E. Hastings Moore

* Wikipedia, [E. H. Moore](https://en.wikipedia.org/wiki/E._H._Moore)

## Selected writings

The first use of the word "[[field]]" to refer to the algebraic structure of a commutative division ring is found in:

* {#Moore93} [[E. Hastings Moore]] (1893), "A doubly-infinite system of simple groups", Bulletin of the American Mathematical Society, 3 (3): 73–78, $[$[doi:10.1090/S0002-9904-1893-00178-X](https://doi.org/10.1090%2FS0002-9904-1893-00178-X), [MR:1557275](https://mathscinet.ams.org/mathscinet-getitem?mr=1557275)$]$

[[!redirects E. H. Moore]]
[[!redirects E. Hastings Moore]]
[[!redirects Eliakim Hastings Moore]]


E. J. Brody

## Selected writings

On classification of [[lens spaces]]:

* {#Brody60} [[E. J. Brody]], _The topological classification of the lens spaces_, Ann. of Math. (2) 71 (1960), 163–184. MR0116336 (22 #7125) Zbl 0119.18901


E. Jan Post

*  R. M. Kiehn: *[About E. J. Post](http://www22.pair.com/csdc/car/carfre56.htm)* (2010)

## Selected writings

On the [[constitutive relation|constitutive map]] (and introducing that terminology as now used for [[electromagnetism]] in [[dielectric media]]) in [[pre-metric electromagnetism]]:

* [[E. Jan Post]], *The constitutive map and some of its ramifications*, Annals of Physics **71** 2 (1972) 497-518 [doi:10.1016/0003-4916(72)90129-7]

category: people

[[!redirects E. J. Post]]


E. John Lemmon

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

## Selected writings

On [[modal logic]]:

* [[E. John Lemmon]] with [[Dana Scott]], *An Introduction to Modal Logic -- The "Lemmon Notes"*, B. Blackwell (1977) [[ark:/13960/t3gz25k3h](https://archive.org/details/introductiontomo0000lemm/)]

category: people

[[!redirects John Lemmon]]


E10

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## Idea

A hyperbolic [[Kac-Moody Lie algebra]] in the E-series

... [[E6]], [[E7]], [[E8]], [[E9]], $E_{10}$, [[E11]], ...

## Properties

### As U-duality group of 1d M-theory
 {#AsUDualityOf1dMTheory}

$E_{10}$ is conjectured (e.g. [Nicolai 08](#Nicolai08)) to be the [[U-duality]] group (see there) of [[M-theory]] [[KK-compactification|compactified]] to 1 dimension (see also [[F/M-theory on elliptically fibered Calabi-Yau 5-folds]]).

[[!include U-duality -- table]]

## References

### General

Lecture notes include

* [[Hermann Nicolai]], _Infinite dimensional symmetries_ (2009) ([pdf](http://www.itp.uni-hannover.de/~lechtenf/Events/Lectures/nicolai.pdf))

The fact that every simply laced hyperbolic Kac-Moody algebra is a sub Lie algebra of $E_{10}$ is due to

* Sankaran Viswanath, _Embeddings of hyperbolic Kac-Moody algebras into $E_{10}$_ ([pdf](http://math.iisc.ernet.in/~svis/ehk.pdf))

### Relation to supergravity

Discussion of $E_{10}$ as [[U-duality]] of [[supergravity]]/[[M-theory]]:

The case of [[E10]] is discussed for [[boson|bosonic]] degrees of freedom in

* [[Thibault Damour]], [[Marc Henneaux]], [[Hermann Nicolai]], _$E(10)$ and a 'small tension expansion' of M
theory_, Phys. Rev. Lett. 89, 221601 (2002) ([arXiv:hep-th/0207267](http://arxiv.org/abs/hep-th/0207267));

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _$E(10)$ and $SO(9,9)$ invariant supergravity_, JHEP 0407,
041 (2004) ([arXiv:hep-th/0407101](http://arxiv.org/abs/hep-th/0407101))

and for fermionic degrees of freedom in [[supergravity|supersymmetric]] [[quantum cosmology]] in

* {#Damour14} [[Thibault Damour]], _Quantum supersymmetric Bianchi IX  Cosmology and its hidden Kac-Moody Structure_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[DamourSQC14.pdf:file]])

Review includes

* {#Nicolai08} [[Hermann Nicolai]], _Wonders of $E_{10}$ and $K(E_{10})$_ (2008) ([pdf](http://ipht.cea.fr/Pisp/pierre.vanhove/Paris08/talk_PDF/nicolai.pdf))

* {#Nicolai14} [[Hermann Nicolai]], _On Exceptional Geometry and Supergravity_, talk at [Gravitation, Solitons and Symmetries](http://www.lestudium-ias.com/#!registration-conference-21-may-gibbons/ci2z) ([[NicolaiTalk14.pdf:file]])

Discussion of [[phenomenology]]:

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _Standard model fermions and $K(E_{10})$_ ([arXiv:1504.01586](https://arxiv.org/abs/1504.01586))

* {#MeissnerNicolai18a} Krzysztof A. Meissner, [[Hermann Nicolai]], _Standard Model Fermions and Infinite-Dimensional R-Symmetries_, Phys. Rev. Lett. 121, 091601 (2018) ([arXiv:1804.09606](https://arxiv.org/abs/1804.09606))

* {#MeissnerNicolai18b} Krzysztof A. Meissner, [[Hermann Nicolai]], _Planck Mass Charged Gravitino Dark Matter_, Phys. Rev. D 100, 035001 (2019) ([arXiv:1809.01441](https://arxiv.org/abs/1809.01441))


E11

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## Idea

A hyperbolic [[Kac-Moody Lie algebra]] in the E-series

... [[E6]], [[E7]], [[E8]], [[E9]], [[E10]], $E_{11}$, ...

## Properties

### As U-duality group of 0d supergravity

$E_{11}$ is conjectured ([West 01](#West01)) to be the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 0 dimensions.

[[!include U-duality -- table]]

### Fundamental representation and brane charges
 {#FundamentalRepresentationAndBraneCharges}

The first [[fundamental representation]] of $E_{11}$, usually denoted $l_1$, is argued, since ([West 04](#West04)), to contain in its decomposition into [[representations]] of $GL(11)$ the representations in which the [[charges]] of the [[M-branes]] and other [[BPS states]] transform.

According to ([Nicolai-Fischbacher 03, first three rows of table 2 on p. 72](#NicolaiFischbacher03), [West 04](#West04), [Kleinschmidt-West 04](#KleinschmidtWest04)) and concisely stated for instance in ([West 11, (2.17)](#West11)), the level decomposition of $l_1$ under $GL(11)$ starts out as so:

$$
  l_1
  \simeq
  \underset{level\,0}{
  \underbrace{
    \mathbb{R}^{10,1}
  }}
  \oplus
  \underset{level\,1}{
  \underbrace{
    \wedge^2 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus
  \underset{level\,2}{
  \underbrace{
    \wedge^5 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus
  \underset{level\,3}{\underbrace{
    \wedge^7 (\mathbb{R}^{10,1})^\ast \otimes_s (\mathbb{R}^{10,1})^\ast
    \oplus
    \wedge^8 (\mathbb{R}^{10,1})^\ast
  }}
  \oplus
  \cdots
$$

Here the $level \leq 2$-truncation happens to coincide with the bosonic [[body]] underlying the [[M-theory super Lie algebra]] and via the relation of that to [[BPS charges]] in [[11-dimensional supergravity]]/[[M-theory]], the [[direct sums|direct summands]] here have been argued to naturally correspond to

* level 0: [[momentum]]

* level 1: [[M2-brane]] charge

* level 2: [[M5-brane]] charge

* level 3: [[dual graviton]] charge ([West 11, section N](#West11)) (has two components [^ThanksToKleinschmidt])

[^ThanksToKleinschmidt]: private communication with [[Axel Kleinschmidt]]

## References

### General

* [[Peter West]], section 16.7 of _[[Introduction to Strings and Branes]]_

* {#NicolaiFischbacher03} [[Hermann Nicolai]], Thomas Fischbacher, _Low Level Representations for E10 and E11_ ([arXiv:hep-th/0301017](http://arxiv.org/abs/hep-th/0301017))

* H. Mkrtchyan, R. Mkrtchyan, _$E_{11},K_{11}$ and $EE_{11}$_ ([arXiv:hep-th/0407148](http://arxiv.org/abs/hep-th/0407148))

### Relation to supergravity
 {#ReferencesRelationToSupergravity}

Literature discussing $E_{11}$ [[U-duality]] and in the context of [[exceptional generalized geometry]] of [[11-dimensional supergravity]].

Review includes

* [[Peter West]], section 17.5 of _[[Introduction to Strings and Branes]]_

* [[Fabio Riccioni]], _$E_{11}$ and M-theory_, talk at [Strings07](http://www.ift.uam.es/strings07/) ([pdf slides](http://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2007/riccioni.pdf))

* [[Fabio Riccioni]], [[Peter West]], _The $E_{11}$ origin of all maximal supergravities_, JHEP 0707:063,2007 ([arXiv:0705.0752](http://arxiv.org/abs/0705.0752), [spire](http://inspirehep.net/record/749966/))

* [[Paul Cook]], _Connections between Kac-Moody algebras and M-theory_ PhD thesis ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

* {#West16} [[Peter West]], _A brief review of E theory_ ([arXiv:1609.06863](http://arxiv.org/abs/1609.06863))

Original articles include the following:

The observation that $E_{11}$ seems to neatly organize the structures in [[11-dimensional supergravity]]/[[M-theory]] is due to

* {#West01} [[Peter West]], _$E_{11}$ and M theory_, Class. Quant. Grav., 18:4443–4460, 2001. ([arXiv:hep-th/0104081](http://arxiv.org/abs/hep-th/0104081))

A precursor to ([West 01](#West01)) is

* [[Bernard Julia]], _Dualities in the classical supergravity limits_ ([arXiv:hep-th/9805083](http://arxiv.org/abs/hep-th/9805083))

as explained in ([Henneaux-Julia-Levie 10](#HenneauxJuliaLevie10)).

The derivation of the [[equations of motion]] of [[11-dimensional supergravity]] and maximally supersymmetric [[5d supergravity]] from a [[vielbein]] with values in the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] is due to

* {#West11} [[Peter West]], _Generalised geometry, eleven dimensions and $E_{11}$_, J. High Energ. Phys. (2012) 2012: 18 ([arXiv:1111.1642](http://arxiv.org/abs/1111.1642))

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ must be a symmetry of strings and branes_, Physics Letters B Volume 759, 10 August 2016, Pages 663–671 ([arXiv:1512.01644](https://arxiv.org/abs/1512.01644))

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ in $11d$_, Physics Letters B Volume 758, 10 July 2016, Pages 278–285 ([arXiv:1601.03974](https://arxiv.org/abs/1601.03974))

This way that elements of [[cosets]] of the [[semidirect product]] $E_{11}$ with its [[fundamental representation]] may encode [[equations of motion]] of [[11-dimensional supergravity]] follows previous considerations for [[Einstein equations]] in

* [[Abdus Salam]], J. Strathdee, _Nonlinear realizations. 1: The Role of Goldstone bosons, Phys. Rev. 184 (1969) 1750,

* [[Chris Isham]], [[Abdus Salam]], J. Strathdee, _Spontaneous, breakdown of conformal symmetry_, Phys. Lett. 31B (1970) 300.

* A. Borisov, V. Ogievetsky, _Theory of dynamical affine and conformal symmetries as the theory of the gravitational field_, Theor. Math. Phys. 21 (1973) 1179-1188 ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3902&option_lang=eng))

* V. Ogievetsky, _Infinite-dimensional algebra of general covariance group as the closure of the finite dimensional algebras of conformal and linear groups_, Nuovo. Cimento, 8 (1973) 988.

