Hellmuth Kneser

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hellmuth_Kneser)

## Selected writings

On [[topological manifolds]] and raising the question of [[triangulation theorems]]:

* [[Hellmuth Kneser]], *Die Topologie der Mannigfaltigkeiten*, Jahresbericht der Deutschen Mathematiker-Vereinigung (1926), Volume: 34, page 1-13 ([eudml:145701](https://eudml.org/doc/145701))

category: people


Helmholtz operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Helmholtz operator_ (going back to [Helmholtz 87](#Helmholtz87), [Sonin](#Sonin), with modern version due to [Vinogradov 78](#Vinogradov78), [Tulczyjew 80](#Tulczyjew80)) is a map from [[partial differential equations]] (on [[sections]] of some [[bundle]]) to [[differential operators]] (on sections of that bundle) with the property that locally its [[kernel]] consists precisely of those [[PDEs]] which are _variational_ in that there is a [[local Lagrangian density]] such that the PDE is the [[Euler-Lagrange equation]] which says that the [[variational derivative]] of this Lagrangian (equivalently: of the [[action functional]] that it induces) vanishes.

In modern terminology this says that together with the [[Euler-Lagrange operator]] the Helmholtz operator constitutes a [[chain complex]] of [[abelian sheaves]] (namely of [[differential forms]] on a [[jet bundle]]) which is locally [[long exact sequence|exact]]. In fact this extends in both directions to a locally [[long exact sequence]] of forms on the jet bundle, called the _[[Euler-Lagrange complex]]_. See there for more.

## Details

A quick way to write the Helmholtz operator $H$ is as follows: If $\mathcal{E}$ denotes a [[partial differential equation]] and $L\mathcal{E}$ its linearization ([[evolutionary derivative]]), and $(L\mathcal{E}^\ast$ its [[formal adjoint differential operator]], as a [[differential operator]] then

$$
  H[\mathcal{E}]
    =
  L\mathcal{E}
   -
  (L\mathcal{E})^\ast
  \,.
$$

This means that the PDE $\mathcal{E}$ is locally variational precisely if its linearization is formally self-adjoint.

## Properties

For proof that every [[Euler-Lagrange equation]] is in the kernel of the Helmholtz operator see _[[geometry of physics -- A first idea of quantum field theory]]_ [this prop.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint).

## References

The Helmholtz operator originates in

* {#Helmholtz87} [[Hermann von Helmholtz]], _Über die physikalische Bedeutung des Princips der kleinsten Wirkung_, Journal für reine und angewandte Mathematik 100 (1887), 137–166

* {#Sonin} Sonin, ...

where it is considered for linear [[ordinary differential equations]]. The modern general incarnation of the Helmholtz condition is due to

* {#Vinogradov78} [[Alexandre Vinogradov]], _A spectral sequence associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints_, Soviet Math. Dokl. 19 (1978), 144–148.

* {#Tulczyjew80} W. M. Tulczyjew, _The Euler-Lagrange resolution_, in Lecture Notes in Mathematics No. 836, Springer-Verlag, New York, 1980, pp. 22–48.

Review includes

* [[Ian Anderson]],  _Aspects of the inverse problem to the calculus of variations_.
Archivum Mathematicum, vol. 24 (1988), issue 4, pp. 181-202 ([web](http://dml.cz/dmlcz/107326))

* {#Anderson89} [[Ian Anderson]], _The variational bicomplex_, Utah State University 1989 ([[AndersonVariationalBicomplex.pdf:file]])


Helmut Bender

* [website](http://www.uni-kiel.de/algebra/bender/)

category: people


Helmut G. F. Winkler

* [obituary pdf](https://www.geosociety.org/documents/gsa/memorials/v13/Winkler-HGF.pdf)

* [Wikipedia entry](https://www.geosociety.org/documents/gsa/memorials/v13/Winkler-HGF.pdf)

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Helmut_G._F._Winkler)

## Selected writings

On [[Bravais lattices]] and [[crystallographic groups]]:

* [[Helmut G. F. Winkler]], *Hundert ]ahre Bravais Gitter*, Die Naturwissenschaften **37** 17 (1950) $[$[doi:10.1007/BF00738360](https://doi.org/10.1007/BF00738360), [pdf](https://link.springer.com/content/pdf/10.1007/BF00738360.pdf)$]$

category: people


Helmut Hasse

* [Wikipedia entry](https://en.wikipedia.org/wiki/Helmut_Hasse)

## Selected writings

Introducing the [[Hasse-Schmidt derivative]]:

* [[Helmut Hasse]]: *Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von [[Friedrich Karl Schmidt|F. K. Schmidt]] in Jena.)*, Reine Angew. Math.  **177** (1937) 215-223 [[paper](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0177?tify={%22pages%22:[219],%22view%22:%22export%22})]

## Related entries

* [[local-global principle]]
* [[Hasse-Schmidt derivative]]
* [[Dedekind-Hasse norm]]

category: people


Helmut Hofer

* [IAS page](http://www.math.ias.edu/people/faculty/hofer), [description](http://www.ias.edu/people/faculty-and-emeriti/hofer)

* [Wikipedia page](http://en.wikipedia.org/wiki/Helmut_Hofer)

## Selected writings

Introducing [[polyfolds]] for [[symplectic topology]]:

* [[Helmut Hofer]],  _A general Fredholm theory and applications_ ([arXiv:math/0509366](http://arxiv.org/abs/math/0509366))

* [[Helmut Hofer]], Kris Wysocki, Eduard Zehnder,

  * _A general Fredholm theory I: A splicing-based differential geometry_ ([arXiv/0612604](http://arxiv.org/abs/math/0612604))

  * _A general Fredholm theory II: implicit function theorems_ ([arXiv:0705.1310](http://arxiv.org/abs/0705.1310))

  * _A general Fredholm theory III: Fredholm functors and polyfolds_ ([arXiv:0810.0736](http://arxiv.org/abs/0810.0736))

* [[Helmut Hofer]], Kris Wysocki, Eduard Zehnder, _Integration theory for zero sets of polyfold Fredholm sections_ ([arXiv/0711.0781](http://arxiv.org/abs/0711.0781))

* [[Helmut Hofer]], _Polyfolds and a general Fredholm theory_ ([arXiv/0809.3753](http://arxiv.org/abs/0809.3753))

* [[Helmut Hofer]], _Polyfolds and Fredholm Theory_,  Oxford University Press (2017) ([doi:10.1093/oso/9780198784913.003.0004](http://dx.doi.org/10.1093/oso/9780198784913.003.0004))

Lecture notes:

* Joel W. Fish, [[Helmut Hofer]], _Lectures on Polyfolds and Symplectic Field Theory_ ([arXiv:1808.07147](https://arxiv.org/abs/1808.07147))

Introducing [[Hofer's metric]] on [[Hamiltonian diffeomorphisms]]:

* [[Helmut Hofer]], _On the topological properties of symplectic maps_, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, Volume 115 Issue 1-2, 1990 ([doi:10.1017/S0308210500024549](https://doi.org/10.1017/S0308210500024549))

category: people


Helmut Neufeld

* [webpage](https://ufind.univie.ac.at/en/person.html?id=3804)

## Selected writings

On [[chiral perturbation theory]] with [[leptons]]:

