  In this paper, we describe recent work towards the mirror P=W conjecture,
which relates the weight filtration on a cohomology of a log Calabi--Yau
manifold to the perverse Leray filtration on the cohomology of the homological
mirror dual log Calabi--Yau manifold, taken with respect to the affinization
map. This conjecture extends the classical relationship between Hodge numbers
of mirror dual compact Calabi--Yau manifolds, incorporating tools and ideas
which appear in the fascinating and groundbreaking works of de Cataldo, Hausel,
and Migliorini, and de Cataldo and Migliorini. We give a broad overview of the
motivation for this conjecture, recent results towards it, and describe how
this result might arise from the SYZ formulation of mirror symmetry. This
interpretation of the mirror P=W conjecture provides a possible bridge between
the mirror P=W conjecture and the well-known P=W conjecture in nonabelian Hodge
theory.

Algebraic Geometry