803, ES.1803 Lecture: MWF1 (54100) Recitation: TR10 (26322, 35308) or TR11 (26322, 35308) or TR12 (4145, 35308) or TR1 (35308, 4145, 4163) or TR2 (4163, 36112) or TR3 (36112) +final Study of differential equations, including modeling physical systems. Solution of firstorder ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams. Fall: L. Demanet Spring: G. Staffilani, D. Jerison No textbook information available 18.034 Differential Equations
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Prereq: None. Coreq: Calculus II (GIR) Units: 507 Credit cannot also be received for 18.03, 18.036, CC.1803, ES.1803 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 existence and uniqueness of solutions. J. Lauer 18.04 Complex Variables with Applications
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Prereq: Calculus II (GIR); 18.03 or 18.034 Units: 408 Credit cannot also be received for 18.075 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 differential equations. H. Cheng 18.05 Introduction to Probability and Statistics
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Prereq: Calculus I (GIR) Units: 408 Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression. J. Orloff 18.06 Linear Algebra
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Prereq: Calculus II (GIR) Units: 408 Credit cannot also be received for 18.700 Lecture: MWF11 (26100) Recitation: T9 (E17136) or T10 (E17136, 36112) or T11 (E17136) or T12 (E17136) or T1 (E17139) or T2 (E17139) +final Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to leastsquares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB. Compared with 18.700, more emphasis on matrix algorithms and many applications. Fall: A. Postnikov Spring: G. Strang No textbook information available 18.062J Mathematics for Computer Science
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(Same subject as 6.042J) Prereq: Calculus I (GIR) Units: 507 Lecture: TR2.304 (32123) Recitation: WF10 (36144) or WF1 (34302) or WF2 (34302) or WF3 (34302) or WF11 (36144) or WF12 (36144) or WF1 (36144) or WF2 (36144) or WF3 (36144) or WF10 (36155) or WF11 (36155) or WF12 (34302) or WF4 (34302, 36144) +final Elementary discrete mathematics for computer science and engineering. Emphasis on mathematical definitions and proofs as well as on applicable methods. Topics: formal logic notation, proof methods; induction, wellordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics such as: recursive definition and structural induction; state machines and invariants; recurrences; generating functions. F. T. Leighton, A. R. Meyer, A. Moitra No textbook information available 18.075 Methods for Scientists and Engineers
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Prereq: Calculus II (GIR); 18.03 Units: 309 Credit cannot also be received for 18.04 Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; SturmLiouville theory; partial differential equations; heat equation; and wave equations. H. Cheng 18.085 Computational Science and Engineering I
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Prereq: Calculus II (GIR); 18.03 or 18.034 Units: 309 Lecture: T3,TR10 (TR10 IN 1190) (54100) Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundaryvalue problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications. Fall: G. Strang Spring: L. Demanet No textbook information available 18.086 Computational Science and Engineering II
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Prereq: Calculus II (GIR); 18.03 or 18.034 Units: 309 Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, NavierStokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods. Information: G. Strang 18.089 Review of Mathematics
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Prereq: Permission of instructor Units: 507 Oneweek review of onevariable calculus (18.01), followed by concentrated study covering multivariable calculus (18.02), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances. Information: G. Staffilani No textbook information available 18.094J Teaching CollegeLevel Science and Engineering
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(Same subject as 1.95J, 5.95J, 6.982J, 7.59J, 8.395J) (Subject meets with 2.978) Prereq: None Units: 202 [P/D/F] Lecture: R911 (4149) Participatory seminar focuses on the knowledge and skills necessary for teaching science and engineering in higher education. Topics include theories of adult learning; course development; promoting active learning, problemsolving, and critical thinking in students; communicating with a diverse student body; using educational technology to further learning; lecturing; creating effective tests and assignments; and assessment and evaluation. Students research and present a relevant topic of particular interest. Appropriate for both novices and those with teaching experience. J. Rankin No textbook information available 18.095 Mathematics Lecture Series
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Prereq: Calculus I (GIR) Units: 204 [P/D/F] Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session. Information: G. Staffilani 18.098 Independent Study
