Kazhdan-Lusztig conjectures and shadows of Hodge theory
  We give an informal introduction to the authors' work on some conjectures of Kazhdan and Lusztig, building on work of Soergel and de Cataldo-Migliorini. This article is an expanded version of a lecture given by the second author at the Arbeitstagung in memory of Friedrich Hirzebruch. 
Ben Elias, Geordie Williamson
math.RT