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The primary theme of this talk is geodesics. First I’ll introduce the Calculus of Variations and explain the variational approach to geodesics from Riemannian geometry. A novel result on the length of the shortest closed geodesic on 2-spheres will be presented. Next, I’ll consider these ideas in a data setting, where data diffusion has been a powerful tool for studying manifold geometry. I’ll discuss NeuralFIM, a diffusion-based method that learns a differentiable representation of the data, allowing for computation of the Fisher Information Metric (FIM). One can then use this Riemannian metric to compute volumes and geodesics on the data manifold. NeuralFIM is joint work with the Krishnaswamy Lab.