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 surface of a sphere is an example of a 2-dimensional manifold.

More generally, for any natural number $k$, a $k$-dimensional manifold is a topological space $X$ such that any neighorhood of any point of $X$ is homeomorphic to an open set of $\mathbb{R}^{k}$ with the Euclidean metric.
A generalization to infinite dimension

Let $\mathbb{R}^{\mathbb{N}}$ be the set of infinite sequences $\left(x_{i}\right)$, where each $i$ is an element of $\mathbb{N}$ and $x_{i}$ is an element of $\mathbb{R}$. We extend the set of the real numbers $\mathbb{R}$ by ajoining two elements denoted by $-\infty$ and $\infty$, which satisfy these conditions:

for each element \$x...