Morse Theory, Discrete Morse Theory and Applications

By employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fáry-Milnor theorem. We also discuss discrete Morse theory, a subject of applied mathematics developed by Robin Forman in the 1990s, and its application. We focus on elucidating especially the inherent similarity between classical Morse theory and discrete Morse theory