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18 157: Introduction to Microlocal Analysis
3.00 Credits
Massachusetts Institute of Technology
The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject.
Prerequisite:
Prereq: 18.155
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18 157 - Introduction to Microlocal Analysis
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18 158: Topics in Differential Equations
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: 18.157
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18 158 - Topics in Differential Equations
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18 175: Theory of Probability
3.00 Credits
Massachusetts Institute of Technology
Sums of random independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.
Prerequisite:
Prereq: 18.125
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18 175 - Theory of Probability
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18 177: Topics in Stochastic Processes
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: 18.175
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18 177 - Topics in Stochastic Processes
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18 199: Graduate Analysis Seminar
3.00 Credits
Massachusetts Institute of Technology
Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.
Prerequisite:
Prereq: Permission of instructor
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18 199 - Graduate Analysis Seminar
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18 238: Geometry and Quantum Field Theory
3.00 Credits
Massachusetts Institute of Technology
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.
Prerequisite:
Prereq: Permission of instructor
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18 238 - Geometry and Quantum Field Theory
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18 276: Mathematical Methods in Physics
3.00 Credits
Massachusetts Institute of Technology
Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.
Prerequisite:
Prereq: 18.745 or some familiarity with Lie theory
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18 276 - Mathematical Methods in Physics
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18 303: Linear Partial Differential Equations: Analysis and Numerics
3.00 Credits
Massachusetts Institute of Technology
Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. MATLAB is introduced and used in homework for simple examples.
Prerequisite:
Prereq: 18.06 or 18.700
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18 303 - Linear Partial Differential Equations: Analysis and Numerics
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18 304: Undergraduate Seminar in Discrete Mathematics
3.00 Credits
Massachusetts Institute of Technology
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.
Prerequisite:
Prereq: 18.310 or 18.310C; 18.06, 18.700, or 18.701; or permission of instructor
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18 304 - Undergraduate Seminar in Discrete Mathematics
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18 305: Advanced Analytic Methods in Science and Engineering
3.00 Credits
Massachusetts Institute of Technology
Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations.
Prerequisite:
Prereq: 18.04, 18.075, or 18.112
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18 305 - Advanced Analytic Methods in Science and Engineering
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