This talk focuses on two instances in which scientific fields outside mathematics benefit from incorporating the geometry of the data. In each instance, the application area motivates the need for new mathematical approaches and algorithms and leads to interesting new questions. (1) A method to determine and predict drug treatment effectiveness for patients based off their baseline information. This motivates building a function adapted diffusion operator on high-dimensional data X when the function F can only be evaluated on large subsets of X, and defining a localized filtration of F and estimation values of F at a finer scale than it is reliable naively. (2) The current empirical success of deep learning in imaging and medical applications, in which theory and understanding is lagging far behind. By assuming the data lie near low-dimensional manifolds and building local wavelet frames, we improve on existing theory that breaks down when the ambient dimension is large (the regime in which deep learning has seen the most success).