Infinite-dimensional Riemannian manifolds naturally arise in the study of shape spaces, yet the precise definition of these spaces remains an ongoing challenge, driven by real-world applications.Over the past two decades, computational anatomy has provided a strong momentum for both theoretical and numerical advances for mathematicians. Today, the emerging field of spatial transcriptomics offers an unprecedented view of gene expression within the spatial context of tissues, shedding light on cellular organization, tissue architecture, and disease pathology. In this talk, I will discuss some mathematical questions that need to be addressed in this evolving landscape.