In the past 20 years, an increasingly clear picture connecting knot invariants, automorphic forms, and curve counting on Calabi-Yau 3-folds has emerged. Much of the geometry in this picture can be explained using various moduli spaces of sheaves on curves, such as Hitchin fibers or Hilbert schemes of points. By combining Chen’s recent proof Goresky-Kottwitz-MacPherson’s purity hypothesis for anisotropic affine Springer fibers for GL_n with the Macdonald formula for curves with planar singularities (following Maulik-Yun and Migliorini-Shende), we show that the Hilbert schemes of points on locally planar unibranch curves satisfy a cohomological purity property. This result has applications to refined BPS invariants on Calabi-Yau 3-folds, after Hosono-Saito-Takahashi, Kiem-Li, and Maulik-Toda. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.