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How Riemannian manifolds converge
 PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON METRIC AND DIFFERENTIAL GEOMETRY IN TIANJING AND BEIJING
, 2010
"... This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of subm ..."
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This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: GromovHausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence.
How Riemannian Manifolds Converge: a Survey
 Progress in Mathematics
, 2010
"... ABSTRACT: This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequence ..."
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ABSTRACT: This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: GromovHausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence. 1.
Conjugate Points in Length Spaces
, 709
"... In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting ..."
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In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We next focus on CBA(κ) spaces, proving Rauchtype comparison theorems. In particular, much like the Riemannian setting, we prove that there are no ultimate conjugate points less than π apart in a CBA(1) space. We also prove a relative Rauch comparison theorem to estimate the distance between nearby geodesics. We close with applications and open problems.
Existence and nonexistence of halfgeodesics on S2
"... In this paper we study halfgeodesics, those closed geodesics that minimize on any subinterval of length l(γ)/2. For each nonnegative integer n, we construct Riemannian manifolds diffeomorphic to S2 admitting exactly n halfgeodesics. Additionally, we construct a sequence of Riemannian manifolds, ea ..."
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In this paper we study halfgeodesics, those closed geodesics that minimize on any subinterval of length l(γ)/2. For each nonnegative integer n, we construct Riemannian manifolds diffeomorphic to S2 admitting exactly n halfgeodesics. Additionally, we construct a sequence of Riemannian manifolds, each of which is diffeomorphic to S2 and admits no halfgeodesics, yet which converge in the GromovHausdorff sense to a limit space with infinitely many halfgeodesics.
My research is a combination of Classical Riemannian Geometry, Metric Geometry, Geometric
"... Manifolds. In this description, I begin with my early work and then organize the rest by topic. 1 Thesis and Postdoc Years: I completed my dissertation with Jeff Cheeger at Courant in 1996 and then had a one year postdoc with ShingTung Yau at Harvard. At this time I focused on complete noncompact R ..."
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Manifolds. In this description, I begin with my early work and then organize the rest by topic. 1 Thesis and Postdoc Years: I completed my dissertation with Jeff Cheeger at Courant in 1996 and then had a one year postdoc with ShingTung Yau at Harvard. At this time I focused on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Yau proved that such manifolds have at least linear volume growth [Yau1]: V ol(Bp(R)) lim inf = V0> 0. (1) R→ ∞ R Bishop proved they have at most Euclidean volume growth [Bi1]. Cheeger and Colding had just begun working together on their notion of almost rigidity, in which they proved, among other things, that annular regions in Riemannian manifolds with nonnegative Ricci curvature and almost Euclidean volume growth were close in the GromovHausdorff sense to cones over spheres [ChCo1]. Their technique involved studying the Laplacian of functions of the distance function and constructing explicit maps to the cones. In my thesis, I applied their technique to Busemann functions which are defined using rays, γ, as follows: bγ(x) = lim R − d(x, γ(R)). (2) R→∞ I proved that if a Riemannian manifold with nonnegative Ricci curvature has linear volume growth: lim sup
Measuring the stability of the universe
"... When using partial differential equations to determine the heat in a region based upon initial data and boundary conditions, the precision in the resulting data is as good as the precision in the measurements in the initial conditions. This is because the heat equation is a stable differential equat ..."
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When using partial differential equations to determine the heat in a region based upon initial data and boundary conditions, the precision in the resulting data is as good as the precision in the measurements in the initial conditions. This is because the heat equation is a stable differential equation. Predicting weather, on the other hand, is far less reliable because the equations are unstable. The famous butterfly effect occurs. There are also questions of how one measures stability: what does it mean for the data to be close? When using Fourier series, for example, one does not get uniform approximations of data. Instead the L2 norm is used to measure proximity: roughly this is the area between the approximated function and it’s truncated Fourier series. Spikes may occur as observed in the Gibbs Phenomenon. These spikes interfere with signal processing and must be controlled. Geometric Analysis and Measuring Stability In Geometric Analysis we are concerned not only with the values of functions but with the distortion of spaces called manifolds. When Einstein’s equations are used to predict the behavior of the universe, a galaxy or the solar system, they are predicting the curvature of the universe, the distance between points and the paths of geodesics (or rays of light). One must immediately