Explore discrete Morse theory and its applications in combinatorial group theory and topological data analysis through this one-hour conference talk. Delve into a refined version of discrete Morse theory that guarantees Whitehead simple homotopy equivalence and provides explicit descriptions of attaching maps for critical cells in simplified complexes. Discover how this theoretical framework applies to problems in combinatorial group theory, including potential counterexamples to the Andrews-Curtis conjecture. Learn about the extension of this method to filtrations of CW-complexes and its role in efficiently computing the persistent fundamental group of point clouds using group presentations. Gain insights from the speaker's joint work with Kevin Piterman and the related research paper on Morse theory for group presentations.