Geometric Analysis and Topology Seminar

Microlocal Analysis Over the Maslov Cycle

Speaker: Alan Weinstein, Berkeley

Location: Warren Weaver Hall 517

Date: Friday, September 21, 2012, 11 a.m.

Synopsis:

In the lagrangian grasmannian $$\Lambda$$ of lagrangian subspaces in $$T^*\mathbb{R}^n$$, the elements which have nonzero intersection with the fibre over $$0$$ form a codimension $$1$$ cooriented subvariety $$\Sigma$$ with singular set of codimension $$3$$ in $$\Lambda$$. $$\Sigma$$ is called the Maslov cycle, as it is dual to the Maslov class in $$H^1(\Lambda,\mathbb{Z})$$. According to a much more general result of Givental, $$\Sigma$$ is the image under the cotangent projection of a smooth, conic lagrangian submanifold $$\mathcal{S}$$ in the cotangent bundle of $$\Lambda$$ with the zero section removed. In this talk, I will describe a distribution (i.e. generalized function) $$\phi$$ on $$\Lambda$$ whose singular support is $$\Sigma$$ and whose wave-front set is $$\mathcal{S}$$. $$\phi$$ is, in fact, a so-called Fourier integral distribution attached to $$\mathcal{S}$$. I will make some remarks on the Maslov class of $$\mathcal{S}$$, which determines the bundle where the principal symbol of $$\phi$$ takes its values, and on the regularity properties of $$\phi$$. Finally, I will explain how the results above fit into a larger program of describing "impossible operations" on distributions as generalized functions on spaces of distributions.