  We develop a geometric framework that characterizes the synchronization
problem --- the problem of consistently registering or aligning a collection of
objects. The theory we formulate characterizes the cohomological nature of
synchronization based on the classical theory of fibre bundles. We first
establish the correspondence between synchronization problems in a topological
group $G$ over a connected graph $\Gamma$ and the moduli space of flat
principal $G$-bundles over $\Gamma$, and develop a discrete analogy of the
renowned theorem of classifying flat principal bundles with fix base and
structural group using the representation variety. In particular, we show that
prescribing an edge potential on a graph is equivalent to specifying an
equivalence class of flat principal bundles, of which the triviality of
holonomy dictates the synchronizability of the edge potential. We then develop
a twisted cohomology theory for associated vector bundles of the flat principal
bundle arising from an edge potential, which is a discrete version of the
twisted cohomology in differential geometry. This theory realizes the
obstruction to synchronizability as a cohomology group of the twisted de Rham
cochain complex. We then build a discrete twisted Hodge theory --- a fibre
bundle analog of the discrete Hodge theory on graphs --- which geometrically
realizes the graph connection Laplacian as a Hodge Laplacian of degree zero.
Motivated by our geometric framework, we study the problem of learning group
actions --- partitioning a collection of objects based on the local
synchronizability of pairwise correspondence relations. A dual interpretation
is to learn finitely generated subgroups of an ambient transformation group
from noisy observed group elements. A synchronization-based algorithm is also
provided, and we demonstrate its efficacy using simulations and real data.
