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Abstract
This dissertation presents three different problems in recovering underlying structure from data in diverse scientific fields, each also elucidating a different flavor of applied mathematical approaches. The first is a problem in manifold learning inspired by cryo-electron microscopy. We consider the problem of recovering a one-dimensional curve embedded in high dimensional space from a high noise observations centered around the curve. We prove a sample complexity lower bound to demonstrate an inherent difficulty in recovering the underlying curve from data, then provide an asymptotically optimal algorithm based on the method of moments. This work demonstrates how analytic and algebraic techniques can be applied to summary statistics to navigate the challenges of high dimensional, noisy data. The second problem we study is an optimal transport problem on discrete spin systems. We formulate a relaxation of the Glauber dynamics on the Ising model as an optimal transport problem in discrete time and space with an exponentially large number of variables. We then devise numerical techniques for tractably solving this algorithm at the scale of over one billion variables. This work highlights numerical and algorithmic approaches in applied mathematics. Finally, we move to the world of computational chemistry and develop a machine learning framework that simultaneously serves as a generative and foundational model for 3D conformers of small molecules. Our model is able to synthesize information across a large library of molecules and effectively learn what the manifold of plausible molecular conformers looks like, as well as learn mappings from conformers to molecular properties of interest. At the core of our method is a novel molecular representation that models conformers not as discrete point clouds but as dense fields. This work exemplifies the role of data-driven techniques in complex scientific domains. We hope this dissertation represents the rich diversity of domains and techniques in applied mathematics.