Course 18 Mathematics General Mathematics 18.UR Undergraduate Research -------------------------------------------------------- Prereq.: -- U (1, 2) Units arranged [P/F] -------------------------------------------------------- Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the Departmental Coordinator. A.Nadim 18.01 Calculus -------------------------------------------------------- Prereq.: -- U (1, 2) 5-0-7 -------------------------------------------------------- Differentiation and integration of functions of one variable, with applications. Concepts of function, limits, and continuity. Differentiation rules, application to graphing, rates, approximations, and extremum problems. Mean-value theorem. Definite and indefinite integration. Fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Approximation of definite integrals, improper integrals, and l'Hpital's rule. F. P.Peterson 18.011 Calculus -------------------------------------------------------- Prereq.: Assumes substantial prior knowledge of calculus U (1) 5-0-7 -------------------------------------------------------- Calculus of one variable, emphasizing applications. Quick review of differentiation, followed by intensive study of integration and infinite series, including, as time permits, special topics selected from: perturbation and iteration procedures, stability, summation techniques, asymptotics, numerical analysis, and other techniques. Practice in mathematical formulation of scientific problems and approximate methods of solution. D. J. Benney 18.012 Calculus with Theory -------------------------------------------------------- Prereq.: -- U (1) 5-0-7 -------------------------------------------------------- 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. D. McDoniel 18.02 Calculus -------------------------------------------------------- Prereq.: 18.01 or 18.011 or 18.012 U (1, 2) 5-0-7 -------------------------------------------------------- Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, approximation techniques. Multiple integrals with applications. Vector fields, line and surface integrals, exact differentials, Green's theorem, Divergence Theorem, Stokes's Theorem. Additional topics: linear algebra (term 1), infinite series (term 2). Term 1: H. Rogers, Jr. Term 2: H. Cheng 18.021 Calculus -------------------------------------------------------- Prereq.: 18.01 or 18.011 U (1, 2) 5-0-7 -------------------------------------------------------- Continues 18.011. Calculus of several variables. Vector algebra, analytic geometry, planetary motion, orbit stability, partial differentiation, functions of several variables. Taylor series, extremal problems, linear programming examples, numerical methods, multiple integrals, approximate and asymptotic methods of evaluation, applications, vector calculus, gradient, curl, theorems of Stokes, Green, and Gauss, conservation laws, fluid motion. Term 1: H. P.Greenspan Term 2: A. Nadim 18.022 Calculus with Theory -------------------------------------------------------- Prereq.: 18.012 U (2) 5-0-7 -------------------------------------------------------- Continues 18.012. 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. D. McDoniel 18.03 Differential Equations -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.012 U (1, 2, S) 4-0-8 SCI DIST -------------------------------------------------------- Examples of initial-value problems in science and engineering associated with single equations and systems of first-order equations. Methods of solution include graphical constructions, series, Laplace transforms, matrices, numerical integration and the phase plane. Emphasizes formulation of natural phenomena in terms of differential equations and interpretation of the solutions. Term 1: D. Jerison, Z. Zhou Term 2: A. P.Mattuck 18.032 Differential Equations (New) -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.012 U (2) 4-0-8 SCI DIST -------------------------------------------------------- Covers essentially the same material as 18.03 with more emphasis on theory. First order equations, separation, initial value problems. Systems, linear equations, independence of solutions, undetermined coefficients. Singular points and periodic orbits for planar systems. Information: D. S. Jerison. 18.04 Complex Variables with Applications -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (1, 2) 4-0-8 -------------------------------------------------------- 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 and Laplace transforms. 18.04 and 18.075 may not both be taken for credit. Term 1: S. H. Strogatz Term 2: A. Toomre 18.05 Introduction to Probability and Statistics -------------------------------------------------------- Prereq.: 18.01 or 18.011 or 18.012 U (1, 2) 4-0-8 SCI DIST -------------------------------------------------------- Elementary introduction, with applications to the life sciences. Descriptive statistics. Relative frequency. Probability models. Combinatorics. Binomial, geometric, hypergeometric, and Poisson experiments. Random variables. Estimation. Hypothesis testing. Confidence regions. Normal distribution methods. Term 1: R. M.Dudley Term 2: W. Olbricht 18.06 Linear Algebra -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1, 2, S) 4-0-8 SCI DIST -------------------------------------------------------- Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, positive definite matrices. Applications to Gauss elimination with pivoting, least-squares approximations, stability of differential equations, linear programming, and game theory. Compared with 18.710, more emphasis on matrix calculations and applications. Information: G. Strang. 18.063 Introduction to Algebraic Systems -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1) 4-0-8 SCI DIST -------------------------------------------------------- Introduction to algebraic systems, primarily for students interested in computer and information sciences, with emphasis on finite systems. Reviews elementary set theory, natural numbers, modular arithmetic, induction, counting arguments. Elementary number theory and group theory. Applications to fast arithmetic, cryptography, combinatorics. Elementary graph theory. Introduction to rings and fields. Finite fields: coding theory, Hamming and BCH codes. Information: A. P. Mattuck. 18.075 Advanced Calculus for Engineers (A except II, VI, VIII, XII, XIII, XVI, XVIII, XXII) -------------------------------------------------------- Prereq.: 18.03 or 18.032 G (1, 2, S) 3-0-9 -------------------------------------------------------- Functions of a complex variable; calculus of residues. Ordinary differential equations; integration by power series; Bessel and Legendre functions. Expansion in series of orthogonal functions, including Fourier series. 18.075 and 18.04 may not both be taken for credit. Information: D. S. Jerison. 18.076 Advanced Calculus for Engineers (A except II, VI, XVI, XVIII, XXII) -------------------------------------------------------- Prereq.: 18.075 G (1, 2, S) 3-0-9 -------------------------------------------------------- Vector analysis: orthogonal curvilinear coordinates. Calculus of variations. Solution of classical partial differential equations of mathematical physics, including applications of conformal mapping and the Laplace transformation. Information: D. S. Jerison. 18.085 Mathematical Methods for Engineers I (A) -------------------------------------------------------- Prereq.: 18.03 or 18.032 G (1, 2, S) 3-0-9 -------------------------------------------------------- Review of linear algebra, applications to networks, structures, and estimation, Lagrange multipliers, differential equations of equilibrium, Laplace's equation and potential flow, boundary-value problems, minimum principles and calculus of variations, Fourier series, discrete Fourier transform, convolution, applications. Term 1: G. Strang Term 2: S. H. Strogatz 18.086 Mathematical Methods for Engineers II (A) -------------------------------------------------------- Prereq.: 18.03 or 18.032 G (1, 2, S) 3-0-9 -------------------------------------------------------- Scientific computing: Fast Fourier Transform, finite differences, finite elements, spectral method, numerical linear algebra. Complex variables and applications. Initial-value problems: stability or chaos in ordinary differential equations, wave equation vs heat equation, conservation laws and shocks, dissipation and dispersion. Optimization: network flows, linear programming, simplex vs Karmarkar. Information: G. Strang. 18.089 Review of Mathematics -------------------------------------------------------- Prereq.: -- G (S) Units arranged -------------------------------------------------------- Reviews calculus and differential equations. Primarily for students in Course XIII-A. Degree credit allowed only in special circumstances. Information: D. S. Jerison. 