arXiv research

Compactness proven for CMC surfaces with bounded topology and boundary length.

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

Enhanced bounds on rho-invariants for 3-manifolds.

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

The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

The paper sets limits on neural network sizes based on dataset shapes.

Lower bound for complexity of finding flex points on cubic curves.

Topological normal generation proved for mapping class groups of certain surfaces.

We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…

In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.

New principles prove precompactness of domains with lower Ricci curvature bound.

Study bounds topological entropy of toroidal attractors.

Constructs flows on manifolds with small curvature, proving Euclidean topology.

Study uses equivariant topology to measure distances between G metric spaces.

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman's theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, co…

Study on scalar curvature bounds and manifold topological complexity.

Study shows local topologies of certain geometric spaces.

Topology helps estimate chromatic numbers of random graphs on spheres.

New findings on knots that are both topologically and rationally slice.

Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.

The paper constructs metrics on tori with Ricci bounds and shows Gromov-Hausdorff limits are not always manifolds.

Lower bound on stretch factor for periodic maps.

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample t…

We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length co…

Proves spheres with bounded curvatures must contain a unit ball.

Curvature of 2D subsets preserved in their space.

Study new bounds on TC of spaces with subgroup inclusions.

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow bo…

Improved lower bounds on volumes of hyperbolic 3-manifolds with specific topologies.

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded …

Using deep analytic methods, Cheeger and Gromov showed that for any smooth (4k-1)-manifold there is a universal bound for the von Neumann $L^2$ $ρ$-invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological (4k-1)-manifolds, using $L^2$-signatures of boun…

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produce…

The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

In established network architectures, shortcut connections are often used to take the outputs of earlier layers as additional inputs to later layers. Despite the extraordinary effectiveness of shortcuts, there remain open questions on the mechanism and characteristics. For example, why are shortcuts powerful? Why do sh…

Proves conjecture on graph configuration spaces' complexity.

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4-genus and the Seifert genus …

We characterize functions which are growth types of Riemannian manifolds of bounded geometry.

In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.

Identifies submanifolds as topological spheres in hyperbolic space.

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the nature of optimal upper bounds in real fewnomial theory. We use a powerful recen…

Study shows equality in Hodge Laplacian bound occurs only on spheres.

Study probabilistic category and complexity bounds, comparing with classical invariants.

In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…