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This talk focuses on two instances in which scientific fields
outside mathematics benefit from incorporating the geometry of the data.
In each instance, the applications 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 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 lies 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).