Further developments of the proposed $E_{11}$ formulation of [[M-theory]] include

* [[Peter West]], _$E_{11}$, ten forms and supergravity_, JHEP0603:072,2006 ([arXiv:hep-th/0511153](http://arxiv.org/abs/hep-th/0511153))

* [[Fabio Riccioni]], [[Peter West]], _Dual fields and $E_{11}$_, Phys.Lett.B645:286-292,2007 ([arXiv:hep-th/0612001](http://arxiv.org/abs/hep-th/0612001))

* [[Fabio Riccioni]], [[Peter West]], _E(11)-extended spacetime and gauged supergravities_, JHEP 0802:039,2008 ([arXiv:0712.1795](http://arxiv.org/abs/0712.1795))

* [[Fabio Riccioni]], Duncan Steele, [[Peter West]], _The E(11) origin of all maximal supergravities - the hierarchy of field-strengths_, 	JHEP 0909:095 (2009) ([arXiv:0906.1177](http://arxiv.org/abs/0906.1177))

* [[Eric Bergshoeff]], I. De Baetselier, T. Nutma, _E(11) and the Embedding Tensor_ ([arXiv:0705.1304](http://arxiv.org/abs/0705.1304), [poster](http://mms.technologynetworks.net/posters/0364.pdf))

* Guillaume Bossard, [[Axel Kleinschmidt]], Jakob Palmkvist, [[Christopher Pope]], [[Ergin Sezgin]], _Beyond $E_{11}$_ ([arXiv:1703.01305](https://arxiv.org/abs/1703.01305))

Discussion of the [[semidirect product]] of $E_{11}$ with its $l_1$-[[representation]], and arguments that the [[charges]] of the [[M-theory super Lie algebra]] and in fact further brane charges may be identified inside $l_1$ originate in

* {#West04} [[Peter West]], _$E_{11}$, $SL(32)$ and Central Charges_, Phys.Lett.B575:333-
342,2003 ([arXiv:hep-th/0307098](http://arxiv.org/abs/hep-th/0307098))

and was further explored in

* {#KleinschmidtWest04} [[Axel Kleinschmidt]], [[Peter West]], _Representations of $G^{+++}$ and the role of space-time_, JHEP 0402 (2004) 033 ([arXiv:hep-th/0312247](http://arxiv.org/abs/hep-th/0312247))

* [[Paul Cook]], [[Peter West]], _Charge multiplets and masses for $E(11)$_, JHEP 11 (2008) 091 ([arXiv:0805.4451](http://arxiv.org/abs/0805.4451))

* [[Peter West]], _$E_{11}$ origin of Brane charges and U-duality multiplets_, JHEP 0408 (2004) 052 ([arXiv:hep-th/0406150](http://arxiv.org/abs/hep-th/0406150))

Relation to [[exceptional field theory]] is discussed in

* Alexander G. Tumanov, [[Peter West]], _$E_{11}$ and exceptional field theory_ ([arXiv:1507.08912](http://arxiv.org/abs/1507.08912))

Relation to [[Borcherds superalgebras]] is discussed in

* Pierre Henry-Labordere, [[Bernard Julia]], Louis Paulot, _Borcherds symmetries in M-theory_, JHEP 0204 (2002) 049 ([arXiv:hep-th/0203070](http://arxiv.org/abs/hep-th/0203070))

* {#HenneauxJuliaLevie10} [[Marc Henneaux]], [[Bernard Julia]], Jérôme Levie, _$E_{11}$, Borcherds algebras and maximal supergravity_ ([arxiv:1007.5241](http://arxiv.org/abs/1007.5241))

* {#Palmkvist12} Jakob Palmkvist, _Tensor hierarchies, Borcherds algebras and $E_{11}$_, JHEP 1202 (2012) 066 ([arXiv:1110.4892](http://arxiv.org/abs/1110.4892))

On [[E11]]-[[exceptional field theory]]:

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))


E6

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## Idea

One of the [[exceptional Lie groups]].

## Properties

### As automorphisms of the exceptional Jordan algebra

The [[group]] of [[determinant]]-preserving [[linear map|linear]] [[isomorphisms]] of the vector space underlying the [[octonionic Albert algebra]] is  ${E}_{6(-26)}$. (see e.g. ([Manogue-Dray 09](#ManogueDray09))).

This may be written as $SL(3,\mathbb{O})$. ([Dray-Manogue 09a (16)](#DrayManogue09a))

<img src="https://ncatlab.org/nlab/files/MapOfE6.png" width="500">

> graphics grabbed from ([Dray-Manogue 09a, p. 12](#DrayManogue09a))

### Outer automorphism

The [[Dynkin diagram]] of $E_6$ has 2-fold symmetry, corresponding to a nontrivial [[outer automorphism]] of order 2. $E_6$ acts on the [[octonionic projective plane]] and its outer automorphism gives rise to a projective duality in its geometry.

### As part of the ADE pattern

[[!include ADE -- table]]

### As U-duality group of 5d SuGra

$E_6$ is the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 5 dimensions.

[[!include U-duality -- table]]

## Related concepts

* [[exceptional geometry]], [[exceptional generalized geometry]]

* [[G2]], [[F4]],

  **E6**, [[E7]],  [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$

## References

* {#DrayManogue09a} [[Tevian Dray]], [[Corinne Manogue]], *Octonionic Cayley spinors and $E_6$* ([pdf](http://www.physics.usu.edu/wheeler/fieldtheory/octonions/draymanoguecayleyspinors.pdf))

* {#ManogueDray09} [[Corinne Manogue]], [[Tevian Dray]], _Octonions, $E_6$, and Particle Physics_, J.Phys.Conf.Ser.254:012005,2010 ([arXiv:0911.2253](http://arxiv.org/abs/0911.2253))

A description of $E_{6(-26)}$ as the [[special linear group]] in dimension 3 with values in the [[octonions]] is given in

* [[Corinne Manogue]], [[Tevian Dray]], _Octonions and the Structure of $E_6$_ Comment. Math. Univ. Carolin., 51:193–207, 2010.

* Aaron Wangberg, Tevian Dray, _$E_6$, the Group: The structure of $SL(3,\mathbb{O})$_ ([arXiv:1212.3182](http://arxiv.org/abs/1212.3182))

Cohomological properties are discussed in

* Mamoru Mimura, Yuriko Sambe, Michishige Tezuka, _Cohomology mod 3 of the classifying space of the exceptional Lie group $E_6$, I : structure of Cotor_ ([arXiv:1112.5811](http://arxiv.org/abs/1112.5811)),

  _Cohomology mod 3 of the classifying space of the exceptional Lie group $E_6$, II : The Weyl group invariants_ ([arXiv:1201.3414](http://arxiv.org/abs/1201.3414))

See also

* Wikipedia, _[$E_6$](http://en.wikipedia.org/wiki/E6_%28mathematics%29)_


E7

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## Idea

One of the [[exceptional Lie groups]].

## Definition
 {#Definition}

Consider the [[vector space]]

$$
   W \coloneqq \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast
$$

of [[dimension]] $56$. This is naturally a [[symplectic vector space]] with [[symplectic form]] $\omega$ given by the natural pairing between [[linear 2-forms]] and [[bivectors]].

In addition, consider on this space the quartic form $q \colon W \to \mathbb{R}$ which sends an element $v = (\{v^{a b}, w_{a b}\}) \in W$ to

$$
  q(v)
   \coloneqq
   v^{a b } w_{b c} v^{c d} w_{d a}
   -
   \tfrac{1}{4} v^{a b} w_{a b} v^{c d} w_{c d}
   +
   \tfrac{1}{96}
   \left(
     \epsilon_{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8}
     v^{a_1 a_2}  v^{a_3 a_4}  v^{a_5 a_6}  v^{a_7 a_8}
     +
     \epsilon^{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8}
     w_{a_1 a_2}  w_{a_3 a_4}  w_{a_5 a_6}  w_{a_7 a_8}
   \right)
  \,.
$$

Now $E_{7(7)} \subset GL(56,\mathbb{R})$ is the [[subgroup]] of the [[general linear group]] acting on $W$ which preserves both the symplectic form $\omega$ as well as the quartic form $q$. See also [below](#SmallestFundamentalRepresentation).

This presentation is due to [Cartan](#Cartan), for review see [Cremmer-Julia 79, appendix B](#CremmerJulia79), [Pacheco-Waldram 08, B.1](#PachecoWaldram08). A construction via [[octonions]] is due to ([Freudenthal 54](#Freudenthal54)),  one via [[quaternions]] is due to ([Wilson 2014](#Wilson14)).

## Properties

### Representation

#### $\mathbf{56}$ -- The smallest fundamental representation
 {#SmallestFundamentalRepresentation}

The smallest [[fundamental representation]] of $E_7$ is the defining one (from the definition [above](#Definition)), of [[dimension]] $56$. Under the [[special linear group|special linear]] [[subgroup]] $SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes as (e.g. [Cacciatori et al. 10, section 4](#CacciatoriEtAl10), also [Pacheco-Waldram 08, appendix B](#PachecoWaldram08))

$$
  \mathbf{56}
    \simeq
  \mathbf{28} \oplus \mathbf{28}^\ast
    \simeq
   \wedge^2 \mathbb{R}^8 \oplus \wedge^2 (\mathbb{R}^8)^\ast
  \,.
$$

Under the further subgroup inclusion $SL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R}) \hookrightarrow E_7$ this decomposes further as

$$
  \mathbf{56}
    \simeq
  \underset{\simeq \wedge^2 \mathbb{R}^8}{\underbrace{\mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast}}
   \oplus
  \underset{\simeq \wedge^2 (\mathbb{R}^8)^\ast}{\underbrace{\wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7}}
  \,,
$$

where $\wedge^2 (\mathbb{R}^7) \subset \wedge^2 (\mathbb{R}^8)^\ast$ is regarded as the subspace of 2-forms with vanishing 8-components, and where  $\wedge^6 \mathbb{R}^7$ is the [[Poincaré duality|Poincaré dual]] to the complementary subspace of $\wedge^2 (\mathbb{R}^8^\ast)$ of 2-forms with non-trivial 8-component.

This is due to [Cartan](#Cartan), for review see [Cremmer-Julia 79, appendix B](#CremmerJulia79), [Pacheco-Waldram 08, B.1](#PachecoWaldram08).

#### $\mathbf{133}$ -- The adjoint representation
 {#AdjointRepresentation}

The [[adjoint representation]] $\mathbf{133}$ of $E_7$ decomposes under $SL(8,\mathbb{R})$ as ([Pacheco-Waldram 08 (B.7)](#PachecoWaldram08))

$$
  \mathfrak{e}_7
  =
  \mathbf{133}
  \simeq
  (\mathbb{R}^8 \otimes (\mathbb{R}^8)^\ast)_{traceless}
  \oplus
  \wedge^4 (\mathbb{R}^8)^\ast
  \,.
$$

In this decomposition the subspace corresponding to the subalgebra $\mathfrak{su}(8) \hookrightarrow \mathfrak{e}_8$ is the vector space

$$
  \mathfrak{su}(8)
  \simeq
  \mathfrak{so}(8)
  \oplus
  (\wedge^4 (\mathbb{R}^8)^\ast)_-
  \,,
$$

where the first summand denotes the skew-symmetric matrices, and the second summand the [[Hodge duality|Hodge anti-self dual]] 4-forms ([Pacheco-Waldram 08 (B.29) (B.30) and below (2.34)](#PachecoWaldram08)).

Under $GL(7,\mathbb{R}) \hookrightarrow SL(8,\mathbb{R})$ the full adjoint representation decomposes further into ([Pacheco-Waldram 08 (B.21)](#PachecoWaldram08))

$$
  \mathbf{133}
  \simeq
  \left(\mathbb{R}^7 \otimes (\mathbb{R}^7)^\ast\right)
  \oplus
  \left(\wedge^6 \mathbb{R}^7 \oplus \wedge^6 (\mathbb{R}^7)^\ast\right)
  \oplus
  \left(
     \wedge^3 \mathbb{R}^7 \oplus \wedge^3 (\mathbb{R}^7)^\ast
  \right)
  \,.
$$

Here $\wedge^6 (\mathbb{R}^7)^\ast \simeq \mathbb{R}^7$ is the $(-,8)$-component of $\mathbb{R}^7 \oplus (\mathbb{R}^7)^\ast$ and dually, while the $(8,8)$-component carries no information by tracelessness; and $\wedge^3 (\mathbb{R}^7)^\ast$ is the $(-,-,-,8)$-component of $\wedge^4 (\mathbb{R}^8)^\ast$, while $\wedge^3 \mathbb{R}^7$ is the 7-dimensional [[Poincaré duality|Poincaré dual]] of the complement of the $(-,-,-,8)$-component ([Pacheco-Waldram 08 (B.22)](#PachecoWaldram08)).

