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18 104: Seminar in Analysis
3.00 Credits
Massachusetts Institute of Technology
Students present and discuss material from books or journals. Several items will be based on the classic work: Polya and Szego's "Problems and Theorems from Analysis." More recent work will also be used depending on each individual's background. Instruction and practice in written and oral communication provided. Enrollment limited.
Prerequisite:
Prereq: 18.100
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18 104 - Seminar in Analysis
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18 112: Functions of a Complex Variable
3.00 Credits
Massachusetts Institute of Technology
Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.
Prerequisite:
Prereq: 18.100; 18.06, 18.700, or 18.701
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18 112 - Functions of a Complex Variable
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18 116: Riemann Surfaces
3.00 Credits
Massachusetts Institute of Technology
Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory.
Prerequisite:
Prereq: 18.112
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18 116 - Riemann Surfaces
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18 117: Topics in Several Complex Variables
3.00 Credits
Massachusetts Institute of Technology
Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.
Prerequisite:
Prereq: 18.112, 18.965
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18 117 - Topics in Several Complex Variables
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18 125: Real and Functional Analysis
3.00 Credits
Massachusetts Institute of Technology
Introductions to set theory and general topology as needed in analysis. Lebesgue's integration theory. Introduction to functional analysis, Banach and Hilbert spaces.
Prerequisite:
Prereq: 18.100
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18 125 - Real and Functional Analysis
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18 135: Geometric Analysis
3.00 Credits
Massachusetts Institute of Technology
A quick description of Riemannian symmetric spaces. Spherical functions and Harish-Chandra's c-function. Fourier transforms and Radon transforms on Riemannian symmetric spaces X. Applications to invariant differential equations, in particular the multitemporal wave equation on X. Eigenspace representations.
Prerequisite:
Prereq: 18.745 or 18.755
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18 135 - Geometric Analysis
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18 137: Topics in Geometric Partial Differential Equations
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: Permission of Instructor
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18 137 - Topics in Geometric Partial Differential Equations
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18 152: Introduction to Partial Differential Equations
3.00 Credits
Massachusetts Institute of Technology
Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.
Prerequisite:
Prereq: 18.100; 18.06, 18.700, or 18.701
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18 152 - Introduction to Partial Differential Equations
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18 155: Differential Analysis
3.00 Credits
Massachusetts Institute of Technology
No course description available.
Prerequisite:
Prereq: 18.102 or 18.103
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18 155 - Differential Analysis
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18 156: Differential Analysis
3.00 Credits
Massachusetts Institute of Technology
Fall: Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Spring: Variable coefficient elliptic, parabolic and hyperbolic partial differential equations. 18.112 recommended for 18.155.
Prerequisite:
Prereq: 18.155
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18 156 - Differential Analysis
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