Speaker:  Junliang Shen (Yale University)

Title: Fourier transforms and perverse filtrations for abelian fibrations.

Abstract: The perverse filtration captures interesting homological information of algebraic maps. In recent years, perverse filtrations are found to share surprising connections to studies of non-abelian Hodge theory (the P=W conjecture), enumerative geometry (refined BPS invariants), and planar singularities (DAHA, knot invariants). In this talk, I will explain a theory of Fourier transform for abelian fibrations, which provides a uniform explanation of certain mysterious features predicted by the connections mentioned above. This Fourier theory can be viewed as an extension of the Beauville decomposition from abelian schemes to certain abelian fibrations with singular fibers. In particular, I will discuss how/why such an extension is possible when singular fibers break (most) symmetries of the geometry.

Based on joint work with Davesh Maulik and Qizheng Yin.