Taken together this means that under $GL(7,\mathbb{R})$ the subspace $\mathbb{su}(8) \hookrightarrow \mathfrak{e}_8$ is that spanned by

1. $\mathfrak{so}(7)$-elements;

1. sums of a 3-form with its 8d-Hodge+7d-Poincaré-dual 3-vector;

1. sums of a 6-form with its dual 6-vector

hence is

$$
  \mathfrak{su}(8)
  \simeq
  \mathfrak{so}(8) \oplus \wedge^3 \mathbb{R}^7 \oplus \wedge^6 \mathbb{R}^7
  \,.
$$

Hence the [[tangent space]] to the [[coset]] $E_{7(7)}/(SU(8)/\mathbb{Z}_2)$ may be identified as

$$
  \mathfrak{e}_7/\mathfrak{su}(8)
  \simeq
  \odot^2 (\mathbb{R}^7)^\ast
  \oplus
  \wedge^3 (\mathbb{R}^7)^\ast
  \oplus
  \wedge^6 (\mathbb{R}^7)^\ast
  \,.
$$

### As part of the ADE pattern

[[!include ADE -- table]]

### As U-Duality group of 4d SuGra
 {#AsUDualityGroup}

$E_{7(7)}$ is the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] on a 7-dimensional fiber to [[4-dimensional supergravity]] (e.g. [[M-theory on G2-manifolds]]).

Specifically, ([Hull 07, section 4.4](#Hull07), [Pacheco-Waldram 08, section 2.2](#PachecoWaldram08)) identifies the [[vector space]] underlying the $SL(7,\mathbb{R})$-decomposition of the [smallest fundamental representation](#SmallestFundamentalRepresentation)

$$
  \mathbf{56}
  \simeq
  \mathbb{R}^7 \oplus \wedge^2 (\mathbb{R}^7)^\ast \oplus \wedge^5 (\mathbb{R}^7)^\ast \oplus \wedge^6 \mathbb{R}^7
  \,.
$$

as the [[exceptional tangent bundle]]-structure to the 7-dimensional fiber space which one obtains as discussed at _[M-theory supersymmetry algebra -- As an 11-dimensional boundary condition](M-theory+supersymmetry+algebra#AsAn11DimensionalBoundaryCondition)_. Here $\mathbb{R}^7$ is the ordinary [[tangent space]] itself, $\wedge^2 (\mathbb{R}^\ast)^7$ is interpreted as the local incarnation of the possible [[M2-brane]] [[charges]], $\wedge^5 (\mathbb{R}^\ast)^7$ the [[M5-brane]] charges and $\wedge^6 \mathbb{R}^7$ as the charges of [[Kaluza-Klein monopoles]].

[[!include U-duality -- table]]

## Related concepts

* [[exceptional geometry]], [[exceptional generalized geometry]]

* [[G2]], [[F4]],

  [[E6]], **E7**,  [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$

## References

### General

The description of the defining fundamental $\mathbf{56}$-representation of $E_{7(7)}$ is due to

* {#Cartan} [[Eli Cartan]], Thesis, in Oeuvres complètes T1, Part I, Gauthier-Villars, Paris 1952

and recalled for instance in

* {#CremmerJulia79} [[Eugene Cremmer]], [[Bernard Julia]], appendix B of _The $SO(8)$ Supergravity_, Nucl. Phys. B 159 (1979) 141 ([spire](http://inspirehep.net/record/140465?ln=en))

See also

* Robert B. Brown, _Groups of type $E _7$_, Jour. Reine Angew. Math. 236 (1969), 79-102.

A construction via the [[octonions]] is due to

* {#Freudenthal54} [[Hans Freudenthal]], _Beziehungen der $\mathfrak{e}_7$ und $\mathfrak{e}_8$ zur Oktavenebene_, I, II, Indag. Math. 16 (1954), 218–230, 363–368. III, IV, Indag. Math. 17 (1955), 151–157, 277–285. V — IX, Indag. Math. 21 (1959), 165–201, 447–474. X, XI, Indag. Math. 25 (1963) 457–487 ([dspace](http://dspace.library.uu.nl/handle/1874/7436))

reviewed in

* [[John Baez]], section 4.5 _[E7](http://math.ucr.edu/home/baez/octonions/node18.html)_ of _The Octonions_ ([arXiv:math/0105155](http://arxiv.org/abs/math/0105155))

* {#CacciatoriEtAl10} Sergio L. Cacciatori, Francesco Dalla Piazza, Antonio Scotti, _E7 groups from octonionic magic square_ ([arXiv:1007.4758](http://arxiv.org/abs/1007.4758))

A [[quaternion|quaternionic]] construction is given in

* {#Wilson14} [[Robert Wilson]], _A quaternionic approach to $E_7$_, Proc. Amer. Math. Soc. 142 (2014), 867-880. doi:[10.1090/S0002-9939-2013-11838-1](http://dx.doi.org/10.1090/S0002-9939-2013-11838-1), ([pre-publication version](http://www.maths.qmul.ac.uk/~raw/pubs_files/E7quat2.pdf)), ([talk notes](http://www.maths.qmul.ac.uk/~raw/talks_files/E7quattalk2.pdf)).

See also

* Wikipedia, _[E7](http://en.wikipedia.org/wiki/E%E2%82%87)_

On the "intermediate" $E_{7 \tfrac{1}{2}}$:

* J. M. Landsberg, L. Manivel, *The sextonions and $E_{7\tfrac{1}{2}}$*, Advances in Mathematics
**201** 1 (2006) 143-179 [[arXiv:math/0402157](https://arxiv.org/abs/math/0402157), [doi:10.1016/j.aim.2005.02.001](https://doi.org/10.1016/j.aim.2005.02.001)]

* Kimyeong Lee, Kaiwen Sun, Haowu Wang, *On intermediate Lie algebra $E_{7 \tfrac{1}{2}}$* [[arXiv:2306.09230](https://arxiv.org/abs/2306.09230)]

See also:

* [[Tevian Dray]], [[Corinne Manogue]], [[Robert Wilson]], *A New Division Algebra Representation of $E_7$* [[arxiv:2401.10534](https://arxiv.org/abs/2401.10534)]

### In view of U-duality

The hidden [[E7]]-[[U-duality]] symmetry of the [[KK-compactification]] of [[11-dimensional supergravity]] on a 7-dimensional fiber to [[4d supergravity]] was first noticed in ([Cremmer-Julia 79](#CremmerJulia79)) and then expanded on in

* [[Bernard de Wit]], [[Hermann Nicolai]], _D = 11 Supergravity With Local SU(8) Invariance_, Nucl. Phys. B 274, 363 (1986) ([spire](http://inspirehep.net/record/227409?ln=en)),  _Local SU(8) invariance in $d = 11$ supergravity_ ([spire](http://inspirehep.net/record/218601?ln=en))

The proposal to make this hidden $E_7$-symmetry manifest via [[exceptional generalized geometry]] is due to

* {#Hull07} [[Chris Hull]], _Generalised Geometry for M-Theory_, JHEP 0707:079 (2007) ([arXiv:hep-th/0701203](http://arxiv.org/abs/hep-th/0701203))

* {#PachecoWaldram08} Paulo Pires Pacheco, [[Daniel Waldram]], appendix B of _M-theory, exceptional generalised geometry and superpotentials_ ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

Further discussion includes

* {#GLSW} [[Mariana Graña]], [[Jan Louis]], Aaron Sim, [[Daniel Waldram]], section 3.1 of  _$E_{7(7)}$ formulation of $N=2$ backgrounds_ ([arXiv:0904.2333](http://arxiv.org/abs/0904.2333))


E8

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#### Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Lie group]] called $E_8$ is the largest-dimensional one of the five [[exceptional Lie groups]].

## Properties

### As part of the ADE pattern

[[!include ADE -- table]]

### Homotopy groups

The first nontrivial [[homotopy group]] of the [[topological space]] underlying $E_8$ is

$$
  \pi_3(E_8) \simeq \mathbb{Z}
$$

as for any compact Lie group. Then the next nontrivial homotopy group is

$$
  \pi_{15}(E_8) \simeq \mathbb{Z}
  \,.
$$

This means that all the way up to the 15 [[coskeleton]] the group $E_8$ looks, [[homotopy theory|homotopy theoretically]] like the [[Eilenberg-MacLane space]] $K(\mathbb{Z},3) \simeq B^3 \mathbb{Z} \simeq B^2 U(1) \simeq B \mathbb{C}P^\infty$.

### Subgroups
  {#Subgroups}

The [[subgroup]] of the [[exceptional Lie group]] [[E8]] which corresponds to the [[Lie algebra]]-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the [[semi-spin group]] [[SemiSpin(16)]]

$$
  SemiSpin(16)
  \;\subset\;
  E_8
$$

On the other hand, the [[special orthogonal group]] $SO(16)$ is _not_ a [[subgroup]] of $E_8$ (e.g. [McInnes 99a, p. 11](SemiSpin16#McInnes99a)).

### Invariant polynomials

By the above discussion of homotopy groups, it follows (by [[Chern-Weil theory]]) that the first [[invariant polynomials]] on the [[Lie algebra]] $\mathfrak{e}_8$ are the quadratic [[Killing form]] and then next an octic polynomial. That is described in ([Cederwall-Palmkvist](#CederwallPalmkvist)).

### As U-duality of 3d SuGra

$E_8$ is the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 3 dimensions.

[[!include U-duality -- table]]

## Related entries

The group $E_8$ plays a role in some exceptional [[differential geometry]]/[[differential cohomology]]. See for instance

* [[exceptional generalized geometry]], [[supergravity C-field]], [[Hořava-Witten theory]], [[heterotic string theory]]

* [[G2]], [[F4]],

  [[E6]], [[E7]],  **E8**, [[E9]], [[E10]], [[E11]], $\cdots$

## References
 {#References}

### General

Surveys:

* [[Skip Garibaldi]], _$E_8$, the most exceptional group_ ([arXiv:1605.01721](https://arxiv.org/abs/1605.01721))

* Wikipedia, _[E8](http://en.wikipedia.org/wiki/E8_%28mathematics%29)_

An introductory survey with an eye towards the relation to the [[octonions]] is given in [section 4.6](http://math.ucr.edu/home/baez/octonions/node19.html) of

* [[John Baez]], _The Octonions_ ([web](http://math.ucr.edu/home/baez/octonions/octonions.html))

### Homotopy groups
 {#HomotopyGroupsReferences}

The lower [[homotopy groups]] of $E_8$ are a classical result due to

* [[Raoul Bott]] and H. Samelson, _Application of the theory of Morse to symmetric spaces_ , Amer.
J. Math., 80 (1958), 964-1029.

The higher homotopy groups are discussed in

* Hideyuki Kachi, _Homotopy groups of compact Lie groups $E_6$, $E_7$ and $E_8$_ Nagoya Math. J. Volume 32 (1968), 109-139. ([project EUCLID](http://projecteuclid.org/euclid.nmj/1118797372))

See also

* [[André Henriques]], [MO comment](http://mathoverflow.net/questions/52286/how-are-the-classifying-space-of-e-8-and-k-mathbbz-4-related/52321#52321)

### Invariant polynomials

The octic [[invariant polynomial]] of $E_8$ is discussed in

* [[Martin Cederwall]], Jakob Palmkvist, _The octic $E_8$ invariant_ J.Math.Phys.48:073505 (2007) ([arXiv:hep-th/0702024](http://arxiv.org/abs/hep-th/0702024))
{#CederwallPalmkvist}

### More

On [[string bordism]] of the [[classifying space]] of $E_8$:

* [[Michael Hill]], *The $String$  bordism of $B E_8$ and $B E_8 \times B E_8$ through dimension 14*, Ill. J. Math. **53** 1 (2009) 183-196 [[doi:10.1215/ijm/1264170845](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-53/issue-1/The-string-bordism-of-BE_8-and-BE_8times-BE_8-through-dimension/10.1215/ijm/1264170845.full)]


E8 lattice

\tableofcontents

## Idea

The *$E_8$ lattice* is an [[exceptional structure|exceptional]] [[lattice (discrete subgroup)|lattice inside]] $\mathbb{R}^8$.

## Definition

The $E_8$-lattice may be presented as the [[linear span]] with [[integer]] [[coefficients]] of the following [[linear basis|basis vectors]] in the [[real vector space]] $\mathbb{R}^8$:

 - $(1,-1,0,0,0,0,0,0)$

 - $(0,1,-1,0,0,0,0,0)$

 - $(0,0,1,-1,0,0,0,0)$

 - $(0,0,0,1,-1,0,0,0)$

 - $(0,0,0,0,1,-1,0,0)$

 - $(0,0,0,0,0,1,-1,0)$

 - $(0,0,0,0,0,1,1,0)$

 - $\big(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\big)$

## Properties
 {#Properties}

### Relation to sphere packing
 {#RelationToSpherePacking}

If [[7-spheres]] of [[radius]] $\frac{\sqrt{2}}{2}$ are centered around every point of the $E_8$ lattice, then each sphere touches 240 other spheres.

A [[proof]] that this arrangement gives the optimal [[sphere packing]] in 8 dimensions was given in [Viazovska (2016)](#Viazovska16). (The analogous statement holds in 24 dimensions with respect to the [[Leech lattice]] [[CKMRV17](sphere+packing#CKMRV17)].)