18.091 Calculus Workshop (New) -------------------------------------------------------- Prereq.: Permission of Instructor U (1) Units arranged [P/F] -------------------------------------------------------- Workshop presenting material supplementing 18.01. Limited enrollment. Information: D. S. Jerison. 18.092 Calculus Workshop (New) -------------------------------------------------------- Prereq.: Permission of Instructor U (2) Units arranged [P/F] -------------------------------------------------------- Workshop presenting material supplementing 18.02. Limited enrollment. Information: D. S. Jerison. 18.093 Tutoring in Mathematics -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1, 2) Units arranged [P/F] -------------------------------------------------------- For undergraduates who are teaching mathematics recitation. Limited enrollment, based on positions available. Permission must be secured in advance to register for this subject. Information: D. S. Jerison. 18.097 Writing Mathematics -------------------------------------------------------- Prereq.: 18.02, 18.03 U (1, 2) 3-0-3 -------------------------------------------------------- Seminar on writing mathematics, designed to help math majors satisfy Phase II of the Institute Writing Requirement. For majors with little or no experience in technical writing. Mathematical content geared to sophomore/junior level. A satisfactory final paper may be submitted to fulfill Phase II of the Writing Requirement. In-class hours will vary from term to term. Information: S. Kleiman. 18.098 Mathematics Lecture Series (New) -------------------------------------------------------- Prereq.: 18.01 or equivalent U (IAP) 2-0-4 [P/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. May be repeated for credit. Information: M. Haiman. 18.099 Independent Activities -------------------------------------------------------- Prereq.: -- U (1, IAP, 2) Units arranged -------------------------------------------------------- For undergraduates desiring credit for studies during IAP or for special individual reading on an undergraduate level during the regular terms. Specific programs and credit arranged in consultation with individual faculty members and subject to departmental approval. J. R. MunkresAnalysis 18.100 Analysis I (A except XVIII) -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (1, 2) 3-0-9 -------------------------------------------------------- Two options offered, both covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Both options show the utility of abstract concepts and teach understanding and construction of proofs. Option A chooses less abstractdefinitions and proofs, and gives applications where possible.Option B is more abstract and for students with moremathematical maturity. Places greater emphasis on point-set topology.Information: R. B. Melrose. 18.101 Analysis II (A except XVIII) -------------------------------------------------------- Prereq.: 18.100, 18.701 or 18.710 U (1) 3-0-9 -------------------------------------------------------- Continues 18.100, in the direction of manifolds and global analysis. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. 18.901 helpful but not required. J. R. Munkres 18.103 Fourier Analysis--Theory and Applications (A except XVIII) -------------------------------------------------------- Prereq.: 18.100 U (2) 3-0-9 -------------------------------------------------------- Continues 18.100. Roughly half the subject devoted to the theory of the Lebesgue integral and half to Fourier series and Fourier integrals. E. Getzler 18.104 Seminar in Analysis -------------------------------------------------------- Prereq.: 18.100 Acad Year 1990-91: Not offered Acad Year 1991-92: U (1) 3-0-9 -------------------------------------------------------- Seminar for mathematics majors. Students present and discuss the subject matter, taken from current journals or books. Topics vary from year to year. Information: R. B. Melrose. 18.115 Functions of a Complex Variable (A) -------------------------------------------------------- Prereq.: 18.100 G (1) 3-0-9 -------------------------------------------------------- Exponential and trigonometric functions, Cauchy integral formula, holomorphic and meromorphic functions. Infinite series and products, the gamma function. Harmonic functions, conformal mapping, Dirichlet's problem. S. Helgason 18.116 Topics in Complex Variables (A) -------------------------------------------------------- Prereq.: 18.115 G (2) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Typical topics: introduction to Riemann Surface Theory, function-theoretic and geometric approaches to Teichmller theory. S. K. Yeung 18.117 Topics in Several Complex Variables (A) -------------------------------------------------------- Prereq.: 18.115, 18.125 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Pseudoconvex domains and plurisubharmonic functions. Complete metrics and inversion of the Laplacian. Existence and approximation theorems for holomorphic functions via L2- estimates for the -operator. R. B. Melrose 18.125 Measure and Integration (A) -------------------------------------------------------- Prereq.: 18.100 G (1) 3-0-9 -------------------------------------------------------- Basic set theory and general topology (review and extension). General theory of Lebesgue integration. The basic theorems of Tonelli-Fubini, Radon-Nikodym, Riesz-Fischer, et al.; Banach, Lp and Hilbert spaces. R. M. Dudley 18.126 Functional Analysis (A) -------------------------------------------------------- Prereq.: 18.125 G (2) 3-0-9 -------------------------------------------------------- General theory of Hilbert and Banach spaces. Examples, including Sobolev spaces and Lp. The Fourier transform. Boundedness and compactness of operators. Spectral theory for self-adjoint operators. Applications to linear partial differential equations. Z. Zhou 18.128 Geometric Measure Theory (A) -------------------------------------------------------- Prereq.: 18.125 G (2) 3-0-9 -------------------------------------------------------- Hausdorff measure, rectifiable sets, structure theory, and the co-area formula. D. W. Stroock 18.135 Geometric Analysis (A) -------------------------------------------------------- Prereq.: 18.125 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Harmonic analysis on Rn. Spherical harmonics. Non-Euclidean Fourier analysis. Paley-Wiener type theorems, group-theoretic potential theory. Eigenfunctions, entire functionals, and hyperfunctions. Radon transforms and applications. S.Helgason 18.152 Introduction to Differential Equations -------------------------------------------------------- Prereq.: 18.100 Acad Year 1990-91: U (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Cauchy problem for ordinary differential equations. Integration of vector fields in the plane. The wave equation and Riemann function. Dirichlet problem for Laplace's operator. Information: R. B. Melrose. 18.155 Distributions and Differential Equations (A) -------------------------------------------------------- Prereq.: 18.103 G (1) 3-0-9 -------------------------------------------------------- Treats the basic theory of distributions with applications to linear partial differential equations. Fourier transform, temperate distributions, Sobolev spaces, and constant coefficient operators. Convolution, fundamental solutions, and the Malgrange-Ehrenpreis theorem. H. Smith 18.156 Introduction to Microlocal Analysis (A) -------------------------------------------------------- Prereq.: 18.155, 18.965 Acad Year 1990-91: G (1) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Examines singularities of distributions. Distributions singular across a submanifold, singular points of ordinary differential equations, noncharacteristic boundary-value problems. Pseudodifferential operators, regularity of elliptic differential operators, wavefront set, and microdistributions. Darboux's theorem. Hamilton-Jacobi theory, the Maslov bundle. Lagrangian distributions, Fourier integral operators, and the Cauchy problem for hyperbolic equations. R. B. Melrose 18.157 Partial Differential Equations (A) -------------------------------------------------------- Prereq.: 18.155, 18.156 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Examines general classes of operators and problems in the theory of linear partial differential operators. Boundary-value problems for elliptic operators, hypoelliptic operators with double characteristics. Operators of real principal type. Uniqueness for the Cauchy problem. Spectral theory. Information: R. B. Melrose. 