The P=W conjecture

Simpson’s non-abelian Hodge correspondence establishes a real analytic isomorphism between two significant moduli spaces in algebraic geometry: the Hitchin moduli space and the character variety. This correspondence implies an identification between their corresponding cohomologies. This identification does not preserve the mixed Hodge structures. The P=W conjecture, now proven, arose from the investigation of the relationship between non-abelian Hodge correspondence and deeper topological structures, such as the perverse filtration versus the weight filtration. The thesis will focus on the proof P=W conjecture given in arXiv:2209.05429, which employs techniques stemming from the theory of cohomological Hall algebras of zero-dimensional sheaves on a smooth surface.

Prerequisites: Familiarity with scheme theory, basic representation theory, and homology theory.

Reference: The paper arXiv:2209.05429.