  This is a simple reading report of professor Weiping Zhang's lectures. In
this article we will mainly introduce the basic ideas of Witten deformation,
which were first introduced by Edward Witten on, and some applications of it.
The first part of this article mainly focuses on deformation of Dirac operators
and some important analytic facts about the deformed Dirac operators. In the
second part of this article some applications of Witten deformation will be
given, to be more specific, an analytic proof of Poincar$\acute{e}$-Hopf index
theorem and Real Morse Inequilities will be given. Also we will use Witten
deformation to prove that the Thom Smale complex is quasi-isomorphism to the
de-Rham complex (Witten suggested that Thom Smale complex can be recovered from
his deformation and his suggestion was first realized by Helffer and
Sj$\ddot{o}$strand, the proof in this article is given by Bismut and Zhang).
And in the last part an analytic proof of Atiyah vanishing theorem via Witten
deformation will be given.
