18.238 Geometry and Quantum Field Theory ( ) Prereq: Permission of instructor Units: 3-0-9 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
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18.276 Mathematical Methods in Physics ( ) Prereq: 18.745 or some familiarity with Lie theory Units: 3-0-9 Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects. V. W. Guillemin
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18.303 Linear Partial Differential Equations ( ) Prereq: Calculus II (GIR) ; 18.03 or 18.034 Units: 3-0-9 URL: http://math.mit.edu/18.303 Lecture: TR1-2.30 ( 2-139 ) +final The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications. D. Z. Freedman
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18.304 Undergraduate Seminar in Discrete Mathematics ( ) Prereq: 18.310 or 18.310C ; 18.06 , 18.700 , or 18.701 ; or permission of instructor Units: 3-0-9 Credit cannot also be received for 18.316 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. D. J. Kleitman
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18.305 Advanced Analytic Methods in Science and Engineering ( ) Prereq: 18.04 , 18.075 , or 18.112 Units: 3-0-9 URL: http://math.mit.edu/18.305/ Lecture: MWF11 ( 2-151 ) A comprehensive treatment of the advanced methods of applied mathematics. Designed to strengthen the mathematical abilities of graduate students and train them to think on their own. Expansion around singular points: special functions; the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques. H. Cheng
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18.306 Advanced Partial Differential Equations with Applications ( ) Prereq: 18.03 or 18.034 ; 18.04 , 18.075 , or 18.112 Units: 3-0-9 URL: http://www-math.mit.edu/18.306/index.html Lecture: MW9.30-11 ( 2-135 ) Concepts and techniques for solving partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Characteristics and shocks. Separation of variables, transform methods, Green's functions. Asymptotics, geometrical optics theory. Dimensional analysis, self-similarity, traveling waves. Singular perturbation and boundary layers. Solitons. Variational methods. Free-boundary problems. Applications in diverse fields, such as fluid dynamics, electrochemistry, materials science, optics, and finance. R. R. Rosales
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18.307 Integral Equations ( ) Prereq: 18.04 , 18.075 , or 18.112 Units: 3-0-9 Emphasis on concepts and techniques for solving integral equations from an applied mathematics perspective. Selection of material from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; group theory. Examples from fluid and solid mechanics, acoustics, quantum mechanics, and other applications. Information: R.R. Rosales
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18.308 Wave Motion ( ) Prereq: Permission of instructor Units: 3-0-9 Dispersive and non-dispersive waves in fluids, with emphasis on nonlinear effects and applications to Geophysical Flows and Nonlinear Acoustics. Stability of shear and stratified flows; surface and internal waves; nonlinear resonant interactions; solitons and solitary wave interactions; characteristics, nonlinear breaking, hydraulic jumps, and bores; weakly nonlinear theory; dispersive wave turbulence; weakly nonlinear geometrical optics and modulation; wave boundary layer phenomena. Information: D. J. Benney
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18.310 Principles of Applied Mathematics ( ) Prereq: Calculus II (GIR) Units: 3-0-9 Credit cannot also be received for 18.310C URL: http://math.mit.edu/~shor/PAM/18.310.html Subject Cancelled Study of illustrative topics in discrete applied mathematics including sorting algorithms, information theory, coding theory, secret codes, generating functions, linear programming, probability theory, and game theory. Information: P.W. Shor
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18.310C Principles of Applied Mathematics ( ) Prereq: Calculus II (GIR) Units: 3-0-9 Credit cannot also be received for 18.310 Study of illustrative topics in discrete applied mathematics, including sorting algorithms, information theory, coding theory, secret codes, generating functions, linear programming, probability theory, and game theory. Instruction and practice in written communication provided. Same content as 18.310, but assignments are structured with an additional focus on writing. Enrollment limited. P. W. Shor, M. X. Goemans
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18.311 Principles of Applied Mathematics ( ) Prereq: Calculus II (GIR) ; 18.03 or 18.034 Units: 3-0-9 Introduction to fundamental concepts in continuous applied mathematics. Continuum limit and conservation laws. Kinematic waves. Diffusion. Linear and nonlinear waves. Hyperbolic equations: method of characteristics, expansion fans, shock dynamics, shock structure. Green functions. Fourier series and transform. Dimensional analysis. Similarity solutions. Boundary layers. Applications from traffic flow, gas dynamics, water waves, granular flow, ion transport, etc. R. R. Rosales
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18.312 Algebraic Combinatorics ( ) Prereq: 18.701 or 18.703 Units: 3-0-9 URL: http://www-math.mit.edu/~apost/courses/18.312/ Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks. R. P. Stanley