18.158 Topics in Differential Equations (A) -------------------------------------------------------- Prereq.: 18.125 Acad Year 1990-91: G (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. Topics: Rings of pseudodifferential operators on manifolds with corners, singular limits in geometric problems. R. B.Melrose 18.175 Theory of Probability (A) -------------------------------------------------------- Prereq.: 18.125 G (2) 3-0-9 -------------------------------------------------------- Ergodic theorems, laws of large numbers, convergence of probability measures, central limit theorems, stochastic processes, Brownian motion, martingales, strong Markov properties. R. M.Dudley 18.177 Stochastic Processes (A) -------------------------------------------------------- Prereq.: 18.175 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Topics in stochastic processes, such as Gaussian, Markov, diffusion and empirical processes. Content varies from year to year; may be repeated for credit. D. W. Stroock 18.199 Graduate Analysis Seminar (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (2) 3-0-21 -------------------------------------------------------- Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance. R. B. Melrose 18.238 Geometry and Quantum Field Theory (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. I. M. Singer 18.248 String Theory (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1) 3-0-9 -------------------------------------------------------- Topics include representation theory of infinite dimensional algebras of geometric origin, semi-infinite cohomology, differential equations on modular spaces, braid and Teichmller group representations. Some familiarity with elementary algebraic geometry or representation theory is helpful. Topics vary from year to year; may be repeated for credit. A. A. Beilinson 18.276 Mathematical Methods in Physics (A) -------------------------------------------------------- Prereq.: -- G (2) 3-0-9 -------------------------------------------------------- Recent developments in physics require mathematical techniques not usually covered in standard graduate subjects; e.g., Kaehlerian geometry, Kac-Moody algebras, Teichmller theory, and the theory of Bieberbach groups. Topic for 1990-91: spectral problems associated with periodic lattices in Rn and applications to solid state physics. Topics for following terms: string theory, Yang-Mills and the topology of four-manifolds. Content varies from year to year; may be repeated for credit. V. W.Guillemin 18.284 Introduction to Functions of a Complex Variable (A except XVIII) -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (1) 3-0-9 -------------------------------------------------------- A deeper and more extensive treatment of complex variables than 18.04, with more challenging problems. Mathematical rigor is, however, not stressed. Branch points and branch cuts. Cauchy's theorem, singularities, residues, contour integrals, conformal mapping. Schwarz-Christoffel transformation, analytic continuation, harmonic function, the Mittag-Leffler theorem. H. Cheng 18.295J General Relativity (A) -------------------------------------------------------- (Same subject as 8.962J) Prereq.: 18.06, 8.06, 8.312 G (2) 3-0-9 -------------------------------------------------------- The physical background and mathematical formulation of the general theory of relativity. Begins with tensor calculus, differential forms, and Riemannian geometry. Extensive discussion of stellar and black hole solutions. Finishes with the Penrose process, black hole thermodynamics, and some cosmology. N. P. WarnerApplied Mathematics 18.301 Introduction to Physical Mathematics I -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (1) 3-0-9 -------------------------------------------------------- Discussion of mathematical techniques motivated by and applied to scientific problems. Particle mechanics and differential equations. Random walk and diffusion equation. Stability analysis. Linearization. Perturbation theory, regular, Poincar, multiple scales. Phase plane, energy methods, limit cycles. Floquet theory. Separation of variables, heat and Laplace's equation. Orthogonal expansion. Fourier analysis. 18.04 or 18.284 helpful. R. R.Rosales 18.302 Introduction to Physical Mathematics II -------------------------------------------------------- Prereq.: 18.301, 18.04 or 18.284 U (2) 3-0-9 -------------------------------------------------------- Formulation and classification of partial differential equations of mathematical physics and continuum mechanics. Solution of the latter equations by separation of variables, Green's functions, integral transform methods, and perturbation techniques. Sturm-Liouville eigenvalue problems. Special functions. Variational problems. R. R. Rosales 18.305 Methods of Applied Mathematics I (A) -------------------------------------------------------- Prereq.: 18.04 or 18.075 or 18.284 or 18.302 G (1) 3-0-9 18.306 Methods of Applied Mathematics II (A) -------------------------------------------------------- Prereq.: 18.04 or 18.075 or 18.284 or 18.302 G (2) 3-0-9 -------------------------------------------------------- A comprehensive treatment of the advanced methods of applied mathematics. Term 1: asymptotic behavior of ordinary differential and difference equations; asymptotic evaluation of integrals; regular and singular perturbation methods; boundary-layer techniques; WKB method; multiple scales. Term 2: partial differential equations; transform methods; characteristics, initial and boundary-value problems; Green's functions; singular perturbation problems; nonlinear wave propagation. 18.305: A.Nadim 18.306: H. Cheng 18.307 Methods of Applied Mathematics III (A) -------------------------------------------------------- Prereq.: 18.04 or 18.075 or 18.284 or 18.302 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Selection of material from the following topics: calculus of variations (the first variation and the second variation); integral equations (Volterra equations; Fredholm equations, the Hilbert-Schmidt theorem); the Hilbert Problem and singular integral equations of Cauchy type; Wiener-Hopf Method and partial differential equations; Wiener-Hopf Method and integral equations; group theory. Information: H. Cheng. 18.308 Wave Motion (A) -------------------------------------------------------- Prereq.: 18.305 G (2) 3-0-9 -------------------------------------------------------- Topics selected from the following: geometrical optics, linear and weakly nonlinear. The particle-wave duality. Geometrical shock dynamics. Modulation theory; formal point of view, the average Lagrangian, group velocity, wave action, and average conservation laws. Multiple phases, small divisor problem, resonant interactions, integro-differential equations. Inverse Scattering Transform, solitons, small dispersion limit. May be repeated for credit. R.R. Rosales 18.310 Principles of Applied Mathematics -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1) 3-0-9 -------------------------------------------------------- Introductory survey of fundamental concepts in applied mathematics: optimization, random process, coding, computer algorithms. This independent half of the complete sequence emphasizes the ideas and topics that relate to a "discrete mathematical approach: computation, combinatorics, probability, linear programming. D. J. Kleitman 18.311 Principles of Applied Mathematics -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (2) 3-0-9 -------------------------------------------------------- Introductory survey of fundamental concepts in applied mathematics: propagation, stability, equilibrium, optimization. This independent half of the complete sequence emphasizes the ideas and topics that relate to a "continuous mathematical approach: diffusion, waves, instabilities, characteristics, and first-order partial differential equations, with applications to traffic problems, fluid flow, and other problems in classical mathematical physics. A. Toomre 18.313 Probability -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (2) 4-0-8 SCI DIST -------------------------------------------------------- Development of theory and applications of probabilistic concepts for scientists and engineers. Emphasizes formulation and solution of probabilistic problems by the algebra of random variables. Topics: sample space, Bernoulli and Poisson processes, uniform process, generating functions and Laplace transforms, discrete and continuous-parameter Markov chains. Introduces the Central Limit Theorem and the foundations of probability. 18.313 and 18.440 may not both be taken for credit. G.-C. Rota 18.314 Combinatorial Analysis -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (2) 3-0-9 -------------------------------------------------------- Combinatorial problems and methods for their solution. Emphasizes problems of enumeration, generating function techniques, and construction of bijections. Additional topics drawn from graph theory, matchings and network flows, partial orders, permutation groups and Polya theory. Prior experience with abstraction and proofs required. R. P. Stanley 18.315 Combinatorial Theory (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1) 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. Topics in the past have included enumeration, generating functions, partially ordered sets, Mbius functions, incidence geometries matroids, matching theory, Ramsey theory, graphs. G.