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Aidan Herderschee | https://ncatlab.org/nlab/source/Aidan+Herderschee |
* [Institute page](https://www.ias.edu/scholars/aidan-herderschee)
* [InSpire page](https://inspirehep.net/authors/1869780)
* [GoogleScholar page](https://scholar.google.com/citations?user=FI0WW9kAAAAJ&hl=en)
## Selected writings
On [[graviton]] [[scattering amplitudes]] in the [[BFSS matrix model]] and [[soft graviton theorems]]:
* [[Aidan Herderschee]], [[Juan Maldacena]], *Three Point Amplitudes in Matrix Theory* [[arXiv:2312.12592](https://arxiv.org/abs/2312.12592)]
* [[Aidan Herderschee]], [[Juan Maldacena]], *Soft Theorems in Matrix Theory* [[arXiv:2312.15111](https://arxiv.org/abs/2312.15111)]
category: people |
Airy function | https://ncatlab.org/nlab/source/Airy+function |
#Contents#
* table of contents
{:toc}
## Idea
An **Airy function** is a special function satisfying the [[ordinary differential equation]]
$$
y''(x) - x \cdot y(x) = 0
$$
and with the Airy integral representation
$$
Ai(x) = \frac{1}{\pi}\int_0^\infty cos\left(\frac{t^3}{3}+tx\right) dt
= \frac{1}{2\pi}\int_{-\infty}^\infty exp\left(\frac{i t^3}{3}+i t x\right) dt
$$
## Properties
This function appears often in the study of oscillating ([[path integral|path]]) [[integrals]], e.g. in [[semiclassical approximation]] to [[quantum mechanics]] and in the geometric approximation to [[wave mechanics]]/[[optics]]. Its asymptotics is important in the study of the singular behaviour of [[light]] in the vicinity of [[caustics]].
The asymptotic expansions for the Airy function have sharp changes at certain lines, observed by G. G. Stokes, and present often in the stationary phase method (cf. [[semiclassical approximation]]). This is called the _[[Stokes phenomenon]]_ and is a special case of the [[wall crossing]].
Airy function appears in the subject of integrable models, related to Painleve transcendents and also in the study of Hermitean [[random matrices]] (work of Tracy and Widom). Airy function has also a remarkable role in the [[Kontsevich]]'s solution to the [[Witten conjecture]].
## References
* [wikipedia](http://en.wikipedia.org/wiki/Airy_function)
* G. B. Airy, _On the intensity of light in the neighbourhood of a caustic_, Trans. Camb. Phil. Soc., 6 (1838), 379-403.
* [[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_, Comm. Math. Phys. 147 (1992), no. 1, 1--23, [euclid](http://projecteuclid.org/euclid.cmp/1104250524)
* C. A. Tracy, H. Widom, _Level-spacing distributions and the Airy kernel_, Physics Letters B 305 (1-2): 115–118 (1993) [hep-th/9210074](http://arxiv.org/abs/hep-th/9210074), <a href="http://dx.doi.org/10.1016/0370-2693(93)91114-3">doi</a>; _Level-spacing distributions and the Airy kernel_, Commun. in Math. Physics 159 (1): 151–174 (1994) [euclid](http://projecteuclid.org/euclid.cmp/1104254495) [doi](http://dx.doi.org/10.1007/BF02100489), [MR1257246](http://www.ams.org/mathscinet-getitem?mr=1257246); _On orthogonal and symplectic matrix ensembles_, Commun. in Math. Phys. __177__ (3): 727–754 (1996) [doi](http://dx.doi.org/10.1007/BF02099545), [MR1385083](http://www.ams.org/mathscinet-getitem?mr=1385083)
See also sec. 7.2 in
* Alain Connes, Caterina Consani, _The universal thickening of the field of real numbers_, [arxiv/1202.4377](https://arxiv.org/abs/1202.4377)
For generalizations see the references
* R. N. Fernandez, V. S. Varadarajan, _Matrix Airy functions for compact Lie groups_, Internat. J. Math. __20__ (2009), no. 8, 945–977, [doi](http://dx.doi.org/10.1142/S0129167X09005595), [MR2554728](http://www.ams.org/mathscinet-getitem?mr=2554728)
* R. N. Fernandez, V. S. Varadarajan, D. Weisbart, _Airy functions over local fields_, Lett. Math. Phys. __88__ (2009), no. 1-3, 187–206, [MR2010d:11138](http://www.ams.org/mathscinet-getitem?mr=2512146), [doi](http://dx.doi.org/10.1007/s11005-009-0311-x)
* Marco Bertola, Boris Dubrovin, Di Yang, _Simple Lie algebras and topological ODEs_, [arxiv/1508.03750](http://arxiv.org/abs/1508.03750)
category: analysis
[[!redirects Airy functions]]
[[!redirects Airy integral]]
|
Aise Johan de Jong | https://ncatlab.org/nlab/source/Aise+Johan+de+Jong |
* [website](http://www.math.columbia.edu/~dejong/)
de Jong is running a wiki on [[algebraic stack]]-theory:
* [[The Stacks Project]]
category: people
[[!redirects Johan de Jong]]
|
AJ Tolland | https://ncatlab.org/nlab/source/AJ+Tolland |
* [website](http://www.math.sunysb.edu/~ajt/)
[[!redirects A. J. Tolland]]
category: people |
Akash Singh | https://ncatlab.org/nlab/source/Akash+Singh |
* [spire page](https://inspirehep.net/authors/1919675?ui-citation-summary=true)
* [institute page](https://web.iisermohali.ac.in/Faculty/anoshjoseph/hep_group/members.html)
## Selected writings
On the [[AdS-QCD correspondence]] (embedding the "hard-wall model" into [[type IIB string theory]]):
* [[Akash Singh]], [[K. P. Yogendran]], *Phases of a 10-D Holographic hard wall model* [[arXiv:2208.09387](https://arxiv.org/abs/2208.09387)]
category: people |
Akhil Mathew | https://ncatlab.org/nlab/source/Akhil+Mathew | * [Webpage](http://math.berkeley.edu/~amathew/)