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Prereq: Permission of instructor Units arranged [P/D/F] Studies or special individual reading arranged in consultation with individual faculty members and subject to departmental approval. Information: G. Staffilani 18.099 Independent Study
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Prereq: Permission of instructor Units arranged TBA. Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval. Information: G. Staffilani No textbook information available Analysis18.100A Real Analysis
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Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR) Units: 309 Credit cannot also be received for 18.100B, 18.100C Lecture: MWF1 (4163) +final No textbook information available 18.100B Real Analysis
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Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR) Units: 309 Credit cannot also be received for 18.100A, 18.100C Lecture: TR9.3011 (4237) +final No textbook information available 18.100C Real Analysis
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Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR) Units: 4011 Credit cannot also be received for 18.100A, 18.100B Limited enrollment. Lecture: MWF10 (4153) Recitation: F12 (4257) or F3 (4257) +final Three options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. Option A: Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. Option B: More demanding; for students with more mathematical maturity. Places more emphasis on pointset topology and nspace. Option C: 15unit (4011) variant of Option B, with further instruction and practice in written communication. Enrollment limited in Option C. Enrollment limited. Fall: 18.100A: A. P. Mattuck 18.100B: R. B. Melrose 18.100C: Information: R. B. Melrose Spring: 18.100A: R. Bezrukavnikov 18.100B: J.L. Kim 18.100C: Information: R. B. Melrose Textbooks (Fall 2014) 18.101 Analysis and Manifolds
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Prereq: 18.100; 18.06, 18.700, or 18.701 Units: 309 Lecture: MWF11 (E17129) +final Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required. V. W. Guillemin No textbook information available 18.102 Introduction to Functional Analysis
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Prereq: 18.100; 18.06, 18.700, or 18.701 Units: 309 Normed spaces, completeness, functionals, HahnBanach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of Lp spaces. Hilbert space. Compact, HilbertSchmidt and trace class operators. Spectral theorem. R. B. Melrose 18.103 Fourier Analysis: Theory and Applications
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Prereq: 18.100; 18.06, 18.700, or 18.701 Units: 309 Lecture: MWF2 (E17129) Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. L. Guth No textbook information available 18.104 Seminar in Analysis
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Prereq: 18.100 Units: 309 Limited enrollment. Lecture: MWF1 (E17129) Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited. V. W. Guillemin No textbook information available 18.112 Functions of a Complex Variable
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Prereq: 18.100; 18.06, 18.700, or 18.701 Units: 309 Lecture: MWF12 (4149) Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of nonEuclidean geometry. CauchyGoursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions. T. Mrowka No textbook information available 18.116 Riemann Surfaces
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Prereq: 18.112 Units: 309 Riemann surfaces, uniformization, RiemannRoch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory. T. S. Mrowka 18.117 Topics in Several Complex Variables
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Prereq: 18.112, 18.965 Units: 309 Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds. V. W. Guillemin 18.125 Real and Functional Analysis
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Prereq: 18.100 Units: 309 Introductions to set theory and general topology as needed in analysis. Lebesgue's integration theory. Introduction to functional analysis, Banach and Hilbert spaces. D. W. Stroock 18.135 Geometric Analysis
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Prereq: 18.745 or 18.755 Units: 309 A quick description of Riemannian symmetric spaces. Spherical functions and HarishChandra's cfunction. Fourier transforms and Radon transforms on Riemannian symmetric spaces X. Applications to invariant differential equations, in particular the multitemporal wave equation on X. Eigenspace representations. S. Helgason 18.137 Topics in Geometric Partial Differential Equations
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Prereq: Permission of Instructor Units: 309 Lecture: TR9.3011 (E17139) Topics vary from year to year. T. Colding No textbook information available 18.152 Introduction to Partial Differential Equations
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Prereq: 18.100; 18.06, 18.700, or 18.701 Units: 309 Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, realworld examples and applications, such as the BlackScholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems. W. Minicozzi 18.155 Differential Analysis
