Exponential motivic homotopy theory, foliations and applications

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This is a proposal submitted to the DFG within the SPP 1786. The project described here consists of two related topics. The first is the construction and exploration of exponential motivic homotopy theory, a variant of motivic homotopy theory for varieties with potentials. The second is centered around the geometric and homotopical theory of foliations and higher differential Galois theory, and its application to the study of motives and algebraic cycles. Exponential connections and the twisted de Rham complex play a central role in both areas.