## Selected writings
On [[stable homotopy theory]]:
* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])
* [[Akhil Mathew]], *The Galois group of a stable homotopy theory* ([arXiv](http://arxiv.org/abs/1404.2156))
On [[Hurewicz cofibrations]]:
* {#Mathew10a} [[Akhil Mathew]], *Cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/07/cofibrations/))
* {#Mathew10b} [[Akhil Mathew]], *Examples of cofibrations*, 2010 ([web](https://amathew.wordpress.com/2010/10/08/examples-of-cofibrations/))
## Related $n$Lab entries
* [[Galois group]]
category: people |
Akihiko Sonoda | https://ncatlab.org/nlab/source/Akihiko+Sonoda |
## Selected writings
Interpretation in [[holographic QCD]] of the [[Schwinger effect]] as exhibited by the [[DBI-action]] on [[flavor branes]]:
* {#HashimotoOkaSonoda14a} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Magnetic instability in AdS/CFT : Schwinger effect and Euler-Heisenberg Lagrangian of Supersymmetric QCD_, J. High Energ. Phys. 2014, 85 (2014) ([arXiv:1403.6336](https://arxiv.org/abs/1403.6336))
* {#HashimotoOkaSonoda14b} [[Koji Hashimoto]], [[Takashi Oka]], [[Akihiko Sonoda]], _Electromagnetic instability in holographic QCD_, J. High Energ. Phys. 2015, 1 (2015) ([arXiv:1412.4254](https://arxiv.org/abs/1412.4254))
Review:
* [[Akihiko Sonoda]], _Electromagnetic instability in AdS/CFT_, 2016 ([spire:1633963](http://inspirehep.net/record/1633963), [[Sonoda16.pdf:file]])
category: people |
Akikazu Hashimoto | https://ncatlab.org/nlab/source/Akikazu+Hashimoto |
* [webpage](https://www.physics.wisc.edu/people/akihashimoto)
## Selected writings
On the [[non-abelian DBI action]] for [[intersecting branes]] with non-abelian [[gauge enhancement]] on their worldvolume:
* [[Akikazu Hashimoto]], [[Washington Taylor]], _Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action_, Nucl. Phys. B503: 193-219, 1997 ([arXiv:hep-th/9703217](https://arxiv.org/abs/hep-th/9703217))
category: people
[[!redirects Aki Hashimoto]] |
Akira Furusaki | https://ncatlab.org/nlab/source/Akira+Furusaki |
* [Research group page](https://cems.riken.jp/en/laboratory/qmtrt)
* [GoogleScholar page](https://scholar.google.com.au/citations?user=JAdEuOIAAAAJ&hl=en)
## Selected writings
Precursor to the [[K-theory classification of topological phases of matter]]:
* [[Andreas P. Schnyder]], [[Shinsei Ryu]], [[Akira Furusaki]], [[Andreas W. W. Ludwig]], *Classification of topological insulators and superconductors in three spatial dimensions*, Phys. Rev. B **78** 195125 (2008) $[$[doi:10.1103/PhysRevB.78.195125](https://doi.org/10.1103/PhysRevB.78.195125), [arXiv:0803.2786](https://arxiv.org/abs/0803.2786)$]$
category: people
|
Akira Kono | https://ncatlab.org/nlab/source/Akira+Kono |
* [ResearchGate page](https://www.researchgate.net/scientific-contributions/Akira-Kono-21367148)
## Selected writings
On [[unstable topological K-theory]]:
* {#HK03} [[Hiroaki Hamanaka]], [[Akira Kono]]: *On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$*, Journal of Mathematics of Kyoto University **43** 2 (2003) 333-348 [[doi:10.1215/kjm/1250283730](https://doi.org/10.1215/kjm/1250283730)[
* [[Hiroaki Hamanaka]], [[Akira Kono]]: *An application of unstable K-theory*, Journal of Mathematics of Kyoto University **44** 2 (2004) 451-456 [[doi:10.1215/kjm/1250283560](https://doi.org/10.1215/kjm/1250283560)]
* [[Hiroaki Hamanaka]]: *Adams $ e $-invariant, Toda bracket and $[X, U (n)]$*, Journal of Mathematics of Kyoto University **43** 4 (2003) 815-827 [[doi:10.1215/kjm/1250281737](https://doi.org/10.1215/kjm/1250281737)]
On [[algebraic topology]] with focus on [[complex oriented cohomology|complex oriented]] [[Whitehead-generalized cohomology]]:
* [[Dai Tamaki]], [[Akira Kono]], _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))
## Related entries
* [[generalized (Eilenberg-Steenrod) cohomology]]
* [[Brown representability theorem]]
* [[multiplicative cohomology theory]]
category: people |
Akshay Venkatesh | https://ncatlab.org/nlab/source/Akshay+Venkatesh | [Website](https://www.ias.edu/math/people/faculty/venkatesh)
## Selected writings
On the [[Mordell conjecture]]:
* {#LawrenceVenkatesh18} [[Brian Lawrence]], [[Akshay Venkatesh]]: _Diophantine problems and $p$-adic period mappings_, Invent. math. **221** (2020) 893–999 [[arxiv:1807.02721](https://arxiv.org/abs/1807.02721), [doi:10.1007/s00222-020-00966-7](https://doi.org/10.1007/s00222-020-00966-7)]
On the [[relative Langlands program]]
* [[David Ben-Zvi]], [[Yiannis Sakellaridis]], [[Akshay Venkatesh]], _Relative Langlands duality_ ([pdf](https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf)).
category: people
|
AKSZ sigma-model | https://ncatlab.org/nlab/source/AKSZ+sigma-model |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
What is called the _AKSZ formalism_ -- after the initials of its four authors -- Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]] -- is a technique for constructing [[action functional]]s in [[BV-BRST formalism]] for [[sigma model]] [[quantum field theories]] whose
[[target space]] is an [[symplectic Lie n-algebroid]] $(\mathfrak{P}, \omega)$.
