arXiv research

The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.

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…

The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.

Hodge decomposition extended to $H^1$ space on non-compact manifolds.

Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be $k$-manifolds $(k = 1, 2)$, which generalize characterizations in the codimens…

Decomposition theory explores topological spaces and their quotient spaces.

The aim of this paper is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this paper, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) deco…

Study classifies equidistant decompositions in 2D spaces.

We extend cell decomposition to moduli space of convex projective structures.

In this paper an extended CPR decomposition theorem for Finsler symmetric spaces of semi-negative curvature in the context of reductive structures is proven. This decomposition theorem is applied to give a geometric description of the complexification of some infinite dimensional homogeneous spaces.

Extends geometric decompositions to arbitrary meshes and forms.

Examines respectful decompositions of Lie algebras.

Generalized Robertson-Walker (GRW) spaces constitute a quite important family in Lorentzian geometry, and it is an interesting question to know whether a Lorentzian manifold can be decomposed in such a way. It is well known that the existence of a suitable vector field guaranties the local decomposition of the manifold…

Study of quaternionic hyperbolic space bisectors and their decompositions.

The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…

We introduce Fenchel-Nielsen coordinates on Teicmüller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geom…

This paper and its companion arXiv:1002.4564 have been replaced by arXiv:1602.05139. We give a general simple definition of JSJ decompositions by means of a universal maximality property. The JSJ decomposition should not be viewed as a tree (which is not uniquely defined) but as a canonical deformation space of trees. …

We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (of genus $g$ hyperbolic surfaces with $n$ boundary components) by lengths of $6g-6+3n$ geodesics. We find a large family of suitable sets of $6g-6+3n$ geodesics, each set forming a special structure called "admissible double pants decomposition". For …

Metric spaces uniquely split into Hilbert and non-line-split parts.

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, a…

Study cohomotopy sets of simply connected 7-manifolds using suspension decompositions.

Classifies CR submanifolds in complex hyperbolic spaces.

We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…

New algorithm improves dynamic mode decomposition for high-dimensional data.

In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the r…

This paper studies compactifications of moduli spaces involving closed Riemann surfaces. The first main result identifies the homeomorphism types of these compactifications. The second main result introduces orbicell decompositions on these spaces using semistable ribbon graphs extending the earlier work of Looijenga.

New upper bound for geodesic complexity derived from cut locus decompositions.

The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.

CDFD analyzes circularity and directionality in weighted directed networks.

The study describes handle decompositions and Kirby diagrams for line arrangements.

The paper describes decompositions of geometric measures on Anosov homogeneous spaces.

Enhanced loop space decomposition for specific Poincaré complexes.

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincaré inequality, the space $\Mone$, defined by Hajłasz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homoge…

This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

Study on $L^p$ cohomology and Hodge decomposition for ALE manifolds.

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…

Abstract: Rational decomposition of homeomorphism spaces for manifolds.

In this paper we study the Föllmer-Schweizer decomposition of a square integrable random variable $ξ$ with respect to a given semimartingale $S$ under restricted information. Thanks to the relationship between this decomposition and that of the projection of $ξ$ with respect to the given information flow, we characteri…

Paper identifies key function spaces for ReLU networks based on Fisher information.

New tensor network decompositions improve CNN performance.

The abstract discusses applications of Menke's JSJ decomposition to symplectic fillings of various 3-manifolds.

New conditions for GRW space-times to be perfect-fluid space-times.

Study homotopy groups of open books and their pages, pages, and bindings.

New method for learning with non-Euclidean data using decomposable kernels.