Enumerative approach to perverse filtrations (не стандартное время доклада!)

Based on an observation by de Cataldo and Migliorini, we perform a calculation of the perverse filtration on the cohomology of the moduli spaces of Higgs bundles in the odd degree, rank-$2$ case, and we reproduce the result of Hausel, de Cataldo and Migliorini on the $P=W$ conjecture. The calculation has a number of interesting features, in particular, the appearance of a mysterious Bethe pole. This is joint work with Tamas Hausel and Simone Chiarello.