integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
Integration is a process by which local data over a manifold or similar is accumulated to an integral.
In its simplest form, this is a limiting notion of the process of forming sums, known then as Riemann integration or Lebesgue integration, where the integration pairs a function on a space against a measure which indicates, roughly, how much a local contribution of the function contributes to the whole accumulation process. Direct variants and refinements of this kind of integration is integration of differential forms and similar.
The integration of differential forms induces a more general notion of integration, namely integration in differential cohomology and hence integration in generalized cohomology. Here the choice of a measure is replaced by a choice of orientation in generalized cohomology.
analytic integration | cohomological integration |
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measure | orientation in generalized cohomology |
Lebesgue integration, of differential forms | push-forward in generalized cohomology/in differential cohomology |
A sum $\sum_{s \in S} a_s$ is defined over a domain $S$ which is, as a rule, a discrete set. This set is also typically fixed in the sense that no subdivision is assumed in its definition, except sometimes for convergence purposes. Thus a general sum takes two arguments: a set $S$ and a function on $S$ and outputs a value. In the case of sum of an infinite series the order may matter as the limiting procedure is taken into account.
An integral is a generalization of a sum where the domain is typically not a discrete set, but some mathematical object, which may be (typically not in a single fixed way, but in many possible ways) approximated or split into pieces. The function to be integrated over the range, is some object which is thought of as living over the range, some sort of a cocycle or a distribution. The basic property from the integral is that it should behave nicely (typically additively in some sense) with respect to combining its values on pieces in the range and that, for a reasonable set of subdivisions the result should be the same (sometimes after passing to a limit).
For a fixed range, integral is typically an operator/functional on a set/space of allowed objects over the range.
Integration in most narrow sense is a process involved in defining or computing integrals.
Many integrals are supposed to be inverse to differentiation procedures of various kinds. Indeed, if integral is a generalization of a sum, then a difference between two partial sums is a value to be added at a step of summation, and its generalization is some sort of differentiation. Of course, the initial value has to be determined as the differentiation gives just a step of the addition.
In some cases, one solves a differential equation by reducing it to a relation of the form $d F = g$ and then $F = \int g$. One says that the equation is solvable in quadratures. Thus the integration is used in examples of solving differential equations and differential relations, hence finding objects satisfying some differential constraints is often also considered a sort of integration. For example, finding integral curves of vector fields and more generally finding integral submanifolds of distributions, is also called an integration. In this vain, also a Lie group is a global object which integrates a Lie algebra (indeed, infinitesimally this reduces to solving the Maurer-Cartan equations). For resolving differential relations there are solvability conditions/obstructions/constraints which are often of cohomological nature. There is sometimes a relation to rational homotopy theory.
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We list a bunch more notions of integration. Should eventually be turned into something more coherent…
See also measure theory (and measurable space, measure space) which is a basis for many kinds of integrals, especially the Lebesgue integral.
Basic kinds of integrals in (super)analysis: the Daniell integral?, the Riemann integral (possibly upper or lower?, improper?, or generalised), the Lebesgue integral, the Henstock integral (same as the generalised Riemann integral or the improper Lebesgue integral), the Stieltjes? variations of the above, the Berezin integral, line integrals.
Special examples of the above include the Batalin-Vilkovisky integral, the Kontsevich integral, the Selberg integral,the elliptic Selberg integral.
Integration is involved in integral transforms, integral transforms on sheaves, in various formulas for pairings, e.g. of chains and cochains …. Some statements involving integrals include the Stokes theorem.
A special topic includes some infinite-dimensional versions including the well-defined Wiener integral? and the more problematic path integral, cf. also
The basic problem with the path integral comes from the fact that there is no translation-invariant Lebesgue measure on an infinite-dimensional real vector space with a finite nonzero value on the unit ball.
differential forms are involved in integration over differentiable manifolds, see at integration of differential forms ; more generally, one can integrate currents (cf. geometric measure theory
Lie integration (the name comes from its relation to the integration of differential equations and finding integral curves of vector fields and flows)
stratified versions related to Grothendieck rings and valuations: Euler integration, motivic integration
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Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
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$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
An introduction to the basic notions of integration of differential forms, with an eye towards applications in physics is in section 3 of
Discussion of integration similarly with an eye towards applications in physics but from a more general abstract perspective is in
A proof of the Riesz representation theorem in constructive mathematics is given in