In 1968, D. Atkinson proved in a series of papers the existence of functions satisfying all known constraints of the S-matrix bootstrap for the 2-to-2 S-matrix of gapped theories. To date, this is the only result of this sort, while a contrario no current technology allows to generate, even numerically, fully consistent S-matrices in d>2. Beyond the mathematical results themselves, the proof, based on establishing the existence of a fixed point of a certain map, also suggests a procedure to be implemented numerically and which would produce fully consistent S-matrix functions via iterating dispersion relations, and using as an input a quantity related to the inelasticity of a given scattering process. In this talk, I will report on some work being finalised, done in collaboration with A. Zhiboedov, about analytical and numerical aspects of developing and implementing this scheme. I will review basic concepts of the S-matrix program and show some of our results on non-perturbative scalar, phi^4-like S-matrices in 4, describe their properties and compare to other approaches in the literature. If time allows, I will present some results in 3 dimensions and discuss subtle aspects of the high energy (Regge behaviour) of the S-matrices.

The monodromy relations of scattering amplitudes in string theory provide an elegant formalism to understand some mysterious properties of tree-level field theory amplitudes, like the color-kinematics duality. This duality has been instrumental in tremendous progress on the computations of loop-amplitudes in quantum field theory, but a loop-level generalisation of the stringy monodromy construction has been lacking for many years. In this talk I will first describe some of these recent developments in the domain of scattering amplitudes in gauge and gravity theories. I’ll then review the monodromies of open string worldsheets and how the lead at tree-level to deepening the understanding of the gauge theory perturbative expansion. Then I will describe in a non-technical manner our results and how we managed to extend these relations to all loops in string and field theory. I’ll finish by discussing implications for the loop expansion in general, and how to extend in principle these results to gravity. I will assume no prior knowledge of the audience in modern scattering amplitudes methods.

Remarkable formulae for the tree-level S-matrix of gauge and gravity theories were recently discovered by Cachazo-He-Yuan, based on a localisation procedure in the moduli space of Riemann surfaces. These formulae are now known to arise from new string-like chiral models. These naturally produces loop-level amplitudes, also localised in the moduli space of higher genus curves. However, the resolution of the localisation equations is immensely more difficult than at tree-level, and it remained as an open question as to what these formulae computed, and more generally if these string models made sense at the quantum level. In this talk I'll describe a resolution of the localisation equations at one-loop in a particular kinematical regime, and show agreement with known amplitudes. Then I'll describe a much more powerful method, based on an integration by parts which degenerates the torus down to a sphere (plus two new points). The model is then solved like at tree-level. I'll briefly discuss extension to higher loops in conclusion.

A systematic understanding of the low energy limit of string theory scattering amplitudes is essential for conceptual and practical reasons. In this talk, I shall report on a recent work where this limit has been analyzed using tropical geometry. Our result is that the field theory amplitudes arising in the low energy limit of string theory are written in a very compact form as integrals over a single object, the tropical moduli space. This picture provides a general framework where the different aspects of the low energy limit of string theory scattering amplitudes are systematically encompassed; the Feynman graphs structure and the ultraviolet regulation mechanism. I shall then give examples of application of the formalism, in particular at genus two, and discuss open issues. No knowledge of tropical geometry will be assumed and the topic shall be introduced during the talk.