Title: Geometry and topology of singularities via Hodge theory - with a view toward secant varieties

 

Abstract: Algebraic geometry is the study of algebraic varieties that are defined as the solution sets of systems of polynomial equations. The topology of algebraic varieties plays a central role in algebraic geometry, particularly in the classification of singularities. Among the most studied singularity classes are the so-called Du Bois and rational singularities, and in recent years their higher analogues have emerged as natural extensions and have become active areas of research. In this talk, I will explain how these notions give rise to refined measures of singularities that connect algebro-geometric and topological viewpoints through Hodge-theoretic techniques. I will also discuss their applications to the study of singularities of secant varieties, a classical and important class of varieties with a history spanning more than a century and broad impact across mathematics.