The optimal sphere packing constant in 8 dimensions is $\frac{\pi^4}{384} \approx 0.2537$

## References

See also:

* Wikipedia, *[E8 lattice](https://en.wikipedia.org/wiki/E8_lattice)*

Discussion in relation to [[sphere packing]]:

* {#Viazovska16} [[Maryna Viazovska]], _The sphere packing problem in dimension 8_, Annals of Mathematics, **185** 3 (2017) 991-1015 [[arXiv:1603.04246](https://arxiv.org/abs/1603.04246), [doi:10.4007/annals.2017.185.3.7](https://doi.org/10.4007/annals.2017.185.3.7)]

Review:

* Ursula Whitcher, *Eight-dimensional spheres and the exceptional $E_8$*, AMS MathVoices Feature Column (Sep. 2022) [[web](https://mathvoices.ams.org/featurecolumn/2022/09/01/eight-dimensional-spheres-and-the-exceptional-e_8/)]

[[!redirects E8-lattice]]


E8 manifold

#Contents#
* table of contents
{:toc}

## Idea

The **$E_8$ manifold** is the unique [[compact]], [[simply connected]] topological 4-[[manifold]] with [[intersection pairing|intersection form]] the [[E8]] [[lattice in a vector space|lattice]].

The $E_{8}$ manifold has no [[smooth structure]], nor even is [[triangulation theorem|triangulable]] as a [[simplicial complex]].

(Stub for the moment)

##References

* [Wikipedia entry](https://en.wikipedia.org/wiki/E8_manifold)

* [[Robion Kirby]] and [[Paul Melvin]], _The E8–manifold, singular fibers and handlebody decompositions_, [Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest](https://arxiv.org/pdf/math/9911253.pdf), pp. 233 - 258.


E9

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=--
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#### Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One of the affine [[Kac-Moody groups]] in the E-series.

* [[E6]], [[E7]],  [[E8]], **E9**, [[E10]], [[E11]], $\cdots$

## Properties

### As U-duality group of 2d supergravity

$E_9$ is the [[U-duality]] group (see there) of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 2 dimensions.

[[!include U-duality -- table]]

## Related concepts

* [[exceptional Lie group]]

* [[exceptional geometry]], [[exceptional generalized geometry]]

* [[U-duality]] group of [[11-dimensional supergravity]] [[KK-compactification|compactified]] to 2 dimensions.

## References

* [[Hermann Nicolai]], [[Henning Samtleben]], _On $K(E_9)$_ ([arXiv:hep-th/0407055](http://arxiv.org/abs/hep-th/0407055))

* {#EHKNT07} [[François Englert]], [[Laurent Houart]], [[Axel Kleinschmidt]], [[Hermann Nicolai]], Nassiba Tabti, _An E9 multiplet of BPS states_, JHEP 0705:065 (2007) [[arXiv:hep-th/0703285](http://arxiv.org/abs/hep-th/0703285)]


Eanna Flanagan

[website](https://physics.cornell.edu/eanna-flanagan)

## Selected writings
 {#SelectedWritings}

On [[MOND]]:

* [[Eanna Flanagan]], *Khronometric theories of modified Newtonian dynamics* ([arXiv:2302.14846](https://arxiv.org/abs/2302.14846))

category: people


Eastin-Knill theorem

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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *Easti-Knill theorem* ([Eastin-Knill 09](#EastinKnill09)) characterizes the (in)compatibility between [[quantum error correction]] and universality of [[quantum computing|quantum gates]].

## References

Due to:

* {#EastinKnill09} [[Bryan Eastin]], [[Emanuel Knill]], *Restrictions on Transversal Encoded Quantum Gate Sets*,  Phys. Rev. Lett. 102, 110502 (2009) ([arXiv:0811.4262](https://arxiv.org/abs/0811.4262))

See also:

* Wikipedia, *[Eastin-Knill theorem](https://en.wikipedia.org/wiki/Eastin%E2%80%93Knill_theorem)*

Further development:

* {#WVSB20} Paul Webster, Michael Vasmer, Thomas R. Scruby, Stephen D. Bartlett, *Universal Fault-Tolerant Quantum Computing with Stabiliser Codes* ([arXiv:2012.05260](https://arxiv.org/abs/2012.05260))


Eberhard Freitag

* [Personal website](http://www.rzuser.uni-heidelberg.de/~t91/)
* [Wikipedia entry](http://de.wikipedia.org/wiki/Eberhard_Freitag)

**Eberhard Freitag** (1942- ) is a German mathematician who specializes in [[modular forms]]. Holding a chair at Heidelberg University since 1977 he has published several books notably one together with R. Kiehl on _[[etale cohomology|Étale cohomology]] and the [[Weil conjecture]]_ (1988) and a standard textbook on [[complex analysis]] in two volumes.

Lecture notes and other texts mainly related to complex analysis are freely available from his [homepage](http://de.wikipedia.org/wiki/Eberhard_Freitag) e.g.:

* E. Freitag, _Lecture on [[Kähler manifolds]]_ , 2003. ([pdf](http://www.rzuser.uni-heidelberg.de/~t91/skripten/kaehler.pdf))

* E. Freitag, _Lecture on [[Hodge theory]]_ , 2010.  ([pdf](http://www.rzuser.uni-heidelberg.de/~t91/skripten/hodge.pdf))

* E. Freitag, _[[Riemann surface|Riemann Surfaces]] - [[sheaf theory|Sheaf theory]], Riemann Surfaces, Automorphic forms_, 2013. ([pdf](http://www.rzuser.uni-heidelberg.de/~t91/skripten/riemfl.pdf))

category:people


Eberhard Kirchberg

Eberhard Kirchberg (1946-2022)

* [obituary](https://www.mathematik.hu-berlin.de/de/personen/nachrufe/kirchberg_nachruf_hu-final.pdf)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=22057)

## Selected writings

On [[continuous bundles of C-star algebras]]:

* [[Eberhard Kirchberg]], [[Simon Wassermann]], *Operations on continuous bundles of $C^\ast$-algebras*, Mathematische Annalen **303** 4 (1995) 677-698 [[eudml:165387](https://eudml.org/doc/165387), [[KirchbergWassermann-ContBndlsOfCStarAlgs.pdf:file]]]

category: people


Eberhard Klempt

* [spire page](https://inspirehep.net/authors/1002652)

## Selected writings

On [[quantum chromodynamics]]:

* [[Franz Gross]], [[Eberhard Klempt]] et al., *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

category: people


Eberhard Zeidler

__Eberhard Zeidler__ worked on nonlinear [[functional analysis]] and was the director of the Max Planck Institute for Mathematics in the Natural Sciences in Leipzig, Germany.

* [website](http://www.mis.mpg.de/zeidler/eberhard-zeidler.html)

## Publications

Zeidler is authoring a textbook series on [[quantum field theory]]:

* _Quantum field theory. A bridge between mathematicians and physicists_ Springer (2009) ([web]([http://www.mis.mpg.de/zeidler/qft.html))

  _I: Basics in mathematics and physics._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1124.81002&format=complete)),

   (on basics of [[classical mechanics]] [[quantum mechanics]])

  _II: Quantum electrodynamics_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1155.81005&format=complete))

  (on [[quantum electrodynamics]])

category: people


Ebubechukwu Ilo-Okeke

* [Institute page](https://research.shanghai.nyu.edu/centers-and-institutes/physics/people/ebubechukwu-ilo-okeke)

## Selected writings

On [[quantum optics]] in [[quantum information theory]]:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], Section 1 of: *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

category: people


Eckard Meinrenken > history

[[Eckhard Meinrenken]]


Eckart Viehweg

* [Wikipedia entry](http://de.wikipedia.org/wiki/Eckart_Viehweg)

## Selected writings

On [[Deligne cohomology]]:

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_,  Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

category: people


Eckhard Meinrenken

* [webpage](http://www.math.toronto.edu/mein/)

## Selected writings

On [[geometric quantization]] of [[compact topological space|compact]] [[Kähler manifolds]]:

* {#BordemannMeinrenkenSchlichenmaier93} [[Martin Bordemann]], [[Eckhard Meinrenken]], [[Martin Schlichenmaier]], _Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$_,  	Commun. Math. Phys. **165** (1994) 281-296 [[arXiv:hep-th/9309134](http://arxiv.org/abs/hep-th/9309134), [doi:10.1007/BF02099772](https://doi.org/10.1007/BF02099772)]

On the [[basic bundle gerbe]] over a [[compact Lie group|compact]] [[simple Lie group]]:

* [[Eckhard Meinrenken]], _The basic gerbe over a compact simple Lie group_, Enseign. Math. (2) 49 (2003), no. 3-4, 307-333 [[arXiv:math/0209194](https://arxiv.org/abs/math/0209194), [e-periodica](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::562)]

On [[Lie groups]] and [[Lie algebras]]:

* [[Eckhard Meinrenken]], _Lie groups and Lie algebas_, Lecture notes 2010 ([pdf](http://www.math.toronto.edu/mein/teaching/lie.pdf))

On [[Clifford algebras]]:

* [[Eckhard Meinrenken]], _Clifford algebras and Lie groups_ ([pdf](http://www.math.toronto.edu/mein/teaching/clif_main.pdf))

On the image in [[K-theory]] of [[Chan-Paton gauge fields]] over [[conjugacy classes]] giving the [[Verlinde ring]]/the [[positive energy representations]] of the [[loop group]]:

* [[Eckhard Meinrenken]], _On the quantization of conjugacy classes_,  	Enseign. Math. (2) 55 (2009), no. 1-2, 33-75 ([arXiv:0707.3963](http://arxiv.org/abs/0707.3963))

On [[twisted K-homology]]:

* [[Eckhard Meinrenken]], _Twisted K-homology and group-valued moment maps_ (2010) ([arXiv:1008.1261](http://arxiv.org/abs/1008.1261))

On [[smooth function|smooth]] [[transformation groups]]/[[G-spaces]] ([[Lie group]]-[[group actions]] on [[smooth manifolds]] in relation to [[Chern-Weil theory]] and [[equivariant de Rham cohomology]]:

* {#Meinrenken03} [[Eckhard Meinrenken]], *Group actions on manifolds*, Lecture Notes 2003 ([pdf](http://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf), [[Meinrenken_GroupActionsOnManifolds.pdf:file]])

On [[equivariant de Rham cohomology]]:

* {#Meinrenken06} [[Eckhard Meinrenken]], _Equivariant cohomology and the Cartan model_, in: _Encyclopedia of Mathematical Physics_, Pages 242-250 Academic Press 2006 ([pdf](http://www.math.toronto.edu/mein/research/enc.pdf), [doi:10.1016/B0-12-512666-2/00344-8](https://doi.org/10.1016/B0-12-512666-2/00344-8))

On [[geometric quantization by push-forward]] and [[Guillemin-Sternberg geometric quantization conjecture]]:

* [[Eckhard Meinrenken]], _Symplectic surgery and the $Spin^c$-Dirac operator_, Adv. Math. 134 (1998), 240-277.

* [[Hans Duistermaat]], [[Victor Guillemin]], [[Eckhard Meinrenken]], [[Siye Wu]], _Symplectic reduction and Riemann-Roch for circle actions_, Mathematical Research Letters 2, 259–266 (1995) ([pdf](http://intlpress.com/_newsite/site/pub/files/_fulltext/journals/mrl/1995/0002/0003/MRL-1995-0002-0003-A-003.pdf))

On [[moment maps]]:

* [[Anton Alekseev]], Anton Malkin, [[Eckhard Meinrenken]], _Lie group valued moment maps_, J. Differential Geom. Volume 48, Number 3 (1998), 445-495. [euclid](http://projecteuclid.org/euclid.jdg/1214460860), [MR1638045](http://www.ams.org/mathscinet-getitem?mr=1638045)

* [[Eckhard Meinrenken]], _Lectures on group-valued moment maps and Verlinde formulas_, 35 pages, January 2012, [pdf](http://www.math.toronto.edu/mein/research/NotreLectures.pdf)

## Related entries

* [[deformation quantization of the 2-sphere]]

category: people


Eckmann-Hilton argument

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#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--

# The Eckmann--Hilton argument
* table of contents
{: toc}

## Statements

In its original form, the _Eckmann--Hilton argument_ shows that a [[monoid object]] or [[group object]] [[internalization|internal]] to the [[category of monoids]] or [[Groups]] is necessarily [[abelian group|commutative]].

In other words this says that if a [[set]] is equipped with a [[pair]] of [[monoid]] [[structures]], such that one is a [[homomorphism]] for the other, then the two structures actually coincide and the resulting monoid is [[commutative monoid|commutative]].

