  The identification of interesting substructures within jets is an important
tool for searching for new physics and probing the Standard Model at colliders.
Many of these substructure tools have previously been shown to take the form of
optimal transport problems, in particular the Energy Mover's Distance (EMD). In
this work, we show that the EMD is in fact the natural structure for comparing
collider events, which accounts for its recent success in understanding event
and jet substructure. We then present a Shape Hunting Algorithm using
Parameterized Energy Reconstruction (SHAPER), which is a general framework for
defining and computing shape-based observables. SHAPER generalizes N-jettiness
from point clusters to any extended, parametrizable shape. This is accomplished
by efficiently minimizing the EMD between events and parameterized manifolds of
energy flows representing idealized shapes, implemented using the
dual-potential Sinkhorn approximation of the Wasserstein metric. We show how
the geometric language of observables as manifolds can be used to define novel
observables with built-in infrared-and-collinear safety. We demonstrate the
efficacy of the SHAPER framework by performing empirical jet substructure
studies using several examples of new shape-based observables.