* M. Knecht, [[Helmut Neufeld]], H. Rupertsberger, P. Talavera, _Chiral Perturbation Theory with Virtual Photons and Leptons_, Eur. Phys. J. C12:469-478, 2000 ([arXiv:hep-ph/9909284](https://arxiv.org/abs/hep-ph/9909284))

* [[Helmut Neufeld]], _Chiral Perturbation Theory with Photons and Leptons_, PiN Newslett. 15:189-192, 1999 ([arXiv:hep-ph/9912462](https://arxiv.org/abs/hep-ph/9912462))

On [[kaon]] [[decays]]:

* V. Cirigliano, [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], J. Portoles, _Kaon Decays in the Standard Model_, Rev. Mod. Phys. 84 (2012) 399 ([arXiv:1107.6001](https://arxiv.org/abs/1107.6001))

and in relation to the [[chiral anomaly]]:

* [[Gerhard Ecker]], [[Helmut Neufeld]], [[Antonio Pich]], _Non-leptonic kaon decays and the chiral anomaly_, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

category: people


Helmut Röhrl

* [Wikipedia entry (English)](https://en.m.wikipedia.org/wiki/Helmut_R%C3%B6hrl)

* [Wikipedia entry (Deutsch)](https://de.m.wikipedia.org/wiki/Helmut_R%C3%B6hrl)

## Selected writings

On [[categorical algebra]]:

* [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

On [[totally convex spaces]]:

* [[Dieter Pumplün]], [[Helmut Röhrl]]: _Banach Spaces and totally convex spaces I_, Comm. Alg. **12** 8 (1984) 953-1019 [[doi:10.1080/00927878408823035](https://doi.org/10.1080/00927878408823035)]

* [[Dieter Pumplün]], [[Helmut Röhrl]], _Banach Spaces and totally convex spaces II_, Comm. Alg. **13** 5 (1985) 1047-1113 [[doi:10.1080/00927878508823205](https://doi.org/10.1080/00927878508823205)]

On [[totally convex algebras]]:

* [[Dieter Pumplün]], [[Helmut Röhrl]], *Totally convex algebras*, Comment. Math. Univ. Carolin. **33** 2 (1992) 205–235
[[pdf](http://emis.maths.tcd.ie/journals/CMUC/pdf/cmuc9202/pump.pdf), [dml:10338.dmlcz/118490](https://dml.cz/handle/10338.dmlcz/118490)]

category: people

[[!redirects H. Röhrl]]


Helmut Schwichtenberg

* [personal page](https://www.mathematik.uni-muenchen.de/~schwicht/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Helmut_Schwichtenberg)

## Selected writings

On [[proof theory]] with emphasis on [[natural deduction]] and [[sequent calculi]]:

* [[Anne Sjerp Troelstra]], [[Helmut Schwichtenberg]], _Basic Proof Theory_ , Cambridge University Press (2000, 2012) [[doi:10.1017/CBO9781139168717](https://doi.org/10.1017/CBO9781139168717)]

On [[constructive analysis]]:

* [[Helmut Schwichtenberg]], *Constructive analysis with witnesses* (2013) [[pdf](https://www.mathematik.uni-muenchen.de/~schwicht/seminars/semss13/constr13.pdf), [[Schwichtenberg-ConstructiveAnalysis.pdf:file]]]

* [[Helmut Schwichtenberg]], *Program Extraction in Constructive Analysis*, in *Logicism, Intuitionism, and Formalism*, Synthese Library, **341** Springer (2009) [[doi:10.1007/978-1-4020-8926-8_13](https://doi.org/10.1007/978-1-4020-8926-8_13)]

category: people


This is a page dedicated to helping all those who're struggling with understanding the classic book:

* B. Bakalov & A. Kirillov, _Lectures on tensor categories and modular functors_ AMS, University Lecture Series, (2000) ([web](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html))

The procedure is: go to the [nForum Help me! I'm trying to understand Bakalov and Kirillov](http://ncatlab.org/nlab/show/Help%20me!%20I%27m%20trying%20to%20understand%20Bakalov%20and%20Kirillov) page, or to [Math Overflow](http://mathoverflow.net/). Type in your question and get someone to answer it. Then put down the answer here in final form for future generations.

### Errata

* pg 112, bottom of proof of Theorem 5.3.8, "the Dehn twist axiom MF7" should read MS7, I believe. [[Bruce Bartlett]]

### How does the "s" map work? ([nForum link](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1099&page=1))

[[Bruce Bartlett]]: I don't understand the $s$ map in Example 5.1.11 in the online version of the book. It's supposed to be the automorphism
 \[
 s : \Gamma_{1,1} \rightarrow \Gamma_{1,1}
\]
of the torus with one boundary circle which is the analogue of a map which I _do_ understand, the automorphism
\[
 s_t : \Gamma_{1,0} \rightarrow \Gamma_{1,0}
\]
of the ordinary torus (no boundary circle) to itself, which sends
\[
 (\theta, \phi) \mapsto (\phi, -\theta).
\]
But the map $s_t$ doesn't seem to work if there is a boundary circle involved. Because an automorphism of $\Gamma_{1,1}$ is supposed to be the identity on the boundary circle, but the map $s_t$ isn't: it rotates the boundary circle a quarter rotation (right?). The reason I say so is the following picture: we think of the torus as $\mathbb{R}^2 / \mathbb{Z}^2$, as Bakalov and Kirillov encourage us to do. Then the presence of the boundary circle can be thought of as having a little tangent vector pointing to the right at all the lattice positions (this uses their alternative Definition 5.1.10 the extended surface category). The map $(x,y) \mapsto (y, -x)$ rotates the plane by a quarter rotation and takes the lattice to itself, hence it descends to the quotient. But this map rotates the tangent vector by a quarter rotation, and we're supposed to fix the tangent vectors!

So how does the map $s$ work?

Answer: [[Domenico Fiorenza]] explained it as follows: consider the torus with one boundary circle as a square made of elastic material with with opposite sides identified, with an open disc missing in the center. Then turn the square through one quarter of a rotation, while keeping the boundary circle fixed. That's the $s$ map.

If you picture it at the level of the plane (i.e. before we mod $R^2$ out by $Z^2$), then you need to imagine a whole bunch of circles ("knobs") located at the integer lattice points of the plane. Get a whole bunch of friends (one for each knob), and then count down "3,2,1,turn!" and turn the knobs through a quarter revolution clockwise (this distorts the elastic material from which the plane is made). Then rotate the whole thing rigidly one quarter of a revolution counterclockwise. The resultant map is the $s$-map at the level of the plane.

category: reference


hemicompact space

## Definition ##

A [[topological space]] $X$ is called hemicompact iff there exists a sequence $(K_n)_{n\in\mathbb{N}}$ of compact subsets $K_n\subseteq X$ such that $X=\bigcup_{n\in\mathbb{N}} K_n$ and $K_n \subseteq int(K_{n+1})$ holds.

## Properties ##

If $X$ is hemicompact and $(K_n)$ a sequence as in the definition, then every compact subset $K\subseteq X$ is contained in one of the $K_n$. In other words: $\lbrace K_n \mid n\in\mathbb{N}\rbrace$ is [[cofinal]] in the set of all compact subsets (partially ordered by inclusion)


hemisphere

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Every [[n-sphere]] may be [[covering|covered]] by two [[disks]] which each cover "half" of it (or a little more). These are the _hemi-$n$-spheres_.

