For a given null-cobordant Riemannian n-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n.This construction relies on another of independent interest. Take X and Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces: for example, any simply connected Lie group. Then two homotopic L-Lipschitz maps f, g : X → Y are homotopic via a CL-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y .A beautiful example of this paradoxical state of affairs is the result of Nabutovsky that despite the result of Smale (proved inter alia in the proof of the high-dimensional Poincaré conjecture) that every smooth codimension one sphere in the unit n-disk (n > 4) can be isotoped to the boundary, the minimum complexity of the embeddings required in the course of such an isotopy (measured by how soon normal exponentials to the embedding intersect) cannot be bounded by any recursive function of the original complexity of the embedding. Effectively, an easy isotopy would give such a sphere a certificate of its own simple connectivity, which is known to be impossible.In other situations, such as those governed by an h-principle, a hard logical aspect of this sort does not arise. In this paper we introduce some tools of quantitative algebraic topology which we hope can be applied to showing that various geometric problems have solutions of low complexity.As a first, and, we hope, typical example, we study the problem, emphasized by Gromov, of trying to understand the work of Thom 1 on cobordism. Given a closed smooth (perhaps oriented) manifold, the cobordism question is whether it bounds a compact (oriented) manifold. The answer to this is quite checkable: it is determined by whether the cycle represented by the manifold in the relevant (i.e. Z or Z/2Z) homology of a Grassmannian (where the manifold is mapped in via the Gauss map classifying the manifold's stable normal bundle) is trivial.This raises two questions: the first is how the geometry of a manifold is reflected in the algebraic topological problem, and second is, how difficult is it to find the nullhomotopy predicted by the algebraic topology. As a test of this combined problem, Gromov suggested the following question: Given a manifold, assume away small scale problems by giving it a Riemannian metric whose injectivity radius is at least 1, and whose sectional curvature is everywhere between −1 and 1. These properties can be achieved through a rescaling. A manifold possessing these properties will be said to have bounded local geometry. The geometric complexity of such a manifold can be measured by its volume.If M is a smooth compact manifold, without a specified me...

We completely characterize isoperimetric regions in R n with density e h , where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.

During the early stage of biomarker discovery, high throughput technologies allow for simultaneous input of thousands of biomarkers that attempt to discriminate between healthy and diseased subjects. In such cases, proper ranking of biomarkers is highly important. Common measures, such as the area under the receiver operating characteristic (ROC) curve (AUC), as well as affordable sensitivity and specificity levels, are often taken into consideration. Strictly speaking, such measures are appropriate under a stochastic ordering assumption, which implies, without loss of generality, that higher measurements are more indicative for the disease. Such an assumption is not always plausible and may lead to rejection of extremely useful biomarkers at this early discovery stage. We explore the length of a smooth ROC curve as a measure for biomarker ranking, which is not subject to directionality. We show that the length corresponds to a divergence, is identical to the corresponding length of the optimal (likelihood ratio) ROC curve, and is an appropriate measure for ranking biomarkers. We explore the relationship between the length measure and the AUC of the optimal ROC curve. We then provide a complete framework for the evaluation of a biomarker in terms of sensitivity and specificity through a proposed ROC analogue for use in improper settings. In the absence of any clinical insight regarding the appropriate cutoffs, we estimate the sensitivity and specificity under a two-cutoff extension of the Youden index and we further take into account the implied costs. We apply our approaches on two biomarker studies that relate to pancreatic and esophageal cancer.

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Abstract. We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + . If the manifold is orientable, then for any > 0 we show that, if we can contract a curve γ traversed twice through curves of length bounded by L, then we can also contract γ through curves bounded in length by L + .Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.