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.


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.