Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
An iterated integral is a expression involving nested integrals, such as
where the grouping parentheses are usually left out. Iterated integrals are the subject of Fubini theorems.
Iterated integrals include nested integration of differential forms of the kind as it appears in the formulation of parallel transport for nonabelian-value connection form (known as the Dyson formula in physics).
Applied to higher degree forms iterated integrals serve to express generalized transgression of differential forms to loop spaces and other mapping spaces realized as diffeological spaces.
The notion of iterated integration of differential forms originates in informal observation like the Dyson formula for parallel transport. It was formalized in the context of differential geometry on diffeological spaces (Chen spaces) (especially loop spaces) in
Kuo Tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217–246. JSTOR
Kuo Tsai Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975) 83–98. doi:10.1090/S0002-9947-1975-0377960-0
Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83(5):831–879, 1977 doi:10.1090/S0002-9904-1977-14320-6
Essentially these formulas are used in
for expressing higher holonomy of certain flat infinity-connections given by infinity-representations of tangent Lie algebroids.
A review of this and more discussion in the context of a higher Riemann-Hilbert correspondence is in
A relation to algebraic cycles is discussed in
Chen iterated integrals in the context of stochastic analysis
Kapranov’s work on “noncommutative Fourier transform” uses some ideas from Chen’s work:
Mikhail Kapranov, Noncommutative geometry and path integrals, In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics 270, doi arXiv:math.QA/0612411
Mikhail Kapranov, Free Lie algebroids and the space of paths, Sel. math., New ser. 13, 277 (2007) doi arXiv:math:AG/0702584
Mikhail Kapranov, Membranes and higher groupoids, arXiv:1502.06166
Iterated sums and iterated integrals over semirings, where the case of tropical semiring is a central, with applications (including in machine learning):
Last revised on August 30, 2024 at 11:58:02. See the history of this page for a list of all contributions to it.