The [[action functional]] of AKSZ theory is that of [[∞-Chern-Simons theory]] induced from the [[Chern-Simons element]] that correspondonds to the [[invariant polynomial]] $\omega$. Details on this are at [∞-Chern-Simons theory -- Examples -- AKSZ theory](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory).
## Examples
* to a [[Poisson Lie algebroid]] corresponds the [[Poisson sigma-model]];
* the a [[Courant algebroid]] corresponds the [[Courant sigma-model]];
in particular to a [[semisimple Lie algebra]] corresponds [[Chern-Simons theory]].
* [[BF-theory]]+[[topological Yang-Mills theory]],
Also, the [[A-model]] and the [[B-model]] topological 2d [[sigma-models]] are examples.
## Definition
A [[sigma-model]] [[quantum field theory]] is, roughly, one
* whose fields are maps $\phi : \Sigma \to X$ to some space $X$;
* whose [[action functional]] is, apart from a
[[kinetic action|kinetic term]], the [[transgression]]
of some kind of [[cocycle]] on $X$
to the [[mapping space]] $\mathrm{Map}(\Sigma,X)$.
Here the terms "space", "maps" and "cocycles"
are to be made precise in a suitable context.
One says that $\Sigma$ is the _[[worldvolume]]_,
$X$ is the _[[target space]]_ and the cocycle is
the _[[background gauge field]]_ .
For instance the ordinary charged [[particle]]
(for instance an electron) is described by a $\sigma$-model
where $\Sigma = (0,t) \subset \mathbb{R}$ is the abstract
_[[worldline]]_, where $X$ is a smooth
([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]]
(for instance our [[spacetime]]) and where the background cocycle
is a [[circle bundle]] with [[connection on a bundle|connection]]
on $X$ (a degree-2 cocycle in [[ordinary differential cohomology]] of $X$, representing a background _[[electromagnetic field]]_ :
up to a kinetic term the action functional is the [[holonomy]] of the connection over a given [[curve]] $\phi : \Sigma \to X$.
The $\sigma$-models to be considered here are
_higher_ generalizations of this example,
where the background gauge field is a cocycle of higher degree
(a [[connection on an infinity-bundle|higher bundle with connection]])
and where the worldvolume is
accordingly higher dimensional -- and where $X$ is allowed to be
not just a manifold but an approximation to a
_higher [[orbifold]] (a [[smooth ∞-groupoid]]).
More precisely, here we take the [[category]] of [[space]]s
to be [[dg-geometry|smooth dg-manifolds]].
One may imagine that we can equip this
with an [[internal hom]]
$\mathrm{Maps}(\Sigma,X)$ given by $\mathbb{Z}$-graded objects.
Given [[dg-geometry|dg-manifolds]] $\Sigma$ and $X$
their canonical degree-1 vector fields
$v_\Sigma$ and $v_X$ acting on the mapping space from the
left and right. In this sense their linear combination
$v_\Sigma + k \, v_X$ for some $k \in \mathbb{R}$
equips also $\mathrm{Maps}(\Sigma,X)$ with the structure of
a differential graded smooth manifold.
Moreover, we take the "cocycle" on $X$ to be a
graded [[symplectic structure]] $\omega$,
and assume that there is a kind of Riemannian structure on $\Sigma$
that allows to form the [[transgression]]
$$
\int_\Sigma \mathrm{ev}^* \omega
:=
p_! \mathrm{ev}^* \omega
$$
by [[integral transform|pull-push]] through the canonical
[[correspondence]]
$$
\mathrm{Maps}(\Sigma,X)
\stackrel{p}{\leftarrow}
\mathrm{Maps}(\Sigma,X) \times \Sigma
\stackrel{ev}{\to}
X
\,,
$$
where on the right we have the [[evaluation map]].
Assuming that one succeeds in making precise sense of all this
one expects to
find that $\int_\Sigma \mathrm{ev}^* \omega$ is in turn a symplectic
structure on the mapping space. This implies that the vector field
$v_\Sigma + k\, v_X$ on mapping space has a
[[Hamiltonian]]
$\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X))$.
The grade-0 components $S_{\mathrm{AKSZ}}$ of $\mathbf{S}$
then constitute a functional on the space
of maps of graded manifolds $\Sigma \to X$. This is
the **AKSZ action functional** defining the AKSZ $\sigma$-model
with target space $X$ and background field/cocycle $\omega$.
In ([AKSZ](#AKSZ)) this procedure is indicated only somewhat vaguely.
The focus of attention there is a discussion, from this perspective,
of the action functionals
of the 2-dimensional $\sigma$-models called the
_[[A-model]]_ and the _[[B-model]]_ .
In ([Roytenberg](#Roytenberg)), a more detailed discussion of the general construction is given, including an explicit
and general formula for $\mathbf{S}$ and hence for $S_{\mathrm{AKSZ}}$ .