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Prereq: 18.102 or 18.103 Units: 309 Lecture: TR1112.30 (4159) No textbook information available 18.156 Differential Analysis
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Prereq: 18.155 Units: 309 Fall: Review of Lebesgue integration. L^{p} spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Spring: Variable coefficient elliptic, parabolic and hyperbolic partial differential equations. 18.112 recommended for 18.155. Fall: R. B. Melrose Spring: L. Guth 18.157 Introduction to Microlocal Analysis
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Prereq: 18.155 Units: 309 The semiclassical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semiclassical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject. R. B. Melrose 18.158 Topics in Differential Equations
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Prereq: 18.157 Units: 309 Topics vary from year to year. Information: R. B. Melrose 18.175 Theory of Probability
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Prereq: 18.125 Units: 309 Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. S. Sheffield 18.176 Stochastic Calculus
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Prereq: 18.175 Units: 309 A rigorous introduction to stochastic calculus. Topics include Brownian motion and continuous martingales, diffusions and Levy processes, Ito calculus, martingale representation and quadratic variation, Girsanov's theorem, Bessel processes, general existence and uniqueness theory for stochastic differential equations, applications to partial differential equations, and a brief overview of applications to finance and statistical physics. A. Guionnet 18.177 Topics in Stochastic Processes
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Prereq: 18.175 Units: 309 Lecture: TR12.30 (E17136) Topics vary from year to year. Fall: J. Miller Spring: A. Guionnet No textbook information available 18.199 Graduate Analysis Seminar
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Prereq: Permission of instructor Units: 309 Studies original papers in differential analysis and differential equations. Intended for first and secondyear graduate students. Permission must be secured in advance. V. W. Guillemin 18.238 Geometry and Quantum Field Theory
() Not offered regularly; consult department Prereq: Permission of instructor Units: 309 A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory. Information: P. I. Etingof 18.276 Mathematical Methods in Physics
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Prereq: 18.745 or some familiarity with Lie theory Units: 309 Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects. V. G. Kac Applied Mathematics18.303 Linear Partial Differential Equations: Analysis and Numerics
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Prereq: 18.06 or 18.700 Units: 309 Lecture: MWF1 (4159) Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finitedifference and finiteelement techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. MATLAB is introduced and used in homework for simple examples. S. G. Johnson No textbook information available 18.304 Undergraduate Seminar in Discrete Mathematics
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Prereq: 18.310 or 18.062; 18.06, 18.700, or 18.701; or permission of instructor Units: 309 Credit cannot also be received for 18.316 Limited enrollment. Lecture: MWF2 (E17128) Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited. Fall: P. Csikvari Spring: Information: M. X. Goemans No textbook information available 18.305 Advanced Analytic Methods in Science and Engineering
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Prereq: 18.04, 18.075, or 18.112 Units: 309 Lecture: MWF11 (E17128) Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the twoscale method and the method of renormalized perturbation; singular perturbation and boundarylayer techniques; WKB method on partial differential equations. H. Cheng No textbook information available 18.306 Advanced Partial Differential Equations with Applications
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Prereq: 18.03 or 18.034; 18.04, 18.075, or 18.112 Units: 309 Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, 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. Freeboundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc. R. R. Rosales 18.310 Principles of Discrete Applied Mathematics
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Prereq: Calculus II (GIR) Units: 4011 Credit cannot also be received for 18.310A Limited enrollment. Lecture: MWF12 (4163) Recitation: R10 (E17136) or R12 (E17136) or R1 (E17139) or R3 (E17139) Study of illustrative topics in discrete applied mathematics, including sorting algorithms, probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited. J. Fox, P. W. Shor No textbook information available 18.310A Principles of Discrete Applied Mathematics
