Linear Logic is a theory born from categorical semantics, following the discovery by Girard of models of computation in which programs are interpreted by analytic maps. This analogy between programs and analytic maps came to fruition in the 2000s, with the discovery by Ehrhard and Regnier of the differential lambda calculus and of the Taylor expansion of programs. This field of research also lead to the discovery of differential categories, by Blute, Cockett and Seely.
The first half of this talk will be a tutorial on the connection between programs and analytic maps through a concrete example: the category of probabilistic coherence spaces.
In the second half of this talk, I will explain the limits of the theory of differentiation developed in the 2000s, because as it turns out the category of probabilistic coherence spaces is not a differential category. I will then introduce coherent differentiation, a generalization of differential categories that is able to capture probabilistic coherence spaces.
This talk is the second part of a two talk series, but each talk can be followed independently.