Fundamentals of Algebraic Microlocal Analysis: A Comprehensive Introduction to the Subject

Algebraic microlocal analysis is a branch of mathematics that studies the behavior of differential operators on manifolds. It is a powerful tool for studying partial differential equations and representation theory.

Fundamentals of Algebraic Microlocal Analysis (Chapman & Hall/CRC Pure and Applied Mathematics Book 217)

by Goro Kato

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Language	:	English
File size	:	15962 KB
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Sheaves of Microfunctions

One of the key concepts in algebraic microlocal analysis is the sheaf of microfunctions. A sheaf of microfunctions is a sheaf of modules over the ring of smooth functions on a manifold. It is a way of representing the singularities of a differential operator.

Sheaves of microfunctions can be used to study the behavior of differential operators in a number of ways. For example, they can be used to determine the index of a differential operator, and to study the asymptotic behavior of solutions to differential equations.

The Fourier Transform

The Fourier transform is another important tool in algebraic microlocal analysis. The Fourier transform is a linear operator that transforms a function from the time domain to the frequency domain. It is a powerful tool for studying the behavior of differential operators, as it can be used to convert differential operators into multiplication operators.

The Fourier transform can be used to study the behavior of differential operators in a number of ways. For example, it can be used to determine the spectrum of a differential operator, and to study the asymptotic behavior of solutions to differential equations.

The Wave Front Set

The wave front set is a subset of the cotangent bundle of a manifold. It is a way of representing the singularities of a distribution.

The wave front set can be used to study the behavior of differential operators in a number of ways. For example, it can be used to determine the regularity of a differential operator, and to study the asymptotic behavior of solutions to differential equations.

Applications to Partial Differential Equations

Algebraic microlocal analysis has a number of applications to partial differential equations. For example, it can be used to study the existence and uniqueness of solutions to partial differential equations, to study the regularity of solutions to partial differential equations, and to study the asymptotic behavior of solutions to partial differential equations.

Applications to Representation Theory

Algebraic microlocal analysis also has a number of applications to representation theory. For example, it can be used to study the irreducible representations of a Lie group, to study the character theory of a Lie group, and to study the automorphic forms of a Lie group.

Algebraic microlocal analysis is a powerful tool for studying differential operators, partial differential equations, and representation theory. It is a subject with a rich history and a wide range of applications.

Fundamentals of Algebraic Microlocal Analysis (Chapman & Hall/CRC Pure and Applied Mathematics Book 217)

by Goro Kato

5 out of 5

Language	:	English
File size	:	15962 KB
Screen Reader	:	Supported
Print length	:	316 pages

FREE