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Prereq: Calculus II (GIR) Units: 309 Credit cannot also be received for 18.310 Study of illustrative topics in discrete applied mathematics, including sorting algorithms, probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. M. X. Goemans 18.311 Principles of Continuum Applied Mathematics
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Prereq: Calculus II (GIR); 18.03 or 18.034 Units: 309 Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasiequillibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment. R. R. Rosales 18.312 Algebraic Combinatorics
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Prereq: 18.701 or 18.703 Units: 309 Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks. P. Csikvari 18.314 Combinatorial Analysis
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Prereq: Calculus II (GIR); 18.06, 18.700, or 18.701 Units: 309 Lecture: MWF11 (E17122) +final Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful. R. P. Stanley No textbook information available 18.315 Combinatorial Theory
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Prereq: Permission of instructor Units: 309 Lecture: WF1.303 (E17139) Content varies from year to year. A. Postnikov No textbook information available 18.316 Seminar in Combinatorics
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Prereq: Permission of instructor Units: 309 Credit cannot also be received for 18.304 Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class. J. Fox 18.318 Topics in Combinatorics
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Prereq: Permission of instructor Units: 309 Topics vary from year to year. C. Lee 18.325 Topics in Applied Mathematics
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Prereq: Permission of instructor Units: 309 Topics vary from year to year. L. Demanet 18.330 Introduction to Numerical Analysis
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Prereq: Calculus II (GIR); 18.03 or 18.034 Units: 309 Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in Fortran, C, or MATLAB helpful. H. Reid 18.335J Introduction to Numerical Methods
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(Same subject as 6.337J) Prereq: 18.03 or 18.034; 18.06, 18.700, or 18.701 Units: 309 Advanced introduction to numerical linear algebra and related numerical methods. Topics include direct and iterative methods for linear systems, eigenvalue and QR/SVD factorizations, stability and accuracy, floatingpoint arithmetic, sparse matrices, preconditioning, and the memory considerations underlying modern linear algebra software. Starting from iterative methods for linear systems, explores more general techniques for local and global nonlinear optimization, including quasiNewton methods, trust regions, branchandbound, and multistart algorithms. Also addresses Chebyshev approximation and FFTs. MATLAB is introduced for problem sets. S. G. Johnson No textbook information available 18.336J Fast Methods for Partial Differential and Integral Equations
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(Same subject as 6.335J) Prereq: 6.336, 16.920, 18.085, 18.335, or permission of instructor Units: 309 Lecture: TR9.3011 (E17128) Unified introduction to the theory and practice of modern, near lineartime, numerical methods for largescale partialdifferential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterflybased methods; multiresolution approaches, such as multigrid algorithms and hierarchical lowrank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging. A. Townsend No textbook information available 18.337J Parallel Computing
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(Same subject as 6.338J) Prereq: 18.06, 18.700, or 18.701 Units: 309 Interdisciplinary introduction to parallel computing and modern big data analysis using Julia. Covers scientific computing topics such as dense and sparse linear algebra, Nbody problems, and Fourier transforms, and geometric computing topics such as mesh generation and mesh partitioning. Focuses on application of these techniques to machine learning algorithms in big data applications. Provides direct experience with programming traditionalstyle supercomputing as well as working with modern cloud computing stacks. Designed to separate the realities and myths about the kinds of problems that can be solved on the world's fastest machines. A. Edelman 18.338 Eigenvalues of Random Matrices
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Prereq: 18.701 or permission of instructor Units: 309 Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semicircle and MarcenkoPastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of MATLAB hepful, but not required. A. Edelman 18.352J Theoretical Environmental Analysis
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(Same subject as 12.009J) Prereq: Physics I (GIR), Calculus II (GIR); Coreq: 18.03 Units: 309 Analyzes cooperative processes that shape the natural environment, now and in the geologic past. Emphasizes the development of theoretical models that relate the physical and biological worlds, the comparison of theory to observational data, and associated mathematical methods. Topics include carbon cycle dynamics; ecosystem structure, stability and complexity; mass extinctions; biospheregeosphere coevolution; and climate change. Employs techniques such as stability analysis; scaling; null model construction; time series and network analysis. D. H. Rothman 18.353J Nonlinear Dynamics: Chaos
