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 C ␣ metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the neighborhoods that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on novel estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient, that hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications. spectral geometry ͉ nonlinear dimensionality reduction I n many recent applications, one attempts to find local parametrizations of data sets. A recurrent idea is to approximate a high dimensional data set, or portions of it, by a manifold of low dimension. A variety of algorithms for this task have been proposed (1-8). Unfortunately, such techniques seldomly come with guarantees on their capabilities of indeed finding local parametrization (but see, for example, refs. 8 and 9) or on quantitative statements on the quality of such parametrizations. Examples of such disparate applications include document analysis, face recognition, clustering, machine learning (10-13), nonlinear image denoising and segmentation (11), processing of articulated images (8), and mapping of protein energy landscapes (14). It has been observed in many cases that the eigenfunctions of a suitable graph Laplacian on a data set provide robust local coordinate systems and are efficient in dimensional reduction (1,4,5). The purpose of this paper is to provide a partial explanation for this phenomenon by proving an analogous statement for manifolds as well as introducing other coordinate systems via heat kernels, with even stronger guarantees. Here, we should point out the 1994 paper of Bérard et al. (15) where a weighted infinite sequence of eigenfunctions is shown to provide a global coordinate system. (Points in the manifold are mapped to ᐉ 2 .) To our knowledge, this was the first result of this type in Riemannian geometry. If a given data set has a piece that is statistically well approximated by a low dimensional manifold, it is then plausible that the graph eigenfunctions are well approximated by the Laplace eigenfunctions of the manifold. One of our results is that, with the normalization that the volume of a d-dimensional manifold M equals one, any suitably embedded ball B r (z) in M has the property that one can find (exactly) d eigenfunctions that are a ''robust''coordinate system on B cr (z) (for a constant c depending on elementary properties of M). In addition, these eigenfunctions, which depend on z and r, ''blow up'' the ball B cr (z) to diameter at least one. In other words, one can find d eigenfunctions ...

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 so by extending and improving results of Peter Jones and Kate Okikiolu for sets in R d . Their results formed the basis of quantitative rectifiability in R d . We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales around most points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analog of the Square function. This allows us to conclude that for a given set K, the ℓ 2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90,Oki92] are independent of the ambient dimension. Mathematics Subject Classification (2000): 28A75

We prove a global implicit function theorem. In particular we show that any Lipschitz map f : R n × R m → R n (with n-dim. image) can be precomposed with a bi-Lipschitz mapḡ :such that f •ḡ will satisfy, when we restrict to a large portion of the domain E ⊂ R n × R m , that f •ḡ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the mapḡ distorts R n+m in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a C 1 map from [0, 1] 3 onto [0, 1] 2 with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function f : [0, 1] n → R D gives rise to a large (in an appropriate sense) subset E ⊂ [0, 1] n such that f | E is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of R n . This extends results of P. Jones and G. David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors-David regular spaces lying in R D having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in R D . This was previously known only for D ≥ 2n+1 by a result of G. David and S. Semmes.Mathematics Subject Classification (2000): 53C23 54E40 28A75 (42C99)

We show that if a subset K in the Heisenberg group (endowed with the Carnot-Carathéodory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of [FFP07] except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of [Sch07b] as well as a new curvature inequality in the Heisenberg group.

A measure is 1-recti able if there is a countable union of nite length curves whose complement has zero measure. We characterize 1-recti able Radon measures µ in n-dimensional Euclidean space for all n ≥ in terms of positivity of the lower density and niteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which µ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between µ and 1-dimensional Hausdor measure H . We also characterize purely 1-unrecti able Radon measures, i.e. locally nite measures that give measure zero to every nite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L variant of P. Jones' traveling salesman construction, which is of independent interest.