Multicellular life forms, comprehensive collection of cells sharing the same genetic blueprint, follow robust developmental paths and achieve vast functions as a result of the complex interactions and heterogeneities of the cells. The emerging single-cell omics technologies have created a new lens enabling the dissection of the heterogeneity in biological systems at single-cell resolution and with high throughput. To extract insights from these complex and high-dimensional data resources, a fundamental task is to infer the low-dimensional underlying structures such as landscapes of cell states and developmental trajectories. Various approaches have been developed for this task and could lead to significantly different results and conclusions. There is a lack of robust and multiscale method to thoroughly examine the underlying structure with a spectrum of parameters and metrics. This proposal will use the emerging powerful topological and geometric data analysis to examine the space of all meaningful structures instead of deriving an individual structure. This project will develop a modern education plan for students in mathematical biology, where they are exposed to the advanced topological and geometric methods that are especially suitable for complex and high-dimensional biological data, to help them adapt to modern data-driven biology.