In first-passage percolation (FPP), one places weights (t_e)
on the edges of Z^d and considers the induced metric. Optimizing paths
for this metric are called geodesics, and infinite geodesics are
infinite paths all whose finite subpaths are geodesics. It is a major
open problem to show that in two dimensions, with i.i.d. continuous
weights, there are no bigeodesics (doubly-infinite geodesics). In this
talk, I will describe work on bigeodesics in arbitrary dimension using
``geodesic graph'' measures introduced in '13 in joint work with J.
Hanson. Our main result is that these measures are supported on graphs
with no doubly-infinite paths, and this implies that bigeodesics cannot
be constructed in a translation-invariant manner in any dimension as
limits of point-to-hyperplane geodesics. Because all previous works on
bigeodesics were for two dimensions and heavily used planarity and
coalescence, we must develop new tools based on the mass transport
principle. Joint with G. Brito (Georgia Tech) and J. Hanson (CUNY).