This talk will discuss how to study singular rational inner functions (RIFs) using their zero set behaviors. In the two-variable setting, zero sets can be used to define a quantity called contact order, which helps quantify derivative integrability and non-tangential regularity. In the three-variable and higher setting, the RIF singular sets (and corresponding zero sets) can be much more complicated. We will discuss what holds in general, what holds for simple three-variable RIFs, and some examples illustrating why some of the nice two-variable behavior is lost in higher dimensions. This is joint work with James Pascoe and Alan Sola.[-]

This talk will describe ongoing joint work with Paul Cadman and Duco van Straten, based on the PhD thesis of the former. Givental and Varchenko used the period mapping to pull back the intersection form on the Milnor fibre of an irreducible plane curve singularity $C$, and thereby define a symplectic structure on the base space of a miniversal deformation. We show how to combine this with a symmetric basis for the module of vector fields tangent to the discriminant, to produce involutive ideals $I_k$ which define the strata of parameter values $u$ such that $\delta(C_u)\leq k$. In the process we find an unexpected Lie algebra and a still mysterious canonical deformation of the module structure of the critical space over the discriminant. Much of this work is experimental - a crucial gap in understanding still needs bridging.[-]