From the [[nPOV]], we may want to think of the statement in this way:

+-- {: .num_prop #2cat}
###### Proposition

Let $C$ be a [[2-category]] and $x \in C$ an [[object]]. Write $Id_x$ for the [[identity]] [[morphism]] of $X$ and $End(Id_x)$ for the set of [[endomorphism|endo]]-[[2-morphism|2-morphisms]] on $X$. Then:

* [[horizontal composition]] and [[vertical composition]] define the same [[monoid object]] structure on $End(Id_x)$;

* this is a [[commutative monoid]].
=--

On the face of it, this is a special case of the general situation, although in fact every case is an example for appropriate $C$.

A more general version is this:  If a set is equipped with two binary operations with [[identity elements]], as long as they commute with each other in the sense that one is (with respect to the other) a homomorphism of sets with binary operations, then everything else follows:

1.  the other is also a homomorphism with respect to the first;
1.  the identities are the same;
1.  the operations are the same;
1.  the operation is commutative;
1.  the operation is associative.

This can also be [[internalisation|internalised]] in any [[symmetric monoidal category]].

## Proofs

[[string diagram|String diagrams]] allow an almost trivial proof. Since there is only one object, and the only 1-morphism is the identity, the diagram of $a \circ b$ (vertical composition) is simply two dots labelled $a, b$ arranged vertically. This diagram can be morphed continuously to a horizontal arrangement, which is the diagram for $a * b$ (horizontal composition). This is then morphed to $a$ below $b$, which is the diagram for $b \circ a$.

A [[pasting diagram]]-proof of \ref{2cat} is depicted in [Cheng](#Cheng) below.  Here we prove the $6$-element general form in $Set$.

+-- {: .proof}
###### Proof
The basic equation that we have (that one operation $*$ is a homomorphism with respect to another operation $\circ$) is
$$ (a \circ b) * (c \circ d) = (a * c) \circ (b * d) .$$
In $End(Id_x)$, this is the [[exchange law]].

We prove the list of results from above in order:

1.  Simply read the basic equation backwards to see that $\circ$ is a homomorphism with respect to $*$.

1.  Then
$$ 1_\star = 1_\star * 1_\star = (1_\star \circ 1_\circ) * (1_\circ \circ 1_\star) = (1_\star * 1_\circ) \circ (1_\circ * 1_\star) = 1_\circ \circ 1_\circ = 1_\circ ,$$
so the identities are the same; we will now write this identity simply as $1$.

1.  Now
$$ a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b ,$$
so the operations are the same; we will write them both with concatenation.

1.  Then
$$ a b = (1 a) (b 1) = (1 b) (a 1) = b a ,$$
so this operation is commutative.

1.  Finally,
$$ (a b) c = (a b) (1 c) = (a 1) (b c) = a (b c) ,$$
so the operation is associative.
=--

If you start with a monoid object in $Mon$, then only (4&5) need to be shown; the others are part of the hypothesis.  This classic form of the Eckmann--Hilton argument may be combined into a single calculation:
$$ a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b = (1 * a) \circ (b * 1) = (1 * b) \circ (a * 1) = b * a ,$$
where the desired results involve the first, middle, and last expressions.

## Corollaries

A $2$-[[k-tuply monoidal n-category|tuply monoidal]] $0$-[[0-category|category]], if defined as a [[pointed object|pointed]] [[k-tuply connected n-category|simply connected]] [[bicategory]], is also the same as an abelian monoid.

A $2$-tuply monoidal $1$-[[1-category|category]], if defined as a pointed simply connected [[tricategory]], is the same as a [[braided monoidal category]].

Every [[homotopy group]] $\pi_n$ for $n \geq 2$ is [[abelian group|abelian]].

## Variation

There are variations on the Eckmann-Hilton argument that do not assume units. For example, if a set is equipped with two symmetric $(a * b = b * a)$ and idempotent $(a * a = a)$ binary operations that commute with each other, then the operations coincide.

$$ a * b = (a * b) + (a * b) = (a * b) + (b * a) = (a + b) * (b + a) = (a + b) * (a + b) = a + b.$$

For example, we might consider the algebraic theory of [[convex space|convex spaces]], and the algebraic theory of [[semilattice|semilattices]]. These theories both contain symmetric idempotent operations: in the theory of convex spaces, take the operation $a * b=c_{0.5}(a,b)$. Thus there can be no commutative algebraic theory that includes these two theories without conflating them. Furthermore, in any conflated theory, all the dyadic rationals are the same, e.g.

$$c_{0.25}(a,b)=c_{0.5}(c_{0.5}(a,b),b)=(a*b)*b=a*(b*b)=a*b=c_{0.5}(a,b).$$

This is relevant in computer science because probability is modelled by the free convex spaces monad, and non-determinism is modelled by the free semilattice monad. These monads are both [[commutative monad|commutative monads]], but there can be no commutative monad that contains both these monads non-degenerately.
In particular, the [[convex powerset of distributions monad]] is not commutative.

## Related concepts

* [[internalization]]

* [[group object]]

* [[H-space]]

* [[exchange law]]

## References

Due to

* [[Beno Eckmann]], [[Peter Hilton]], Theorem 1.12 in: *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221 ([doi:10.4064/fm-50-2-207-221](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory))

reviewed (somewhat imperfectly) in:

* [[Beno Eckmann]], [[Peter Hilton]], Theorem 5.4.2 in: _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

Expositions:

* {#Catsters} [[The Catsters]]

  _Eckmann-Hilton 1_ ([video](http://www.youtube.com/watch?v=Rjdo-RWQVIY))

  _Eckmann-Hilton 2_ ([video](http://www.youtube.com/watch?v=wnRqo7UHa-k))

* [[Eugenia Cheng]], ([picture](http://eugeniacheng.com/ehclock/))
{#Cheng}

* [[Andrew Stacey]], [animation](https://web.archive.org/web/20150326221729/http://www.math.ntnu.no/~stacey/documents/eckmannhilton.mpg) (mpeg)
{#Stacey}

Formulation in [[homotopy type theory]]:

* [[UF-IAS-2012|Univalent Foundations Project]], Thm. 2.1.6 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* 2013 ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* [[Kristina Sojakova]], *Syllepsis in Homotopy Type Theory*, 2021 ([[Sojakova_Syllepsis_20210728.pdf:file]])

Generalization to [[higher algebra]]/[[higher category theory]] via [[operads]]/[[(∞,1)-operads]]:

* {#Batanin} [[Michael Batanin]], _The Eckmann-Hilton argument and higher operads_ ,  Adv. Math.  217  (2008),  no. 1, 334--385; ([arXiv:math.CT/0207281](http://arxiv.org/abs/math/0207281))

* [[Tomer Schlank]], [[Lior Yanovski]], *The ∞-Categorical Eckmann-Hilton Argument*, Algebr. Geom. Topol. 19 (2019) 3119-3170 ([arXiv:1808.06006](https://arxiv.org/abs/1808.06006), [doi:10.2140/agt.2019.19.3119](https://doi.org/10.2140/agt.2019.19.3119))

[[!redirects Eckmann-Hilton argument]]
[[!redirects Eckmann-Hilton arguments]]
[[!redirects Eckmann–Hilton argument]]
[[!redirects Eckmann–Hilton arguments]]
[[!redirects Eckmann--Hilton argument]]
[[!redirects Eckmann--Hilton arguments]]


Eckmann-Hilton duality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Beno Eckmann|Eckmann]] and [[Peter Hilton|Hilton]] noticed that a number of notions and theorems in the [[homotopy theory]] of [[pointed spaces]] can be dualized in the sense of certain expanding dictionary:

* [[homotopy]] -- [[cohomology]]

* [[fibration]]  -- [[cofibration]]

* [[mapping cylinder]] -- [[mapping cocylinder]]

* [[suspension]] $\Sigma X$ -- [[loop space]] $\Omega X$

* [[homotopy lifting property]] -- [[homotopy extension property]]

The "duality" in their work has been just a heuristic rather than a systematic algorithm to [[duality|dualize]] constructions and theorems. It has a little flavour of working with [[adjoint functors]], though it does not reduce to Kan adjointness. It has also the flavour of the usual arrow reversal duality, as is commented below.

Later some versions have been made rigorous, the most well-known being Fuks duality in homotopy theory.

## Fuks duality

D. B. Fuks considers the category $End(k Top_*)$ of [[endofunctors]] on the category $k Top_*$ of pointed [[compactly generated space|compactly generated]] Hausdorff spaces. He enriches this category in $k Top_*$ as follows: Let $S,T$ be two endofunctors, then the set $\{S\to T\}$ of isomorphism classes of natural transformations $S\to T$ is equipped with the weakest topology in which all maps $\alpha_X$, $X\in k Top_*$, of taking components $\{S\to T\}\mapsto Map(S(X),T(X))\in kTop_*$ are [[continuous map|continuous]]. Recall that the topology on the set of maps $S(X)\to T(X)$ in $k Top_*$ is obtained by first taking the [[compact-open topology]] and then performing the [[kaonization]] of the space so obtained.

Notice that if $[-,A]$ is a [[representable functor]] in the classical homotopy category of $k Top_*$, then for any endofunctors $S$ and $T$ we have

$$\{[-,A]\to S\} = S A,$$

$$ [ \{S\to T\},A ] = \{S\to [T(-),A]\} = \{ A \wedge S\to T\}.$$

A **Fuks duality** is any functor $D: End(k Top_*)\to End(k Top_*)$ such that

1. for any two functors $S,T\in End(k Top_*)$, $\{S\to D T\}$ is binaturally homeomorphic to
$\{D S\to T\}$

2. $D\Sigma =\Omega$, $D\Omega = \Sigma$.

+-- {: .un_theorem}
###### Theorem (D. B. Fuks)
There exists a unique Fuks duality functor. Moreover it is given by $DS:A\mapsto \{S\to [-,A]\}$ where $[,]$ denotes the homotopy classes and $\{,\}$ is defined as above.
=--

Notice that $D^2 = D\circ D$ is **not** isomorphic to the [[identity functor|identity]] in general, though it is identical when applied to a number of natural endofunctors in homotopy theory. Fuks duality can be extended suitably to functors of many arguments.

This duality to some extent resembles the usual categorical duality for objects by **arrow reversal**. The reason is seen in the formula for $D S$ above, which can be interpreted as arrow reversal for representable functors.
In a more perfect world, like [[model categories]], the axioms are self-dual and a true categorical duality of the usual categorical kind holds for basic model-categorical notions. On the other hand Eckmann--Hilton duality is not only about the notions entering the axioms of model categories but there are dualities among a number of interesting homotopy theoretical functors in topological context.

One can see this in the fact that [[cohomology group]]s (for [[integral cohomology|ordinary cohomology]] using the [[Eilenberg-Mac Lane spectrum]]) consist of [[homotopy]] classes of maps into a space with a single nontrivial [[homotopy group]], while homotopy groups consist of homotopy classes of maps from a space with a single nontrivial cohomology group:

* $H^n(X, \mathbb{Z}) \cong [X, K(\mathbb{Z},n)]$
* $\pi_n(X) \cong [S^n, X]$

Here, $K(\mathbb{Z},n)$ is an [[Eilenberg-Mac Lane space]], whose only interesting homotopy group is $\pi_n(K(\mathbb{Z},n)) = \mathbb{Z}$, while $S^n$ is a [[sphere]], whose only interesting ordinary cohomology group is the $H^n(S^n,\mathbb{Z}) = \mathbb{Z}$.

Note, though, that the arrow reversal duality does not hold perfectly in [[Top]] (the arrow reversal "dual" theorems are not necessarily true).
For example, a [[pullback]] of a cofibration by a fibration is a cofibration, but a pushforward of a fibration by a cofibration is not a fibration.

## References

* D. B. Fuks, On duality in homotopy theory, Soviet Math. Dokl. 2 (1961), 1575--1578.

* D. B. Fuks, Eckmann--Hilton duality and the theory of functors in the category of topological spaces, 1966 Russ. Math. Surv. 21 1--33 [doi](http://dx.doi.org/10.1070/RM1966v021n02ABEH004149), [free Russian original pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=5847&what=fullt&option_lang=rus)

* R. Nakagawa, On Fuks homotopy duality, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 8 (1963), 93--98.

