I will construct several cohomology theories for pairs (X,f) consisting of an algebraic variety X and a regular function f on X, and explain how to produce a universal cohomology theory for such pairs. This universal cohomology theory takes its values in the tannakian category of Exponential Motives. I will then show by means of concrete examples how to compute some interesting tannakian fundamental groups of exponential motives, and explain their relevance for transcendence questions.

Prep material:

Read the introduction and the examples in ch. 12.

http://www.fields.utoronto.ca/sites/default/files/uploads/Exponential_mo...