Explore two applications of topological data analysis in medicine and chemistry using persistent homology. Delve into a study on quantifying lung structure in Chronic Obstructive Pulmonary Disease (COPD) using CT scans, developing topological characteristics for patient classification and disease stratification. Then, examine the topological and geometric structure of chemical compounds to understand water solubility, crucial for medication design. Learn how these advanced analytical methods can provide valuable insights in complex medical diagnostics and pharmaceutical development.