Witten Deformation and Its Application toward Morse Inequalities

TL;DR

This thesis provides an analytical proof of Morse inequalities for closed smooth manifolds using Witten's approach, employing PDE techniques to analyze harmonic oscillators.

Contribution

It offers a novel analytical proof of Morse inequalities based on PDE methods, expanding the understanding of their relation to harmonic oscillators.

Findings

Proof of Morse inequalities using PDE techniques

Reduction to eigenspaces of harmonic oscillators

Application of Witten's approach to smooth manifolds

Abstract

In this undergraduate thesis, we present an analytical proof of the Morse inequalities for closed smooth $n$-manifolds following Witten's approach. Using techniques from PDE theory, the proof is reduced to study the eigenspaces and eigenvalues of harmonic oscillators on $\mathbb{R}^n$.