-C. Rota 18.316 Seminar in Combinatorics (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (2) 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. Readings from current research papers in combinatorics. Topic to be chosen and presented by the class. D. J. Kleitman 18.318 Topics in Combinatorics (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (2) 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. 1990-1991: Topics chosen from applications of algebra to combinatorics; matroids; umbral calculus. G.-C. Rota 18.325 Topics in Applied Mathematics (A) -------------------------------------------------------- Prereq.: -- G (1) 3-0-9 -------------------------------------------------------- Nonlinear oscillations and chaos, with applications to physics, engineering, and biology. Emphasizes analytical methods and concrete examples. Students solve difference and differential equations on the computer. Topics: phase plane; bifurcation theory; Lyapunov stability theory; nonlinear oscillators; coupled oscillators in biology and physics; chaos: iterated mappings; period doubling; Lorenz model; driven pendulum; Melnikov's method; fractal basin boundaries. Permission of instructor required for undergraduates. S. H. Strogatz 18.330 Introduction to Numerical Analysis -------------------------------------------------------- Prereq.: 18.03 or 18.032 U (2) 3-0-9 -------------------------------------------------------- Introduces basic techniques for efficient solution of numerical problems in science and engineering. Root finding, integration, function approximations, differential equations, direct and iterative methods in matrix theory, analysis of numerical stability. D. S. Henningson 18.335 Numerical Methods of Applied Mathematics I (A) -------------------------------------------------------- Prereq.: 18.06 G (1) 3-0-9 18.336 Numerical Methods of Applied Mathematics II (A) -------------------------------------------------------- Prereq.: 18.302 G (2) 3-0-9 -------------------------------------------------------- Advanced introduction to theory and application of numerical methods. Term 1: Fundamental methods for various problems including linear equations, quadrature, nonlinear equations, matrix eigenvalues, Fourier transforms, and ordinary differential equations. Term 2: numerical solution of differential equations, especially of time-dependent partial differential equations by finite-difference and spectral methods, together with the associated theory of accuracy, stability, and convergence. 18.335 A. Toomre 18.336 L.Van Dommelen 18.337 Topics in Numerical Analysis (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Reading and problem solving on a topic in numerical analysis not normally covered by 18.335 or 18.336, to be agreed upon by students and instructor. Possibilities include: optimization and nonlinear equations, approximation theory, sparse matrices, parallel algorithms, boundary-value problems, finite-element methods, numerical methods in complex analysis. Topic varies from term to term; may be repeated for credit. L. N. Trefethen 18.350 Experiments in Fluid Mechanics -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: U (2) 3-6-3 -------------------------------------------------------- A "get your hands wet approach to fluid mechanics for the theoretically inclined in which students, assisted by laboratory staff, set up and carry out a number of exploratory and diagnostic experiments. These fairly simple demonstrations of basic fluid processes and phenomena are designed to motivate further study by providing the intuition and experience that underlie theoretical models, approximations, and analyses. Enrollment is limited. H.P. Greenspan 18.354 Fluid Mechanics -------------------------------------------------------- Prereq.: 18.04 or 18.075 or 18.302 U (1) 3-0-9 18.355 Fluid Mechanics (A) -------------------------------------------------------- Prereq.: 18.354 G (2) 3-0-9 -------------------------------------------------------- A study of the basic concepts of fluid dynamics: conservation laws of mass, momentum, and energy; equation of state; vorticity and circulation theorems; boundary-layer theory; instability and transition; waves; compressible flows and shocks; convection. Multiphase fluids, and other selected topics of current research interest. 18.354: D. S. Henningson 18.355: W. V. R.Malkus 18.356 Rotating Fluids (A) -------------------------------------------------------- Prereq.: 18.305, 18.354 G (2) 3-0-9 -------------------------------------------------------- General theory of rotating fluids; transient flows; effects of viscosity, stratification, compressibility and nonlinear interactions; wave motion and stability theory; multiphase flows and centrifugal separation of mixtures. Application to laboratory, technological, and geophysical problems. H. P. Greenspan 18.357 Seminar in Fluid Dynamics (A) -------------------------------------------------------- Prereq.: 18.355 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Nonlinear phenomena in fluid flow, especially stability and wave mechanics. Current topics of research interest. Emphasis varies from year to year. May be repeated for credit. D. J.Benney 18.358 Hydrodynamic Stability and Turbulence (A) -------------------------------------------------------- Prereq.: 18.354 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Linear stability theory of incompressible and compressible flows. Nonlinear stability theory; modes of transition, the advent of aperiodicity. Upper-bound and statistical theories of turbulence. Statistical stability and the turbulent transport of heat and momentum. Properties of convection and shear turbulence. W. V. R.Malkus 18.359 Hydrodynamics of Fluid-Particle Systems (A) -------------------------------------------------------- Prereq.: 18.354 G (1) 3-0-9 -------------------------------------------------------- Hydrodynamics of liquids containing particles, droplets, or bubbles. Topics drawn from the following list: motion of particles and droplets in Stokes flow, translational and rotational Brownian motion, rheology of suspensions, gravitational and centrifugal sedimentation, multiphase flow theory and mixture models, coarse graining methods, macromolecular hydrodynamics and viscoelasticity, colloidal and interfacial phenomena. A. Nadim 18.375 Dynamics of Galaxies (A) -------------------------------------------------------- Prereq.: 8.06, 18.076, or 18.302 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Types and main properties of galaxies. Orbits, equilibria, and instabilities of large aggregates of stars. Wave mechanics in disks: density waves, swing amplification, global modes, resonances, bars; bending waves, forced warps. Evolution of groups: tidal encounters, orbital decay, mergers, refueling the central engines. Effects of dark halos. A. Toomre 18.395 Group Theory with Applications to Physics (A) -------------------------------------------------------- Prereq.: 18.302 or 18.305 or 8.321 G (1) 3-0-9 -------------------------------------------------------- Selection of topics from the theory of finite groups, Lie groups, and group representations, presented with some applications to quantum mechanics and particle physics. D. Z. Freedman 18.396 Topics in Theoretical Physics (A) -------------------------------------------------------- Prereq.: 8.20, 6.017 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Emphasizes major points of theoretical physics. For students who want to understand fully the fundamentals, not just to know formulae and terminology. No advanced background beyond quantum mechanics and special relativity required. Topics include: action principle in classical mechanics, classical fields, symmetry and groups, canonical quantization, path integrals, gauge field theories, quantum electrodynamics, quantum chromodynamics, weak interaction and the Weinberg-Salam model, renormalization, and the theory of gravitation. H. ChengTheoretical Computer Science 18.400J Automata, Computability, and Complexity -------------------------------------------------------- (Same subject as 6.045J) Prereq.: 18.063 or 18.310 U (2) 4-0-8 -------------------------------------------------------- See description under subject 6.045J. N. A. Lynch 18.404J Theory of Computation (A except XVIII) -------------------------------------------------------- (Same subject as 6.840J) Prereq.: 18.063 or 18.310 U (1) 4-0-8 -------------------------------------------------------- A more extensive and theoretical treatment of the material in 6.045J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory, Kolmogorov complexity. Time and spare measures on computation, completeness, hierarchy theorems, inherently complex problems. M.Sipser 18.405J Advanced Complexity Theory (A) -------------------------------------------------------- (Same subject as 6.841J) Prereq.: 6.840J/18.404J G (2) 3-0-9 -------------------------------------------------------- Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Relativization. Definitions of randomness. Approaches to the P = NP? and related questions. M.Karchmer 18.406 Concrete Complexity Theory (A) (New) -------------------------------------------------------- Prereq.: 18.404J/6.840J Acad Year 1990-91: G (1) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- A combinatorial treatment of complexity theory through concrete models of computation. Topics include communication complexity, decision trees, branching programs, algebraic computation, time-space tradeoffs, randomness as a resource. Alternate years. M. Karchmer 18.409 Topics in Theoretical Computer Science (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Study of an area of current interest in theoretical computer science. Topic varies from term to term; may be repeated for credit. Information: F. T. Leighton. 