arXiv research

The study proves unique path lifting properties and their implications on quotient spaces and covering maps.

We characterize the existence of horizontal path lifts for general connections on arbitrary fiber bundles with a new property that also gives fresh insight into linear and $G$-connections.

Proof of wall-crossing formula using spectral networks.

The paper is devoted to introduce some notions extending the unique path lifting property from a homotopy viewpoint and to study their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting property and find all possible relationships between them. Moreover, we supplemen…

We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $π_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?

A new method to rescale ReLU neural networks based on path-lifting.

For a finite simplicial graph $Γ$, let $G(Γ)$ denote the right-angled Artin group on the complement graph of $Γ$. In this article, we introduce the notions of "induced path lifting property" and "semi-induced path lifting property" for immersions between graphs, and obtain graph theoretical criteria for the embedabilit…

Novel geodesic results on affine and Lorentzian manifolds.

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spa…

An inverse limit of a sequence of covering spaces over a given space $X$ is not, in general, a covering space over $X$ but is still a lifting space, i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fi…

Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm P}^1$-structures. In this note, we summarize Thurston's parametrization of $\ma…

The paper proves metrizability and dynamics of Weil bundles.

Generalizes abelianization for framed local systems over surfaces.

Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.

Develops a lifting theory for exponential maps in semi-Riemannian geometry.

Study of gauge theory and parallel transport in Lie 2-group bundles over Lie groupoids.