  Analytical and numerical methods are developed to analyze the quantum nature
of the big bang in the setting of loop quantum cosmology. They enable one to
explore the effects of quantum geometry both on the gravitational and matter
sectors and significantly extend the known results on the resolution of the big
bang singularity. Specifically, the following results are established for the
homogeneous isotropic model with a massless scalar field: i) the scalar field
is shown to serve as an internal clock, thereby providing a detailed
realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac
observables and semi-classical states are constructed rigorously; iii) the
Hamiltonian constraint is solved numerically to show that the big bang is
replaced by a big bounce. Thanks to the non-perturbative, background
independent methods, unlike in other approaches the quantum evolution is
deterministic across the deep Planck regime. Our constructions also provide a
conceptual framework and technical tools which can be used in more general
models. In this sense, they provide foundations for analyzing physical issues
associated with the Planck regime of loop quantum cosmology as a whole.
