Please note the day and time for this special AG seminar. 

The characteristic 0 Non Abelian Hodge Theory (0NAHT) entails that the moduli space of semistable Higgs bundles of degree zero on a curve is diffeomorphic to the moduli space of semistable flat connections.
The characteristic p Non Abelian Hodge Theory (pNAHT) entails that the moduli stack of Higgs bundles is a twisted form of the moduli stack of flat connections.
The technical cores of 0NAHT and pNAHT are very different, and the comparison between the two theories is intriguing. For example, 0NAHT respects semistability, does pNAHT also respect semistability?
In joint work with Mark de Cataldo and Michael Groechenig (vector bundle case), and joint work with Andres Fernandez Herrero (principal bundle case), we answer this question affirmatively, and establish a version of pNAHT on the level of moduli spaces of semistable objects. Moreover, we show that the two moduli spaces have isomorphic intersection cohomologies.