Algebra Topology Seminar: Generalized Morse Theory for Maps to Two Dimensional Manifolds: PL-immersions of Polygons and Protein

Motivated by basic questions from robotics and molecular biology, we consider certain configuration spaces, and some associated maps to two dimensional Euclidian space. We are able to understand the singular sets for a natural subset of these maps. For this subset, we apply variants of Bott-Morse theory to determine the structure of inverse images - in the configuration spaces - of curves and points in the plane. In turn, these results answer key questions about the structure of spaces of length preserving immersions of polygons into Euclidean space and provide insights into the process of protein folding