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Applications of gram matrices of polytopes


Research objectives and content
To study, by using the space of Gram matrices, whether the dihedral angles determine hyperbolic polyhedra up to isometry. Also if there is some local or infinitesimal rigidity.
To study some geometric properties of the space of dihedral angles, such as some special kind of convexity.
To study the complexity of our characterization of Gram matrices in order to give a practical criterion to decide whether a combinatorial sphere is polytopal or not.
To study whether a metric on the topological sphere can be realized as the induced metric under a convex embedding of the sphere in some geometric space.
To apply our techniques to some aspects of Kleinian groups, for instance, how the geometry of the convex hull boundarie determines the group. Training content (objective, benefit and expected impact)
In order to develop the above objectives I would need to acquire new techniques, in which there are experts at the University of Warwick. With this, I expect to considerably develop the above objectives. Links with industry / industrial relevance (22)


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