This presentation focuses on the field of Topological Data Analysis, Computational Topology, and Persistent Homology. The research explores the need for Multiparameter Persistent Homology and addresses the challenges it faces in terms of decomposition, particularly when compared to the 1-parameter case. To address these challenges, Vineyards are introduced as a representation tool for Multiparameter Persistent Homology and their stability is mathematically proven. The study also delves into Sparse Filtration, highlighting its usefulness in approximating the Cech complex. Furthermore, an algorithm to approximate the Rips complex, developed by the research team I worked with, is presented. Results indicate limitations in its effectiveness as an approximation and shed light on the need for optimization.
