相关论文: Generalised subbundles and distributions: A compre…
A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…
This work is dedicated to studying holomorphic distributions on Grassmann manifolds and smooth quadric hypersurfaces. In special, we prove, under certain conditions, when the tangent and conormal sheaves of a distribution splits as a sum of…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
Understanding how torsion theories are described and constructed is crucial to the study of torsion theory. Mutations of torsion theories have been studied as a method of constructing another torsion theory from a given one. We have already…
We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some…
Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics -- first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the…
We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M \equiv TM \oplus T^*M$…
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation…
We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such…
In this study, Hamiltonian and Lagrangian theories, which are mathematical models of mechanical systems, are structured on the horizontal and the vertical distributions of tangent and cotangent bundles. In the end, the geometrical and…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing…
We study fiber bundles where the fibers are not a group $G$, but a free $G$-space with disjoint orbits. These bundles closely resemble principal bundles, hence we call them semi-principal bundles. The study of such bundles is facilitated by…
Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to…
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general…
This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper…
Recent critiques of the semantic conception of scientific theories suggest that a theory is not best formulated as a collection of models satisfying some set of kinematical or dynamical conditions. Thus it has been argued that additional…
A fibre bundle viewpoint of gauge field theories is reviewed with focus on a possible quantum interpretation. The fundamental quantum properties of non-separability of state spaces is considered in the context of defining the connection on…
It is shown that certain lower semi-continuous maps from a paracompact space to the family of closed subsets of the bundle space of a Banach bundle admit continuous selections. This generalization of the theorem of Douady, dal…