arXiv subjects

Filip Bár

Publications and source records attributed to Filip Bár.

5 recordsLinked to original sources

arXiv2024

Towards a geometric theory of integration

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 a geometric theory of integration in the context of Synthetic Differential Geometry (SDG) by analysing the differential aspect of the integration process. Starting from two heuristic principles that combine the idea of differential forms as infinitesimal measures while formalising the process of taking infinitesimal differences at the same time we derive a general notion of differential form as an equivariant map from infinitesimal $n$-cuboids to the base ring coordinatising a line. Besides the familiar differential forms introduced by Cartan we discover two new types. We also discover a new differential operator besides the exterior derivative. Analogous to the relationship between the exterior derivative and the Stokes-Cartan integral theorem, this new operator is linked to the generalised Fundamental Theorem of Calculus in higher dimensions, as discussed in prior research. This shows that the Fundamental Theorem is an integral theorem like Stokes-Cartan, but for one of the new types of differential forms.

math.DG↗

arXiv2024

The Fundamental Theorem of Calculus in higher dimensions

We generalise the Fundamental Theorem of Calculus to higher dimensions. Our generalisation is based on the observation that the antiderivative of a function of $n$-variables is a solution of a partial differential equation of order $n$ generalising the classical case. The generalised Fundamental Theorem of Calculus then states that the $n$-dimensional integrals over $n$-dimensional axis-parallel rectangular hypercuboids is given by a combinatorial formula evaluating the antiderivative on the vertices of the hypercuboid.

math.GM↗

arXiv2023

Gluing of infinitesimal models of algebraic theories

Categories of models of algebraic theories have good categorical properties except for gluing. Building upon insights and examples from Synthetic Differential Geometry, we introduce a generalisation of models of algebraic theories to infinitesimal models. We demonstrate that the category of infinitesimal models retains most of the good categorical properties, but with a stark improvement in the behaviour of gluing. This makes infinitesimal models an interesting natural construction with the ability to interpolate between algebra and geometry.

math.CT↗

arXiv2023

Second-order infinitesimal groups and affine connections

This paper presents new research in infinitesimal algebra by introducing the concept of an infinitesimal group and exploring its properties and ramifications. The author investigates first- and second-order subgroups of Lie groups and demonstrates the use of the second-order infinitesimal group structure to define a Lie bracket of points intrinsic to the Lie group. This construction allows for the derivation of a second-order Baker-Campbell-Hausdorff formula for the infinitesimal group operation and provides a means to reconstruct the Lie bracket of the Lie algebra of a Lie group. The author also characterises all second-order infinitesimal group structures on KL vector spaces as deformations of vector addition by bilinear maps. The main contribution of the paper is the generalisation of the previously established correspondence between symmetric affine connections and second-order infinitesimally affine structures to manifolds with non-symmetric affine connections via second-order infinitesimal groups.

math.DG↗

arXiv2020

Affine connections and second-order affine structures

Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure of an infinitesimally affine space. This structure is comprised of two pieces of data: a sequence of symmetric and reflexive relations defining the tuples of mutual infinitesimally close points, called an infinitesimal structure, and an action of affine combinations on these tuples. For smooth manifolds the only natural infinitesimal structure that has been considered so far is the one generated by the first neighbourhood of the diagonal. In this paper we construct natural infinitesimal structures for higher-order neighbourhoods of the diagonal and show that on any manifold any symmetric affine connection extends to a second-order infinitesimally affine structure.

math.DG↗