  This is a survey article on the recent progress in understanding the
Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture, especially on the
effect of quantum corrections, via Witten-Morse theory using the program first
depicted by Fukaya to obtain an explicit relation between differential
geometric operations, e.g. wedge product of differential forms and Lie-bracket
of the Kodaira-Spencer complexes, with combinatorial structures, e.g. Morse
$A_\infty$ structures and scattering diagrams.
