Jaroslav Trnka

Into the Amplituhedron

A bstractWe initiate an exploration of the physics and geometry of the amplituhedron, starting with the simplest case of the integrand for four-particle scattering in planar N=4

Anatomy of the amplituhedron

mathcal{N}=4

Positive amplitudes in the amplituhedron

SYM. We show how the textbook structure of the unitarity double-cut follows from the positive geometry. We also use the geometry to expose the behavior of the multicollinear limit, providing a direct motivation for studying the logarithm of the amplitude. In addition to computing the two and three-loop integrands, we explore various lower-dimensional faces of the amplituhedron, thereby computing non-trivial cuts of the integrand to all loop orders.

Singularity Structure of Maximally Supersymmetric Scattering Amplitudes

A bstractWe initiate an exploration of on-shell functions in N=4

Local integrand representations of all two-loop amplitudes in planar SYM

mathcal{N}=4

Boundary configurations, graphs, and permutations

SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV amplitudes and showing that they can always be expressed as a positive sum of differently ordered Parke-Taylor tree amplitudes. This is understood in terms of an extended notion of positivity in G(2, n), the Grassmannian of 2-planes in n dimensions: a single on-shell diagram can be associated with many different “positive” regions, of which the familiar G+(2, n) associated with planar diagrams is just one example. The decomposition into Parke-Taylor factors is simply a “triangulation” of these extended positive regions. The U(1) decoupling and Kleiss-Kuijf (KK) relations satisfied by the Parke-Taylor amplitudes also follow naturally from this geometric picture. These results suggest that non-planar MHV amplitudes in N=4