arXiv research

Lower bounds on cone density for nontrivial complements in low dimensions.

Paper compares dimension reduction methods using topological analysis on EEG data.

Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions $5$, $6$, and $7$ and obtain topological characterizations…

We study the topology of smectic defects in two and three dimensions. We give a topological classification of smectic point defects and disclination lines in three dimensions. In addition we describe the combination rules for smectic point defects in two and three dimensions, showing how the broken translational symmet…

Proposes a new metric space example showing non-constant topological dimension.

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of…

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].

This paper classifies 13D manifolds based on Bazaikin spaces.

The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.

New gravitational solitons and infinite topological manifolds found.

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…

Paper estimates neural network size needed for topology learning.

Study numerical invariants under retraction maps between topological spaces.

The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.

Ellipsoids approach Gaussian distribution in high dimensions.

Study topological hyperbolicity of moduli spaces of elliptic surfaces.

The abstract applies waist inequality to dynamical systems and entropy.

The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n >2, as shown by Malament and Hawking-King-McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spa…

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

Connected sums defined for codimension two locally flat submanifolds in higher dimensions.

The paper shows dense and residual sets of continuous maps with positive metric mean dimension.

Develops topological concepts for Morrey-Sobolev bundles in high dimensions.

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…

Paper infers intrinsic dimension from quasi-convex measurements.

This paper evaluates fractal dimension and persistent homology for neural network generalization.

Study on stable Hamiltonian topology finds non-density of certain structures.

This paper introduces two-dimensional diagrams that are slight generalizations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as w…

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

We describe off-shell $\mathcal{N}=1$ M-theory compactifications down to four dimensions in terms of eight-dimensional manifolds equipped with a topological $Spin(7)$-structure. Motivated by the exceptionally generalized geometry formulation of M-theory compactifications, we consider an eight-dimensional manifold $\mat…

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\…

CMS formulation solves Poincare conjecture for all dimensions.

Proves existence and uniqueness of vacuum black hole solutions in higher dimensions.

Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms …

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

Explains research on 3D dynamics and manifold topology.

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are als…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…

In higher dimensions, Schottky spaces have unique topological properties.

Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $…

Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…

FibeRed reduces complex data dimensions while preserving topology.

The Gannon-Lee singularity theorems give well-known restrictions on the spatial topology of singularity-free (i.e., nonspacelike geodesically complete), globally hyperbolic spacetimes. In this paper, we revisit these classic results in the light of recent developments, especially the failure in higher dimensions of a c…

Recent work by physicists on gravity in two dimensions has a natural generalization to four dimensions, formulated in terms of an analogue of Segal's category [defined for the study of conformal field theory].

The goal of this article is to survey recent developments in the theory of contact structures in dimension three.

This paper proposes a new method for automatically selecting the optimal kernel bandwidth in density estimation.