18.410J Introduction to Algorithms -------------------------------------------------------- (Same subject as 6.046J) Prereq.: 6.001, 18.063 or 18.310 U (1, 2) 4-0-8 -------------------------------------------------------- See description under subject 6.046J. C. E. Leiserson 18.414J Theory of Algorithms (A except XVIII) -------------------------------------------------------- (Same subject as 6.851J) Prereq.: 18.06 or 18.710, 18.063 or 18.310 U (2) 3-0-9 -------------------------------------------------------- Techniques for design and analysis of algorithms, emphasizing mathematical methods and proofs. Proof-oriented version of 6.046J/ 18.410J. Topics: Data structures, sorting, selection, hashing. Solving recurrences. Upper and lower bounds. Dynamic programming. Divide and conquer. Graph algorithms: spanning trees, matching, shortest paths, max flow. Matrix operations. Fast Fourier transform. Integer and polynomial arithmetic. Permutation group membership. Primality testing. Linear programming. Parallel algorithms. R. L.Rivest 18.415J Advanced Algorithms (A) -------------------------------------------------------- (Same subject as 6.854J) Prereq.: 18.414J/6.851J, 18.06 or 18.710 G (1) 3-0-9 -------------------------------------------------------- Continuation of 18.414J/6.851J, emphasizing fundamental algorithms and advanced methods of algorithmic design and analysis. Advanced graph algorithms (matching, network flow, and the traveling salesman problems). Linear programming. Basis reduction, integer programming, polynomial factorization, diophantine approximation. M. Goemans 18.419 Seminar in Theoretical Computer Science (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (2) 3-0-9 -------------------------------------------------------- A seminar on advanced topics in theoretical computer science. Current literature presented by students and instructors with a view toward preparing students for research in theoretical computer science, and for developing the skills needed to present such results effectively. May be repeated for credit. M. Sipser 18.421J Algorithmic Algebra and Number Theory -------------------------------------------------------- (Same subject as 6.047J) Prereq.: 18.06 or 18.710, 18.063 or 18.310 or 18.703 Acad Year 1990-91: U (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Emphasis on constructing efficient algorithms for classical problems in algebra and number theory. Integer and polynomial GCD computation, modular arithmetic, Chinese remainder theorem, Jacobi symbol computation, primality testing, extracting square roots mod primes, integral lattices, factorization of polynomials over the rationals, simultaneous diophantine approximations, solving binary quadratic and cubic modular equations, application to public-key cryptography. Information: H. Rogers, Jr. 18.423J Computability, Logic, and Programming -------------------------------------------------------- (Same subject as 6.044J) Prereq.: 18.063 or 18.310 Acad Year 1990-91: Not offered Acad Year 1991-92: U (1) 3-0-9 -------------------------------------------------------- See description under subject 6.044J. A. R. Meyer 18.425J Cryptography and Cryptanalysis (A) -------------------------------------------------------- (Same subject as 6.875J) Prereq.: 6.046J/18.410J or 6.851J/18.414J or 6.047J/18.421J G (2) 3-0-9 -------------------------------------------------------- See description under subject 6.875J. S. Micali 18.426J Advanced Topics in Cryptography (A) -------------------------------------------------------- (Same subject as 6.876J) Prereq.: Permission of Instructor G (1) 3-0-9 -------------------------------------------------------- See description under subject 6.876J. S. Goldwasser 18.427J Program Semantics and Verification (A) -------------------------------------------------------- (Same subject as 6.830J) Prereq.: 6.821, 6.044J/18.423J or 6.045J/ 18.400J or 6.840J/18.404J Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- See description under subject 6.830J. A. R. Meyer 18.428J Machine Learning (A) -------------------------------------------------------- (Same subject as 6.858J) Prereq.: 6.034, 6.041, or 6.045 (or equivalents) G (1) 3-0-9 -------------------------------------------------------- See description under subject 6.858J. R. L. Rivest 18.431J Graph Algorithms -------------------------------------------------------- (Same subject as 6.048J) Prereq.: 18.310 or 18.314 or 18.063 Acad Year 1990-91: Not offered Acad Year 1991-92: U (1) 3-0-9 -------------------------------------------------------- Emphasizes the design and analysis of efficient algorithms for graph theoretic problems. Includes fast algorithms for minimum spanning trees, shortest paths, network flow, vertex and edge connectivity, maximum cardinality matching, planarity testing, distributed and parallel algorithms. Alternate years. B.Awerbuch, C. E. Leiserson 18.433 Combinatorial Optimization and Linear Programming -------------------------------------------------------- Prereq.: 18.06 or 18.710, 18.310 U (2) 3-0-9 -------------------------------------------------------- A thorough treatment of linear programming theory, Dantzig's Simplex method, and duality theory. Linear programming applications to game theory, approximation algorithms, and NP-complete problems. Assignment problem, transportation problem, and min-cost network flow problem. Ellipsoid method and its implications for combinatorial optimization. Integer programming. M. Goemans 18.435J Theory of Parallel and VLSI Computation (A) -------------------------------------------------------- (Same subject as 6.848J) Prereq.: 6.046J/18.410J or 6.851J/18.414J G (1) 3-0-9 -------------------------------------------------------- Introduces parallel computation and very large scale integration. Design and analysis of systolic algorithms for routing, sorting, arithmetic, and graph problems on arrays, trees, hypercubes, and other fixed-connection networks. Network transformations, broadcast simulation, retiming. Packet routing and nonblocking networks. Mathematical models of hardware. Lower bounds, P-completeness, area-time trade-offs. Layout, placement, routing. 3D models, volume/area universal networks, fat-trees. Parallel programming on a connection machine. Survey of other parallel architectures. F. T. Leighton 18.436J Advanced Parallel and VLSI Computation (A) -------------------------------------------------------- (Same subject as 6.849J) Prereq.: 18.435J/6.848J G (2) 3-0-9 -------------------------------------------------------- Advanced topics in theory of parallel computation and very large scale integration. Parallel matching and related graph problems. Methods for removing randomness from algorithms. Automatic parallelization of straight-line code. AKS, columnsort, and universality. Packet routing. Fault tolerance. Network embedding problems. Network simulations. Current research topics. Alternate years. F. T. Leighton, C. E. Leiserson 18.437J Distributed Algorithms (A) -------------------------------------------------------- (Same subject as 6.852J) Prereq.: -- G (1) 3-0-9 -------------------------------------------------------- See description under subject 6.852J. N. Lynch 18.438J Distributed Network Protocols and Graph Algorithms -------------------------------------------------------- (Same subject as 6.855J) Prereq.: 18.410J/6.046J G (2) 3-0-9 -------------------------------------------------------- Design and analysis of efficient distributed algorithms in communication networks, with emphasis on graph algorithms methods. Models of communication networks: static, dynamic, synchronous, asynchronous. Complexity measures and trade-offs between them. Efficient simulations of stronger models by weaker models. Graph algorithms. Routing with compact tables. On-line algorithms; resource management, mobile users communication, deadlock resolution, etc. Protocols in dynamic networks. May be repeated for credit. B.AwerbuchApplied Mathematics: Statistics 18.440 Probability and Random Variables -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1, 2) 4-0-8 SCI DIST -------------------------------------------------------- Topics in applications. Probability spaces, random variables, distribution functions, expected value. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Mean, variance, moments, and generating functions. Conditional probability, Bayes theorem, joint distributions, and distributions of transformed random variables. Tchebychev inequality, law of large numbers, and central limit theorem. Multivariate normal distribution, covariances and correlation. Applications to statistics and decision theory. 18.440 and 18.313 may not both be taken for credit. Information: P. J. Huber. 18.441 Statistical Inference (A except XVIII) -------------------------------------------------------- Prereq.: 6.041 or 18.440 or 18.313 U (2) 3-0-9 -------------------------------------------------------- Introduces statistical inference. Decision theory, hypothesis testing, point and interval estimation. Bayesian methods, maximum-likelihood and likelihood-ratio tests. Chi-square goodness of fit tests. Comparison of populations by parametric and nonparametric methods. Analysis of variance, regression, and correlation. Sequential analysis if time permits. Treatment more mathematical than that of 18.05 and more detailed in its treatment of statistics. D.Khoshnevisan 18.443 Statistics for Applications (A except XVIII) -------------------------------------------------------- Prereq.: 18.440 or 18.313 or 6.041 U (1) 3-0-9 -------------------------------------------------------- A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Chi-square goodness of fit, regression, correlation, time-series analysis, analysis of variance and experimental design. Treatment more oriented toward application and less toward theory than 18.441. W. Olbricht 18.445J Introduction to Stochastic Processes (A) -------------------------------------------------------- (Same subject as 15.071J) Prereq.: 18.313 or 18.440 or 6.041 G (1, 2) 3-0-9 -------------------------------------------------------- Introduces the theory and application of stochastic processes. Empirical phenomena for which stochastic processes provide models. Markov-chains. Markov processes. Renewal theory. Semi-Markov processes. Queueing theory and Brownian motion. Term 1: D.Khoshnevisan Term 2: J. Keilson 18.446 Applied Time-Series Analysis (A) -------------------------------------------------------- Prereq.: 18.441 or 18.443 or 15.075 G (2) 3-0-9 -------------------------------------------------------- Statistical techniques commonly used to analyze time-series data. Topics: estimation of trends and seasonal adjustment, stationary series--autocorrelation and spectrum. Estimation and interpretation of spectra. ARIMA models and fitting them to data. Analysis of bivariate series--cross correlation and cross spectrum. Emphasizes learning techniques by using them on actual data. P. J. Huber 18.448 The Analysis of Categorical Data (A) -------------------------------------------------------- Prereq.: 18.441 or 18.443 or 15.075 G (1) 3-0-9 -------------------------------------------------------- Theory and application of log-linear models to multiway contingency tables and other data sets where the dependent variable is categorical. Topics: the Poisson distribution, one-way, two-way, and multiway frequency tables, logit regression, and maximum-likelihood estimation and computations. W. Olbricht 18.454 Sampling, Simulation, and Monte Carlo (A) -------------------------------------------------------- Prereq.: 18.440 or 18.313 or 6.041 G (1) 3-0-9 -------------------------------------------------------- Introduction to principles and techniques of sampling for the purpose of a survey. Includes simple random sampling, stratified sampling, systematic sampling, and cluster sampling. Discussion of statistical background of Monte Carlo methods and simulation--prominent parts of experimental mathematics with wide applicability. Includes variance reduction, conditional Monte Carlo, control variates, antithetic variates, regression methods, Monte Carlo optimization, application to statistical inference problems. D.Khoshnevisan 18.455 Analysis of Variance and Design of Experiments (A) -------------------------------------------------------- Prereq.: 18.06, 18.441 or 18.443 or 15.075 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Detailed presentation and use of the classical models of analyses of variance (ANOVA): one-way classification, two-way classifications, block designs, nested designs, latin squares. Factorial designs. Estimation. Tests of hypothesis, simultaneous confidence intervals. Presentation of regression, and analysis of covariance (ANOCOVA). Properties of the multivariate normal and related distributions, linear models, general linear hypothesis. Finally, effect of departure from assumptions and nonparametric analogs of ANOVA. Information: P. J. Huber. 18.456J Applied Multivariate Methods (A) -------------------------------------------------------- (Same subject as 15.079J) Prereq.: 18.06, 18.441 or 18.443 or 15.075 G (2) 3-0-9 -------------------------------------------------------- Theory and application of commonly used techniques involving multivariate data. Attention devoted to specific applications, and to computational facilities for applying the methods. Selects topics from the following: multivariate regression, discriminate analysis, and pattern classification. Cluster analysis, factor analysis, and principal components. Multidimensional scale analysis. Contingency tables. P. J. Huber, R. E. Welsch 18.457J Statistical Modeling (A) -------------------------------------------------------- (Same subject as 15.077J) Prereq.: 18.06, 18.441 or 18.443 or 15.075 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- See description under subject 15.077J. R. E. Welsch 18.458 Robust Statistics and Nonparametric Methods (A) -------------------------------------------------------- Prereq.: 18.440, 18.441 or 18.443 or 15.075 G (2) 3-0-9 -------------------------------------------------------- Overview of robust statistical theory, including asymptotic minimax, infinitesimal (bounded- influence) aspects, robust covariances, and robust regression. Nonparametric methods that give useful and valid results under a very wide class of underlying distributions -- particularly useful for social scientists and biologists. Topics: Wilcoxon test, sign test, Wilcoxon-Mann-Whitney test, U-statistics theorems, optimal linear rank tests, Kruskal-Wallis test, rerandomization tests. Information: P. J. Huber. 18.465 Topics in Statistics (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1) 3-0-9 -------------------------------------------------------- Topics for 1990-91 include data types and structures; model formulation, fitting, simulation, and validation. Linear and nonlinear models: interpolation, forecasting, diagnostics, robustness, variable selection. Smoothing, data transformations, logistic and nonparametric regression. P. J. Huber 18.466 Mathematical Statistics (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of alternative statistical procedures. Sequential analysis, empirical Bayes methods. May be repeated for credit. Information: P. J. Huber. For additional related subjects in Statistics, see: Civil Engineering: 1.03, 1.06, 1.15, 1.151, 1.202, 1.205, and 1.732 Electrical Engineering and Computer Science: 6.041, 6.231, 6.262, 6.264J, 6.431, 6.432, and 6.435 Economics: 14.30, 14.31, 14.381, 14.382, 14.383, 14.384, 14.385, and 14.388 Management: 15.034, 15.061, 15.065, 15.074, 15.075, 15.076, 15.077J, 15.306, and 15.832 Mathematics: 18.05, 18.175, 18.177, and 18.313 See also: 2.061, 2.845, 5.72, 7.011, 8.044, 8.08, 10.816, 11.220, 11.222, 16.37, 16.371, 17.842, 17.846, 17.850, 22.38, 22.40J, and HST 191 Logic 18.504 Seminar in Logic -------------------------------------------------------- Prereq.: -- U (1) 3-0-9 -------------------------------------------------------- Seminar for mathematics majors. Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Topic for 1990-91: Gdel completeness and incompleteness theorems. E. Hrushovski 18.510 Introduction to Mathematical Logic and Set Theory -------------------------------------------------------- Prereq.: -- Acad Year 1990-91: U (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Propositional and predicate logic. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Gdel incompleteness theorem. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Constructible sets. This subject introduces logic and set theory as a foundation for mathematics, and is especially recommended to students enrolled in theoretical mathematics subjects. 18.510 and 18.511 are offered in alternate years; they may not both be taken for credit. S. D. Friedman 18.511 Introduction to Mathematical Logic and Recursion Theory -------------------------------------------------------- Prereq.: -- Acad Year 1990-91: Not offered Acad Year 1991-92: U (1) 3-0-9 -------------------------------------------------------- Propositional and predicate logic. