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][1].) 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 ca *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][2] of $L^2(X)$ determined by this operator. 







  [1]: http://en.wikipedia.org/wiki/Gelfand_representation
  [2]: http://www3.nd.edu/~lnicolae/RandCrVal_p2.pdf