I should let you in a revolutionary point of view proposed by I.M. Gelfand more than seven decades ago. More precisely he observed that a compact topological $X$ space is completely determined by the algebra $C(X)$ of continuous complex valued functions on it. (This is a commutative Banach algebra, but I will not dwell on this, referring you instead to this Wikipedia article.) He noticed that the space $X$ can be identified as a set with the set of maximal ideal of $C(X)$, called the *maximal spectrum* of $C(X)$. This spectrum can then be equipped with a natural topology making it homeomorphic to $X$.

The point of this result is that you can *read* the topology of $X$ from the space of continuous functions on $X$. Moreover any Banach algebra morphism $T:C(X)\to C(Y)$ is determined by a continuous map

$$F: Y\to X. $$

More precisely $Tu= u\circ F$, $\forall u\in C(X)$.

This point of view lead to the development of schemes by Grothendieck and to the creation of non-commutative geometry by Alain Connes.

If you are interested in more refined properties of the space $X$, then you need to add additional structure to the ring of functions on $X$. If for example, $X$ is a compact submanifold of some Euclidean space $\mathbb{R}^n$, then $X$ is equipped with a Riemann metric giving it some shape (think ellipsoid vs. round sphere). The metric on $X$ defines a natural (unbounded) operator on $L^2(X)$, the Laplace-Beltrami operator in the paper you quote.

It can be proved that the metric on $X$, hence its shape, is completely determined by the spectral decomposition of $L^2(X)$ determined by this operator.