Abstract The de Rham-Hodge theory is a landmark of the 20 th Century's mathematics and
has had a great impact on mathematics, physics, computer science, and engineering. This …

Relationalism about space is a venerable doctrine that is enjoying renewed attention among
philosophers and physicists. Relationalists deny that space is ontologically prior to matter …

We relate L p convergence of metric tensors or volume convergence to a given smooth
metric to intrinsic flat and Gromov–Hausdorff convergence for sequences of Riemannian …

The investigation of the Hamiltonian dynamical counterpart of phase transitions, combined
with the Riemannian geometrization of Hamiltonian dynamics, has led to a preliminary …

Different arguments led us to surmise that the deep origin of phase transitions has to be
identified with suitable topological changes of potential-related submanifolds of …

In this thesis we study issues related to the convergence theory of Riemannian manifolds
with boundary. First, we establish a compactness theorem for the class of compact, uniformly …

The positive mass theorem states that the total mass of a complete asymptotically flat
manifold with nonnegative scalar curvature is nonnegative; moreover, the total mass equals …

We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets
of manifolds and orbifolds of bounded curvature. The distortion and dimension of such …

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and
the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic …

This dissertation explores in particular a few ways in which similarity bears on general
relativity, our best physical theory of space and time. The first way concerns the role of the …