## Related concepts

* [[equator]], [[antipode]], [[meridian]]

* [[stereographic projection]]

* [[clutching construction]]

## References

* Wikipedia, _[Hemisphere](https://en.wikipedia.org/wiki/Hemisphere)_

[[!redirects hemisphere]]

[[!redirects hemispheres]]

[[!redirects hemi-n-sphere]]
[[!redirects hemi-n-spheres]]


Hendrik Casimir

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hendrik_Casimir)

## related $n$Lab entries

* [[Casimir effect]]

* [[Casimir element]]

* [[rigid body dynamics]]

category: people


Hendrik Lenstra

* [website](http://math.berkeley.edu/~hwl/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Hendrik_Lenstra)

category: people


Hendrik Lorentz

> not to be confused with [[Ludvik Lorenz]]

***

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hendrik_Lorentz)

## related $n$Lab entries

* [[Lorentz group]]

* [[Lorentz force]]

* [[Lorentzian manifold]]

category: people


Hendryk Pfeiffer

* [arXiv page](https://arxiv.org/a/pfeiffer_h_1.html)

* [InSpire page](https://inspirehep.net/authors/993522)

## Selected writings

On [[higher gauge theory]] for [[principal 2-bundles]] [[connection on a 2-bundle|with connection]]:

* [[Florian Girelli]], [[Hendryk Pfeiffer]], *Higher gauge theory -- differential versus integral formulation*, J. Math. Phys. **45** (2004) 3949-3971 [[hep-th/0309173](http://arxiv.org/abs/hep-th/0309173), [doi:10.1063/1.1790048](https://doi.org/10.1063/1.1790048)]

category: people


Henning Basold

* [personal page](https://liacs.leidenuniv.nl/~basoldh/)

## Selected writings

On [[higher inductive types]] in [[homotopy type theory]]:

* [[Henning Basold]], [[Herman Geuvers]], [[Niels van der Weide]], *Higher Inductive Types in Programming*, Journal of Universal Computer Science **23** 1 (2017) 63-88 [[pdf](https://www.jucs.org/jucs_23_1/higher_inductive_types_in/jucs_23_01_0063_0088_basold.pdf), [slides pdf](https://nmvdw.github.io/pubs/Agda25.pdf)]

category: people


Henning Bruhn

* [Institute page](https://www.uni-ulm.de/mawi/mawi-or/mitarbeiter/prof-dr-henning-bruhn-fujimoto/)

## Selected writings

On the [[axiomatization]] of [[infinite set|infinite]] [[matroids]]:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

category: people


Henning Hohnhold

* [Maths Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=113277)

## Selected writings

On [[supermanifolds]] form the point of view of [[functorial geometry]]:

* {#HohnholdSolzTeichner11} [[Henning Hohnhold]], [[Stephan Stolz]], [[Peter Teichner]], _Super manifolds: an incomplete survey_, Bulletin of the Manifold Atlas (2011) 1–6 ([pdf](https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/Survey-Journal.pdf))

* [[Henning Hohnhold]], [[Matthias Kreck]], [[Stephan Stolz]], [[Peter Teichner]], Sections 2-3 of: _Differential forms and 0-dimensional supersymmetric field theories_, Quantum Topology Volume 2, Issue 1 (2011) pp. 1–41 ([doi:10.4171/QT/12](http://dx.doi.org/10.4171/QT/12))

category: people


Henning Krause

[[Henning Krause]] is a German mathematician (Padeborn, Bielefeld) specialized in algebra including representation theory of finite dimensional associative algebras and quivers, study of abelian and triangulated categories, including localization theory and spectra of abelian and (tensor) triangulated categories.

* [webpage](http://www2.math.uni-paderborn.de/people/henning-krause.html), [publications](http://www2.math.uni-paderborn.de/people/henning-krause/publications.html)

## Selected writings

On [[localization]] of [[triangulated categories]]:

* {#Krause08} [[Henning Krause]], *Localization theory for triangulated categories*, in proceedings of *[Workshop on Triangulated Categories, Leeds 2006](https://www.commalg.org/2006/08/13/workshop-on-triangulated-categories-leeds/)* [[arXiv:0806.1324](https://arxiv.org/abs/0806.1324)]

See also:

* _The spectrum of a module category_, Memoirs AMS vol. __149__, n. 707 (2001), [gBooks](http://books.google.hr/books?id=ax_P331Qfu0C)

[[!redirects H. Krause]]


Henning Samtleben

* [website](http://perso.ens-lyon.fr/henning.samtleben/henning_samtleben/home.html)

## Selected writings

On [[D=3 supergravity|D=3]] [[gauged supergravity]] and introducing the concept of [[tensor hierarchies]]:

* [[Bernard de Wit]], [[Henning Samtleben]], _Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems_, Fortsch. Phys. 53 (2005) 442-449 ([arXiv:hep-th/0501243](https://arxiv.org/abs/hep-th/0501243))

* [[Bernard de Wit]], [[Henning Samtleben]], _The end of the $p$-form hierarchy_, JHEP 0808:015, 2008 ([arXiv:0805.4767](https://arxiv.org/abs/0805.4767))

On [[tensor hierarchies]] in [[gauged supergravity]] and possible relation to [[U-duality]] and [[M-theory]]:

* [[Bernard de Wit]], [[Hermann Nicolai]], [[Henning Samtleben]], _Gauged Supergravities, Tensor Hierarchies, and M-Theory_, JHEP 0802:044, 2008 ([arXiv:0801.1294](https://arxiv.org/abs/0801.1294))

Introducing a class of [[Lagrangian densities]] for [[D=6 N=(1,0) SCFTs]] with [[non-abelian group|non-abelian]] [[gauge group]] has been proposed in

* {#SSW11} [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, _$(1,0)$ superconformal models in six dimensions_,  J. High Energ. Phys. 2011, 62 (2011) ([arXiv:1108.4060](https://arxiv.org/abs/1108.4060), doi:10.1007/JHEP12(2011)062)

reviewed in

* [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, Linus Wulff, _New superconformal models in six dimensions: Gauge group and representation structure_, PoS CORFU2011 (2011) 071 ([arXiv:1204.0542](https://arxiv.org/abs/1204.0542), [spire:1102894](http://inspirehep.net/record/1102894))

On [[E9]] [[exceptional field theory]]:

* Guillaume Bossard, Franz Ciceri, Gianluca Inverso, [[Axel Kleinschmidt]], [[Henning Samtleben]], _$E_9$ exceptional field theory II. The complete dynamics_ ([arXiv:2103.12118](https://arxiv.org/abs/2103.12118))

On [[D=6 supergravity]] with [[number of supersymmetries|$\mathcal{N} = (4,0)$]]:

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Toward Exotic 6D Supergravities*, Phys. Rev. D **103** (2021) 046002 [[arXiv:2007.11644](https://arxiv.org/abs/2007.11644), [doi:10.1103/PhysRevD.103.046002](https://doi.org/10.1103/PhysRevD.103.046002)]

* [[Yannick Bertrand]], [[Stefan Hohenegger]], [[Olaf Hohm]], [[Henning Samtleben]], *Supersymmetric action for 6D $(4,0)$ supergravity*, JHEP (2022) [[arXiv:2206.04100](https://arxiv.org/abs/2206.04100)]

