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We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity…

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a…

We study the (in)dependence of additivity and homogeneity conditions in the definition of linear mappings between vector spaces over the same scalar field. Unlike other works on the subject, dealing with particular fields like real or…

The thesis studies Frobenius-type theorems in non-smooth settings. We extend the definition of involutivity to non-Lipschitz subbundles using generalized functions. We prove the real Frobenius Theorem with sharp regularity on log-Lipschitz…

The recent introduction of the gradient flow has provided a new tool to probe the dynamics of quantum field theories. The latest developments have shown how to use the gradient flow for the exploration of symmetries, and the definition of…

As an analogy of superalgebra of multivector fields with the Schounte bracket, we introduce a non-trivial superbracket on differential forms of manifold. We show properties of this new superalgebra. We extend this superalgebra by adding one…

In this paper, we study infinite dimensional holomorphic vector fields on sequence spaces, having a fixed point at $0$. Under suitable hypotheses we prove the existence of analytic invariant submanifolds passing through the fixed point. The…

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order…

We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural…

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the…

In this paper we continue the study of bi-conformal vector fields started in {\em Class. Quantum Grav.} {\bf 21} 2153-2177. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions $\lie P_{ab}=\phi…

Close insight into mathematical and conceptual structure of classical field theories shows serious inconsistencies in their common basis. In other words, we claim in this work to have come across two severe mathematical blunders in the very…

We present a thorough study of the differential geometry of weightings and develop the theory of weightings for vector bundles, Lie groupoids, and Lie algebroids. We begin by extending the work of Loizides and Meinrenken on weighted…

This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment, we laid out the natural formulation of bosonic variational field theory via the…

We generalize Turaev's definition of torsion invariants of pairs (M,x), where M is a 3-dimensional manifold and x is an Euler structure on M (a non-singular vector field up to homotopy relative to bM and local modifications in int(M).…

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…

We consider possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants. A particular case of volume preserving diffeomrphism is the most important, since it has…

We study spaces of ultradifferentiable functions which contain Gevrey classes. Although the corresponding defining sequences do not satisfy Komatsu's condition (M.2)', we prove appropriate continuity properties under the action of…

We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with…