CONTENTS

1. Submanifolds of $\mathbb{R}^n$: geometry and optimization problems (gradient descent on submanifolds).
2. Riemannian Geometry: metrics and connections (Fisher metric), vectorfields-differential forms duality (gradient descent v2), geodesics and distances (dimensionality reduction algorithms: ISOMAP, LLE. …)
3. Curvatures: geometric meaning, Laplace-Beltrami operator and its eigenvalues (Eigenmap and diffusion algorithms for dimensionality reduction).
4. Discretization: sampling on manifolds (graphs, triangulations, meshes…), brief overview of Hamiltonian dynamics and Monte Carlo gradient descent.