All Questions
Tagged with index-theory pseudo-differential-operators
6 questions
21
votes
2
answers
2k
views
Applications of Atiyah-Singer using pseudodifferential operators
Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
13
votes
1
answer
1k
views
Atiyah-Singer for pseudodifferential operators via heat kernel?
The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
10
votes
0
answers
409
views
Between Being a Connection and Being an Elliptic Operator
Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
9
votes
2
answers
1k
views
Understanding the analytic index map of the Atiyah-Singer index theorem
Hi,
I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn.
I do not understand why the analytic index map ...
3
votes
0
answers
68
views
Pseudodifferential calculus for the Diffeomorphism Invariant Geometry
In the paper "Local Index Formula in Noncommutative Geometry" Connes and Moscovici build the spectral triple $(A,H,D)$ where $A=C^{\infty}_c(P) \rtimes \Gamma$ where $\Gamma$ is an arbitrary subgroup ...
0
votes
0
answers
109
views
Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...