CHAPTER 1
Tempered distributions and the Fourier transform
Microlocal analysis is a geometric theory of distributions, or a theory of geometric distributions. Rather than study general distributions w
CHAPTER 11
Hochschild homology
11.1. Formal Hochschild homology
The Hochschild homology is dened, formally, for any associative algebra. Thus
if A is the algebra then the space of formal k -chains, fo
CHAPTER 10
K-theory
This is a brief treatment of K-theory, enough to discuss, and maybe even prove,
the Atiyah-Singer index theorem. I am starting from the smoothing algebra discussed earlier in Chapt
CHAPTER 9
The wave kernel
Let us return to the subject of good distributions as exemplied by Dirac
delta functions and the Schwartz kernels of pseudodierential operators. In fact
we shall associate a
CHAPTER 8
Elliptic boundary problems
Summary
Elliptic boundary problems are discussed, especially for operators of Dirac type.
We start with a discussion of manifolds with boundary, including function
CHAPTER 7
Scattering calculus
7.1. Scattering pseudodierential operators
There is another calculus of pseudodierential operators which is smaller than
the traditional calculus. It arises by taking amp
CHAPTER 6
Pseudodierential operators on manifolds
In this chapter the notion of a pseudodierential on a manifold is discussed.
Some preliminary material on manifolds is therefore necessary. However th
CHAPTER 5
Microlocalization
5.1. Calculus of supports
Recall that we have already dened the support of a tempered distribution in
the slightly round-about way:
(5.1)
if u S (Rn ), supp(u) = cfw_x Rn ;
CHAPTER 4
Isotropic calculus
The algebra of isotropic pseudodierential operators on Rn has global properties very similar to the algebra of pseudodierential operators on a compact manifold discussed b
CHAPTER 3
Residual, or Schwartz, algebra
The standard algebra of operators discussed in the previous chapter is not really
representative, in its global behaviour, of the algebra of pseudodierential o
CHAPTER 2
Pseudodierential operators on Euclidean space
Formula (1.92) for the action of a dierential operator (with coecients in
C (Rn ) on S (Rn ) can be written
P (x, D)u = (2 )n
ei(xy) P (x, )u(y
CHAPTER 12
The index theorem and formula
Using the earlier results on K-theory and cohomology the families index theorem of Atiyah and Singer is proved using a variant of their embedding proof. The
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