arXiv research

This article announces the completion of the classification of rank 4 locally projective polytopes and their quotients. There are seventeen universal locally projective polytopes (nine nondegenerate). Amongst their 441 quotients are a further four (nonuniversal) regular polytopes, and 152 nonregular but section regular…

Study of line congruences for Appell's rank-4 hypergeometric functions.

A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.

Found a basis and presentation for a specific algebra.

A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge s…

Study of singular curves in a specific type of hyperbolic distribution.

The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.

New method classifies special Vinberg cones of rank 4.

The study explores properties and mutations in oriented matroids, proving new results on Euclidean and non-Euclidean structures.

We introduce a construction turning some Coxeter and Davis realizations of buildings into systolic complexes. Consequently groups acting geometrically on buildings of triangle types distinct from $(2,4,4)$, $(2,4,5)$, $(2,5,5)$, and various rank $4$ types are systolic.

By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is a $Ω$-stable submanifold with parallel mean curvature, when $Ω$ is the Kähler calibration of rank 4 of $\mathbb{C}^3$.

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a pre…

We give a construction of $G_2$ and $Spin(7)$ instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the $G_2$ case, we show that, in the asymptotically conical model, the conn…

Study compares two subriemannian structures on S7, finding non-isometric and non-isospectral properties.

We study $\mathrm{Spin}(7)$-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons-Hawking type ansatz that describes such geometries in terms of a symmetric $4\times4$-matrix of functions. This description leads to the first known $\mathrm{Spin}(7)$-manifolds w…

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $π$ is defined on the tubular neighbourhood of the singular locus $Z_ω$ of the 2-form $ω$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $…

This paper confirms Singer's conjecture for rank 4 in specific generic degrees.

Study extends symmetries of sphere points to surface mapping classes.

Study the complexity of horizontality in 4-torus vector bundles.

We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in …

The complexity of horizontality in twistor spaces on tori is infinite.

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this…

Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs. We co…

Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic …

The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.

Novel M-theory approach classifies topological phases of matter.

The paper explores properties of Lie algebra g2 and related geometric structures.

Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the ca…