Higher stacks in analytic geometry (Part I/II)

I give a survey of my work with Kai Behrend on the construction of the category of higher analytic stacks. I will show how to associate to any differential graded Banach algebra its higher analytic stack of quasi-invertible elements. This generalizes the theorem that the invertible elements of a Banach algebra form a Lie group. This is part 1 of 2 talks.