On [[KK-reduction]] of [[D=10 supergravity]] and [[D=11 supergravity]] on [[squashed sphere|squashed]] [[7-spheres]] via [[exceptional field theory]]:

* Bastien Duboeuf, [[Emanuel Malek]], [[Henning Samtleben]]
*Kaluza-Klein Spectrometry beyond Consistent Truncations: The Squashed $S^7$*,  J. High Energ. Phys. **2023** 62 (2023) [[arXiv:2212.01135](https://arxiv.org/abs/2212.01135), doi:10.1007/JHEP04(2023)062]

## Related entries

* [[supergravity]]

* [[exceptional field theory]]

category: people


Henri Cartan

\begin{imagefromfile}
    "file_name": "HenriCartan_Portrait.jpg",
    "float": "right",
    "width": 200,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 25
    },
    "caption": "from [Tu 2020, p. 31](equivariant+de+Rham+cohomology#Tu20)"
\end{imagefromfile}

**Henri Cartan** (son of [[Élie Cartan|Élie Joseph Cartan]]) was a French mathematician, one of the founders of modern [[homological algebra]] (including aparatus of [[spectral sequence]]s), complex [[analytic geometry]] (several complex variables), [[sheaf theory]], [[cohomology]] theory.

* [Wikipedia entry](http://en.wikipedia.org/wiki/Henri_Cartan)

* [[John Frank Adams]], _The work of M. H. Cartan in its relation with homotopy theory_, Colloque analyse et topologie,Astérisque no. 32-33 (1976), p. 29-41 ([numdam:AST_1976__32-33__29_0/](http://www.numdam.org/item/AST_1976__32-33__29_0/))

## Selected writings

* *Séminaire Henri Cartan* 1948-1964 ([numdam:SHC](http://www.numdam.org/actas/SHC/))

On [[Cartan calculus]] and [[equivariant de Rham cohomology]] ([[Cartan's magic formula]], [[Cartan's map]], ...):

* {#Cartan50} [[Henri Cartan]], _La transgression dans un groupe de Lie et dans un espace fibré principal_, Colloque de topologie (espaces fibrés). Bruxelles, 1950

On [[fiber bundles]], [[principal bundles]] and introducing [[Chern-Weil theory]]:

* [[Henri Cartan]], Section 7 of: _Cohomologie réelle d'un espace fibré principal différentiable. I : notions d'algèbre différentielle, algèbre de Weil d'un groupe de Lie _,   Séminaire Henri Cartan, Volume 2  (1949-1950), Talk no. 19, May 1950  ([numdam:SHC_1949-1950__2__A18_0](http://www.numdam.org/item/?id=SHC_1949-1950__2__A18_0))

  \linebreak

  [[Henri Cartan]], Section 7 of: _Notions d'algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie_, in: Centre Belge de Recherches Mathématiques, _Colloque de Topologie (Espaces Fibrés) Tenu à Bruxelles du 5 au 8 juin 1950_, Georges Thon 1951 ([GoogleBooks](https://books.google.de/books/about/Colloque_de_topologie_espaces_fibres.html?id=sagqHQAACAAJ&redir_esc=y))

  \linebreak

  (These two articles have the same content, with the same section outline, but not the same wording. The first one is a tad more detailed.)

On [[homological algebra]]:

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

## Related $n$Lab entries

* [[Cartan model]] for [[equivariant de Rham cohomology]]

category: people


Henri Epstein

## writings

* Henri Epstein, [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211.

## related $n$Lab entries

* [[renormalization]]

* [[causal perturbation theory]],

* [[lcoally covariant perturbative quantum field theory]]

category: people


Henri Gillet

__Henri Gillet__ is a mathematician at the University of Illinois at Chicago (UIC) working mainly in arithmetic geometry, algebraic geometry and algebraic k-theory.

* [web](http://www.math.uic.edu/~henri)
* Publication lists at uic: [automatic](http://www.math.uic.edu/people/publications?mscsid=henri), [personal](http://www.math.uic.edu/~henri/Pubs.html).

## Publications

Some articles of relevance to $n$Lab:

* The $K$-theory of twisted complexes, in "Applications of algebraic $K$-theory to algebraic geometry and number theory", Part I, II (Boulder, Colo., 1983), 159–191, Contemp. Math. __55__, AMS 1986, [MR862636](http://www.ams.org/mathscinet-getitem?mr=862636_
* $K$-theory and intersection theory. Handbook of $K$-theory. Vol. 1, 2, 235–293, Springer, Berlin, 2005.
* $K$-theory and algebraic geometry. Algebraic $K$-theory and its applications (Trieste, 1997), 206–233, World Sci. 1999
* _Arithmetic intersection theory on Deligne-Mumford stacks_, in: _Motives and algebraic cycles_, pages 93--109.  Amer. Math. Soc. 2009, [MR2562454](http://www.ams.org/mathscinet-getitem?mr=2562454), [arxiv/0904.1424](http://arxiv.org/abs/0904.1424)
* H. Gillet, [[Christophe Soulé]], _Descent, motives and $K$-theory_ J. Reine Angew. Math. __478__ (1996), 127–176h, [doi](http://dx.doi.org/10.1515/crll.1996.478.127), [alg-geom/9507013](http://arxiv.org/abs/alg-geom/9507013); _Filtrations on higher algebraic $K$-theory_, Algebraic $K$-theory (Seattle, WA, 1997), 89–148, Proc. Sympos. Pure Math. __67__, Amer. Math. Soc. 1999, [K-th archive](http://www.math.uiuc.edu/K-theory/0327/index.html); _Motivic weight complexes for arithmetic varieties_, J. Algebra __322__ (2009), no. 9, 3088–3141, [doi](http://dx.doi.org/10.1016/j.jalgebra.2009.07.015)
* [[Spencer Bloch]], Henri Gillet, [[Christophe Soulé]], _Non-Archimedean Arakelov theory_, J. Algebraic Geom. __4__ (1995), no. 3, 427–485, [MR96g:14019](http://www.ams.org/mathscinet-getitem?mr=1325788)

_Comparing algebraic and topological K-theory_, in _Higher Algebraic K-Theory: an overview_ Lecture Notes in Mathematics Volume 1491, 1992, pp 55-99

## Related entries

* [[comparison between algebraic and topological K-theory]]

[[!redirects H. Gillet]]


Henri Lebesgue

* [Wikipedia entry](https://en.wikipedia.org/wiki/Henri_Lebesgue)

## related $n$Lab entries

* [[Lebesgue measure]]

* [[Lebesgue integration]]

* [[Lebesgue number lemma]]

* [[Lebesgue space]]

category: people

[[!redirects Lebesgue]]


Henri Lombardi

* [personal page](http://hlombardi.free.fr/)

## Selected writings

On [[commutative algebra]] via [[constructive mathematics]]:

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

category: people


Henri Moscovici

* [website](https://math.osu.edu/people/moscovici.1)

## related $n$Lab entries

* [[spectral triple]]

category: people


Henri Orland

* [GoogleScholar page](https://scholar.google.com/citations?user=QA6AmkQAAAAJ&hl=de)

## Selected writings

On [[solid state physics]] (with an emphasis on [[path integral]] methods):

* [[John W. Negele]], [[Henri Orland]], *Quantum Many-Particle Systems*, Westview Press (1988, 1998) [[doi:10.1201/9780429497926](https://doi.org/10.1201/9780429497926)]

category: people


Henri Poincaré

* [Wikipedia entry](http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9)

