Motivic invariants of moduli stacks of Higgs bundles and bundles with connections: results and speculations

We review some results and techniques from our papers devoted to the computation of motivic classes of stacks of parabolic Higgs budles and bundles with connections on a curve. In the last section we present some directions for future work, as well as some speculations. The latter include a generalization of the P=W conjecture inspired by the work of Maxim Kontsevich and the third author on the Riemann--Hilbert correspondence for complex symplectic manifolds as well as our running project on the motivic classes of the moduli stacks of nilpotent pairs on the formal disk and geometric Satake correspondence for double affine Grassmannians.