The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
In many problems arising in the computational sciences datasets come equipped with some underlying notion of similarity. Metric geometry provides a natural point of view which is applicable in the realm of non-smooth data.
One idea is to represent datasets as metric measure spaces and then, at the same time, regard the collection D of all datasets as a metric space in itself. The Gromov-Wasserstein distance --a variant of the Gromov-Hausdorff distance based on ideas from mass transport-- provides an intrinsic metric on D which exhibits non-negative curvature in the Alexandrov sense.
I will describe the construction of the GW metric, its properties, and discuss how the representation of datasets as metric measure spaces gives rise to a number of stable invariants which are counterparts to concepts emerging in algebraic topology and differential geometry, including analogues to homology and notions of curvature that are valid beyond the smooth case.
Abstract: In this talk we will discuss some new boundary regularity estimates for Riemannian manifolds with boundary that come from control over integral norms of various curvature quantities. This leads to a new collection of compactness theorems. As an application, we are able to prove some geometric stability theorems for Riemannian manifolds with boundary. One such theorem allows us to give a partial answer to the following question. Suppose a convex 3-manifold with boundary has intrinsic boundary close to the round sphere, and suppose further that the ambient curvature satisfies a smallness condition on its square integral norm. Is its geometry close to the geometry of the unit Euclidean ball?
A simplicial complex is a collection of vertices, edges, triangles, and simplexes of higher dimensions, and one can think of it as a generalization of a graph. In a geometric complex, we start with a set of vertices in a metric space, and add higher dimensional faces according to certain geometric properties of these vertices. Choosing vertices at random thus yields a random topological space with many interesting features. We wish to study the homology of such spaces, and in particular - their Betti numbers (number of connected components and 'holes' or 'cycles').
In this talk we present a few ways to construct a simplicial complex, and discuss the limiting behavior of these complexes as the number of vertices goes to infinity. We will show a wide variety of phenomena related to the appearance and vanishing of homology, and to the limiting distribution of the Betti numbers. Finally, we will also discuss the relevance of these results to topological manifold learning.
The problem of isometrically immersing a given Riemannian surface into Euclidean 3-space is locally solvable in the real-analytic category, but aside from topological restrictions little is known about the existence of global embeddings or the size of the space of non-rigid deformations a given surface has. We analyze the problem as an exterior differential system, and find that there are 4 distinct metrics (up to scale) for which the system is Darboux-integrable. This means that, in theory, all isometric immersions for these metrics can be produced by solving linear systems of ODE. For three of these metrics (those with positive curvature) there is also a Weierstrass-type representation, so that immersions can be determined by quadrature given a holomorphic curve on an affine quadric in C^3.
This is joint work with Jeanne Clelland, Ben McKay and Peter Vassiliou.