The main part of the project describes a new isospectral construction technique (Anticommutator Technique), which provides the first isospectral pairs of metrics on the most simple manifolds: on balls and spheres. The most striking examples are constructed on suitable spheres, where one of the members of an isospectral pair is a homogeneous metric, while the other is locally inhomogeneous. This demonstrates the surprising fact that no information about the isometries is encoded in the spectrum of the Laplacian acting on functions. These investigations also extend to the Laplacian spectrum of forms. Related questions are also considered. One of them is construction of Brownian-motion-equivalent spaces (Isothermal Metrics). This equivalence relation is much stronger then the isospectrality property, yet it does not determine the local geometry. The same statement is true regarding the metrics with equivalent density functions (Isodasyc Metrics).
The old argument between Relativity and Quantum Physics is easily discovered in the depth of these questions. In Relativity, the whole Physics is derived from a curved space. Actually, Physics is identified with the complete Geometry of this curved space. Einstein put his idea this way: "There is no such thing as Physics. Everything is Geometry." Contrary to Relativity, the Quantum Physics uses only particular aspects of Geometry such as the spectra of several operators or the Brownian Motion defined by a metric space. Einstein suggested, however, that the Brownian Motion may determine the complete local geometry. This reflects the extent of the confusion about the following question: "How much Geometry is used by the Quantum Physics?" The proposed investigations demonstrate, for the first time, how little information about Geometry is used by Quantum Physics. For instance, Quantum Physics completely ignores the isometries of the spaces, which otherwise form the central piece of a theory developed in Geometry.