## Selected writings

Introducing modern foundations of [[topology]] and basic concepts of  [[algebraic topology]] (such as [[singular homology]]) and [[homotopy theory]] (such as the [[fundamental group]]):

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique. (2). 1: 1–123  (1895) (Orig.: [gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7), Engl.: [[John Stillwell]]: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]])

On the future of mathematics

* {#Poincaré1908} [[Henri Poincaré]], *The future of mathematics*, Revue generale des Sciences pures et appliquees, Paris, 19th year, No. 23, December, 1908 ([pdf](https://sites.math.rutgers.edu/courses/535/535-f08/Poincare.pdf))

## Related $n$Lab entries

* [[Poincaré lemma]]

* [[Poincaré conjecture]]

* [[Poincaré group]],

  * [[Poincaré spinor group]]

   * [[Poincaré Lie algebra]]

* [[Liouville-Poincaré 1-form]]

* [[Poincaré duality]], [[Poincaré duality algebra]]

* [[Poincaré sphere]], [[Poincaré homology sphere]]

* [[Poincaré-Hopf theorem]]

* [[Poincaré line bundle]]

category: people

[[!redirects Poincaré]]
[[!redirects Poincare]]
[[!redirects Henri Poincare]]
[[!redirects Henri Poincaré's]]


Henri Verschelde

* [webpage](https://biblio.ugent.be/person/801000442621)

## Selected writings

On [[weight systems]] on [[chord diagrams]] as [[observables]] in  [[Jackiw-Teitelboim gravity]] and, under [[AdS2/CFT1]], the [[Sachdev-Ye-Kitaev model]]:

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _The Schwarzian Theory - A Wilson Line Perspective_, JHEP 1812 (2018) 022 ([arXiv:1806.07765](https://arxiv.org/abs/1806.07765))

* [[Andreas Blommaert]], [[Thomas Mertens]], [[Henri Verschelde]], _Fine Structure of Jackiw-Teitelboim Quantum Gravity_, JHEP 1909 (2019) 066 ([arXiv:1812.00918](https://arxiv.org/abs/1812.00918))

category: people


Henriette Elvang

* [personal webpage](https://sites.google.com/a/umich.edu/h-elvang/home)

* [institute page](https://lsa.umich.edu/physics/people/faculty/elvang.html)

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Henriette_Elvang)

## Selected writings

Finding [[black rings]] in [[D=5 supergravity]]:

* [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042,2005 ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

On reducing these to [[black holes]]:

* {#ElvangEmparanMateosReall05} [[Henriette Elvang]], [[Roberto Emparan]], [[David Mateos]], [[Harvey Reall]], _Supersymmetric 4D Rotating Black Holes from 5D Black Rings_, JHEP0508:042,2005 ([arXiv:hep-th/0504125](http://arxiv.org/abs/hep-th/0504125))

On [[scattering amplitudes]] in [[perturbative quantum field theory]]:

* [[Henriette Elvang]], [[Yu-tin Huang]], *Scattering Amplitudes*, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)]

category: people


Henrik Forssell

* [webpage](http://folk.uio.no/jonf/)

## Publications

* [[Henrik Forssell]], [[Håkon Robbestad Gylterud]], [[David Spivak]], *Type theoretical databases*, Journal of Logic and Computation, Volume 30, Issue 1, January 2020, Pages 217–238, ([doi:10.1093/logcom/exaa009](https://doi.org/10.1093/logcom/exaa009), [arXiv:1406.6268](https://arxiv.org/abs/1406.6268))

category: people


Henrik Johansson

* [institute page](https://www.physics.uu.se/research/theoretical-physics/people/henrik-johansson/)

## Selected writings

On [[scattering amplitudes]] in [[perturbative quantum gravity|perturbative]] [[D=4 N=8 supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

On the "[[double copy]]" [[KLT relations]]-approach to [[scattering amplitudes]] in [[perturbative quantum gravity]] and [[supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Henrik Johansson]], _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_, Phys. Rev. Lett. **105** 061602 (2010) [[arXiv:1004.0476](https://arxiv.org/abs/1004.0476)]

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Marco Chiodaroli]], [[Henrik Johansson]], [[Radu Roiban]], *Supergravity amplitudes, the double copy and ultraviolet behavior*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.07392](https://arxiv.org/abs/2304.07392)]

category: people


Henrik Schlichtkrull

* [webpage](http://www.math.ku.dk/~schlicht/)

## related $n$Lab entries

* [[differentiable manifold]]

category: people


Henrique Bursztyn

* [webpage](http://w3.impa.br/~henrique/)

category: people


Henry W. Lin

* [institute page](https://sitp.stanford.edu/people/henry-lin)

* [GoogleScholar page](https://scholar.google.com/citations?user=WQ8fqcIAAAAJ)

## Selected writings

On [[AdS2/CFT1 duality]] between [[JT-gravity]] and the [[SYK model]] in terms of [[chord diagrams]]:

* [[Henry W. Lin]], *The bulk Hilbert space of double scaled SYK*, J. High Energ. Phys. **2022** 60 (2022) [[arXiv:2208.07032](https://arxiv.org/abs/2208.07032), doi:10.1007/JHEP11(2022)060]

* [[Henry W. Lin]], [[Douglas Stanford]], *A symmetry algebra in double-scaled SYK* [[arXiv:2307.15725](https://arxiv.org/abs/2307.15725)]

category: people


Hensel's lemma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The result called **Hensel's lemma** is a generalisation of a result due to [[Kurt Hensel]] on solving [[polynomial]] [[equations]] in [[p-adic number]] [[rings]]. It applies to certain [[completion of a ring|complete]] [[topological rings]], and now [[local rings]] that satisfy the conclusion of the lemma (really a theorem) are called _[[Henselian rings]]_.

## Statement

An element $x$ of a [[topological ring]] is called *topologically nilpotent* if 0 is a limit of the sequence $\{x^n\}$. (For example: in any ring $A$ equipped with a (two-sided) [[ideal]] $m$, the elements of $m$ are topologically nilpotent in the $m$-adic topology.)

A topological ring is *linearly topologized* if 0 has a fundamental system of neighbourhoods consisting of ideals.

+-- {: .num_theorem}
###### Theorem
("Hensel's Lemma")

Let $A$ be a complete Hausdorff linearly topologized commutative ring. Let $m$ be a closed ideal of $A$ whose elements are topologically nilpotent. Let $B=A/m$ be the quotient topological ring and $\phi\colon A\to B$ the quotient map. Let $R\in A\{X\}$ be a [[restricted formal power series]], $\overline{P}$ a monic polynomial in $B[X]$ and $\overline{Q}\in B\{X\}$. Suppose that $\overline{\phi}(R) = \overline{P}.\overline{Q}$ and that $\overline{P}$ and $\overline{Q}$ are strongly relatively prime in $B\{X\}$. Then there exists a unique lift of $\overline{P}$ to $P\in A[X]$ and of $\overline{Q}$ to $Q \in A\{X\}$ such that $R=P.Q$. Moreover $P$ and $Q$ are strongly relatively prime in $A\{X\}$, and if $R$ is a polynomial, so is $Q$.

=--

The proof proceeds by considering successively more general statements, starting with various cases in which $A$ is discrete, in which case $R$ and $\overline{Q}$ are polynomials.