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18.314 Combinatorial Analysis ( ) Prereq: Calculus II (GIR) ; 18.06 , 18.700 , or 18.701 Units: 3-0-9 URL: http://math.mit.edu/~apost/courses/18.314/ Lecture: TR9.30-11 ( 2-136 ) +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. S. H. Assaf
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18.315 Combinatorial Theory ( ) Prereq: Permission of instructor Units: 3-0-9 URL: http://math.mit.edu/~apost/courses/18.315/ Lecture: MWF1 ( 2-135 ) Content varies from year to year. Fall 2009: Generating functions. R. P. Stanley
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18.316 Seminar in Combinatorics ( ) Prereq: Permission of instructor Units: 3-0-9 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. Information: D. J. Kleitman
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18.318 Topics in Combinatorics ( ) Prereq: Permission of instructor Units: 3-0-9 URL: http://math.mit.edu/~pak/courses/318.htm Topics vary from year to year. A. Postnikov
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18.319 Combinatorics and Geometry ( ) Prereq: Permission of instructor Units: 3-0-9 URL: http://math.mit.edu/~apost/courses/18.319/ Connections between combinatorics and geometry (and algebra). Discussion of combinatorial problems that arise in algebraic geometry, convex geometry, and algebraic topology. Topics include toric varieties, polytopes and fans, hyperplane arrangements, triangulations and tilings, matroids, topological combinatorics, Schubert calculus. Information: A. Postnikov
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18.325 Topics in Applied Mathematics ( , ) Prereq: Permission of instructor Units: 3-0-9 URL: http://www-math.mit.edu/~jyu/18.325 TBA. Topics vary from year to year. Fall: L. Demanet Spring: Information: M. X. Goemans
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18.330 Introduction to Numerical Analysis ( ) Prereq: Calculus II (GIR) ; 18.03 or 18.034 Units: 3-0-9 URL: http://www-math.mit.edu/~lippert/classes/18.330 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. D. S.-W. Tam
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18.335J Introduction to Numerical Methods ( ) (Same subject as 6.337J ) Prereq: 18.03 or 18.034 ; 18.06 , 18,700, or 18.701 Units: 3-0-9 URL: http://math.mit.edu/~stevenj/18.335 Lecture: MWF2 ( 32-155 ) Advanced introduction to numerical linear algebra and related numerical methods. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB. S. G. Johnson
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18.336 Numerical Methods for Partial Differential Equations ( ) Prereq: 18.330 , 18.335J , Units: 3-0-9 URL: http://web.mit.edu/jcnave/www/courses/18.336.htm Advanced introduction to applications and theory of numerical methods for solution of differential equations, especially of physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods. Topics include finite differences, spectral methods, finite elements, well-posedness and stability, particle methods and lattice gases, boundary and nonlinear instabilities. A. Edelman
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18.337J Parallel Computing ( ) (Same subject as 6.338J ) Prereq: 18.06 , 18.700 , or 18.701 Units: 3-0-9 URL: http://beowulf.csail.mit.edu/18.337/index.html Advanced interdisciplinary introduction to modern scientific computing on parallel supercomputers. Numerical topics include dense and sparse linear algebra, N-body problems, and Fourier transforms. Geometrical topics include partitioning and mesh generation. Other topics include architectures and software systems with emphasis on understanding the realities and myths of what is possible on the world's fastest machines. Programming languages include MPI and Star-P with MATLAB and Python. A. Edelman
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18.338 Eigenvalues of Random Matrices ( ) Prereq: Permission of instructor Units: 3-0-9 URL: http://www.mit.edu/~18.338/ Lecture: TR2.30-4 ( 2-135 ) An introduction to the theory and applications of stochastic eigen-analysis. Theoretical topics include matrix calculus, "free" probability, and stochastic operators. Applications in signal processing, finance, and large stochastic systems are discussed. There is a hands-on emphasis on using the theory to discover new applications. A. Edelman
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18.353J Nonlinear Dynamics I: Chaos ( ) (Same subject as 2.050J , 12.006J ) Prereq: 18.03 or 18.034 ; Physics II (GIR) Units: 3-0-9 URL: http://segovia.mit.edu/12.006/ Lecture: MW2.30-4 ( 56-154 ) 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 II. T. Peacock
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18.354J Nonlinear Dynamics II: Continuum Systems ( ) (H except 18) (Same subject as 12.207J ) Prereq: 18.353J / 12.006J or permission of instructor Units: 3-0-9 General mathematical principles of continuum systems. (1) From microscopic to macroscopic. Examples range from random walkers, to Newtonian mechanics, to option pricing. (2) Singular Perturbations. Examples include boundary layer theory, snow flakes and geophysical flows. (3) Instability. Generalize ideas from 18.353 to continuum systems. Examples from fluid mechanics, solid mechanics, astrophysics and biology. (4) Pattern formation and turbulence. P. M. Reis
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