报告摘要：Geometric apparatuses are frequently inundated with too much structural detail to be computationally tractable, while traditional topological tools often incur too much simplification of the original data to be practically useful. Persistent homology, a new branch of algebraic topology, is able to bridge the gap between geometry and topology. In this talk, I will discuss a few new developments in persistent homology. First, we introduce multiscale-multiresolution persistent homology to describe the topological fingerprints and topological transitions of nano-bio materials. Additionally, multidimensional persistence is developed for topological denoising and revealing the topology-function relationship in biomolecular data. Moreover, molecular topological fingerprints are utilized to resolve ill-posed inverse problems in cryo-EM structure determination. Finally, objective-oriented persistent homology is constructed via the variational principle and differential geometry for proactive feature extracting from big data sets, which leads to topological partial differential equations (TPDEs).