Abstract

Around 2008, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which predicts aconnection between the topology of the Hitchin system and the Hodge theory of the charactervariety under the non-abelian Hodge correspondence. Since then, much effort has been devoted tounderstanding this myterious phenomenon, leading to the discovery of new geometric structures onboth the Higgs side and the character variety side. In this talk, l will first review the P=W conjecturewhich has been proven as a theorem since 2022 by Maulik-Shen and Hausel-Mellit-MinetsSchiffmann. Then l will discuss a recent attempt, in joint work with Davesh Maulik and Qizheng Yinto understand this conjecture using Fourier transform. This approach establishes links betweenderived equivalences and the decomposition theorem, shedding further light on the algebraiccycles/motives associated with the Hitchin system.