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(Same subject as 2.050J, 12.006J) Prereq: 18.03 or 18.034; Physics II (GIR) Units: 309 Lecture: TR12.302 (2105) Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207J/18.354J for Nonlinear Dynamics: Continuum Systems. R. Lagrange Textbooks (Fall 2014) 18.354J Nonlinear Dynamics: Continuum Systems
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(Same subject as 12.207J) Prereq: 18.03 or 18.034; Physics II (GIR) Units: 309 General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential 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. J. Dunkel 18.355 Fluid Mechanics
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Prereq: 18.354, 2.25, or 12.800 Units: 309 Lecture: MW23.30 (E17136) Topics include the development of NavierStokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations. J. W. Bush No textbook information available 18.357 Interfacial Phenomena
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Prereq: 18.354, 18.355, 12.800, 2.25, or permission of instructor Units: 309 Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, waterrepellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology. J. W. Bush 18.369 Mathematical Methods in Nanophotonics
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Prereq: 18.305 or permission of instructor Units: 309 Highlevel approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequencydomain computation, perturbation theory, and coupledmode theories. S. G. Johnson 18.376J Wave Propagation
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(Same subject as 1.138J, 2.062J) Prereq: 2.003, 18.075 Units: 309 Theoretical concepts and analysis of wave problems in science and engineering with examples chosen from elasticity, acoustics, geophysics, hydrodynamics, blood flow, nondestructive evaluation, and other applications. Progressive waves, group velocity and dispersion, energy density and transport. Reflection, refraction and transmission of plane waves by an interface. Mode conversion in elastic waves. Rayleigh waves. Waves due to a moving load. Scattering by a twodimensional obstacle. Reciprocity theorems. Parabolic approximation. Waves on the sea surface. Capillarygravity waves. Wave resistance. Radiation of surface waves. Internal waves in stratified fluids. Waves in rotating media. Waves in random media. T. R. Akylas, R. R. Rosales 18.377J Nonlinear Dynamics and Waves
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(Same subject as 1.685J, 2.034J) Prereq: Permission of instructor Units: 309 A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flowstructure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; selfexcited oscillations; lockin phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Kortewegde Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year. T. R. Akylas, R. R. Rosales 18.384 Undergraduate Seminar in Physical Mathematics
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Prereq: 18.311, 18.353, 18.354, or permission of instructor Units: 309 Limited enrollment. Lecture: TR9.3011 (E17133) Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited. P.T. Brun No textbook information available 18.385J Nonlinear Dynamics and Chaos
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(Same subject as 2.036J) Prereq: 18.03 or 18.034 Units: 309 Lecture: TR1112.30 (E17129) Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, PoincareBendixson theory. Floquet theory. Poincare maps. Averaging. Nearequilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment. R. R. Rosales No textbook information available 18.386 Advanced Nonlinear Dynamics and Chaos
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Prereq: 18.385 or permission of instructor Units: 309 Advanced subject on the modern theory of nonlinear dynamical systems with an emphasis on applications in science and engineering. Invariant manifolds, homoclinic orbits, global bifurcations. Hamiltonian systems, completely integrable systems, KAM theory. Different mechanisms for chaotic dynamics, Shilnikovtype orbits, attractors, horseshoes, symbolic dynamics. Geometric singular perturbation theory. Physical applications. Information: R. R. Rosales 18.395 Group Theory with Applications to Physics
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Prereq: 8.321 Units: 309 Selection of topics from the theory of finite groups, Lie groups, and group representations, motivated by quantum mechanics and particle physics. 8.322 and 8.323 helpful. D. Z. Freedman 18.396J Supersymmetric Quantum Field Theories
