# Research Projects of Domenico Giulini

The research interests of Domenico Giulini and his group mainly focus on conceptual and/or mathematical problems of classical and quantum General Relativity (GR). On the mathematical side they relate to problems in differential geometry, differential topology, algebraic topology, and combinatorial group theory. Recent work in a more physical direction investigates the theoretical basis for existing or planned quantum tests of the Equivalence Principle.

The validity of Einstein's Principle of Equivalence form the basis on which the geometrisation of the gravitational interaction rests. If we believe, as many in the field in fact do, that Quantum Gravity is some form of "Quantum Geometry", it seems mandatory to scrutinise its validity to quantum domains. There are several non-trivial issues associated with this endeavour, one of which is the quest for a proper formulation, or adaptation, of this principle in genuine quantum-mechanical terms. Related recent publications are

In Canonical General Relativity, Einstein's field equations are cast into the form of a Hamiltonian dynamical system with constraints. In many cases, symplectic reduction leads to the cotangent bundle over a space that can be described as Teichmüller Space of Riemannian geometries on some 3-manifold M. It differs from the proper moduli space of 3-dimensional Riemannian structures in that it still supports an action of the mapping-class group of M, which is interpreted as residual gauge symmetry and might give rise to superselection structures. General Relativity does not topologically constrain M and hence it is of interest to study these groups (which are topological but not homotopy invariants) for some variety of M, including reducible ones. More information as well as an extended bibliography is contained in

In classical General Relativity there are basically two paradigmatic sets of exact solutions: Those describing quasi-isolated localised objects, like black holes, which are modelled on asymptotically flat space-times, and cosmological solutions, which are meant to describe the more or less homogeneous Universe on the largest scales. Unlike in linear Newtonian theory, we cannot just superpose solutions to, say, obtain a specific black hole in a specific background cosmology. Corresponding exact solutions are known, but the physical characterisation and interpretation of the local inhomogeneity in terms of proper (quasi-local) geometric quantities is difficult. The simplified case of spherical symmetry may act as a guide, but is itself not entirely trivial if realistic solutions (say with radial currents of mass and heat) are allowed. A review is