  We consider D-branes in string theory and address the issue of how to
describe them mathematically as a fundamental object (as opposed to a solitonic
object) of string theory in the realm in differential and symplectic geometry.
The notion of continuous maps, $k$-times differentiable maps, and smooth maps
from an Azumaya/matrix manifold with a fundamental module to a (commutative)
real manifold $Y$ is developed. Such maps are meant to describe D-branes or
matrix branes in string theory when these branes are light and soft with only
small enough or even zero brane-tension. When $Y$ is a symplectic manifold
(resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold
with an almost complex structure $J$), the corresponding notion of Lagrangian
maps (resp. special Lagrangian maps; associative maps, coassociative maps;
$J$-holomorphic maps) are introduced. Indicative examples linking to symplectic
geometry and string theory are given. This provides us with a language and part
of the foundation required to study themes, new or old, in symplectic geometry
and string theory, including (1) $J$-holomorphic D-curves (with or without
boundary), (2) quantization and dynamics of D-branes in string theory, (3) a
definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds
with a fundamental module with a connection, (4) a theory of fundamental matrix
strings or D-strings, and (5) the nature of Ramond-Ramond fields in a
space-time. The current note D(11.1) is the symplectic/differential-geometric
counterpart of the more algebraic-geometry-oriented first two notes D(1)
([L-Y1]) (arXiv:0709.1515 [math.AG]) and D(2) ([L-L-S-Y], with Si Li and
Ruifang Song) (arXiv:0809.2121 [math.AG]) in this project.
