November 22, 2024
Title: b-geometry and asymptotic analysis
Speaker: Michal Wrochna and Juri Sampieri Bjornsson
Melrose's b-geometry provides a framework for dealing with problems of partial differential equations that arise in singular, degenerate or non compact geometric situations. This talk will give a brief outline, focusing particularly on the role of b-geometry in the asymptotic analysis of PDEs.
In the first part, we will introduce the fundamentals of b-geometry, including the notions of b-metrics and b-vector bundles on manifolds with boundaries.
The second part will shift to an analytical focus, exploring the asymptotic behavior of solutions to the Klein-Gordon equation on asymptotically de Sitter manifolds. We will introduce the essential analytical tools for this, such as b-differential operators and their normal operators, conormal and polyhomogeneous functions, and the Mellin transform.