"Spacetime and Quantum Mechanics form the pillars of our understanding
of modern physics, but there are several indications that these
concepts are approximate and must emerge from deeper principles,
undoubtedly involving new mathematics.
In this talk I will describe some emerging ideas along these lines,
and present a new formulation of some very basic physics-- fundamental
to particle scattering and to cosmology--not following from quantum
evolution in space-time, but associated with simple new mathematical
structures in "positive geometry". The simplest examples of positive
geometries are polytopes old and new, from cyclic polytopes and
Associahedra to ``cosmological" polytopes. Others, such as the
``Amplituhedron", involve generalizations of polytopes into the
Grassmannian. In these examples we can concretely
see how the usual rules of space-time and quantum mechanics can arise,
joined at the hip, from fundamentally geometric and combinatorial origins."