User ezra getzler - MathOverflow

Answer by Ezra Getzler for Why is Lie's Third Theorem difficult?

2011-02-24T07:30:13Z

This question is related to the following rather deep result: if $G$ is a finite dimensional Lie group, then $\pi_2(G)=0$.

The analogous statement fails for Lie algebroids, as does Lie III. The paper of Tseng and Zhu shows that this is no accident.

Lie III may be interpreted as saying the following: the foliation of the space of $\frak{g}$-connections on the 1-simplex associated to infinitesimal gauge transformation has a Hausdorff leaf space.

A connection is a one-form $A\in\mathcal{A}=\Omega^1([0,1],\frak{g})$ with values in the Lie algebra $\frak{g}$. The infinitesimal gauge action is given by the formula

$\delta_XA=dX+[A,X]$,

where $X\in\Omega^0([0,1],\frak{g})$: these vectors span an integrable distribution in the tangent space of $\mathcal{A}$.

(This leaf space is then the simply connected Lie group associated to $G$.) I was taught this point of view by Raoul Bott.

Morphisms of Banach spaces

2011-02-24T07:04:32Z

What is the standard name in English for bounded linear maps $f:E\to F$ between Banach spaces such that the kernel $\ker(f)$ has a complement, and $\text{im}(f)$ is closed, and has a complement?

Apparently, when $f$ is injective or surjective, it is sometimes referred to as an admissible monomorphism or epimorphism, respectively. Is this terminology standard?

Answer by Ezra Getzler for Atiyah-Singer index theorem

2011-02-24T07:20:37Z

You need to understand pseudodifferential operators if you want to understand the original statement of the full Atiyah-Singer index theorem. However, in most applications to differential geometry, only the theorem for twisted Dirac operators is needed. (One of the main results of Atiyah and Singer is that the Bott periodicity theorem - or rather, its generalization to vector bundles, the Thom isomorphism theorem for K-theory - reduces the general case to that of twisted Dirac operators.)

If you want to learn the theory of pseudodifferential operators, I recommend the original papers of Kohn and Nirenberg and Hörmander. This theory is not needed to prove the Atiyah-Singer index theorem: you can get away with the existence of an asymptotic solution of the heat equation. To see this in action, see the paper of McKean and Singer.

One advantage of the heat-kernel approach is that it is well-adapted to study the generalizations of the theory, such as the theory of analytic torsion and the family index theorem.