| The main focus of this thesis is on the geometry of polyhedral partitions of Euclidean spaces and other orientable manifolds and its connections to rigidity theory, convexity, splines, and stochastic geometry. In the first part of the work we introduce a general notion of stress on cell-complexes and explore connections between stresses and liftings (generalization of C01 -splines) for d-dimensional manifolds realized in Rd . For example, we offer new sufficient conditions for the existence of a sharp lifting for a "flat" piecewise-linear realization of a manifold. As an application, two algorithms are given that determine whether a piecewise-linear realization of a d-manifold in Rd admits a lifting to Rd+1 which satisfies given constraints. In the thesis we discuss two generalizations of the famed Maxwell correspondence between stresses on planar frameworks and projections of spatial polyhedra. The former says that the spaces of liftings and d-stresses of a manifold are isomorphic under rather general conditions on the topology of the manifold. The latter is a partial analog of the Maxwell correspondence for spatial frameworks. We also demonstrate connections between stresses and Dirichlet-Voronoi diagrams. In the probabilistic part of the thesis we investigate geometric bootstrap percolation models suggested by Connelly for describing the rigidity/flexibility properties of molecular systems. In these models local rules are of geometric nature as opposed to simple counts used in standard bootstrap percolation models. Both models deal with a relaxation of tension in a 2-dimensional medium. We also discuss possible applications of these results to mathematical chemistry. As a consequence of Maxwell correspondence these models can also be interpreted in terms of the geometry of convex surfaces and polygonal tilings. We find the exact value of the critical probability for both models. In fact, we obtain somewhat stronger results showing that in both cases the relaxation of tension occurs in finite time almost surely. |
