Geometric Integral Definition Using Triangulations Enables Coordinate-Free Riemann Integration and Stochastic Calculus

The challenge of defining integration rigorously, beyond its practical application, continues to drive mathematical innovation. Joshua Lackman, working independently, proposes a novel geometric definition of the integral based on triangulations, moving away from traditional coordinate-dependent approaches. This research offers a coordinate-free version of the Riemann integral and establishes its relevance in complex areas such as Lie algebroids, stochastic integration and field theory, particularly where path integrals rely on lattice structures. By connecting integration to cochains on pair groupoids via the van Est map, Lackman’s work not only incorporates various stochastic integrals more naturally, but also proves a generalization of the fundamental theorem of calculus relating cohomology cap products. This new framework promises to simplify the establishment of integral identities from their combinatorial origins and offers a potentially unifying approach to integration across diverse mathematical disciplines.

Joshua Lackman proposes a novel geometric definition of the integral based on triangulations, moving away from traditional coordinate-dependent approaches. This research offers a coordinate-free version of the Riemann integral and establishes its relevance in complex areas such as Lie algebroids, stochastic integration and field theory, particularly where path integrals rely on lattice structures. By connecting integration to cochains on pair groupoids via the van Est map, Lackman’s work incorporates various stochastic integrals more naturally and proves a generalization of the fundamental theorem of calculus relating cohomology cap products.

This new framework promises to simplify the establishment of integral identities from their combinatorial origins and offers a potentially unifying approach to integration across diverse mathematical disciplines. The definition naturally incorporates the different stochastic integrals, which involve integration over Hölder continuous paths. Furthermore, this definition is well-adapted to establishing integral identities from their combinatorial counterparts, offering a bridge between discrete and continuous mathematics. The construction is based on the observation that, in great generality, the objects of integration are determined by cochains on the pair groupoid.

Abstractly, the definition uses the van Est map to lift a differential form to the pair groupoid, providing a geometric interpretation of the integration process.

Triangulation Defines Integrals on Manifolds

The study pioneers a novel geometric definition of the integral of differential forms, diverging from standard approaches reliant on local coordinates and partitions of unity. Researchers constructed a coordinate-free definition based on triangulations, aiming for a method better suited to explicit computation and integration over non-differentiable maps, particularly relevant to stochastic integration and Lie algebroid morphisms. This work directly addresses the question of what constitutes the most general object integrable within a given manifold.

Central to this methodology is the formulation of integration as a limit of Riemann-like sums over a triangulation, where the integral over M of ω equals the limit as the triangulation’s size approaches zero of a summation over all simplices within the triangulation. To achieve this, the team observed that relevant objects for Riemann-like sums are cochains on the pair groupoid, necessitating the lifting of the differential form ω to a cochain via the van Est map, VE. This innovative approach leverages the orientation of a manifold to define an orientation on the vertices of each simplex, requiring a function Ω invariant under even permutations to evaluate on these oriented vertices.

Scientists rigorously established that this Riemann-like sum converges to the correct integral, proving a generalization of the fundamental theorem of calculus. The theorem demonstrates that the pairing between the fundamental class and the induced simplicial cocycle equals the integral of the lifted form, VE(Ω), over the manifold and its boundary. Experiments employ this theorem to recover the standard fundamental theorem of calculus and Stokes’ theorem through specific choices of triangulation and cocycles. Furthermore, the research demonstrates the method’s applicability to functional integrals, showing that computing a VE-antiderivative for a form reduces the problem of integration over arbitrary maps to a finite summation over a triangulation.

This technique extends beyond differentiable maps, offering a powerful tool for path integrals and integration within the framework of Lie algebroids, ultimately providing a robust and geometrically intuitive approach to integration.

Triangulations Define a Coordinate-Free Riemann Integral Scientists have

Scientists have developed a novel definition of integration, moving beyond standard coordinate-based methods to utilise triangulations instead. This new approach provides a coordinate-free version of the Riemann integral, proving particularly relevant in complex areas like Lie algebroids, stochastic integration, and field theory where path integrals are constructed using lattice structures. The research demonstrates that this definition naturally accommodates diverse stochastic integrals, including those involving Hölder continuous paths, offering a more versatile framework for mathematical operations.

Experiments revealed that the construction relies on cochains on the pair groupoid, effectively determining integrated quantities through abstract means. The team measured the behaviour of cochains and cocycles, identifying naturally occurring examples within the new integration paradigm. Specifically, the Riemann-Stieltjes integral, expressed as a limit of Riemann sums, neatly aligns with this framework, where functions like dg are represented as 1-cochains on the pair groupoid. Results demonstrate that this allows for the integration of differential forms over non-differentiable maps by initially lifting them to cochains, a significant advancement in handling complex geometries.

Further investigations focused on Brownian motion and its implications for stochastic integrals, revealing discrepancies between the Itô and Stratonovich integrals due to higher-order Taylor expansion terms. Scientists recorded that the integration of forms over Wiener paths requires consideration of terms beyond the standard one-form, specifically incorporating elements of the form f(x) dx + g(x) dx2. Measurements confirm that antisymmetric sections, satisfying the fundamental theorem of calculus, are crucial for defining the Stratonovich integral and accurately representing path integrals. The breakthrough delivers a precise definition for integrating these higher-order terms, establishing a limit of Riemann sums as a random variable.

This definition, based on a smooth function Ω satisfying specific Taylor expansion conditions, ensures consistency between different approximation schemes. The work suggests a generalization of the fundamental theorem of calculus, successfully proving the equality of singular and de Rham cohomology cap products with the fundamental class, opening new avenues for theoretical exploration and practical applications in areas requiring precise path integral calculations.

Triangulations, Groupoids and a New Calculus Theorem

This work introduces a novel definition of integration based on triangulations, offering a coordinate-free alternative to the standard approach reliant on local coordinates and partitions of unity. The authors demonstrate that this definition aligns naturally with contexts such as Lie algebroids, stochastic integration, and field theory, particularly where path integrals are defined using lattice structures. By framing integration through the lens of cochains on the pair groupoid, they establish a connection between combinatorial and analytical integral identities.

A central achievement of this research is a generalized form of the fundamental theorem of calculus, proving the equality of singular and de Rham cohomology cap products with the fundamental class. The authors illustrate the applicability of their framework through examples including the Riemann-Stieltjes integral and stochastic integrals like the Itô and Stratonovich integrals, showing how these can be understood as instances of integrating specific cochains. While the authors acknowledge limitations regarding the need for conditions like bounded variation for certain integrals, they suggest future research could explore extending the framework to more general cocycles on local pair groupoids.