Related papers: Towards geometric integration of rough differentia…

We introduce a notion of distributional $k$-forms on $d$-dimensional manifolds which can be integrated against suitably regular $k$-submanifolds. Our approach combines ideas from Whitney's geometric integration [Whi57] with those of sewing…

We build a connection between rough path theory and noncommutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove…

We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated…

We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We…

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on…

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in…

We introduce a geometric approach of integral curves for functional inequalities involving directional derivatives in the general context of differentiable manifolds that are equipped with a volume form. We focus on Hardy-type inequalities…

Integration is the final key step when turning an infinitesimal argument into a result applicable to quantities of finite size. Conceptually, it is about combining infinitesimal contributions to a finite whole. We make a first step towards…

We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration. As an application, we extend an argument of Schwartz to rough differential equations, and prove the…

We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one…

The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used used in explicit computations or approximation schemes. We present a…

This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a…

Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main…

In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of…

Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them…

We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time…

We show that geometric integrals of the type $\int_\Omega f\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $\Omega$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just H\"{o}lder continuous…

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…