One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity is fundamentally a geometric theory and, as such, it ought to be invariant under change of coordinates by the arbitrary element of the diffeomorphism group Diff(M) of the ambient manifold M. The Problem of Observables is an instance of the difficulties associated with general covariance and one that relates to our inability to apply classical quantization recipes such as Canonical Quantization. The Problem of Observables asks whether General Relativity admits a complete set of smooth observables. That is, if the family of all smooth, diffeomorphism invariant, real-valued maps on the space Ein(M) of all Einstein metrics on M is rich enough to separate all physical spacetimes. Currently, the only smooth observables known when M=R^4 are constant maps.
In this series of talks, I will illustrate how one can employ methods from Descriptive Set Theory in order to answer the Problem of Observables in the negative. A sample application is the following: there exists an uncountable collection of non-isometric vacuum spacetimes on which every continuous observable is constant. These results are inspired by old discussions with Marios Christodoulou and are based on recent work with George Sparling. No background in Descriptive Set Theory or General Relativity will be assumed.
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