The on-shell superspace formulation of N=4 SYM allows the writing of all possible scattering processes in one compact object called the super-amplitude.   Famously, the super-amplitude integrand can be extracted from generalized polyhedra called the amplituhedron. In this talk, I will review this construction and present a natural generalization of the amplituhedron that we proved at tree level and conjectured at loop level to correspond to the product of two parity conjugate superamplitudes. The sum of all parity conjugate amplitudes corresponds to a particular limit of the supercorrelator through the Wilson Loop/Amplitude duality.  I will conclude by discussing this connection from a geometrical point of view. This talk is based on the reference arXiv:2106.09372 .

I'll review 2104.13932, where we analyze the supersymmetric index of N=4 SU(N) Super Yang-Mills using the Bethe Ansatz approach, expressing it as a sum and concentrating on some family of contributions to the sum. We show that in the large N limit each term in this family corresponds to the contribution of a different euclidean black hole to the partition function of the dual gravitational theory. By taking into account non-perturbative contributions (wrapped D3-branes), we further show a one to one match between the contributions of the gravitational saddles and this family of contributions to the index, both at the perturbative and non-perturbative levels. I'll end with some new results regarding the Bethe Ansatz expansion and the information one could extract from it.