Brief description:
This course will examine questions arising from the hypothesis that data lies on a manifold. Special emphasis will be placed on learning Riemannian metrics and studying the Laplace Beltrami Operator. Time permitting, we will also consider the question of generating Lie groups using generators in an efficient way, where efficiency refers to the rate at which approximation error diminishes.

References:
1) A graph discretization of the Laplace-Beltrami operator Dmitri Burago, Sergei Ivanov, Yaroslav Kurylev,  J. Spectr. Theory 4 (2014), no. 4, 675-714.
2) C. Fefferman, S. Ivanov, Y. Kurylev, M. Lassas, H. Narayanan. Reconstruction and interpolation of manifolds I: The geometric Whitney problem. arXiv:1508.00674
3) (J. Bourgain and  A. Gamburd) On the spectral gap for finitely generated subgroups of SU(2), Invent. Math. 171 (2008), no. 1, 83–121.