10/16 at 17:30 in 507

Junliang Shen

Title: Perverse filtrations and Fourier transforms

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 the first part of the talk, I will introduce the perverse filtration and some of the connections mentioned above. Then I will explain a theory of Fourier transform which provides a uniform method proving certain features of perverse filtrations of abelian fibrations. This 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 explain how/why such an extension is possible when singular fibers break certain symmetries of the geometry. Based on joint work with Davesh Maulik and Qizheng Yin.