arXiv research

We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem

Maps between metric spaces preserve asymptotic dimension up to a constant.

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…

THEOREM. For every prime $p$ and each $n=2, 3, ... \infty$, there is an action of $G=\prod_{i=1}^{\infty}(Z/ pZ)$ on a two-dimensional compact metric space $X$ with $n$-dimensional orbit space. This theorem was proved in [DW: A.N. Dranishnikov and J.E. West, Compact group actions that raise dimension to infinity, Topol…

I give a construction of compact group action on a finite dimensional space Y, whose orbit space is infinite dimensional.

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer …

Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.

Minimal sphere dimension for equivariant embedding of circles.

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wi…

Study shows free group complexes are Cohen-Macaulay of dimension n-1.

The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…

In this paper we disprove a conjecture stated in [4] on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family $\mathcal{C}$ of closed cones and a set $S$ such that every blow-up of $S$ at every point $x\…

Author simplifies and generalizes p-adic integer action construction.

Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. …

Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.

Generalizes Alexandroff's $V^n$-continua to cohomological dimensions.

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of th…

Enhances anomaly detection in high dimensions with pretrained networks.

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvatu…

The paper analyzes bank decisions in a three-step model, focusing on equity and debt raising.

Manifolds with nonnegative Ricci curvature have nilpotent subgroups with specific geometric properties.

Stein fillability of circle bundles over symplectic manifolds is restricted.

In this paper, we discuss the following conjecture raised by Baum-Douglas: For any first-order elliptic differential operator $D$ on smooth manifold $M$ with boundary $\p M$, $D$ possesses an elliptic boundary condition if and only if $\partial [D]$ = 0 in $K_1(\partial M)$, where $[D]$ is the relative $K$-cycle in $K_…

Characterizes statistical complexity of realizable regression in PAC and online learning.

The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.

We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, r…

Paper studies asymptotic dimension and Assouad-Nagata dimension of graphs and surfaces.

Efficient algorithms learn high-dimensional distributions robustly, independent of dimensionality.

Proves weakly non-collapsed RCD spaces are strongly non-collapsed.

Researchers show Hodge numbers modulo m can be achieved by smooth projective varieties.

Proposes a method to enforce nestedness in subspace learning methods.

A new filtration of the spaces of tri-/univalent graphs B_m^u that occur in the theory of finite-type invariants of knots and 3-manifolds is introduced. Combining the results of the two preceding articles, the quotients of this filtration are modeled by spaces of graphs with two types of edges and four types of vertice…

Study shows how a strip's twisting increases at infinity, affecting its spectrum.

Gradient methods struggle with high dimensions in convex optimization.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

The paper proves a function extension on Kähler manifolds.

Study on Gehring link problem and width of bands in curved manifolds.

New examples of austere submanifolds and hypersurfaces with specific curvature properties.

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent …

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferent…

Raising statistical hurdles may not be justified due to data bias.

Markov random fields (MRFs) are difficult to evaluate as generative models because computing the test log-probabilities requires the intractable partition function. Annealed importance sampling (AIS) is widely used to estimate MRF partition functions, and often yields quite accurate results. However, AIS is prone to ov…

Optimizes exp-concave losses with a new risk bound.

Graphs on surfaces have a 2-dimensional large scale structure.

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using th…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension of the Teichmuller space of S. Since closed geodesics with one double point fall …

We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant $s<\infty$ is of infinite…