Special Guest Lecture

The talk will be based on two groups, the automorphic Galois group and the motivic Galois group. These are given by conjectural formulas that contain much data, but which are also quite explicit. The foundations for their construction are Langlands' Principle of Functoriality and his Reciprocity Conjecture. After introducing the groups, we shall describe an extension of the second one, the mixed motivic Galois group. This group promises to have many interesting applications to Galois representations and periods. We shall describe what we can of a third application, to Feynman amplitudes and fundamental particle physics.