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I will give a brief review of some recent progress regarding mathematical aspects of three closely-related quantities in N=4 SYM theory: half-BPS correlators, the square of amplitudes and energy correlators.

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.

I will review the recent progress on the application of the Amplituhedron framework to study IR finite quantities in the planar N=4 SYM theory. This includes the negative geometry expansion for the Wilson loops and the deformed Amplituhedron which connects to the Coulomb branch amplitudes.

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.

I will review the recent progress on the application of the Amplituhedron framework to study IR finite quantities in the planar N=4 SYM theory. This includes the negative geometry expansion for the Wilson loops and the deformed Amplituhedron which connects to the Coulomb branch amplitudes.

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.

In these lectures, we will discuss two different approaches to on-shell recursion relations that are used to construct scattering amplitudes of various massless theories. The first lecture, based on soft limits of amplitudes, will be on theories resulting from spontaneous (super-)symmetry breaking. The second lecture will focus on the mathematical structures or building blocks that result from BCFW recursion of maximally supersymmetric gluons and gravitons.