  We utilize the deformation theory of algebraic singularities to study charged
matter in compactifications of M-theory, F-theory, and type IIa string theory
on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is
more physical than that of resolution. We describe how two-cycles can be
identified and systematically studied after deformation. For ADE singularities,
we realize non-trivial ADE representations as sublattices of Z^N, where N is
the multiplicity of the codimension one singularity before deformation. We give
a method for the determination of Picard-Lefschetz vanishing cycles in this
context and utilize this method for one-parameter smooth deformations of ADE
singularities. We give a general map from junctions to weights and demonstrate
that Freudenthal's recursion formula applied to junctions correctly reproduces
the structure of high-dimensional ADE representations, including the 126 of
SO(10) and the 43,758 of E_6. We identify the Weyl group action in some
examples, and verify its order in others. We describe the codimension two
localization of matter in F-theory in the case of heterotic duality or simple
normal crossing and demonstrate the branching of adjoint representations.
Finally, we demonstrate geometrically that deformations correctly reproduce the
appearance of non-simply-laced algebras induced by monodromy around codimension
two singularities, showing the reduction of D_4 to G_2 in an example. A
companion mathematical paper will follow.
