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The Surface Signature and Rough Surfaces

Darrick Lee

Abstract

Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel transport factors uniquely through the path signature. Furthermore, the path signature is a central object in the theory of rough paths, which provides an integration theory for highly irregular paths. A fundamental result is Lyons' extension theorem, which allows us to compute the signature of rough paths, and in turn provides a way to compute parallel transport of arbitrarily irregular paths. In this article, we consider the notion of surface holonomy, a generalization of parallel transport to the higher dimensional setting of surfaces parametrized by rectangular domains, which preserves the higher algebraic structures of horizontal and vertical concatenation. Building on work of Kapranov, we introduce the surface signature, which is universal among surface holonomy maps with respect to continuous 2-connections. Furthermore, we introduce the notion of a rough surface and prove a surface extension theorem, which allows us to compute the signature of rough surfaces. By exploiting the universal property of the surface signature, this provides a method to compute the surface holonomy of arbitrarily irregular surfaces.

The Surface Signature and Rough Surfaces

Abstract

Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel transport factors uniquely through the path signature. Furthermore, the path signature is a central object in the theory of rough paths, which provides an integration theory for highly irregular paths. A fundamental result is Lyons' extension theorem, which allows us to compute the signature of rough paths, and in turn provides a way to compute parallel transport of arbitrarily irregular paths. In this article, we consider the notion of surface holonomy, a generalization of parallel transport to the higher dimensional setting of surfaces parametrized by rectangular domains, which preserves the higher algebraic structures of horizontal and vertical concatenation. Building on work of Kapranov, we introduce the surface signature, which is universal among surface holonomy maps with respect to continuous 2-connections. Furthermore, we introduce the notion of a rough surface and prove a surface extension theorem, which allows us to compute the signature of rough surfaces. By exploiting the universal property of the surface signature, this provides a method to compute the surface holonomy of arbitrarily irregular surfaces.

Paper Structure

This paper contains 59 sections, 72 theorems, 478 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Paths
  3. Surfaces
  4. Contributions
  5. Surface Signature.
  6. Rough Surfaces
  7. Related Work
  8. The Path Signature
  9. Surface Holonomy
  10. Crossed Modules and Double Groups
  11. Crossed Modules of Algebras
  12. Surface Holonomy
  13. The Surface Signature
  14. Construction of the Universal 2-Connection
  15. Free Crossed Modules of Associative Algebras
  16. ...and 44 more sections

Key Result

Theorem 1.1

Let $(A_0, \cdot, 1)$ be a unital Banach algebra, and $\alpha \in L(V, A_0)$ be a bounded linear map. Then, there exists a unique map $\widetilde{\alpha}: G_0 \to A_0$ such that

Figures (7)

  • Figure :
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  • ...and 2 more figures

Theorems & Definitions (158)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal lyons_differential_1998
  • Theorem 1.3: Informal
  • Theorem 1.4: Eckmann-Hilton
  • Theorem 1.5: Informal
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8: Informal
  • Theorem 1.9: Informal
  • Remark 1.10
  • ...and 148 more