Motivic Donaldson–Thomas invariants of the moduli stacks of parabolic Higgs bundles and bundles with connections

Using the ideas of motivic integration I will explain how to compute the "number" of semistable Higgs bundles  (maybe with parabolic structure) of fixed rank and degree on a smooth complex curve. Based on that result I am going to discuss a similar problem in the case of bundles with connections. If time permits, I will explain open questions, including a conjectural relation to Satake correspondence for affine Grassmannians over 2-dimensional local rings. This is a joint project with Roman Fedorov and Alexander Soibelman.