When searching for examples satisfying certain geometric properties, it is often convenient to examine manifolds constructed by gluing simple pieces together. One common example of such a construction involves gluing disk bundles together along their common boundary. On the other hand, many geometric phenomena impose strong topological conditions on the underlying manifold, such as the...
The configuration space U(X) over a base space X is the space
of all locally finite point measures on X. The space U(X) being equipped
with the vague topology, the L^2-transportation distance and a point
process, it is a Polish extended metric measure space. In this talk, we
show that U(X), equipped with the Poisson point process, satisfies
synthetic lower Ricci curvature bounds if and...
A hyperbolic surface is a surface, in the intuitive sense, with a geometry that is negatively curved at every point with the same curvature (-1) everywhere. These are not easy to visualize, but there are many of them.
Two interesting things to study on a hyperbolic surface are the dynamics of the geodesic flow (classical mechanics) and the Laplacian differential operator (quantum...
Persistence diagrams are fundamental objects in topological data analysis. They are pictorial representations of persistence homology modules, which in turn describe topological features of a data set when viewed at different scales or levels, i.e. along a filtration. However, given a data set, it is possible to obtain different persistence diagrams depending on the filtration, so it is...
For compact negatively curved Riemannian manifolds, a classical object of study is the exponential growth rate of closed geodesics, which is the same as the exponential growth rate of volume in the universal cover, and also equal to the topological entropy of the geodesic flow. In this talk, we discuss the question of the growth rate of closed geodesics in noncompact regular covering manifolds...