For $\{x^a\}$ a coordinate chart on $X$ that formula is
the following.
+-- {: .num_defn #TheAKSZAction}
###### Definition
For $(X,\omega)$ a [[symplectic Lie n-algebroid|symplectic dg-manifold]] of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$
and $k \in \mathbb{R}$, the
**AKSZ action functional**
$$
S_{\mathrm{AKSZ},k} :
\mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X)
\to
\mathbb{R}
$$
(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)
is
$$
S_{\mathrm{AKSZ},k}
:
\phi
\mapsto
\int_\Sigma
\left(
\frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b
+
k
\,
\phi^* \pi
\right)
\,,
$$
where $\pi$ is the [[Hamiltonian]] for $v_X$ with respect to $\omega$
and where on the right we are interpreting fields as forms on $\Sigma$.
=--
This formula hence defines an infinite class of $\sigma$-models
depending on the target space structure $(X, \omega)$, and on the
relative factor $k \in \mathbb{R}$. In ([AKSZ](#AKSZ)) it was
already noticed that ordinary [[Chern-Simons theory]] is a special
case of this for $\omega$ of grade 2, as is the
[[Poisson sigma-model]]
for $\omega$ of grade 1 (and hence, as shown there, also the [[A-model]]
and the [[B-model]]).
The main example in ([Roytenberg](#Roytenberg)) is spelling out the
general case for $\omega$ of grade 2, which is called the
_[[Courant sigma-model]]_ there.
One nice aspect of this construction is that it follows immediately
that the full Hamiltonian $\mathbf{S}$ on mapping space satisfies
$\{\mathbf{S}, \mathbf{S}\} = 0$. Moreover, using the standard formula
for the internal hom of chain complexes one finds that the cohomology
of $(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X)$ in degree 0
is the space of functions on those fields that satisfy the
[[Euler-Lagrange equations]] of $S_{\mathrm{AKSZ}}$.
Taken together this implies that
$\mathbf{S}$ is a solution of the "master equation"
of a [[BV-BRST complex]] for the quantum field theory
defined by $S_{\mathrm{AKSZ}}$. This is a crucial ingredient for the
quantization of the model, and this is what the AKSZ construction
is mostly used for in the literature.
## Related concepts
* [[sigma-model]]
* [[schreiber:infinity-Chern-Simons theory]]
* [[higher dimensional Chern-Simons theory]]
* [[1d Chern-Simons theory]]
* [[2d Chern-Simons theory]]
* [[3d Chern-Simons theory]]
* [[4d Chern-Simons theory]]
* [[5d Chern-Simons theory]]
* [[6d Chern-Simons theory]]
* [[7d Chern-Simons theory]]
* [[11d Chern-Simons theory]]
* [[string field theory]]
* [[infinite-dimensional Chern-Simons theory]]
* **AKSZ $\sigma$-model**
* [[Poisson sigma-model]]
* [[A-model]], [[B-model]]
* [[Courant sigma-model]]
* [[Chern-Simons theory]]
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]
## References
The original reference is
* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]], _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))
Dmitry Roytenberg wrote a useful exposition of the central idea of the original work and studied the case of the [[Courant sigma-model]] in
* {#Roytenberg} [[Dmitry Roytenberg]], _AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories_ Lett.Math.Phys.79:143-159,2007 ([arXiv:hep-th/0608150](http://arxiv.org/abs/hep-th/0608150)).
Other reviews include
* [[Noriaki Ikeda]], _Deformation of graded (Batalin-Volkvisky) Structures_ in Dito, Lu, Maeda, Weinstein (eds.) _Poisson geometry in mathematics and physics_ Contemp. Math. 450, AMS (2008)
* [[Noriaki Ikeda]], _Lectures on AKSZ Topological Field Theories for Physicists_ ([arXiv:1204.3714](http://arxiv.org/abs/1204.3714))
A cohomological reduction of the formalism is described in
* F. Bonechi, P. Mnëv, [[Maxim Zabzine]], _Finite dimensional AKSZ-BV-theories_ ([arXiv](http://arxiv.org/abs/0903.0995))
That the AKSZ action on bounding manifolds $\partial \hat \Sigma$ is the integral of the graded symplectic form over $\hat \Sigma$ is theorem 4.4 in
* A. Kotov, T. Strobl, _Characteristic classes associated to Q-bundles_ ([arXiv:0711.4106v1](http://arxiv.org/abs/0711.4106v1))
The discussion of the AKSZ action functional as the [[nLab:∞-Chern-Simons theory]]-functional induced from a [[symplectic Lie n-algebroid]] in [[∞-Chern-Weil theory]] is due discussed in
* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 ([arXiv:1108.4378](http://arxiv.org/abs/1108.4378))
In the broader context of smooth [[higher geometry]] this is discussed in section 4.3 of
* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_
Discussion of [[boundary conditions]] for the AKSZ sigma model includes
* [[Peter Bouwknegt]], [[Branislav Jurco]], _AKSZ construction of topological open $p$-brane action and Nambu brackets_, [arxiv/1110.0134](http://arxiv.org/abs/1110.0134)
* {#IkedaXu13} [[Noriaki Ikeda]], Xiaomeng Xu, _Canonical functions and differential graded symplectic pairs in supergeometry and AKSZ sigma models with boundary_ ([arXiv:1301.4805](http://arxiv.org/abs/1301.4805))
The AKSZ model is extended to coisotropic boundary conditions in
* [[Theo Johnson-Freyd]], _Exact triangles, Koszul duality, and coisotropic boundary conditions_ ([arxiv/1608.08598](https://arxiv.org/abs/1608.08598))
