arXiv research

New algebraic structures for Lie 2-algebroids and their connections.

Split Courant algebroids linked to special algebra structures.

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. Further it is shown that non-split structures appear in the first case as deformations of a split reduction and in the …

Link Floer homology detects split links.

This paper generalizes Batchelor's theorem in $C^\infty$-superschemes.

Research shows Khovanov homology can identify split links.

According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known e.g. for Lie algebroids, Dirac structures and generalized complex struc…

It is a classical result that any complex analytic Lie supergroup $\mathcal{G}$ is split \cite{kosz}, that is its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds. We study the question how to find out whether …

A new method for growing neural networks by splitting neurons, improving efficiency.

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

The Ricci flow preserves product structures with instantaneous curvature bounds.

Study of splitting maps in Type I Ricci flows for understanding singular set structure.

The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alt…

Data structure for Heegaard splittings reduces complexity.

Geometric structures over algebras describe geodesics and spaces.

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold $X$, and they allow us to construct a countable family of compact complex non-$\partial\overline…

Given a stabilized Heegaard splitting of a $3$-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard spli…

Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.

A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling Poisson structures which describe the interaction between the tangential and transver…

Low regularity spacetimes split into simpler structures.

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …

Paper develops a splitting principle for RCD spaces, extending manifold properties.

Develops integrators for contact Hamiltonian systems preserving geometric structure.

New proof of Giroux Correspondence for tight contact 3-manifolds.

We study geometric structures of $\mathcal{W}_4$-type in the sense of A. Gray on a Riemannian manifold. If the structure group $\mathrm{G} \subset \SO(n)$ preserves a spinor or a non-degenerate differential form, its intrinsic torsion $Γ$ is a closed 1-form (Proposition \ref{dGamma} and Theorem \ref{Fixspinor}). Using …

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can pe…

New classification of conformal structures with maximal $G_2$ symmetry.

The study examines conditions for weak nearly cosymplectic manifolds to split into products.

The abstract proves a global splitting theorem for Poisson manifolds.

Proves Giroux Correspondence in 3D using Heegaard splittings.

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

The paper focuses on the sparse approximation of signals using overcomplete representations, such that it preserves the (prior) structure of multi-dimensional signals. The underlying optimization problem is tackled using a multi-dimensional split Bregman optimization approach. An extensive empirical evaluation shows ho…

In this work, we propose a novel node splitting method for regression trees and incorporate it into the regression forest framework. Unlike traditional binary splitting, where the splitting rule is selected from a predefined set of binary splitting rules via trial-and-error, the proposed node splitting method first fin…

An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler…

We prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties. The result is deduced from a generalization of a theorem of Möller. Namely, we prove that the image of a certain twisted Abel-Jacobi map lands in the torsion of a factor of the Jacobians. This statement can be viewed as a split…

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We …

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.

New method escapes local optima in neural architecture optimization.

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $rel \partial$ case the surgery on…

Boosting as gradient descent algorithms is one popular method in machine learning. In this paper a novel Boosting-type algorithm is proposed based on restricted gradient descent with structural sparsity control whose underlying dynamics are governed by differential inclusions. In particular, we present an iterative reg…

Differential K-theory gets a $λ$-ring structure.

In this paper we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.

Study cone structures on contact manifolds to understand their geometric properties.

For a genus two Heegaard splitting of a lens space, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting spanned by all vertices of primitive disks. In this work, we describe the complete combinatorial structure of the primitive disk complex fo…

We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the trace is 3), then any irreducible splitting is strongly irreducible and of genu…

Study PSCT manifolds splitting into well-understood factors.

LoBoost improves local conformal prediction for gradient-boosted trees without extra data splits.