In this talk, I will discuss a novel strategy to fit experimental data using an amplitude ansatz satisfying the constraints of Analyticity, Crossing, Unitarity, and UV completeness. The fit strategy requires both the use of S-matrix Bootstrap methods and non-convex Particle Swarm Optimization (PSO) techniques. As a proof of principle, I will focus on $\pi\pi$ scattering. Using this procedure, I will show how to construct numerically a full-fledged scattering amplitude that fits the available experimental and lattice data, and that features all the known QCD spectrum with quantum numbers $I^G=0^+,1^+$ below 1.4 GeV, plus an additional surprise.

The numerical S-matrix Bootstrap aims at establishing non-perturbative universal bounds on physical observables that can be extracted from scattering amplitudes in any dimension. In the first part of the talk, I will review our past explorations of the space of supergravity amplitudes and their connection to String/M theory. I will discuss the universal bounds on the first non-universal correction to sugra amplitudes, and how the extremal solution is compatible with clustering in the Born regime, and with the Quantum Regge growth hypothesis. In the second part of the talk I will report on a first Bootstrap exploration of multi-particle scattering. I will focus on the simplest non-integrable S-matrix describing the scattering of branons on the world-sheet of confining strings in three dimensions.

In this talk I will review some of the recent developments in the S-matrix Bootstrap focussing on applications to effective field theories. As an example, I will apply the bootstrap methods to the supergravity effective field theory in ten dimensions. I will show the improved numerical bootstrap bounds on the first correction to the universal graviton scattering and compare the result with the String Theory predictions. In the last part, I will comment on some new numerical ideas that will boost the explorations in different dimensions and for higher dimensional operators.