An example in [[higher spin gauge theory]] is discussed in
* {#AlkalevGrigorievSkvortsov14} K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, _Uniformizing higher-spin equations_ ([arXiv:1409.6507](http://arxiv.org/abs/1409.6507))
See also
* Theodore Th. Voronov, _Vector fields on mapping spaces and a converse to the AKSZ construction_, [arxiv/1211.6319](http://arxiv.org/abs/1211.6319)
[[!redirects AKSZ sigma-models]]
[[!redirects AKSZ sigma model]]
[[!redirects AKSZ functional]]
[[!redirects AKSZ model]]
[[!redirects AKSZ formalism]]
[[!redirects AKSZ theory]]
[[!redirects AKSZ]]
[[!redirects AKSZ-sigma models]] |
Alain Aspect | https://ncatlab.org/nlab/source/Alain+Aspect |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alain_Aspect)
## Selected writings
On [[experiment|experimental]] tests of [[Bell's inequality]] in [[quantum physics]]:
* {#Aspect15} [[Alain Aspect]], *[Closing the Door on Einstein and Bohr’s Quantum Debate](https://physics.aps.org/articles/v8/123)*, Physics **8** 123 (2015)
category: people |
Alain Bruguières | https://ncatlab.org/nlab/source/Alain+Brugui%C3%A8res |
* [personal page](https://imag.umontpellier.fr/~bruguieres/)
## Selected writings
On [[Hopf monads]] and [[Hopf adjunctions]]:
* [[Alain Bruguières]], [[Alexis Virelizier]], *Hopf monads*, Advances in Mathematics **215** 2 (2007) 679-733 [[doi:10.1016/j.aim.2007.04.011](https://doi.org/10.1016/j.aim.2007.04.011), [arXiv:math/0604180](https://arxiv.org/abs/math/0604180)]
* [[Alain Bruguières]], [[Steve Lack]], [[Alexis Virelizier]], *Hopf monads on monoidal categories*, Adv. Math. __227__ 2 (2011) 745-800 [[arXiv:1003.1920](https://arxiv.org/abs/1003.1920), [doi:10.1016/j.aim.2011.02.008](https://doi.org/10.1016/j.aim.2011.02.008)]
category: people
[[!redirects Alain Bruguieres]]
|
Alain Connes | https://ncatlab.org/nlab/source/Alain+Connes | Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at [IHÉS](http://www.ihes.fr), Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include [[geometry]], [[topology]], especially K-theory and index theory, [[operator algebras]], the connections between noncommutative geometry and number theory, and more recently also the absolute geometry over a [[field with one element]].
#Contents#
* table of contents
{:toc}
## Research
Most recently Connes is studying number theory including connections to the "thermodynamic aspect" via the [[KMS state|KMS states]], to the absolute geometry via the [[field with one element]] and a related approach to the [[Riemann hypothesis]].
Alain Connes uses intuition from [[mathematical physics]] like the notion of KMS state and has introduced the noncommutative extensions of [[standard model of particle physics]] predicting a value for the mass of [[Higgs particle]]. He proposed elements of a unification of [[gravity]] and [[gauge theories]] via [[spectral action]] functionals on spaces of [[spectral triple|spectral triples]].
Connes is most well known for introducing a dominant direction to [[noncommutative geometry]] where his contributions include, most remarkably, the introduction of [[cyclic homology]] and its connections to the [[K-theory]] of spaces and of [[operator algebra|operator algebras]], study of fundamental examples of noncomutative spaces like
the space of leaves of a foliation, noncommutative tori, groupoid operator algebras, spaces of Penrose tilings, introducing noncommutative motives into operator algebras (with [[Matilde Marcolli]]), Baum-Connes hypothesis, the [[local index formula]], combinatorial approach to Feynman diagrams (Connes-Kreimer Hopf algebra of renormalization) and analytic aspects connecting them to the Birkhoff decomposition, introducing Hopf-cyclic homology etc.
As an inspirative and energetic lecturer Connes also directly contibuted to the popularization of noncommutative geometry and its connections to physics.
## Selected writings
Introducing the [[cyclic category]] and [[cyclic objects]] ([[cyclic sets]], [[cyclic spaces]]) for [[cyclic homology]]:
* [[Alain Connes]], _Cohomologie cyclique et foncteurs $Ext^n$_, C.R.A.S. **269** (1983), Série I, 953-958
On the [[equivariant Chern character]]:
* [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))
On [[noncommutative geometry]]:
* Alain Connes, _Noncommutative geometry_, Academic Press 1994, 661 p. [PDF](http://www.alainconnes.org/docs/book94bigpdf.pdf)
* Alain Connes, [[Matilde Marcolli]], _Noncommutative geometry, quantum fields and motives_, draft [pdf](http://www.alainconnes.org/docs/bookwebfinal.pdf)
Introducing the [[Bost-Connes system]]:
* {#BostConnes95} [[Jean-Benoit Bost]]; [[Alain Connes]], *Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory*, Selecta Mathematica, New Series **1** 3 (1995) 411-457 [[doi:10.1007/BF01589495](https://doi.org/10.1007/BF01589495), ISSN 1022-1824, [pdf](https://web.archive.org/web/20110806043925/https://alainconnes.org/docs/bostconnesscan.pdf)]
For more see Connes' official [website](http://www.alainconnes.org/en).