* Encyclopedia of Mathematics: [Eckmann-Hilton duality](http://eom.springer.de/E/e120020.htm)

Some blog discussion on Eckmann-Hilton duality is
[here](http://golem.ph.utexas.edu/category/2008/09/group_cocycles_and_simplices.html#c019087),
[here](http://golem.ph.utexas.edu/category/2008/09/mathematical_miniatures.html#c019103),
and
[here](http://golem.ph.utexas.edu/category/2009/06/cohomology_and_homotopy.html#c024807)

[[!redirects Fuks duality]]
[[!redirects Fuchs duality]]
[[!redirects Eckmann–Hilton duality]]
[[!redirects Eckmann--Hilton duality]]


Ed Morehouse

[Homepage](http://emorehouse.web.wesleyan.edu/)

## Publications

* _Homotopical patch theory_.  [[Carlo Angiuli]], [[Ed Morehouse]], [[Dan Licata]], [[Robert Harper]], [PDF](http://dlicata.web.wesleyan.edu/pubs/amlh14patch/amlh14patch.pdf)

category: people


Ed Segal

* [webpage](http://www2.imperial.ac.uk/~epsegal/)

category: people


Ed Threlfall

* [Spire page](https://inspirehep.net/authors/1047814?ui-citation-summary=true)

## Selected writings

On [[mesons]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Ingo Kirsch]], [[Ed Threlfall]], *Mesons in Gauge/Gravity Duals - A Review*, Eur. Phys. J. A **35** (2008) 81-133 [[arXiv:0711.4467](https://arxiv.org/abs/0711.4467), [doi:10.1140/epja/i2007-10540-1](https://doi.org/10.1140/epja/i2007-10540-1)]

category: people


Edgar Brown

Edgar H. Brown

(1926 - 2021)

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

* [webpage](http://www.brandeis.edu/facultyguide/person.html?emplid=1ea382383b3eafe98ed7fec4c23acf2f354e1c13)

## Selected writings

On [[Whitehead-generalized cohomology theories]], [[homotopy theory]] ([[localization]] and [[homotopy categories]]) and proving the [[Brown representability theorem]]

* {#Brown62} [[Edgar Brown]], _Cohomology theories_, Annals of Mathematics, Second Series 75: 467–484 (1962) ([jstor:1970209](https://www.jstor.org/stable/1970209))

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) ([doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6))

On the [[integral cohomology]] of the [[classifying spaces]] of [[rotation groups]]:

* {#Brown82} [[Edgar H. Brown]], _The Cohomology of $B SO_n$ and $BO_n$ with Integer Coefficients_, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 ([jstor:2044298](https://www.jstor.org/stable/2044298))

On [[rational homotopy theory]], the [[fundamental theorem of dg-algebraic rational homotopy theory]] and its generalization both to [[Borel-equivariant rational homotopy theory]] (for [[covering spaces]] of non-[[nilpotent spaces]]) and to [[real homotopy theory]] (using [[continuous cohomology|continuous]] [[real cohomology]] and [[topological algebra|topological]] [[dgc-algebras]]):

* {#BrownSzczarba95} [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*,  Elsevier, 1995 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009))

based on (including discussion of globally [[Kan fibration|Kan fibrant]] [[simplicial topological spaces|topological]] [[simplicial groups]]):

* {#BrownSzczarba89} [[Edgar H. Brown]], [[Robert H. Szczarba]], *Continuous cohomology and real homotopy type*, Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 ([doi:10.1090/S0002-9947-1989-0929667-6](https://doi.org/10.1090/S0002-9947-1989-0929667-6), [jstor:2001017](https://www.jstor.org/stable/2001017))

* {#BrownSzczarba90} [[Edgar Brown]], [[Robert H. Szczarba]], _Continuous cohomology and Real homotopy type II_ Asterisque 191, Societe Mathematique De France (1990) ([numdam:AST_1990__191__45_0](http://www.numdam.org/book-part/AST_1990__191__45_0))

* [[Edgar Brown]], [[Robert H. Szczarba]], *Real and rational homotopy theory for spaces with arbitrary fundamental group*, Duke Mathematical Journal 71. (1993): 229-316 ([doi:10.1215/S0012-7094-93-07111-6](https://projecteuclid.org/journals/duke-mathematical-journal/volume-71/issue-1/Rational-and-real-homotopy-theory-with-arbitrary-fundamental-groups/10.1215/S0012-7094-93-07111-6.short))

## Related entries

* [[Brown representability theorem]]

* [[Brown-Peterson spectrum]]

* [[Brown-Gitler spectrum]]

* [[orthogonal group]], [[special orthogonal group]], [[SO(3)]]

[[!redirects Edgar H. Brown]]

category: people


Edgar Shaghoulian

* [personal page](https://edgarshaghoulian.com/)

* [institute page](https://scipp.science.ucsc.edu/people/edgar-shaghoulian/)

## Selected writings

Review of the claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _Replica Wormholes and the Entropy of Hawking Radiation_, J. High Energ. Phys. **2020** 13 (2020)  \[doi:10.1007/JHEP05(2020)013, [arXiv:1911.12333](https://arxiv.org/abs/1911.12333)\]

Review:

* [[Ahmed Almheiri]], [[Thomas Hartman]], [[Juan Maldacena]], [[Edgar Shaghoulian]], [[Amirhossein Tajdini]], _The entropy of Hawking radiation_, Rev. Mod. Phys. **93** 35002 (2021) [[arXiv:2006.06872](https://arxiv.org/abs/2006.06872), [doi:10.1103/RevModPhys.93.035002](https://doi.org/10.1103/RevModPhys.93.035002)]

category: people


Edgar Stout

* [personal webpage](http://www.math.washington.edu/People/fac_individ.php?mathid=stout)

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=8442)

category: people


edge

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Graph theory
+-- {: .hide}
[[!include graph theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In just about any concept of [[graph]], an **edge** is something which *connects two [[vertex|vertices]]*.  In a [[directed graph]] an edge is directed, whereas in an undirected graph (the default meaning of "graph" in contemporary combinatorics) it is not.

## Terminological remarks

Edges are sometimes called **arcs** (less common in combinatorics, but sometimes used in other fields) or **lines** (e.g. [Harary, Chapter 2](#HararyGraphTheory)).  The latter is now unusual and considered old-fashioned, but survives in the standard technical term [[line graph]] (which is more common than "edge graph" for the same concept).  In more specialized contexts other terms are used.  For instance, if the [[graph]] is the [[quiver]] underlying a [[category]], then an edge is a [[morphism]]; while if it underlies a [[simplicial set]], then an edge is a 1-[[dimension|dimensional]] [[simplex]].

Historically, the use of "edge" for connections even in abstract graphs probably derives from the natural connections between [[graph theory]] and the theory of [[polyhedra]].  One such connection is [[Ernst Steinitz|Steinitz's]] [[Steinitz's theorem|1916 theorem]] on the one-dimensional skeleta of [[polyhedra]] in three-dimensional [[Euclidean space]].  However, some deplore the geometric connotations of "edge", with its geometric connotations of straightness and rigidity.

## Related concepts

* [[graph]], [[directed graph]], [[quiver]], [[digraph]]

* [[dimer]]

* [[vertex]], [[simplex]], [[dendrex]]

## References

* Frank Harary: Graph Theory. Addison-Wesley. 1969
{#HararyGraphTheory}

[[!redirects edge]]
[[!redirects edges]]
[[!redirects arc]]
[[!redirects arcs]]


edge homomophism > history


edge mode

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], *edge modes* are excitations (of [[electrons]] or [[quasi-particles]]) seen at the [[boundary]] or at a [[corner]] of a  [[crystal|crystalline materials]], where the [[dynamics]] may be different from that in the crystal's [[bulk]].

A notable example are edge modes of [[topological insulators]]: As their name suggests, these [[topological phases of matter]] are in fact [[insulator|insulating]] in their [[bulk]], but on their [[boundary]] they behave like [[conductors]]/[[metals]], due to "edge modes" which may carry [[electric current]].

## References

For the moment see any of the main references at *[[topological insulator]]*.

Otherwise, there is also:

* Lukasz Fidkowski, T. S. Jackson, Israel Klich, *Edge modes in band topological insulators* $[$[arXiv:1101.0320](https://arxiv.org/abs/1101.0320)$]$

(...)

[[!redirects edge modes]]


edge morphism

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[spectral sequence]] $\{E_r^{s,t},d_r\}$ with a "vanishing edge" in the sense that all its terms vanish for $s$ or $t$ smaller or larger some fixed value, then the fact that all [[differentials]] starting or ending on that edge necessarily vanish implies that all terms on the edge project onto or inject into the corresponding terms on the infinity-page, respectively. These are called the _edge homomorphisms_.

Specifically, given a first-quadrant (cohomological) [[spectral sequence]] $(E_r^{p,q})$ there are natural morphisms

$$
  E_2^{n,0} \longrightarrow E^n
$$

and

$$
  E^n \longrightarrow E_2^{0,n}
  \,.
$$

These are called the _edge morphisms_ or _edge maps_ of the spectral sequence.

## Properties

The edge morphisms sit in an [[exact sequence]] of the form

$$
  0 \to E_2^{1,0} \to E^1 \to E_2^{0,1} \stackrel{d_2}{\to} E_2^{2,0} \to E^2
$$

This is often called the _exact sequence of terms of low degree_ or _five term exact sequence_.

e.g. ([Cartan_Eilenberg XV, 5](#CartanEilenberg), [Tamme, 0 2.3.2](#Tamme))

If

$$
  E_r^{p,(0 \leq q \leq n)} = 0
$$

then  $E^{p,0}_2 \simeq E^p$ for $p \lt n$ and also

$$
  0 \to E_2^{1,0} \to E^1 \to E_2^{0,1} \stackrel{d_2}{\to} E_2^{2,0} \to E^2
$$

is [[exact sequence|exact]].

e.g. ([Cartan_Eilenberg XV, 5](#CartanEilenberg), [Tamme, 0 2.3.3](#Tamme))

## References

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, 1956
 {#CartanEilenberg}

* [[Günter Tamme]], section 0 2.3 of _[[Introduction to Étale Cohomology]]_
 {#Tamme}

[[!redirects edge morphisms]]

[[!redirects edge homomorphism]]
[[!redirects edge homomorphisms]]

[[!redirects edge map]]
[[!redirects edge maps]]

[[!redirects exact sequence of terms of low degree]]
[[!redirects exact sequences of terms of low degree]]

[[!redirects five term exact sequence]]
[[!redirects five term exact sequences]]


EDGES

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[astrophysics]] [[experiment]]

## Related entries

* [[LHC]], [[LIGO]], [[RHIC]]

* [[Super-Kamiokande]]

## References

* [MIT Haystack Observatory](https://www.haystack.mit.edu/): _[EDGES: Experiment to Detect the Global EoR Signature](https://www.haystack.mit.edu/ast/arrays/Edges/)_

* Wikipedia, _[Experiment to Detect the Global EoR Signature](https://en.wikipedia.org/wiki/Experiment_to_Detect_the_Global_EoR_Signature)_

Claim of detection of the [[hydrogen line]] in the [[cosmic microwave background]] is due to

* {#BRMMM18} Judd D. Bowman, Alan E. E. Rogers, Raul A. Monsalve, Thomas J. Mozdzen,  Nivedita Mahesh, _An absorption profile centred at 78 megahertz in the sky-averaged spectrum_, Nature volume 555, pages 67–70 (01 March 2018) ([doi:10.1038/nature25792](https://www.nature.com/articles/nature25792))

Discussion of this claim incluces

* {#Falkowski18} [[Adam Falkowski]], _[21cm to dark matter](http://resonaances.blogspot.fr/2018/03/21cm-to-dark-matter.html)_, March 2018


Edinburgh Logical Framework > history

< [[Edinburgh Logical Framework]]


Edmund Husserl

Edmund Husserl was a [[philosophy|philosopher]] who initiated the philosophical movement of [[phenomenology (philosophy)|phenomenology]]. His work has influenced the type theory of [[Per Martin-Löf]] ([Sundholm 12, p. xx](#Sundholm12))

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

* [Stanford Encyclopedia of Philosophy entry](https://plato.stanford.edu/entries/husserl/)

### Selected writings

* 1900/1 [2nd, revised edition 1913], Logical Investigations, trans. J. N. Findlay, London: Routledge 1973.

* 1910, “Philosophy as Rigorous Science,” trans. in Q. Lauer (ed.), Phenomenology and the Crisis of Philosophy, New York: Harper 1965.

* 1913, Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy—First Book: General Introduction to a Pure Phenomenology, trans. F. Kersten. The Hague: Nijhoff 1982 (= Ideas).

* 1929, Formal and Transcendental Logic, trans. D. Cairns. The Hague: Nijhoff 1969.

* 1931, Cartesian Meditations, trans. D. Cairns, Dordrecht: Kluwer 1988.

* 1939, Experience and Judgement, trans. J. S. Churchill and K. Ameriks, London: Routledge 1973.

