ICTS

Join me on a journey through the evolution of scattering amplitudes, where we will explore a selection of significant milestones over the last decades. In this personal overview, I’ll highlight key moments that have shaped the field and its trajectory.

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In these mini-lectures, I will explore the method of differential equations, a powerful tool for computing Feynman integrals. This approach has found widespread applications, from precision calculations in high-energy scattering amplitudes to the study of gravitational wave physics. More recently, it has also been employed in deriving analytic expressions for integrals appearing in cosmological correlators. Beyond facilitating computations, the method of differential equations reveals deep algebraic structures underlying Feynman integrals, making it an essential technique in modern theoretical physics.

In these two lectures we will go over some applications of nonlinear algebra to physics. In the first lecture we will take a look at the CHY scattering equations in order to see what algebraic statistics and theoretical physics have in common. In the second lecture, we will consider a classical algebraic variety, the Grassmannian. We will discuss its basic properties and see several contexts in which the Grassmannian appears in positive geometry and physics.

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In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.

In this talk we will review the amplituhedron, the correlahedron, and the relations between them. We will explore the generalisation of the definition of positive geometry required for it (and also for the loop amplituhedron). We will show the equivalence between the correlahedron and a recently defined geometry for four-point correlators. Finally we will discuss the non maximally nilpotent case.

We prove the equivalence between two traditional approaches to the classical mechanics of a massive spinning particle in special relativity. One is the spherical top model of Hanson and Regge, recast in a Hamiltonian formulation with improved treatment of covariant spin constraints. The other is the massive twistor model, slightly generalized to incorporate the Regge trajectory relating the mass to the total spin angular momentum. We establish the equivalence by computing the Dirac brackets of the physical phase space carrying three translation and three rotation degrees of freedom. Lorentz covariance and little group covariance uniquely determine the structure of the physical phase space. We comment briefly on how to couple the twistor particle to electromagnetic or gravitational backgrounds.

Positivity properties of scattering amplitudes are typically related to unitarity and causality. However, in some cases positivity properties can also arise from deeper underlying structures. In these lectures, we will discuss infinitely many positivity constraints that certain amplitudes and their derivatives obey called completely monotonicity in the mathematics literature.
In the first lecture, we will discuss completely monotone functions and some of their properties. We shall then show why some objects such scalar Feynman integrals admit this property via integral representations. In the second lecture, we will discuss the connection between complete monotonicity and positive geometries.

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