## Related entries
* [[noncommutative geometry]], [[operator algebra]], [[C-star algebra]]
* [[spectral triple]]
* [[Hochschild-Kostant-Rosenberg theorem]]
* [[cyclic object]], [[cycle category]], [[cyclic cohomology]]
* [[renormalization]], [[field with one element]]
* [[Bost-Connes system]]
category: people
[[!redirects A. Connes]]
[[!redirects Connes]]
|
Alain Joye | https://ncatlab.org/nlab/source/Alain+Joye |
* [institute page](https://www-fourier.ujf-grenoble.fr/~joye/)
* [GoogleScholar page](https://scholar.google.fr/citations?user=4NUK5AgAAAAJ&hl=en)
## Selected writings
On [[open quantum systems|open]] [[quantum systems]] ([[quantum probability]], [[quantum noise]], [[quantum channels]], ...):
* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems I -- The Hamiltonian Approach*, Lecture Notes in Mathematics **1880**, Springer (2006) [[doi:10.1007/b128449](https://doi.org/10.1007/b128449)]
* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*, Lecture Notes in Mathematics **1881**, Springer (2006) [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]
* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems III -- Recent Developments*, Lecture Notes in Mathematics **1882**, Springer (2006) [[doi:10.1007/b128453](https://doi.org/10.1007/b128453)]
category: people |
Alain Prouté | https://ncatlab.org/nlab/source/Alain+Prout%C3%A9 | * [Home page](http://www.logique.jussieu.fr/~alp/)
## Related entries
* [[A-infinity-algebra]]
##References:
* Alain Prouté, _Introduction à la Logique Catégorique_, [Lecture Notes](http://www.logique.jussieu.fr/~alp/cours_2010.pdf)
* Alain Prouté, _Modèles minimaux de Baues-Lemaire et Kadeishvili et homologie des fibrations_, [TAC reprints](http://www.tac.mta.ca/tac/reprints/articles/21/tr21abs.html). This is a reprint of his thesis.
category:people |
Alain Valette | https://ncatlab.org/nlab/source/Alain+Valette |
* [webpage](http://www2.unine.ch/alain.valette)
## Selected writings
On [[equivariant K-theory]] and the [[Baum-Connes conjecture]]:
* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))
category: people
|
Alain Verschoren | https://ncatlab.org/nlab/source/Alain+Verschoren | __Alain Verschoren__ was an [[algebraist]] at Antwerp. With [[Fred van Oystaeyen]] he coauthored the first monograph on noncommutative algebraic geometry
_Non-commutative algebraic geometry_ (Springer LNM 887, 1981).
* [webpage](http://www.ua.ac.be/main.aspx?c=alain.verschoren)
Verschoren died suddenly in 2020 at the age of 66, [obituary](https://www.uantwerpen.be/en/about-uantwerp/organisation/in-the-spotlight/alain-verschoren)
* [[Alain Verschoren]], _Compatibility and stability_,
Notas de Matemática [Mathematical Notes], 3. Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. xii+81 pp. ISBN: 84-7684-934-6
* J. Mulet, A. Verschoren, _On compatibility. II._, Comm. Algebra 20 (1992), no. 7, 1897--1905. [doi](http://dx.doi.org/10.1080/00927879208824438)
* M. I. Segura, D. Tarazona, A. Verschoren, _On compatibility_, Comm. Algebra __17__ (1989), no. 3, 677--690.
category: people
[[!redirects A. Verschoren]] |
Alan Carey | https://ncatlab.org/nlab/source/Alan+Carey |
* [website](http://www.cirs-tm.org/researchers/researchers.php?id=251)
## Selected writings
On [[twisted K-theory]]:
* [[Alan Carey]], [[Bai-Ling Wang]], p. 5 of: *Thom isomorphism and Push-forward map in twisted K-theory*, Journal of K-Theory **1** 2 (2008) 357-393 ([arXiv:math/0507414](https://arxiv.org/abs/math/0507414), [doi:10.1017/is007011015jkt011](https://doi.org/10.1017/is007011015jkt011))
On [[twisted differential K-theory]]:
* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))
## Related $n$Lab entries
* [[bundle gerbe]]
* [[Chern-Simons 2-gerbe]]
* [[noncommutative geometry]]
* [[quantum anomaly]]
category: people |
Alan Coley | https://ncatlab.org/nlab/source/Alan+Coley |
* [webpage](https://www.mscs.dal.ca/~aac/)
## related $n$Lab entries
* [[gravity]], [[general relativity]]
* [[mathematical physics]]
* [[inhomogeneous cosmology]]
* [[Penrose-Hawking singularity theorem]]
category: people |
Alan Guth | https://ncatlab.org/nlab/source/Alan+Guth |
* [webpage](http://web.mit.edu/physics/people/faculty/guth_alan.html)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Alan_Guth)
## Selected writings
On [[false vacuum eternal inflation]]:
* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))
## Related entries
* [[cosmic inflation]]
category: people
|
Alan Hatcher > history | https://ncatlab.org/nlab/source/Alan+Hatcher+%3E+history |
> see [[Allen Hatcher]] |
Alan Hatcher > history 2 | https://ncatlab.org/nlab/source/Alan+Hatcher+%3E+history+2 | [[!redirects Alan Hatcher > history 2]]
* [Home page](https://www.math.cornell.edu/~hatcher/)
category:people |
Alan J. Deschner | https://ncatlab.org/nlab/source/Alan+J.+Deschner |
## Selected writings
On [[Sullivan models]] in [[rational homotopy theory]]
* [[Alan J. Deschner]], *Sullivan's theory of minimal models*, MSc thesis, Univ. British Columbia (1976) [[doi:10.14288/1.0080132](http://hdl.handle.net/2429/20052), [pdf](https://open.library.ubc.ca/media/stream/pdf/831/1.0080132/2)]
## Related entries
* [[Sullivan model of n-spheres]]
* [[Sullivan model of complex projective space]]
[[!redirects Alan Deschner]]
[[!redirects Alan Joseph Deschner]]
|
Alan M. Turing | https://ncatlab.org/nlab/source/Alan+M.+Turing | A British mathematician, logician and computer scientist (before computers were invented).