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Elementary recursion theory: enumeration and recursion theorems. Post's Problem. Gdel incompleteness theorem. 18.511 and 18.510 are offered in alternate years; they may not both be taken for credit. Information: S. D. Friedman. 18.515 Mathematical Logic (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1) 3-0-9 18.516 Mathematical Logic (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (2) 3-0-9 -------------------------------------------------------- First-order logic. Compactness and ultraproducts. Lowenheim-Skolem theorems and categoricity. Quantifier elimination. Recursively enumerable sets and definability in arithmetic. Incompleteness and undecidability. Consistency and cut-elimination. Zermelo-Fraenkel set theory. Reflection principles and absoluteness. The constructible universe. Forcing. E. Hrushovski 18.535 Graduate Logic Seminar (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1, 2) 3-0-9 -------------------------------------------------------- Students report on fundamental papers in mathematical logic. Open to all graduate students with an interest in logic. Topics vary from year to year. May be repeated for credit. S. D.Friedman 18.565 Recursion Theory (A) -------------------------------------------------------- Prereq.: 18.516 G (2) 3-0-9 -------------------------------------------------------- Topics in recursion theory chosen from priority arguments, hyperarithmetic theory, ordinal recursion, E-recursion, theory of projective sets. Permission of instructor required for those not having 18.516. G. E. Sacks 18.575 Model Theory (A) -------------------------------------------------------- Prereq.: 18.516 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Topics in model theory chosen from stability theory, O-minimal structures, model-theoretic algebra, models of arithmetic. Permission of instructor required for those not having 18.516. Information: S. D. Friedman. 18.585 Set Theory (A) -------------------------------------------------------- Prereq.: 18.516 G (1) 3-0-9 -------------------------------------------------------- Topics in set theory chosen from large cardinals, combinatorial set theory, forcing, descriptive set theory, fine structure theory. Permission of instructor required for those not having 18.516. S. D. Friedman 18.595 Seminar on Current Topics in Logic (A) -------------------------------------------------------- Prereq.: 18.516 G (1, 2) 3-0-9 -------------------------------------------------------- Analysis of results of current interest in logic. Students present recent developments in the field for general discussion. Uses formal and informal sources. Topics vary from year to year; may be repeated for credit. Term 1: G. E. Sacks Term 2: S. D.Friedman 18.597 Universal Algebra (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Properties shared by algebraic structures, introduction to category theory, Hopf algebras, tensor algebras, supersymmetric algebras, classical invariant theory, equational logic, word problems. G.-C. RotaAlgebra and Number Theory 18.701 Algebra I -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1) 3-0-9 18.702 Algebra II -------------------------------------------------------- Prereq.: 18.701 U (2) 3-0-9 -------------------------------------------------------- More extensive and theoretical than the 18.710-18.703 sequence. Experience with linear equations and matrices helpful. First term: group theory, geometry, and linear algebra. Second term: rings and fields--ideals, polynomial rings, factorization, modules, Jordan form for matrices, extension fields, Galois theory. M.Artin 18.703 Modern Algebra -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (2) 3-0-9 SCI DIST -------------------------------------------------------- A one-term treatment, covering the traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals, unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.710 and 18.703 together cover most of basic algebra. 18.06 or 18.710 should precede 18.703 if both subjects are to be taken. R. P. Stanley 18.704 Seminar in Algebra and Number Theory -------------------------------------------------------- Prereq.: 18.702 or 18.703 Acad Year 1990-91: U (1) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Seminar for mathematics majors. Students present and discuss the subject matter, taken from current journals or books. Topics may vary from year to year. D. McDoniel 18.705 Commutative Algebra (A) -------------------------------------------------------- Prereq.: 18.701-18.702 or 18.710-18.703 G (1) 3-0-9 -------------------------------------------------------- Basic topics in commutative algebra, with a brief introduction to categorical ideas and homological algebra. Modules, localization, noetherian rings, finiteness properties, dimension theory. G. Lusztig 18.706 Noncommutative Algebra (A) -------------------------------------------------------- Prereq.: 18.705 G (2) 3-0-9 -------------------------------------------------------- Topics in noncommutative algebra, selected from such areas as representations of finite groups and enveloping algebras. G.Lusztig 18.710 Abstract Linear Algebra -------------------------------------------------------- Prereq.: 18.02 or 18.021 or 18.022 U (1) 3-0-9 SCI DIST -------------------------------------------------------- An algebraic treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with 18.06, more emphasis on theory and proofs, less on matrix calculations and applications. B. Kostant 18.711 Game Theory -------------------------------------------------------- Prereq.: -- Acad Year 1990-91: Not offered Acad Year 1991-92: U (1) 3-0-9 -------------------------------------------------------- Two-person combinatorial games. Finding winning moves in such games as Nim, Hackenbush, and Dots-and-Boxes. Analysis of positions. Algorithmic and algebraic strategies for reduction of positions. Study of impartial games. Matrix games, continuous games, Arrow's Theorem. No formal prerequisite, but students should be familiar with elements of linear algebra. N. C. Ankeny 18.715 Topics in Homological Algebra (A) -------------------------------------------------------- Prereq.: 18.705 G (2) 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. Topics selected from such areas as cohomology of groups and Lie algebras, K-theory, higher algebraic K-theory, homotopy groups of spheres. M. J. Hopkins 18.725 Algebraic Geometry (A) -------------------------------------------------------- Prereq.: 18.705 Acad Year 1990-91: G (1) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Introduces algebraic geometry. Centered around the basic techniques of the subject, including the theory of schemes, sheaf cohomology, and the Riemann-Roch theorem for curves. S. L.Kleiman 18.727 Topics in Algebraic Geometry (A) -------------------------------------------------------- Prereq.: 18.725 G (1) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Topics for 1990-1991: Hodge theory. Variations of Hodge structures. Moduli spaces of representations of fundamental groups of algebraic varieties. Yang-Mills equations on algebraic varieties. Hodge D-modules. A. A. Beilinson 18.735 Topics in Algebra (A) -------------------------------------------------------- Prereq.: 18.702 or 18.703 G (1, 2) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Topic for term 1: introduction to Hopf algebras. Topic for term 2: Hecke algebras and quantum groups. Term 1: V. Kac Term 2:G. Lusztig 18.737 Linear Algebraic Groups (A) -------------------------------------------------------- Prereq.: 18.705 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Introduces the classification of affine groups over an algebraically closed field via their representations as groups of invertible matrices. D. A. Vogan 18.745 Introduction to Lie Algebras (A) -------------------------------------------------------- Prereq.: 18.701 or 18.703 G (1) 3-0-9 -------------------------------------------------------- Emphasizes theory of Lie algebras and algebraic aspects of Lie theory. Structure of finite-dimensional Lie algebras; Engel and Lie theorems, Cartan subalgebras, Cartan criteria. Structure and classification of semi-simple Lie algebras. Weyl and Levi theorems. Finite-dimensional representations of semi-simple Lie algebras, Weyl character formula. Verma modules. D. A. Vogan 18.747 Infinite-dimensional Lie Algebras (A) -------------------------------------------------------- Prereq.: 18.745 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Highest-weight representations of infinite-dimensional Lie algebras. Character formulas and combinatorics. Connections to the theory of theta functions and modular forms. Infinite wedge representation and algebraic curves. Vertex representations and their relation to soliton equations and to the quantum field theory. Virasoro algebra and the determinantal formula. Connection to critical exponents in statistical mechanics. Information: D. A. Vogan. 