arXiv research

We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension.

The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper metric space and extension dimension of its Higson corona.

Paper defines Bartnik mass for hyperbolic extensions and proves staticity.

Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.

Study optimal holomorphic extensions for jets along submanifolds as tensor powers increase.

Metrics are semipositively curved if they meet a specific asymptotic condition.

Existence proved for static vacuum extensions near Schwarzschild spheres.

This paper constructs charged Riemannian manifolds to test Penrose inequality.

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, C. -Y. Lin used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension and C. Sormani and Lin proved useful results with this …

Study on finiteness property of right-angled Artin groups actions on extension graphs.

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …

Study the geometry of graph product extension graphs.

We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik…

We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to com…

Unified framework for response-adaptive targeting in multi-treatment experiments

The study connects norms and filtrations on section rings of projective manifolds.

Study of torsion forms for positive line bundles.

Study optimal holomorphic extensions on complex manifolds with transitivity property.

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their…

We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group $G$ with a …

Develops a new risk measure for Markov chains' asymptotic behavior.

We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asympt…

Paper proves rigidity of static manifolds and applies to metric extensions.

Extends Floquet-Bloch theory to nilpotent groups for geometric applications.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotica…

Paper extends Chernoff sampling for active testing and parameter estimation, improving neural network and regression models.

We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subcl…

We consider a planar surface Σof infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σand prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the …

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As a…

New bound for group action length without diameter restriction.

Simple proof for sphere mass calculation.

We prove that graph products constructed over infinite graphs with bounded clique number preserve finite asymptotic dimension. We also study the extent to which Dranishnikov's property C, and Dranishnikov and Zarichnyi's straight finite decomposition complexity are preserved by constructions such as unions, free produc…

We study an asymptotic Dirichlet problem for Weyl structures on asymptotically hyperbolic manifolds. By the bulk-boundary correspondence, or more precisely by the Fefferman-Graham theorem on Poincaré metrics, this leads to a natural extension of the notion of Branson's $Q$-curvature to Weyl structures on even-dimension…

Abstract: Necessary conditions for stabilizing subsets in systems are found.

The Singular Asymptotics Lemma by Brüning and Seeley and the Push-Forward Theorem by Melrose lie at the very heart of their respective approaches to singular analysis. We review both and show that they deal with the same basic problem, giving solutions that emphasize different aspects of it. This also points to a possi…

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is…

Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.

We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if $G$ and $H$ are groups with asymptotic property C, then both $G \times H$ and $G * H$ have asymptotic property C. We also prove that a group~$G$ has asymptotic property C i…

Study shows decay of correlations on specific types of flows.

We construct asymptotically flat, scalar flat extensions of Bartnik data $(Σ, γ, H)$, where $γ$ is a metric of positive Gauss curvature on a two-sphere $Σ$, and $H$ is a function that is either positive or identically zero on $Σ$, such that the mass of the extension can be made arbitrarily close to the half area radius…

Study on solutions near isolated singularities in 6D Yamabe equation.

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere. A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik …

The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The comp…

We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. The construction uses an extension to asymptotically cylindrical setting of Joyce's existence result for torsion-free Spin(7)-structures. One source of examples ar…

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…