## References
* [Wikipedia page](https://en.wikipedia.org/wiki/Alan_Turing)
category: people
[[!redirects Alan Turing]]
[[!redirects A. M. Turing]]
|
Alan Martin | https://ncatlab.org/nlab/source/Alan+Martin |
* <a href="https://en.wikipedia.org/wiki/Alan_Martin_(physicist)">Wikipedia entry</a>
## selected writings
* {#HalzenMartin84} [[Francis Halzen]], [[Alan Martin]], _Quarks and Leptons: An Introductory Course in Modern Particle Physics_, Wiley 1984 ([pdf](http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks%20&%20Leptons.pdf))
## related $n$Lab entries
* [[quark]]
* [[lepton]]
* [[QCD]]
* [[standard model of particle physics]]
category: people
|
Alan Mycroft | https://ncatlab.org/nlab/source/Alan+Mycroft |
* [personal page](https://www.cl.cam.ac.uk/~am21/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alan_Mycroft)
* [GoogleScholar page](https://scholar.google.com/citations?user=0ycSEz8AAAAJ&hl=en)
## Selected writings
On [[comonads]] and [[graded modalities]] [[comonads in computer science|in computer science]];
* {#POM13} [[Tomas Petricek]], [[Dominic Orchard]], [[Alan Mycroft]], *Coeffects: Unified Static Analysis of Context-Dependence*, in: *Automata, Languages, and Programming. ICALP 2013*, Lecture Notes in Computer Science **7966** Springer (2013) [[doi:10.1007/978-3-642-39212-2_35](https://doi.org/10.1007/978-3-642-39212-2_35)]
* [[Dominic Orchard]], [[Alan Mycroft]], *A Notation for Comonads*, in: *Implementation and Application of Functional Languages. IFL 2012*, Lecture Notes in Computer Science **8241** [[doi:10.1007/978-3-642-41582-1_1](https://doi.org/10.1007/978-3-642-41582-1_1)]
> (on [[codo-notation]])
category: people
|
Alan Robinson | https://ncatlab.org/nlab/source/Alan+Robinson |
* [personal page](http://homepages.warwick.ac.uk/~maspas/)
* [institute page](https://warwick.ac.uk/fac/sci/maths/people/staff/alan_robinson/)
## Selected writings
Introducing the "[[stable Dold-Kan correspondence]]" (equivalence between [[Eilenberg-MacLane spectrum|$H R$-]][[module spectra]] and [[chain complexes]]) on the level of [[stable homotopy category|stable]] [[homotopy categories]]:
* {#Robinson87} [[Alan Robinson]], *The extraordinary derived category*, Math. Z. **196** 2 (1987) 231-238 [[doi:10.1007/BF01163657](https://doi.org/10.1007/BF01163657)]
category: people |
Alan S. Cigoli | https://ncatlab.org/nlab/source/Alan+S.+Cigoli |
* [institute page](https://www.matematica.unito.it/do/docenti.pl/Alias?alanstefano.cigoli#tab-profilo)
## Selected writings
On [[final functors|finality]] of [[internal functors]] between [[internal groupoids]] in [[exact categories]]:
* [[Alan S. Cigoli]], *A characterization of final functors between internal groupoids in exact categories*, Theory and Applications of Categories **33** 11 (2018) 265-275. [[arXiv:1711.10747](https://arxiv.org/abs/1711.10747), [tac:33-11](http://www.tac.mta.ca/tac/volumes/33/11/33-11abs.html)]
category: people
[[!redirects Alan Cigoli]] |
Alan Weinstein | https://ncatlab.org/nlab/source/Alan+Weinstein |
* [website](http://math.berkeley.edu/~alanw/)
## Selected writings
On [[Lagrangian submanifolds]] in [[symplectic geometry]]:
* [[Alan Weinstein]], _Symplectic Manifolds and Their Lagrangian Submanifolds_, Advances in Math. **6** (1971) 329346 \[<a href="https://doi.org/10.1016/0001-8708(71)90020-X">doi:10.1016/0001-8708(71)90020-X</a>\]
On [[symplectic geometry]]:
* [[Alan Weinstein]], _Symplectic geometry_, Bulletin Amer. Math. Soc. **5** (1981) 1-13 [[doi:10.1090/S0273-0979-1981-14911-9](http://dx.doi.org/10.1090/S0273-0979-1981-14911-9)]
On [[Poisson manifolds]]:
* [[Alan Weinstein]], *The local structure of Poisson manifolds*, J. Differential Geom. **18** 3 (1983) 523-557 [[doi:10.4310/jdg/1214437787](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full)]
On [[geometric quantization]] via [[symplectic groupoids]]:
* {#Weinstein87} [[Alan Weinstein]], _Symplectic groupoids and Poisson manifolds_, Bull. Amer. Math. Soc. (N.S.) **16** (1987) 101-104 [[euclid:bams/1183553676](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-16/issue-1/Symplectic-groupoids-and-Poisson-manifolds/bams/1183553676.full)]
* [[Alan Weinstein]], _Symplectic groupoids, geometric quantization, and irrational rotation algebras_, in:
_Symplectic geometry, groupoids, and integrable systems_ (Berkeley, CA, 1989), Springer (1991) 281-290 [[doi:10.1007/978-1-4613-9719-9_19](https://doi.org/10.1007/978-1-4613-9719-9_19), MR1104934]
