arXiv research

R. B. Melrose's b-calculus provides a framework for dealing with problems of partial differential equations that arise in singular or degenerate geometric situations. This article is a somewhat informal short course introducing many of the basic ideas of this world, assuming little more than a basic analysis and manifo…

Study on automorphisms of complex $b^k$-manifolds, extending previous work.

Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.

Let $X$ be a smooth compact manifold with corners which has two embedded boundary hypersurfaces $\partial_0 X , \partial_1 X$, and a fiber bundle $φ:\partial_0 X \to Y$ is given. By using the method of blowing up, we define a pseudodifferential culculus $Ψ^* _{Φ,b} (X)$ generalizing the $Φ$-calculus of Mazzeo and Melro…

New calculus for pseudodifferential operators on manifolds with cylindrical ends.

M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string the…

Tichler proved that a manifold admitting a smooth closed one-form fibers over a circle. More generally a manifold admitting $k$ independent closed one-forms fibers over a torus $T^k$. In this article we explain a version of this construction for manifolds with boundary using the techniques of $b$-calculus. We explore n…

Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.