Abstract: 
In the first part of this talk I will present one possible point of view on Simpson's nonabelian Hodge theory and present a series of conjectures formulated by Katzarkov, Pantev, Simpson and Toën loosely aiming at the study of the homotopy type of algebraic varieites. Next, I will explain how to use ideas from derived analytic geometry to solve the first of these conjectures, extending Deligne's version of the RiemannHilbert correspondence over derived bases and with coefficients in perfect complexes. Finally, I'll explain how to obtain more general coefficients by using an analytic version of Tannaka duality. This talk is based on the preprints arXiv 1703.03907 and 1812.09300.
