Eugene Stepanov: On the spectra of squared distance matrices of metric measure spaces

In the given metric space we choose randomly points (independently of each other and with the same distribution law provided by the given probability measure). Once a finite number of points is chosen, we compute the matrix of squared distances between these points. We are interested in what kind of information on the metric space and the measure can be retrieved from the spectral information (e.g. the spectra and eigenvectors/eigenspaces amd/or their functions, e.g. just the signatures) about the latter matrices. Although there is a lot of open questions in this direction, something can be said. For instance just the signatures of the squared distance matrices determine in which pseudo-euclidean space can one naturally embed isometrically the original metric space. We discuss these questions as well as their relationships with the problems of reconstruction of geometrical structures in the (big) data.