OUTLINE Introduction A generalization to infinite dimension Is there any relationship between an infinite-dimensional manifold and a finite-dimensional manifold? Manifolds in applications Infection dynamics Topological data analysis Riemannian manifold optimization software library SUMMARY: We define a manifold of an infinite dimension, which is an extension of a finite-dimensional manifold, and we show in what sense an infinite-dimensional manifold is related to any finite-dimensional manifold. We then mention the use of manifolds in applications. Introduction Given any point $x$ on the surface of a sphere and any circle drawn around $x$, the region inside the circle approaches the shape of a 2-dimensional ``flat'' disk as the circle gets smaller. The…