18.755 Introduction to Lie Groups (A) -------------------------------------------------------- Prereq.: 18.100, 18.710 G (1) 3-0-9 -------------------------------------------------------- A general introduction to Lie groups. The role of Lie groups in mathematics and physics. Correspondence with Lie algebras. Homogeneous spaces. Adjoint representation. Invariant differential forms and cohomology. Compact Lie groups. Structure of Lie algebras. R. D. MacPherson 18.756 Analysis on Lie Groups (A) -------------------------------------------------------- Prereq.: 18.755 Acad Year 1990-91: G (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Semi-simple Lie groups and symmetric spaces. Topics in function theory on symmetric spaces, such as Fourier analysis and Radon transform, invariant differential operators and potential theory. Emphasizes connections with classical analysis and representation theory. S. Helgason 18.757 Representations of Lie Groups (A) -------------------------------------------------------- Prereq.: 18.755 G (1) 3-0-9 -------------------------------------------------------- An introduction to the representation theory of compact Lie groups. D. A. Vogan 18.758 Representations of Lie Groups (A) -------------------------------------------------------- Prereq.: 18.757 Acad Year 1990-91: G (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Introduction to unitary representations of semisimple Lie groups: compact groups and the Borel-Weil theorem; parabolic induction; Zuckerman construction; unipotent representations. D.A. Vogan 18.769 Topics in Lie Theory (A) -------------------------------------------------------- Prereq.: Permission of Instructor Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Information: D. A. Vogan. 18.775 Algebraic Number Theory (A) -------------------------------------------------------- Prereq.: 18.705 Acad Year 1990-91: G (1) Acad Year 1991-92: Not offered 3-0-9 18.776 Algebraic Number Theory (A) -------------------------------------------------------- Prereq.: 18.775 Acad Year 1990-91: G (2) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Basic algebraic number theory. Algebraic number fields. Ideals, class numbers, units. Introduction to class field theory. Zeta functions. Values at special points. Other topics as time permits. 18.775: R. D. MacPherson 18.776: H. Grundman 18.781 Theory of Numbers -------------------------------------------------------- Prereq.: 18.701 or 18.703 Acad Year 1990-91: U (1) Acad Year 1991-92: Not offered 3-0-9 -------------------------------------------------------- Primes, congruences, and arithmetic functions. Kronecker's theorem, the geometry of numbers. Quadratic forms and quadratic number fields. N. C. Ankeny 18.785 Analytic Number Theory (A) -------------------------------------------------------- Prereq.: 18.115 Acad Year 1990-91: Not offered Acad Year 1991-92: G (1) 3-0-9 -------------------------------------------------------- Zeta functions for number fields. The prime ideal theorem. Analytic proofs of the finiteness of the class-number, the unit theorem, and discriminant bounds. Abelian characters and L-functions. The Chebatorev density theorem. Results based on various Riemann hypotheses including Artin's primitive root conjecture. Information: H. M. Stark. 18.786 Topics in Number Theory (A) -------------------------------------------------------- Prereq.: Permission of Instructor G (1) 3-0-9 -------------------------------------------------------- Topics vary from year to year; may be repeated for credit. Information: R. B. Melrose. Topology and Geometry 18.901 Introduction to Topology (A except XVIII) -------------------------------------------------------- Prereq.: 18.100 U (1, 2) 3-0-9 -------------------------------------------------------- Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces, connectedness, compactness, continuous functions, separation axioms, function spaces. Metrization theorems, the Tychonoff theorem. Topological groups. Term 1: H.R. Miller Term 2: F. P. Peterson 18.904 Seminar in Topology -------------------------------------------------------- Prereq.: 18.901 U (2) 3-0-9 -------------------------------------------------------- Seminar for mathmatics majors. Students present and discuss the subject matter, taken from current journals or books. Topics may vary from year to year. 1990-91: introduction to algebraic topology. Covers fundamental group, covering spaces, Van Kampen theorem and applications to knot theory, classification of compact surfaces and separation theorems in the plane. F. P. Peterson 18.905 Algebraic Topology (A) -------------------------------------------------------- Prereq.: 18.702 or 18.705; 18.901 G (1) 3-0-9 18.906 Algebraic Topology (A) -------------------------------------------------------- Prereq.: 18.905 G (2) 3-0-9 -------------------------------------------------------- Fundamental group, covering spaces, simplicial homology, simplicial approximation manifolds. Homology and cohomology of topological spaces, universal coefficient theorem, plus additional topics to be chosen by the instructor. J. R. Munkres 18.915 Graduate Topology Seminar (A) -------------------------------------------------------- Prereq.: 18.906 G (1) 3-0-21 -------------------------------------------------------- Study and discussion of important original papers in the various parts of algebraic and differential topology. Open to all students who have had 18.906 or the equivalent, not only prospective topologists. D. M. Kan 18.917 Advanced Topology (A) -------------------------------------------------------- Prereq.: 18.906 G (2) 3-0-9 -------------------------------------------------------- Content varying from term to term, so that graduate students taking the subject in successive terms may have an introduction to several important phases of topology such as homotopy theory, cohomology theory, fibre spaces, K-theory, combinatorial topology, and/or differential topology. Homotopy theory, cobordism theory. H. R. Miller 18.945 Geometry and Topology of Singular Spaces (A) -------------------------------------------------------- Prereq.: 18.705 or 18.905 or 18.965 Acad Year 1990-91: Not offered Acad Year 1991-92: G (2) 3-0-9 -------------------------------------------------------- Examples of singular spaces: Schubert varieties, toric varieties, compactifications of locally symmetric spaces. Techniques for studying singular spaces: stratifications, intersection homology, D-modules, mixed Hodge theory. Topics vary from year to year; may be repeated for credit. R. D. MacPherson 18.950 Elementary Differential Geometry and Differential Topology (Revised Content) -------------------------------------------------------- Prereq.: 18.100 U (1) 3-0-9 -------------------------------------------------------- A sample of some of the theorems of modern differential geometry and differential topology. Typical topics -- differential geometry: Riemannian manifolds, vector bundles. Differential topology: vector fields, Stokes's theorem, degree, Euler number. De Rham cohomology and characteristic classes. I. M. Singer 18.965 Geometry of Manifolds (A) -------------------------------------------------------- Prereq.: 18.101 G (1) 3-0-9 18.966 Geometry of Manifolds (A) -------------------------------------------------------- Prereq.: 18.965 G (2) 3-0-9 -------------------------------------------------------- Differentiable manifolds, vector fields and forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds. 18.966 continues 18.965. Focuses on symplectic and complex geometry. V. W. Guillemin 18.969 Topics in Geometry (A) -------------------------------------------------------- Prereq.: 18.965 G (2) 3-0-9 -------------------------------------------------------- Content varies from year to year; may be repeated for credit. Typical topics: heat kernels, Dirac operators, the index theorem, operator algebras and geometry, cyclic homology, equivariant differential forms. E. Getzler 18.994 Seminar in Geometry -------------------------------------------------------- Prereq.: -- U (2) 3-0-9 -------------------------------------------------------- Seminar for mathematics majors. Students present and discuss the subject matter, taken from current journals or books. Topics may vary from year to year. 1990-91: study of surfaces and Gauss curvature. More advanced topics according to interests of students. R. D. MacPherson 18.999 Mathematical Reading -------------------------------------------------------- Prereq.: -- G (1, 2, S) Units arranged -------------------------------------------------------- Reading of advanced mathematical treatises under supervision of a member of the Department. For graduate students desiring advanced work not provided in regular subjects. S.Helgason