* 1954, The Crisis of European Sciences and Transcendental Phenomenology, trans. D. Carr. Evanston: Northwestern University Press (= Crisis) 1970.

### References

* [[Göran Sundholm]] (2012), _On the Philosophical Work of Per Martin-Löf_. In: Dybjer P., Lindström S., Palmgren E., Sundholm G. (Eds.) _Epistemology versus Ontology Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf._ Logic, Epistemology, and the Unity of Science no. 27 Dordrecht: Springer Netherlands. xvii-xxiv. ([pdf](https://www.cmu.edu/dietrich/philosophy/docs/lectures/Sundholm%20-%20On%20Per%20Martin-Lof.pdf))

category: people


Edmund Robinson

* [Homepage](http://www.eecs.qmul.ac.uk/~edmundr/)

## Selected writings

On [[PIE-limits]]:

* [[John Power]], [[Edmund Robinson]], *A characterization of pie limits*, Math. Proc. Cam. Phil. Soc.  **110** (1991) 33 [[doi:10.1017/S0305004100070092](https://doi.org/10.1017/S0305004100070092)]

category:people


Edmund T. Whittaker

* [Wikipedia entry](https://en.wikipedia.org/wiki/E._T._Whittaker)

## Selected writings

On the history of [[electromagnetism]]:

* [[Edmund T. Whittaker]]:

  *A History of the Theories of Aether and Electricity*

  * Vol. 1: *The Classical Theories*

    *From the age of Descartes to the close of the Nineteenth century*

    (1910, 1951) [[pdf](https://ia804708.us.archive.org/32/items/historyoftheorie00whitrich/historyoftheorie00whitrich.pdf)]

  * Vol. 2: *The Modern Theories 1900-1926*

    (1953) [pdf]

  reprinted by Humanities Press (1973)

  [Wikipedia entry]

## Related entries

* [[Whittaker function]]

* [[pre-metric electromagnetism]]

[[!redirects E. T. Whittaker]]
[[!redirects Edmund Taylor Whittaker]]


Edoardo Lanari

* [personal page](https://edolana.github.io/)

* [personal page (old)](https://sites.google.com/view/edoardo-lanari)

* [GoogleScholar entry](https://scholar.google.com/citations?user=r4Bg57AAAAAJ&hl=en)

## Selected writings

On the [[homotopy hypothesis]] for [[Grothendieck infinity-groupoid|Grothendieck 3-groupoids]]:

* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)]

category: people


Eduard Heine

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

## Related $n$Lab entries

* [[Heine-Borel theorem]]

category: people


Eduard Looijenga

* [webpage](http://www.staff.science.uu.nl/~looij101/Coordinates.html)

* [Wikipedia entry](http://de.wikipedia.org/wiki/Eduard_Looijenga)

## related $n$Lab entries

* [[equivariant elliptic cohomology]]

* [[modular functor]], [[Hitchin connection]]

category: people


Eduard Stiefel

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

## related $n$Lab entries

* [[Stiefel manifold]]

* [[Stiefel-Whitney class]]

category: people


Eduard Čech

Eduard Čech was a Czech mathematician.

He got his PhD degree in 1920 from Charles University in Prague, advised by Karel Petr.

He was the first one to define [[homotopy groups]] (unpublished).
(An apocryphal story told about this is that [[Paul Alexandroff]] expressed
skepticism that these new invariants could be anything but [[homology groups]]
since they were abelian.)

He is also known for [[Čech methods]] in [[homotopy theory]], further developing the [[nerve]] construction by [[Paul Alexandroff]].

* [English Wikipedia page](http://en.wikipedia.org/wiki/Eduard_%C4%8Cech)
* [MacTutor biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Cech.html)

His PhD students include [[Petr Vopěnka]], [[Věra Trnková]], and many others.

## related nLab entries

* [[Čech cohomology]]

* [[Čech methods]]

* [[Čech nerve]]

category: people

[[!redirects Čech]]
[[!redirects Čech]]
[[!redirects Cech]]
[[!redirects Eduard Čech]]
[[!redirects Eduard Čech]]
[[!redirects Eduard Cech]]
[[!redirects Edward Cech]]


Eduardo Dubuc

* website??

## Selected writings

Monograph

* E. Dubuc, _Kan extensions in enriched category theory_, Springer Lecture Notes in Mathematics __145__, 1970 xvi+173 pp.

* Eduardo Dubuc, [[Ross Street]], _Dinatural transformations_, In: Reports of the Midwest Category Seminar IV. Springer Lec. Notes Math. __137__ [doi](https://doi.org/10.1007/BFb0060443)

* E. Dubuc, _Adjoint triangles_, pp.69-81 in LNM 61 Springer Heidelberg 1968 (see [[adjoint triangle theorem]])

On [[quasi-toposes]]:

* {#DubucEspanol06} [[Eduardo Dubuc]], [[Luis Español]], _Quasitopoi over a base category_  ([arXiv:math.CT/0612727](http://arxiv.org/abs/math.CT/0612727))

* E.J. Dubuc, [[Anders Kock|A. Kock]], _On 1-form classifiers_, Commun. Alg.  12 (1984) 1471--1531 [doi](https://doi.org/10.1080/00927878408823064)
* E. J. Dubuc, G.M. Kelly, _A presentation of topoi as algebraic relative to categories or graphs_, J. Algebra __81__ (1983) 420--433

Galois theory

* E. J. Dubuc, _Localic Galois theory_, Adv. Math. __175__:1 (2003) 144--167 doi
* E.J. Dubuc, C. Sanchez de la Vega, _On the Galois Theory of Grothendieck_, arXiv:[math.CT/0009145](https://arxiv.org/abs/math/0009145)

[[localization of a 2-category|Localizaton of 2-categories]]

* M. E. Descotte, E. J. Dubuc, M. Szyld, _A localization of bicategories via homotopies_, Theory and Applications of Categories __35__, 2020, No. 23, 845--874, [TAC](http://tac.mta.ca/tac/volumes/35/23/35-23abs.html) [[zoranskoda:MR4112764]]

Homotopy bicategories

* M. E. Descotte, E. J. Dubuc, M. Szyld, _Model bicategories and their homotopy bicategories_, Adv. Math. __404__B, (2022) 108455 [arXiv:1805.07749](https://arxiv.org/abs/1805.07749) [doi](https://doi.org/10.1016/j.aim.2022.108455)

* [[Maria Emilia Descotte]], [[Eduardo Dubuc]], _A theory of 2-pro-objects (with expanded proofs)_, ([arXiv:1406.5762](https://arxiv.org/abs/1406.5762))

* M.E. Descotte, E.J. Dubuc, M. Szyld, _Sigma limits in 2-categories and flat pseudofunctors_, (v1:On the notion of flat 2-functors) [arXiv:1610.09429](https://arxiv.org/abs/1610.09429) Adv. Math. __333__ (2018) 266--313

The notion of the spectrum of a $C^\infty$-ring and that of $C^\infty$-[[schemes]]; synthetic differential geometry

* E. Dubuc, _$C^\infty$-schemes_ Amer. J. Math. __103__ (1981) [pdf](http://www.math.ist.utl.pt/~jroquet/Dubuc.pdf) [JSTOR](https://www.jstor.org/stable/select/2374046)
* E. Dubuc, _Sur les modèles de la géométrie différentielle synthétique_ [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques __20__ no. 3 (1979) 231--279  [numdam:CTGDC_1979__20_3_231_0](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)
* E. Dubuc, [[Horacio Porta]], _Convenient categories of topological algebras_, Bull. Amer. Math. Soc. __77__:6 (1971), 975--979 [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-77/issue-6/Convenient-categories-of-topological-algebras/bams/1183533170.full); _Convenient categories of topological algebras, and their duality theory_, J. Pure Appl. Alg. __1 __:3 (1971) 281--316.
* [[Marta Bunge]], Eduardo Dubuc, _Archimedian local $C^\infty$-rings and models of synthetic differential geometry_ Cahiers de Topologie et Géométrie Différentielle Catégoriques __27__ no. 3 (1986) 3--22 ([numdam](https://www.numdam.org/item?id=CTGDC_1986__27_3_3_0))
* [[Marta Bunge]], Eduardo Dubuc, _Local concepts in synthetic differential geometry and germ representability_,
chapter in book Mathematical Logic and Theoretical Computer Science, CRC Press 1987
* E. Dubuc, _Axiomatic etal maps and a theory of spectrum_ , JPAA **149** (2000) 15--45

On MV-algebras

* Eduardo J. Dubuca, Yuri A. Poveda, _Representation theory of MV-algebras_, Annals of Pure and Applied Logic __161__:8 (2010) 1024--1046 [doi](https://doi.org/10.1016/j.apal.2009.12.006)
* E. J. Dubuc, Daniele Mundici, _Extending Stone duality to multisets and locally finite MV-algebras_, J.Pure Appl. Alg.
__189__1--3 (2004) 37--59
* R. Cignoli, E.J. Dubuc, D. Mundici, _An MV-algebraic invariant for boolean algebras with a finite-orbit automorphism_, Tatra Mount. Math. Publ, 27 (2003) 23--43

## Related $n$Lab entries

* [[Dubuc topos]]

* [[generalized smooth algebra|C∞-rings]]

* [[synthetic differential geometry]]

* [[Galois topos]]

* [[topological algebra]], [[Gelfand duality]]

* [[2-pro-object]], [[localization of a 2-category]]

category: people
[[!redirects Eduardo J. Dubuc]]
[[!redirects E. J. Dubuc]]


Eduardo Eyras

* [inspire page](https://inspirehep.net/authors/1010498)

## Selected writings

On [[brane intersection]] laws of [[M-branes]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Eduardo Eyras]], [[Bert Janssen]], [[Jan Pieter van der Schaar]], _Intersections involving waves and monopoles in eleven dimensions_, Class. Quantum Grav. **14** (1997) 2757 [[doi:0264-9381/14/10/005](http://iopscience.iop.org/0264-9381/14/10/005)]

On the [[KK-monopole]] in massive [[D=11 supergravity]], [[KK-reduction|reducing]] to the [[D6-brane]] in [[massive type IIA string theory]]:

* [[Eric Bergshoeff]], [[Eduardo Eyras]], [[Yolanda Lozano]], *The massive Kaluza-Klein monopole*, Physics Letters B
**430** 1–2 (1998) 77-86 [doi:10.1016/S0370-2693(98)00501-2, [arXiv:hep-th/9802199](https://arxiv.org/abs/hep-th/9802199)]

* [[Eduardo Eyras]], [[Yolanda Lozano]], *The Kaluza-Klein Monopole in a Massive IIA Background*, Nucl. Phys. B**546** (1999) 197-218 [[arXiv:hep-th/9812188](https://arxiv.org/abs/hep-th/9812188), doi:10.1016/S0550-3213(99)00098-X]

On [[T-duality]] for [[NS5-branes]]:

Discussion of the effect of [[T-duality]] on NS5-branes includes

* [[Eduardo Eyras]], [[Bert Janssen]], [[Yolanda Lozano]], _5-branes, KK-monopoles and T-duality_, Nucl. Phys. B **531** (1998) 275-301 [[arXiv:hep-th/9806169](https://arxiv.org/abs/hep-th/9806169), doi:10.1016/S0550-3213(98)00575-6]

category: people


Eduardo Fradkin

* [personal page](http://eduardo.physics.illinois.edu/homepage/)

## Selected writings

Introducing [[para-fermions]]:

* [[Eduardo Fradkin]], [[Leo P. Kadanoff]], *Disorder variables and para-fermions in two-dimensional statistical mechanics*, Nuclear Physics B **170** 1 (1980) 1-15 $[$doi:10.1016/0550-3213(80)90472-1$]$

Relating [[parafermions]] to the [[affine Lie algebra|affine]] [[su(2)]]-[[current algebra]]/[[WZW model]] ([[su(2)-anyons]]):

* [[Daniel C. Cabra]], [[Eduardo Fradkin]], G. L. Rossini, F. A. Schaposnik, *Non-Abelian fractional quantum Hall states and chiral coset conformal field theories*, International Journal of Modern Physics A **15** 30 (2000) 4857-4870 $[$[doi:10.1142/S0217751X00002354](https://doi.org/10.1142/S0217751X00002354), [arXiv:cond-mat/9905192](https://arxiv.org/abs/cond-mat/9905192)$]$

On [[solid state physics]] via [[quantum field theory]]:

* [[Eduardo Fradkin]], *Field Theories of Condensed Matter Physics*, Cambridge University Press (2013) [ISBN: 9781139015509, [doi:10.1017/CBO9781139015509](https://doi.org/10.1017/CBO9781139015509)]

category: people


Eduardo Hoefel

* [persona page](https://docs.ufpr.br/~hoefel/)

## Selected writings

On [[recognition of relative loop spaces]]:

* {#HLS16} [[Eduardo Hoefel]], [[Muriel Livernet]], [[Jim Stasheff]], _$A_\infty$-actions and recognition of relative loop spaces_, Topology and its Applications **206** (2016) 126-147 [[arXiv:1312.7155](https://arxiv.org/abs/1312.7155), [doi:10.1016/j.topol.2016.03.023](https://doi.org/10.1016/j.topol.2016.03.023)]

category: people


Eduardo Ochs

My main research interest is on what happens when we _drop all equalities between morphisms_ from categorical proofs; if we do that in the right way we are left with all the constructions (the "fun part") and we lose the equations (the "boring part"). The best article I've written so far - ["Internal Diagrams in Category Theory"](http://angg.twu.net/math-b.html#internal-diags-in-ct) (2010) - is an introduction to that.