+-- {: .proof}
###### Proof

First consider the case that $m^2=0$. Let $S,T\in A[X]$ with $S$ monic, $\overline{\phi}(S)=\overline{P}$ and $\overline{\phi}(T) = \overline{Q}$.

\ldots

The second case is that $m$ is a nilpotent ideal: there is some $n\geq 2$ such that $m^n=0$. This is proved by induction on $n$, with the base case covered by the first part of the proof.

\ldots

The third case assumes merely that $A$ is discrete, or equivalently that $m$ is a [[nil-ideal]] (every element in $m$ is nilpotent, with no global bound on the order). This case considers an ideal $n$ generated by coefficients of the polynomials at hand, which is then a finitely-generated ideal contained in a nil-ideal, hence nilpotent. We thus can use the second case.

\ldots

The last case is the general case, where one considers a fundamental system of open neighbourhoods of $0$ by ideals $I$, whence $A/I$ is discrete.

\ldots

=--

For the full proof, see ([Bourbaki](#Bourbaki_CA))

This gives rather simpler looking results in special cases, but all of them boil down to lifting factorisations through a quotient map $A \to A/m$.

The original example is of $A$ being the [[p-adic integers]] $\mathbb{Z}_p$, with the quotient $\mathbb{Z}_p \to \mathbb{F}_p = \mathbb{Z}_p/p\mathbb{Z}_p$.

## Related concepts

* [[Henselian ring]]
* [[Henselian pair]]

## References

The original paper in which a special case of Hensel's lemma appeared, for monic polynomials over the p-adic integers, is

* Kurt Hensel, _Neue Grundlagen der Arithmetik._ Journal für die reine und angewandte Mathematik 127 (1904) 51-84 ([EuDML](https://eudml.org/doc/149178))

and updated to remove monicity in

* Kurt Hensel, _Theorie der algebraischen Zahlen_, 1908. ([Internet Archive](https://archive.org/details/theoriederalgeb01hensgoog))

A proof for more general topological rings is in

* {#Bourbaki_CA} [[Bourbaki]], _Commutative Algebra_, III.4.3

See also for simple examples over the $p$-adic numbers:

* Wikipedia, [Hensel's lemma](https://en.wikipedia.org/wiki/Hensel%27s_lemma)

A viewpoint of Hensel's lemma using étale covers is in Chapitre XI of

* {#Raynaud} Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Volume 169 (1970) doi:[10.1007/BFb0069571](https://dx.doi.org/10.1007/BFb0069571)

For an application/quick explanation see [this Math.SE answer](https://math.stackexchange.com/a/1347065/3835)

A version of Hensel's lemma for arbitrary continuous functions $\mathbb{Z}_p \to \mathbb{Z}_p$ (rather than polynomials or formal power series) is in:

* Hajime Kaneko, Thomas Stoll, _Hensel's lemma for general continuous functions_, arXiv:[1707.01445](https://arxiv.org/abs/1707.01445)


Henselian pair

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that a [[Henselian ring]] is a [[local ring]] $R$ with [[maximal ideal]] $\mathfrak{m}$ satisfying the conclusion of [[Hensel's lemma]]. A **Henselian pair** is a generalisation of this concept to the case where the ring doesn't have to be a local ring, and the maximal ideal is replaced by a more general ideal $I$.

## Definition

A *pair* consists of a ring $R$ and an [[ideal]] $I\subset R$.

+-- {: .num_defn}
###### Definition

A **henselian pair** is a pair $(R, I)$ satisfying

1. $I$ is contained in the [[Jacobson radical]] of $R$, and
1. for any [[monic polynomial]] $f \in R[T]$ and factorization
$\overline{f} = g_0h_0$ with $g_0, h_0 \in R/I[T]$ monic
generating the unit ideal in $R/I[T]$, there
exists a factorization $f = gh$ in $R[T]$ with $g, h$ monic
and $g_0 = \overline{g}$ and $h_0 = \overline{h}$.

There are several equivalent characterizations, see
[the Stacks Project](#Stacks-henselian-pair). Another characterization is the following (see [Gabber](#Gabber)):

+-- {: .num_prop}
###### Proposition

A pair $(R,I)$ is a henselian pair if and only if

1. $I$ is contained in the [[Jacobson radical]] of $R$, and
1. Let $f(t) = (t-1)t^N + g(t)$ where $g(t) \in IR[t]$ has degree $\leq N$. Then $f(t)$ has a (necessarily unique) root in $I+1$.
=--

This characterization shows that the property of $(R,I)$ being a henselian pair essentially depends only on the ideal $I$, regarded as a non-unital ring, and not on $R$. Thus the property of $I$ being henselian may be axiomatized in terms of an additional set of algbraic operations on $I$, sending $(g_0,\dots, g_{N})$ to $i$ where $1+i$ is the root of $(t-1)t^N + g_{N}t^{N} + \dots + g_0$, and the resulting algebraic category is equivalent to a full subcategory of the category of nonunital rings.

=--

## Properties

The category of pairs has as morphisms $(R,I) \to (S,J)$ those ring maps $\phi\colon R\to S$ such that $\phi(I) \subset J$. The category of Henselian pairs is the obvious full subcategory.

+-- {: .num_prop}
###### Proposition

The inclusion map $HenselianPairs \to Pairs$ has a left adjoint
=--

For a proof, see ([Tag 0A02](https://stacks.math.columbia.edu/tag/0A02)) in the Stacks Project. This Henselization functor restricts to the usual one on the subcategory of local rings and local homomorphisms.

## References

* {#Stacks-henselian-pair} Stacks Project [Tag 09XD](https://stacks.math.columbia.edu/tag/09XD)
* Eberhard Scherzler, _On henselian pairs_,  Communications in Algebra
Volume 3 (1975) pp 391-404,  doi:[10.1080/00927877508822052](https://doi.org/10.1080/00927877508822052)
* Michel Raynaud, _Anneaux locaux henséliens_, Lecture Notes in Mathematics Volume **169** 1970 doi:[10.1007/BFb0069571](https://dx.doi.org/10.1007/BFb0069571)
* {#Gabber} Gabber, Ofer, _$K$-Theory of Henselian Local Rings and Henselian Pairs_, Algebraic $K$-Theory, Commutative Algebra, and Algebraic Geometry (Santa Margherita Ligure, 1989), 126:59–70. Contemp. Math. Amer. Math. Soc., Providence, RI, 1992. [mathscinet](https://mathscinet.ams.org/mathscinet-getitem?mr=1156502)


Henselian ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Henselian ring_ is a [[local ring]] for which the conclusion of [[Hensel's lemma]] (classically stated for a [[complete local ring]]) holds.

## Definition

+-- {: .num_defn}
###### Definition

Let $R$ be a local ring with [[maximal ideal]] $m$. Let $k = R/m$ be the residue class [[field]], and for a [[polynomial]] $f \in R[x]$, let $\bar{f} \in k[x]$ obtained by reduction of the coefficients of $f$ modulo $m$. Then $R$ is _Henselian_ if, for any [[monic polynomial|monic]] $f \in R[x]$ such that $\bar{f}$ factorizes as $g_0 h_0$ with $g_0, h_0$ monic and relatively prime in $k[x]$, there exist monic $g, h \in R[x]$ such that $f = g h$ with $\bar{g} = g_0$, $\bar{h} = h_0$, and the ideal $(g, h)$ is the unit ideal in $R[x]$.

