arXiv research

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.

A 3D space of hyperbolic manifolds is connected but not path-connected.

Proves bijection between smooth conformal immersions and immersions.

Study shows almost complex structures with certain tensor properties are prevalent.

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature paths for points in the tangent bundle of the Euclidean plane. We show the exis…

Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geod…

The paper extends symplectic techniques to generalized complex geometry.

The existence of local bases in which the components of derivations of tensor algebras over a differentiable manifold vanish along paths is proved. The holonomicity of these bases is investigated. The obtained results are applied to the case of linear connections. Some relations with the equivalence principle are shown…

Choose two points in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at $x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected component of the space of paths…

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

We investigate the validity of the equivalence principle along paths in gravitational theories based on derivations of the tensor algebra over a differentiable manifold. We prove the existence of local bases, called normal, in which the components of the derivations vanish along arbitrary paths. All such bases are expl…

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact three-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational…

Study on moduli spaces of negatively curved metrics on surfaces.

The study characterizes Nash maps between semialgebraic sets and their properties.

We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph with few connected components; by exploiting prior knowledge, one can indeed improve the prediction performance or o…

Computes sections of a submersion and applies to evasion path problem.

The study examines mean curvature flow and Heegaard surfaces in lens spaces.

This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these proper…

Authors create déjà vu links in Legendrian geometry.

The paper examines ellipticity of specific equations on vector bundles.

Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.

BWFlow improves graph generation by smoothly interpolating graph components.

The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case o…

Study path spaces and their homology, extending loop products and coproducts.

Obstructions found for closed Fedosov star products on symplectic and Kähler manifolds.

The study examines how gamma positivity and PL homeomorphism types affect simplicial spheres.

The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on S^n bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics w…

Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.

Partial covariance factorizes in path diagrams, simplifying analysis.

Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Path signatures reveal community structure in coupled oscillators' dynamics.

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show that the spaces $\mathcal{O}_d$ and $\mathcal{P}_d$ are path connected and the …

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components …

The paper shows non-aspherical path components in G2-moduli spaces.

The paper defines a path metric on a stable component of polynomial families.

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

Path-connectivity of thick laminations on high-genus surfaces.

ScoreMatchingRiesz improves debiased machine learning and policy effects estimation.

Space of hyperbolic surfaces is path-connected.

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$ preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq \infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology…

Analyzes geodesic lengths in sparse networks, deriving a distribution.

Characterizes character varieties of generalized torus knot groups.

We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least one marked point, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.

Classifies 3D manifolds with specific structures and automorphisms.

Let $\mathcal H_c(M)$ stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold $M$. It is shown that $\mathcal H_c(M)$ is perfect and simple under mild assumptions on $M$. Next, conjugation-invariant norms on $\H_c(M)$ are considered and the boundedness of $\m…

Geodesics connect model modes in neural network loss landscapes.

We study the topology of the space of harmonic maps from $S^2$ to \CP 2$. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to$\CP n$for$n\geq 2$. We show that the components …