I work in a campus in the countryside of Brazil that is a kind of (academical) nowhere. In order to cope with the, hm, psychological effects of that I have started, together with a few friends, an informal group to study the psychodynamics of corruption and apathy using tools from the Social Sciences. Our [notes](http://angg.twu.net/psi-cor.html) are mostly in Portuguese at the moment.

The 2D diagrams and derivation trees in my papers and notes are all created with this package: [dednat4](http://angg.twu.net/dednat4.html). Its documentation is precarious and its code is a mess, but its [rewrite](http://angg.twu.net/dednat5/) is 95% finished, and I am migrating to it slowly. If you think you would like to use it, then get in touch to speed things up. 8-)

category: people

[[!redirects Eduardo Ochs]]
[[!redirects EduardoOchs]]


Edvard Musaev

* [webpage](https://sites.google.com/view/emusaev/)

## Selected writings

On [[U-duality]] in [[string theory]] and [[M-theory]]:

* {#Musaev13} [[Edvard Musaev]], _U-dualities in Type II string theories and M-theory_, _Symmetry_ 2019, _11_(8), 993 ([arXiv:1311.3331](https://arxiv.org/abs/arXiv:1311.3331), [doi:10.3390/sym11080993](https://doi.org/10.3390/sym11080993))

On possible structures in [[M2-brane]] dynamics and [[M2-M5-brane bound states]] which could be [[M-theory]]-lifts of the familiar [[integrable system|integrability]] of the [[Green-Schwarz superstring]] on [[AdS spacetime|$AdS_5$]] $\times$ [[5-sphere|$S^5$]]:

* [[Kirill Gubarev]], [[Edvard Musaev]], *Integrability structures in string theory*,  J. High Energ. Phys. **2018** 53 (2018) [[arXiv:2301.06486](https://arxiv.org/abs/2301.06486), doi:10.1007/JHEP12(2018)053]

On [[exotic branes]] in [[exceptional field theory]]:

* {#BermanMusaevOtsuki18} [[David S. Berman]], [[Edvard T. Musaev]], [[Ray Otsuki]], *Exotic Branes in Exceptional Field Theory: $E_{7(7)}$ and Beyond*,  J. High Energ. Phys. **2018** 53 (2018) [[arXiv:1806.00430v4](https://arxiv.org/abs/1806.00430), doi:10.1007/JHEP12(2018)053]

* {#BMO19} [[David Berman]], [[Edvard Musaev]], [[Ray Otsuki]], _Exotic Branes in M-Theory_, contribution to: *[Dualities and Generalized Geometries, Corfu Summer Institute 2018](http://www.physics.ntua.gr/corfu2018/dg.html)* [[arXiv:1903.10247](https://arxiv.org/abs/1903.10247)]

## Related entries

* [[exceptional field theory]], [[embedding tensor]]

* [[gauged supergravity]]

* [[exotic branes]]

category: people

[[!redirects Edvard T. Musaev]]


Edward Becerra

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=163433)

## Selected writings

On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:

* [[Edward Becerra]], [[Bernardo Uribe]], _Stringy product on twisted orbifold K-theory for abelian quotients_, Trans. Amer. Math. Soc. 361 (2009), 5781-5803  ([arXiv:0706.3229](https://arxiv.org/abs/0706.3229), [doi:10.1090/S0002-9947-09-04760-6](https://doi.org/10.1090/S0002-9947-09-04760-6))

category: people


Edward Bierstone

* [institute page](https://www.math.toronto.edu/bierston/)

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

## Selected writings

On [[equivariant bundles]] in [[equivariant differential topology]]:

* [[Edward Bierstone]], _The equivariant covering homotopy property for differentiable $G$-fibre bundles_, J. Differential Geom. 8(4): 615-622 (1973) ([doi:10.4310/jdg/1214431963](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-8/issue-4/The-equivariant-covering-homotopy-property-for-differentiable-G-fibre-bundles/10.4310/jdg/1214431963.full))

category: people


Edward Casey

* [Webpage](http://edwardscasey.com/)

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

category: people


Edward Curtis

* [Maths genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=28298)

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

## Publications

On [[simplicial homotopy theory]]:

*  [[Edward B. Curtis]], _Simplicial homotopy theory_, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, [MR279808](http://www.ams.org/mathscinet-getitem?mr=279808))

## Related $n$Lab entries

* [[simplicial homotopy theory]]

* [[Curtis algorithm]]

* [[Lambda-algebra]]

category:people

[[!redirects Ed Curtis]]
[[!redirects E. B. Curtis]]

[[!redirects Edward B. Curtis]]


Edward Fadell

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=8293)

## Selected writings

On [[configuration spaces of points]]:

* {#FadellNeuwirth62} [[Edward Fadell]], [[Lee Neuwirth]], *Configuration spaces*, Math. Scand. __10__ (1962) 111-118 $[$[doi:10.7146/math.scand.a-10517](https://doi.org/10.7146/math.scand.a-10517), [MR141126](http://www.ams.org/mathscinet-getitem?mr=141126)$]$

* [[Edward Fadell]], [[Sufian Husseini]], _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), xvi+313

On [[loop spaces]] of [[configuration spaces of points]]:

* [[Edward Fadell]], [[Sufian Husseini]], _The space of loops on configuration spaces and the Majer-Terracini index_, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 ([euclid:tmna/1476842829](https://projecteuclid.org/euclid.tmna/1476842829))

category: people


Edward Farhi

* [personal page](https://physics.mit.edu/faculty/edward-farhi/)

* [Wikiepedia entry](https://en.wikipedia.org/wiki/Edward_Farhi)

## Selected writings

On [[adiabatic quantum computation]]:

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Michael Sipser, *Quantum Computation by Adiabatic Evolution* [[arXiv:quant-ph/0001106](https://arxiv.org/abs/quant-ph/0001106)]

* [[Edward Farhi]], [[Jeffrey Goldstone]], [[Sam Gutmann]], Joshua Lapan, Andrew Lundgren, Daniel Preda, *A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem*, Science  **292** 5516 (2001) 472-475 [[doi:10.1126/science.1057726](https://doi.org/10.1126/science.1057726)]

* [[Andrew Childs]], [[Edward Farhi]], [[John Preskill]], *Robustness of adiabatic quantum computation*, Phys.Rev. A **65** (2002) 012322 [[arXiv:quant-ph/0108048](https://arxiv.org/abs/quant-ph/0108048), [doi:10.1103/PhysRevA.65.012322](https://doi.org/10.1103/PhysRevA.65.012322)]

category: people


Edward Frenkel

Edward Frenkel (Russian: Эдуáрд Влади́мирович Фре́нкель) is a Russian-born mathematician who is currently a professor at the University of California, Berkeley.

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

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

category: people


Edward Hook

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=15817)

## Selected writings

On [[equivariant bordism homology theory]]:

* [[Theodor Bröcker]], [[Edward Hook]], _Stable equivariant bordism_, Math Z 129, 269–277 (1972) ([doi:10.1007/BF01187353](https://doi.org/10.1007/BF01187353))

* [[Edward Hook]], _Equivariant cobordism and duality_, Trans. Amer. Math. Soc. 178 (1973), 241-258 ([doi:10.1090/S0002-9947-1973-0321120-4](https://doi.org/10.1090/S0002-9947-1973-0321120-4))

category: people


Edward MacKinnon

* [Institute page](https://philpeople.org/profiles/edward-mackinnon)

## Selected writings

On  the [[interpretation of quantum physics]]:

* [[Edward MacKinnon]], *Interpreting Physics -- Language and the Classical/QuantumDivide*, Springer (2012) [[doi:10.1007/978-94-007-2369-6](https://doi.org/10.1007/978-94-007-2369-6)]

category: people


Edward Marczewski

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

## Selected writings

On [[linear extension of partial orders]]:

* {#Marczewski30} [[Edward Marczewski]], _Sur l'extension de l'ordre partiel_, Fundamenta Mathematicae **16** (1930) 386–389  $[$[dml:212499](https://eudml.org/doc/212499), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm16125.pdf)$]$

On Marczewski-compact [[measurable spaces]], a necessary ingredient for Gelfand-type duality for [[commutative von Neumann algebras]]:

* [[Edward Marczewski]], _On compact measures_, Fundamenta Mathematicae 40:1 (1953), 113--124, [doi](https://doi.org/10.4064/fm-40-1-113-124).

category: people


Edward Nelson

Edward Nelson (May 4, 1932 – September 10, 2014) was a professor in the Mathematics Department at Princeton University.

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

* [former webpage](https://web.math.princeton.edu/~nelson/)

## Selected writings

### Ultrafinitism

Nelson argued for a position in the [[philosophy of mathematics]] known as [[ultrafinitism]]:

* {#Nelson11} [[Edward Nelson]], _Warning Signs of a Possible Collapse of Contemporary Mathematics_ (~2011) [[pdf](https://web.math.princeton.edu/~nelson/papers/warn.pdf), [[Nelson-WarningSigns.pdf:file]]]

### Nelson arithmetic

[Nelson 2011](#Nelson11) also defined a [[first-order theory]] that was weaker than [[Peano arithmetic]], which replaces the [[induction]] schema of induction on [[natural numbers]] with a predicate and two axioms that state that induction on natural numbers for that predicate is true.

He did not give a name to the theory, so it has been called "[[Nelson arithmetic]]" for lack of a better name.

### (In)Consistency of Peano Arithmetic

At one point Nelson claimed to have a [[formal proof]] of [[inconsistency]] of [[Peano arithmetic]] (PA). While the proof was ultimately acknowledged by Nelson to be wrong, given that the suspicion that PA might be inconsistent after all is shared by others (e.g. [[Vladimir Voevodsky]]), it is still interesting to see Nelson's proof strategy via [[Chaitin's incompleteness theorem]]:

* Edward Nelson, _Outline_ (of an attempt to prove inconsistency of PA) (~2011) [[pdf](http://www.thebigquestions.com/nelson.PDF), [[Nelson-Outline.pdf:file]]]

* Edward Nelson, _Elements_, 2015 [[arXiv:1510.00369 \[math.LO\]](https://arxiv.org/abs/1510.00369)]

## See also

* inconsistent aka [[paraconsistent arithmetic]]

category: people


Edward Rezayi

* [institute page](https://www.calstatela.edu/faculty/edward-rezayi)

## Selected writings

On [[su(2)-anyons]] via [[parafermions]]:

* [[Nicholas Read]], [[Edward Rezayi]], *Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level*, Phys. Rev. B **59** (1999) 8084 $[$[doi:10.1103/PhysRevB.59.8084](https://doi.org/10.1103/PhysRevB.59.8084)$]$

category: people


Edward Shuryak

* [webpage](http://dau2.physics.sunysb.edu/~shuryak/)

## Selected writings

On the [[phase diagram]] of [[QCD]]:

* [[Edward Shuryak]], _The Phases of QCD_ ([arXiv:hep-ph/9609249](https://arxiv.org/abs/hep-ph/9609249))

On [[instantons in QCD]]:

* [[Edward Shuryak]], _Physics of QCD Instantons_, Proceedings of the International School of Subnuclear Physics Erice, Sicily, Italy, August – September 2002 ([web](http://www.worldscientific.com/doi/abs/10.1142/9789812796653_0003))

On [[quantum hadrodynamics]] via [[light-front QCD]]:

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front I: Instanton effects and quark-antiquark effective potentials* ([arXiv:2110.15927](https://arxiv.org/abs/2110.15927))

* [[Edward Shuryak]], [[Ismail Zahed]], *Hadronic structure on the light-front II: QCD strings, Wilson lines and potentials* ([arXiv:2111.01775](https://arxiv.org/abs/2111.01775))

## Related $n$Lab entries

* [[QCD]]

* [[instanton in QCD]]

category: people