=--

Often this definition is given just for the case when one of $g_0$, $h_0$ is a linear factor $x - a$, where the idea is that the (simple) root $a$ can be lifted to $R$.

+-- {: .num_defn}
###### Definition

A Henselian ring $R$ with residue class field $k$ is _strictly Henselian_ if $k$ is separably closed.

=--

## Examples

* Any field $k$ is trivially Henselian.

* A complete local ring, such as [[p-adic number]] rings, is Henselian. This includes rings of formal power series over a field, $k[ [x_1, \ldots, x_n] ]$.

* Rings of convergent power series over a [[local field]] are Henselian.

## Properties

+-- {: .num_prop}
###### Proposition

The quotient of a Henselian ring is also Henselian.

=--

+-- {: .proof}
###### Proof

In the first place, a quotient $R'$ of a local ring $R$ is local (and has the same residue class field $k$). Secondly, $R'$ is clearly Henselian since we can lift factorizations $\bar{f} = g_0 h_0$ in $k[x]$ to factorizations in $R[x]$, and then push them down to $R'[x]$.

=--

+-- {: .num_prop}
###### Proposition

If $R$ is local and its reduced ring (i.e., $R$ modulo its ideal of nilpotent elements) is Henselian, then $R$ itself is Henselian.

=--

+-- {: .num_prop}
###### Proposition

If $R$ is Henselian and $S$ is a local ring that is integral over $R$ (meaning that $S$ is an $R$-algebra and each $x \in S$ is an [[integral closure|integral element]] over $R$), then $S$ is Henselian.

=--

## Henselization

Let $LocRing$ denote the category of local rings (commutative of course) and local ring homomorphisms ($f: R \to S$ is _local_ if the pullback along $f$ of the maximal ideal of $S$ is the maximal ideal of $R$).

+-- {: .num_theorem}
###### Theorem

The full subcategory of $LocRing$ consisting of Henselian rings is reflective. The left adjoint to the full inclusion is called _Henselization_.

=--

+-- {: .num_example}
###### Example

Let $R$ be a [[valuation ring|discrete valuation ring]], with $K$ its field of fractions. Let $\hat{R}$ be the $m$-adic completion of $R$ with respect to its maximal ideal. Then the Henselization of $R$ is isomorphic to the subring of $\hat{R}$ whose elements are roots of separable polynomials with coefficients in $K$.

=--

A rule of thumb, as suggested by this example, is that the Henselization is the algebraic part of a local ring completion.

## Related concepts

* [[restricted formal power series]]
* [[Henselian pair]]

## References

An original reference is

* [[Alexandre Grothendieck]],  (1967), [[EGA]]: IV. _Étude locale des schémas et des morphismes de schémas_ , Quatrième partie", Publications Mathématiques de l'IHÉS 32: 5–361. ([web](http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1967__32_))

Lecture notes are in

* [[James Milne]], section 4 of _[[Lectures on Étale Cohomology]]_

Other sources include

* Michel Raynaud, _Anneaux locaux henséliens_, Lecture Notes in Mathematics Volume **169** 1970 doi:[10.1007/BFb0069571](https://dx.doi.org/10.1007/BFb0069571)

* [[Jacob Lurie]], [[Descent Theorems]], section 3.

* Alonso, Lombardi, Perdry, _Henselian local rings_ ([pdf](http://drops.dagstuhl.de/volltexte/2006/437/pdf/05021.PerdryHerve.Paper.437.pdf))

* Krzysztof Jan Nowak, _Remarks on Henselian rings_ ([pdf](http://www.emis.de/journals/UIAM/PDF/46-79-85.pdf));

* [[Ieke Moerdijk]], _Rings of smooth functions and their localizations_ ([pdf](http://nlab.mathforge.org/nlab/files/RingsOfSmoothFunctionsI.pdf))

[[!redirects Hensel ring]]
[[!redirects Hensel rings]]
[[!redirects Henselian rings]]
[[!redirects henselian ring]]
[[!redirects henselian rings]]


Henstock integral

# The Henstock integral
* table of contents
{: toc}

## Idea

The *Henstock integral* (also attributed to *Kurzweil*, *Denjoy*, *Luzin*, and *Perron*, and sometimes called, neutrally but perhaps ambiguously, the *gauge integral*) is a way to define the [[integral]] of a (partial) function $f:\mathbb{R}\to \mathbb{R}$ which applies to more functions than either the [[Riemann integral]] or the [[Lebesgue integral]] and is in some ways better behaved as well.

However, the Lebesgue integral is more commonly used by working mathematicians because it fits more naturally into the general theory of [[measure]], while the Riemann/Darboux integral is more commonly used in introductory calculus courses because its definition is simpler.

## Definition

Let $[a,b]$ be a closed [[interval]] in $\mathbb{R}$ and let $f:[a,b]\to \mathbb{R}$.  A *tagged partition* $P$ of $[a,b]$ is a finite sequence of points $a = u_0 \leq u_1 \leq \dots \leq u_n = b$ together with points $t_i \in [u_i, u_{i+1}]$ for all $i$.  The *Riemann sum* of $f$ over such a tagged partition is

$$ \sum_P f = \sum_{i=0}^{n-1} f(t_i) \cdot (u_{i+1} - u_{i}). $$

Define a *gauge* on $[a,b]$ to be any function $\delta: [a,b] \to (0,\infty]$.  We say that a tagged partition is *$\delta$-fine* if $[u_i, u_{i+1}] \subset [t_i - \delta(t_i), t_i + \delta(t_i)]$.

Finally, we say that $I$ is the **integral** of $f$ on $[a,b]$, written $I = \int_a^b f = \int_{a}^b f(x) \,d x$, if for any $\epsilon\gt 0$ there exists a gauge $\delta$ such that

$$ {\left| {\sum_P f - I} \right|} \lt \epsilon $$

for any $\delta$-fine partition $P$.  If such an $I$ exists, it must be unique, and we say that $f$ is (Henstock) **integrable** on $[a,b]$.

We can also write $\int_a f(x) \,d x$ for the [[semidefinite integral|semidefinite]] Henstock integral $\int_a^x f(t) \,d t$, and write $\int_a f$ for the function $x \mapsto \int_a f(x) \,d x$.

### Comparison to the Riemann definition

If we require a gauge to be a [[constant function]], then we recover the definition of the [[Riemann integral]].

Thus the Henstock integral may be seen as a non-uniform generalization of the Riemann integral.  Whereas specifying a constant $\delta$ is tantamount to picking an [[entourage]] on $[a,b]$, specifying a gauge $\delta$ is tantamount to assigning a [[neighbourhood]] to each point in $[a,b]$.  (Indeed, with either definition of integral, it would be equivalent to replace $\delta$ in the definition with an entourage or an assignment of neighbourhoods.)  Similarly, the definition of [[uniformly continuous function]] becomes that of [[continuous function]] if you change $\delta$ from a constant to a gauge.

### Constructive version

In [[constructive analysis]], we must allow a gauge to take [[lower real number|lower real]] values.  (This is not necessary with the Riemann integral.)  Otherwise, there may not be enough gauges, since these are rarely continuous.  (The definition could also be made constructive by explicitly referring to an assignment of neighbourhoods to points or by replacing $\delta$ with an [[entire relation]].)

