Georgia Institute of Technology College of Sciences

 A central theme in algebraic geometry is understanding how much of the well-behaved theory of smooth varieties survives in the presence of singularities. Du Bois and rational singularities are among the most important classes studied in algebraic geometry due to their nice cohomological behavior. For instance, they preserve features like the Hodge decomposition and vanishing theorems one expects for smooth varieties.
Recently, motivated by developments in Hodge theoretic methods, there has been substantial interest in studying their higher analogs. This talk will survey recent developments connecting these notions to deformation properties and moduli theory, and applications to Calabi-Yau varieties. I will also report on recent joint work with Brian Nugent extending the theory of higher Du Bois and higher rational singularities to pairs — a framework that is both essential in modern birational geometry and a natural setting for studying Hodge theory for open varieties.