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Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

- Peter W. Jones, M. Maggioni, R. Schul
- Mathematics, MedicineProceedings of the National Academy of Sciences
- 12 February 2008

TLDR

Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

- R. Schul
- Mathematics
- 28 February 2006

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do… Expand

An analyst’s traveling salesman theorem for sets of dimension larger than one

- Jonas Azzam, R. Schul
- Mathematics
- 9 September 2016

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $$\beta $$β-numbers. These $$\beta $$β-numbers are geometric quantities… Expand

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

- Jonas Azzam, R. Schul
- Mathematics
- 21 May 2011

We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be… Expand

Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space

- R. Schul
- Mathematics
- 22 February 2007

We prove a quantitative version of the following statement. Given a Lipschitz function f from the k-dimensional unit cube into a general metric space, one can decomposed f into a finite number of… Expand

Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian

- Peter W. Jones, M. Maggioni, R. Schul
- Mathematics
- 13 September 2007

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with… Expand

The traveling salesman problem in the Heisenberg group: upper bounding curvature

We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's… Expand

Quantitative decompositions of Lipschitz mappings into metric spaces

- Guy C. David, R. Schul
- Mathematics
- 24 February 2020

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. We prove that it is always possible to decompose the domain of such a mapping into pieces on which… Expand

A doubling measure on R^d can charge a rectifiable curve

- J. Garnett, R. Killip, R. Schul
- Mathematics
- 13 June 2009

For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0.

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

- Matthew Badger, R. Schul
- Mathematics
- 11 February 2016

Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean… Expand

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