Nilufer Koldan

Witten Deformation: Old and New

In 1982 E. Witten suggested an analytic proof of the Morse inequalities on compact smooth manifolds without boundary by studying deformation of the de Rham complex of the manifold by a Morse function. In their simplest form, these inequalities state that the number $m_p$ of the critical points with index $p$ of a Morse function $f$ can not be less than the Betti number $b_p$ of the underlying manifold: $m_p\ge b_p$, for all $p$. I will describe the main ingredients of Witten's proof of the Morse inequalities.

On manifold with boundary Witten's Laplacian can be defined in the same way, but we need to specify its domain. I will define two different domains which will lead us to the absolute and relative Morse inequalities for manifolds with boundary.

At the end, I will talk about semiclassical asymptotics of small eigenvalues of Witten's Laplacian on smooth compact manifolds with boundary.