Arnold's spectral sequence

In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to canonical form near critical points. It was introduced by Vladimir Arnold in 1975. ^{[1] [2] [3]}

Definition

Arnold's spectral sequence is a computational tool in singularity theory for reducing a function to its canonical or normal form near a critical point. Introduced by Vladimir Arnold in 1975, it analyzes the structure of singularities by using a spectral sequence built from differential forms with singularities.^{[ citation needed ]}

Overview of the spectral sequence

The spectral sequence is constructed on the space of differential forms associated with the function's critical points. The key components are:

• Filtration: A filtering of the differential forms is established based on the increasing order of their poles along the "discriminant loci"—the diagonals where singularities occur.
• -page: The first page of the spectral sequence, denoted , consists of differential forms that exhibit logarithmic singularities along these diagonals.
• Differential: The differentials in the sequence (the maps from one page to the next) encode the algebraic relationships among the singular forms.

By taking successive homology operations, the spectral sequence converges to the desired invariants of the singularity, allowing for its reduction to a simpler, canonical form. ^{[4] [5]}

References

1. Arnol'd, V. I. (1979). "Spectral sequence for reduction of functions to normal form". Functional Analysis and Its Applications. 9 (3): 251–253. doi:10.1007/BF01075605.
2. Goryunov, Victor; Lippner, Gabor (2008). "Simple framed curve singularities". Geometry and topology of caustics. Banach Center Publications. pp. 85–100. doi:10.4064/bc82-0-6.
3. Gazor, Majid; Yu, Pei (January 2012). "Spectral sequences and parametric normal forms". Journal of Differential Equations. 252 (2): 1003–1031. arXiv: 1103.3891 . Bibcode:2012JDE...252.1003G. doi:10.1016/j.jde.2011.09.043.
4. Zhang, Wei; Sun, Tiecheng; Wang, Sen; Cheng, Qing; Haala, Norbert (February 2024). "HI-SLAM: Monocular Real-Time Dense Mapping With Hybrid Implicit Fields". IEEE Robotics and Automation Letters. 9 (2): 1548–1555. arXiv: 2310.04787 . Bibcode:2024IRAL....9.1548Z. doi:10.1109/lra.2023.3347131.
5. Selick, P. (2006). "Spectral Sequences". Encyclopedia of Mathematical Physics. pp. 623–633. doi:10.1016/B0-12-512666-2/00417-X. ISBN   978-0-12-512666-3.