Feynman integrals from the viewpoint of Picard-Lefschetz theory

I will present work in progress with Erik Panzer, Matteo Parisi and Ömer Gürdoğan on the analytic structure of Feynman(esque) integrals: We consider integrals of meromorphic differential forms over relative cycles in a compact complex manifold, the underlying geometry encoded in a certain (parameter dependant) subspace arrangement (e.g. Feynman integrals in their parametric representation). I will explain how the analytic struture of such integrals can be studied via methods from differential topology; this is the seminal work by Pham et al (using tools and methods developed by Leray, Thom, Picard-Lefschetz etc.). Although their work covers a very general setup, the case we need for Feynman integrals has never been worked out in full detail. I will comment on the gaps that have to be filled to make the theory work, then discuss how much information about the analytic structure of integrals can be derived from a careful study of the corresponding subspace arrangement.