## Examples

The [[characteristic function]] $\chi_{\mathbb{Q}}$ of the [[rational numbers]] (as a subset of the real numbers) is a famous example of a function that's Lebesgue integrable but not even locally Riemann integrable.  It is Henstock integrable (with integral $0$) on any $[a,b]$ as follows:  Enumerate the rationals in $[a,b]$ as $(q_i)_{i=0}^\infty$.  Given $\epsilon \gt 0$, let $\delta(x)$ be $b-a$ if $x$ is irrational but $2^{-i-2}\epsilon(b-a)$ if $x$ is the rational $q_i$.  Then $\sum_{x\in[a,b]} \chi_{\mathbb{Q}}(x) \,2\delta(x) = \epsilon$.  Since a $\delta$-fine Riemann sum consists of just some of these terms, with $2\delta(x)$ replaced by a length that might be smaller, the value of the $\delta$-fine Riemann sum is thus always at most $\epsilon$.

The (even) function
$$
x\mapsto \frac{\sin(1/x^3)}{x},\quad x \in \mathbb{R}\setminus\{0\}
$$
is not Riemann or Lebesgue integrable on any interval containing 0, but it has the Henstock integral
$$
\int_{0} \frac{\sin (1/x^3)}{x}\,d x = \frac{1}{3}\left( \pi/2 - Si(1/x^3)\right)
$$
where $Si(t)$ is the [[sine integral]] $\int_0 \frac{\sin(t)}{t}d t $ (which extends to an [[entire function]] on $\mathbb{C}$).  This integral can also be found as an [[improper integral|improper]] Riemann integral.

For an example that is neither Lebesgue integrable nor improperly Riemann integrable (not even locally Riemann integrable), we can let $f(x)$ be $\sin(1/x^3)/x$ for irrational $x$ and $1$ for rational $x$.  This one can still be done as an improper Lebesgue integral.  (Are there any functions that are Henstock integrable but not locally Lebesgue integrable?)

## Properties

### The fundamental theorem of calculus

The Henstock integral satisfies a very nice form of the [[fundamental theorem of calculus]]:

+-- {: .un_theorem}
###### Theorem
If $f$ is differentiable on $[a,b]$, then $f'$ is Henstock integrable on $[a,b]$, and $\int_a^b f'(x) d x = f(b) - f(a)$.
=--

+-- {: .un_theorem}
###### Theorem
If $f$ is Henstock integrable on $[a,b]$, then $F(x) = \int_a f(x) d x$ is differentiable [[almost everywhere]] on $[a,b]$ and $F' = f|_{\dom F'}$.
=--

### Hake's theorem

+-- {: .un_theorem}
###### Theorem
For any $f$ we have
$$ \int_a^b f(x) d x = \lim_{c\to b^-} \int_a^c f(x) d x $$
in the strong sense that if either side exists, then so does the other, and they are equal.
=--

In particular, what is often taken as a *definition* of the [[improper integral|improper]] Riemann integral (of a potentially unbounded function on a finite interval) is actually a *theorem* for Henstock integrals.  (However, we still need improper Henstock integrals to allow $a = -\infty$ or $b = \infty$.)

### Recovery of Riemann and Lebesgue integrals

+-- {: .query}
I need to check some of the claims below, but I\'m out of time right now.  They are definitely correct for the proper integrals.  ---Toby
=--

Recall that $f\colon [a, b] \to \mathbb{R}$ is Riemann integrable iff $f$ is continuous almost everywhere and bounded; in this case, $f$ is also Henstock integrable, and the Riemann integral of $f$ equals its Henstock integral.

More generally, $f$ is improperly Riemann integrable iff $f$ is Henstock integrable and $f$ is locally Riemann integrable at all but finitely many points in $[a, b]$; then the improper Riemann integral of $f$ equals its Henstock integral.

Still more generally, $f\colon \mathbb{R} \to \mathbb{R}$ is improperly Riemann integrable iff $f$ is improperly Henstock integrable (meaning merely that
$$ \lim_{a \to -\infty, b \to \infty} \int_a^b f(x) \,\mathrm{d}x $$
exists using Henstock integrals) and locally Riemann integrable except at a set of [[isolated point]]s; then the improper Riemann integral of $f$ equals its improper Henstock integral.

Finally (and with incomparable generality), $f\colon \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable iff ${|f|}$ is improperly Henstock integrable; then the Lebesgue integral of $f$ equals its improper Henstock integral (which is proper if the [[support]] of $f$ is [[bounded set|bounded]], of course).

[[!redirects Henstock integral]]
[[!redirects Henstock integrals]]
[[!redirects Henstock integrable]]
[[!redirects Henstock-integrable]]
[[!redirects Henstock integrable function]]
[[!redirects Henstock integrable functions]]
[[!redirects Henstock-integrable function]]
[[!redirects Henstock-integrable functions]]

[[!redirects Kurzweil integral]]
[[!redirects Kurzweil integrals]]
[[!redirects Kurzweil integrable]]
[[!redirects Kurzweil-integrable]]
[[!redirects Kurzweil integrable function]]
[[!redirects Kurzweil integrable functions]]
[[!redirects Kurzweil-integrable function]]
[[!redirects Kurzweil-integrable functions]]

[[!redirects Henstock-Kurzweil integral]]
[[!redirects Henstock-Kurzweil integrals]]
[[!redirects Henstock–Kurzweil integral]]
[[!redirects Henstock–Kurzweil integrals]]
[[!redirects Henstock--Kurzweil integral]]
[[!redirects Henstock--Kurzweil integrals]]
[[!redirects Henstock-Kurzweil integrable function]]
[[!redirects Henstock-Kurzweil integrable functions]]
[[!redirects Henstock–Kurzweil integrable function]]
[[!redirects Henstock–Kurzweil integrable functions]]
[[!redirects Henstock--Kurzweil integrable function]]
[[!redirects Henstock--Kurzweil integrable functions]]

[[!redirects Kurzweil-Henstock integral]]
[[!redirects Kurzweil-Henstock integrals]]
[[!redirects Kurzweil–Henstock integral]]
[[!redirects Kurzweil–Henstock integrals]]
[[!redirects Kurzweil--Henstock integral]]
[[!redirects Kurzweil--Henstock integrals]]
[[!redirects Kurzweil-Henstock integrable function]]
[[!redirects Kurzweil-Henstock integrable functions]]
[[!redirects Kurzweil–Henstock integrable function]]
[[!redirects Kurzweil–Henstock integrable functions]]
[[!redirects Kurzweil--Henstock integrable function]]
[[!redirects Kurzweil--Henstock integrable functions]]

[[!redirects Denjoy integral]]
[[!redirects Denjoy integrals]]
[[!redirects Denjoy integrable function]]
[[!redirects Denjoy integrable functions]]

[[!redirects Luzin integral]]
[[!redirects Luzin integrals]]
[[!redirects Luzin integrable function]]
[[!redirects Luzin integrable functions]]

[[!redirects Perron integral]]
[[!redirects Perron integrals]]
[[!redirects Perron integrable function]]
[[!redirects Perron integrable functions]]

[[!redirects gauge integral]]
[[!redirects gauge integrals]]
[[!redirects gauge integrable]]
[[!redirects gauge-integrable]]
[[!redirects gauge integrable function]]
[[!redirects gauge integrable functions]]
[[!redirects gauge-integrable function]]
[[!redirects gauge-integrable functions]]