Shmuel Weinberger

Learn More
Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an(More)
We construct examples of nonresolvable generalized «-manifolds, n > 6 , with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed «-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology. By a generalized n-manifold we will mean a finite-dimensional(More)
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the(More)
This paper is devoted to large scale aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a mix of tools from logic/computer science, and differential geometry and topology, we study the diameter functional and its critical points,(More)
The past couple of decades has seen significant progress in the theory of stratified spaces through the application of controlled methods as well as through the applications of intersection homology. In this paper we will give a cursory introduction to this material, hopefully whetting your appetite to peruse more thorough accounts. In more detail, the(More)
We prove that the answer to the ”zero-in-the-spectrum” conjecture, in its form, suggested by J. Lott, is negative. Namely, we show that for any n ≥ 6 there exists a closed n-dimensional smooth manifold Mn, so that zero does not belong to the spectrum of the Laplace-Beltrami operator acting on the L forms of all degrees on the universal covering M̃ . 1. The(More)
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the(More)
Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2, K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds(More)