Geometry and Topology of Singularities

This interdisciplinary project brings together innovative ideas and techniques from algebraic geometry and complex analytic geometry, singularity theory, commutative algebra and topology.

Given two complex analytic varieties X and X' that are part of an analytic family in a complex affine space we will study the relationships between the different similarities X and X' might share.

Such similarities can be of geometric and topological nature.The synonym for similarity we will use here is equisingularity.

We say X and X' are topologically equisingular if there is a homeomorphism between X and X' that extends to an ambient homomorphism that trivializes the entire family; we say that X and X' are geometrically equisingular if X and X' have similar resolution of singularities, or if a resolution of the entire family induces a resolution of each fiber.

Understanding the topology of either X or X' usually involves a continuous deformation of either one of them to a smooth variety, the existence of which is guaranteed under a special smoothability hypothesis.A stronger version of topological equisingularity, called Lipschitz equisingularity, asks for a homeomorphism between X and X' that satisfies the Lipschitz inequality.The ultimate goal of the project is to gain insight in the relationships between topological (Lipschitz) and geometric equisingularity and develop an approach to study the topology of singularities that are not necessarily smoothable without imposing special dimension (or codimension) assumptions as was done previously.