Maximally supersymmetric string theory compactified on d-dimensional torus is invariant under the non-perturbative U-duality groups E_{d+1}(Z). These duality groups are the discrete version over the integers of the real split forms of the simply laced lie groups of rank d. The interactions of the theory are given by automorphic function under E_{d+1}(Z), given in the simplest cases by the residue of maximal parabolic Eisenstein series. The dimensional reduction of the theory to three dimensions gives rise to the duality group E_8 from which all BPS couplings in higher dimensions can be extracted. We will present the construction of the automorphic form relevant for the supersymmetric protected BPS interactions in the low-energy expansion of maximal supersymmetric effective action. We will show that the support of the non-vanishing Fourier coefficients for these automorphic form is given by the expected BPS conditions one can derive from the supersymmetric conditions.

We will describe the recent progresses in the computation of four-graviton amplitudes at higher-loop order in maximal supergravity and string theory in various dimensions. We will discuss the constraints from dualities and the extended supersymmetry on the structure of the amplitudes. We will explain that the cross-symmetry of gravity, maximal supersymmetry, and the constraints from higher dimensions play an important role in the improved ultraviolet behaviour of maximal supergravity in four dimensions.

We will present a string scenario for localization of gravity in four dimensions. We will discuss the field theory model, proposed by Dvali Gabadadze and Porrati and details its problems and their resolutions. We will explain that this model presents issues well posed in string theory and address their resolution in this context.