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(Same subject as 8.831J) Prereq: Permission of instructor Units: 309 Lecture: TR1112.30 (E17128) Topics selected from the following: SUSY algebras and their particle representations; Weyl and Majorana spinors; Lagrangians of basic fourdimensional SUSY theories, both rigid SUSY and supergravity; supermultiplets of fields and superspace methods; renormalization properties, and the nonrenormalization theorem; spontaneous breakdown of SUSY; and phenomenological SUSY theories. Some prior knowledge of Noether's theorem, derivation and use of Feynman rules, lloop renormalization, and gauge theories is essential. D. Z. Freedman No textbook information available 18.398 Quantum Field Theories
() Not offered regularly; consult department Prereq: Permission of instructor Units: 309 For students who want to have a clear understanding of quantum field theories. Appropriate for students who have not taken such a subject as well as students who have but are not entirely comfortable with the basic concepts and techniques. The topics begin with classical mechanics and end with gauge field theories and the renormalization of the standard model. Information: H. Cheng Theoretical Computer Science18.400J Automata, Computability, and Complexity
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(Same subject as 6.045J) Prereq: 6.042 Units: 408 Provides an introduction to some of the central ideas of theoretical computer science, including circuits, finite automata, Turing machines and computability, efficient algorithms and reducibility, the P versus NP problem, NPcompleteness, the power of randomness, cryptography, computational learning theory, and quantum computing. Examines the classes of problems that can and cannot be solved in various computational models. S. Aaronson 18.404J Theory of Computation
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(Same subject as 6.840J) Prereq: 18.310 or 18.062J Units: 408 Lecture: TR2.304 (E25111) Recitation: F11 (E17136) or F12 (E17136) or F1 (E17128) +final A more extensive and theoretical treatment of the material in 6.045J/18.400J, emphasizing computability and computational complexity theory. Regular and contextfree languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. M. Sipser Textbooks (Fall 2014) 18.405J Advanced Complexity Theory
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(Same subject as 6.841J) Prereq: 18.404 Units: 309 Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomialtime hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudorandomness and derandomizations. Interactive proof systems and probabilistically checkable proofs. D. Moshkovitz 18.409 Topics in Theoretical Computer Science
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Prereq: Permission of instructor Units: 309 Lecture: TR2.304 (2105) Study of areas of current interest in theoretical computer science. Topics vary from term to term. Fall: J. A. Kelner Spring: A. Moitra No textbook information available 18.410J Design and Analysis of Algorithms
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(Same subject as 6.046J) Prereq: 6.006 Units: 408 Lecture: TR1112.30 (26100) Recitation: F10 (26302) or F11 (26302) or F12 (24307) or F1 (24307) or F2 (26302) or F3 (26302) or F11 (36156) or F12 (36156) or F1 (26168) or F2 (26168) +final Techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics include sorting; search trees, heaps, and hashing; divideandconquer; dynamic programming; greedy algorithms; amortized analysis; graph algorithms; and shortest paths. Advanced topics may include network flow; computational geometry; numbertheoretic algorithms; polynomial and matrix calculations; caching; and parallel computing. E. Demaine, M. Goemans Textbooks (Fall 2014) 18.415J Advanced Algorithms
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(Same subject as 6.854J) Prereq: 6.041, 6.042, or 18.440; 6.046 Units: 507 Lecture: MWF2.304 (32155) Firstyear graduate subject in algorithms. Emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Surveys a variety of computational models and the algorithms for them. Data structures, network flows, linear programming, computational geometry, approximation algorithms, online algorithms, parallel algorithms, external memory, streaming algorithms. D. R. Karger No textbook information available 18.416J Randomized Algorithms
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(Same subject as 6.856J) Prereq: 6.854J, 6.041 or 6.042J Units: 507 Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms. D. R. Karger 18.417 Introduction to Computational Molecular Biology
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Prereq: 6.01, 6.006, or permission of instructor Units: 309 Introduces the basic computational methods used to model and predict the structure of biomolecules (proteins, DNA, RNA). Covers classical techniques in the field (molecular dynamics, Monte Carlo, dynamic programming) to more recent advances in analyzing and predicting RNA and protein structure, ranging from Hidden Markov Models and 3D lattice models to attribute Grammars and tree Grammars. Information: B. Berger 18.418 Topics in Computational Molecular Biology
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Prereq: 18.417, 6.047, or permission of instructor Units: 309 Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the SIGACT International Conference on Computational Molecular Biology (RECOMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, and biological networks. Recent research by course participants also covered. Participants will be expected to present either group or individual projects to the class. B. Berger 18.424 Seminar in Information Theory
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Prereq: 18.05, 18.440, or 6.041; 18.06, 18.700, or 18.701 Units: 309 Considers various topics in information theory, including data compression, Shannon's Theorems, and errorcorrecting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited. P. W. Shor 18.425J Cryptography and Cryptanalysis
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(Same subject as 6.875J) Prereq: 6.046J Units: 309 A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives of publickey encryption, digital signatures, pseudorandom number generation, and basic protocols and their computational complexity requirements. S. Goldwasser, S. Micali 18.426J Advanced Topics in Cryptography
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(Same subject as 6.876J) Prereq: 6.875 Units: 309 TBA. Recent results in cryptography, interactive proofs, and cryptographic game theory. Lectures by instructor, invited speakers, and students. S. Goldwasser, S. Micali No textbook information available 18.433 Combinatorial Optimization
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Prereq: 18.06, 18.700, or 18.701 Units: 309 Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NPhard optimization problems. Prior exposure to discrete mathematics (such as 18.310) helpful. M. X. Goemans 18.434 Seminar in Theoretical Computer Science
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Prereq: 18.410 Units: 309 Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited. R. Peng 18.435J Quantum Computation
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(Same subject as 2.111J, 8.370J) Prereq: Permission of instructor Units: 309 Lecture: TR12.30 (56114) +final Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required. I. Chuang, E. Farhi, S. Lloyd, P. Shor No textbook information available 18.436J Quantum Information Science
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(Same subject as 6.443J, 8.371J) Prereq: 18.435 Units: 309 Examines quantum computation and quantum information. Topics include quantum circuits, the quantum Fourier transform and search algorithms, the quantum operations formalism, quantum error correction, CalderbankShorSteane and stabilizer codes, fault tolerant quantum computation, quantum data compression, quantum entanglement, capacity of quantum channels, and quantum cryptography and the proof of its security. Prior knowledge of quantum mechanics required. Information: P. W. Shor 18.437J Distributed Algorithms
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(Same subject as 6.852J) Prereq: 6.046 Units: 309 Lecture: TR1112.30 (4237) Design and analysis of concurrent algorithms, emphasizing those suitable for use in distributed networks. Process synchronization, allocation of computational resources, distributed consensus, distributed graph algorithms, election of a leader in a network, distributed termination, deadlock detection, concurrency control, communication, and clock synchronization. Special consideration given to issues of efficiency and fault tolerance. Formal models and proof methods for distributed computation. N. A. Lynch Textbooks (Fall 2014) 18.438 Advanced Combinatorial Optimization
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Prereq: 18.433 or permission of instructor Units: 309 Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Nonbipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments. M. X. Goemans Probability and Statistics18.440 Probability and Random Variables
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Prereq: Calculus II (GIR) Units: 309 Credit cannot also be received for 6.041, 6.431 Lecture: MWF10 (54100) +final Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Fall: A. Guionnet Spring: J. A. Kelner No textbook information available 18.443 Statistics for Applications
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Prereq: 18.440 or 6.041 Units: 309 Lecture: TR9.3011 (4270) A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chisquare tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics. Fall: L. Wang Spring: R. M. Dudley No textbook information available 18.445 Introduction to Stochastic Processes
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Prereq: 18.440 or 6.041 Units: 309 Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion. L. Wang 18.465 Topics in Statistics
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Prereq: Permission of instructor Units: 309 Lecture: MWF11 (E17133) Topics vary from term to term. R. M. Dudley No textbook information available 18.466 Mathematical Statistics
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Prereq: Permission of instructor Units: 309 Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis. L. Wang 18.472 Topics in Mathematics with Applications in Finance
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