* [[Alan Weinstein]], _Tangential deformation quantization and polarized symplectic groupoids_, in: _Deformation theory and symplectic geometry_ (Ascona, 1996), 301-314, Kluwer (1997) [[ISBN:9780792345251](https://link.springer.com/book/9780792345251), [MR1480730](http://www.ams.org/mathscinet-getitem?mr=1480730)]
On [[formal deformation quantization]]:
* [[Alan Weinstein]], *Deformation quantization*, Séminaire Bourbaki volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Talk no. 789 [[numdam:SB_1993-1994__36__389_0](http://www.numdam.org/item/?id=SB_1993-1994__36__389_0)]
On [[geometric quantization]]:
* Sean Bates, [[Alan Weinstein]], _[[Lectures on the geometry of quantization]]_, AMS (1997) [[pdf](http://www.math.berkeley.edu/~alanw/GofQ.pdf)]
On [[groupoids]]:
* [[Alan Weinstein]], _Groupoids: Unifying Internal and External Symmetry -- A Tour through some Examples_, Notices of the AMS **43** 7 (1996) [[pdf](http://www.ams.org/notices/199607/weinstein.pdf), [[Weinstein_Groupoids.pdf:file]]]
## Related entries
* [[symplectic geometry]]
[[symplectic manifold]], [[symplectic groupoid]]
* [[Lagrangian correspondence]], [[canonical transformation]]
* [[symplectic category]]
* [[geometric quantization]]
* [[geometric quantization of symplectic groupoids]]
* [[n-symplectic manifold]]
* [[volume of a Lie groupoid]]
category: people |
Alastair Craw | https://ncatlab.org/nlab/source/Alastair+Craw |
Alastair Craw is a mathematician at University of Bath.
* [webpage](http://people.bath.ac.uk/ac886/)
## related $n$Lab entries
* [[mirror symmetry]]
* [[McKay correspondence]]
category: people |
Alastair Grant-Stuart | https://ncatlab.org/nlab/source/Alastair+Grant-Stuart |
* [InSpire page](https://inspirehep.net/authors/2016419)
## Selected writings
Formulation of the [[CS/WZW correspondence]] in [[homotopical AQFT]]:
* [[Marco Benini]], [[Alastair Grant-Stuart]], [[Alexander Schenkel]], *The linear CS/WZW bulk/boundary system in AQFT*, Annales Henri Poincaré (2023) [[arXiv:2302.06990](https://arxiv.org/abs/2302.06990)]
category: people |
Alastair Hamilton | https://ncatlab.org/nlab/source/Alastair+Hamilton | __Alastair Hamilton__ is a mathematician at the University of Connecticut.
* [webpage](http://www.math.uconn.edu/~hamilton)
## Selected writings
* On the extension of a TCFT to the boundary of the moduli space. [arXiv/1002.2670](http://arxiv.org/abs/1002.2670).
* Cohomology theories for homotopy algebras and noncommutative geometry (with A. Lazarev). Algebr. Geom. Topol. 9 (2009), 1503--1583, [arxiv/0707.3937](http://arxiv.org/abs/0707.3937)
* Classes on compactifications of the moduli space of curves through solutions to the quantum master equation. Lett. Math. Phys. 89 (2009), no. 2, 115--130.
* Noncommutative geometry and compactifications of the moduli space of curves. Journal of Noncommutative Geometry __4__, 2, pp. 157–188, 2010, [arXiv/0710.4603](http://arxiv.org/abs/0710.4603)
* Graph cohomology classes in the Batalin-Vilkovisky formalism (with A. Lazarev). J. Geom. Phys. __59__ (2009), no. 5, 555--575, [arxiv/0701825](http://arxiv.org/abs/math/0701825)
* Characteristic classes of A-infinity algebras (with A. Lazarev). J. Homotopy Relat. Struct. __3__ (2008), no. 1, 65--111, [math.QA/0608395](http://arxiv.org/abs/math/0608395)
* Symplectic C-infinity algebras (with A. Lazarev). Mosc. Math. J. __8__ (2008), no. 3, 443--475, 615, [arxiv/0707.3951](http://arxiv.org/abs/0707.3951)
* Symplectic A-infinity algebras and string topology operations (with A. Lazarev). Amer. Math. Soc. Transl. (2), Vol. 224, 2008, 147--157, [arxiv/0707.4003](http://arxiv.org/abs/0707.4003)
* A super-analogue of Kontsevich's theorem on graph homology. Lett. Math. Phys. 76 (2006), no. 1, 37--55, [math.QA/0510390](http://arxiv.org/abs/math/0510390)
* On the classification of Moore algebras and their deformations. Homology, Homotopy Appl. __6__ (2004), no. 1, 87--107, [math.QA/0304314](http://arxiv.org/abs/math/0304314)
* Homotopy algebras and noncommutative geometry (with A. Lazarev). [math.QA/0410621](http://arxiv.org/abs/math/0410621)
On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:
* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]
category: people
[[!redirects A. Hamilton]] |
Alastair King | https://ncatlab.org/nlab/source/Alastair+King |
* [webpage](http://people.bath.ac.uk/masadk/)
## related $n$Lab entries
* [[geometric invariant theory]]
* [[McKay correspondence]]
* [[Bridgeland stability condition]]
category: people |
Alastair Wilson | https://ncatlab.org/nlab/source/Alastair+Wilson |
* [personal page](https://alastairwilson.org/)
* [institute page](https://www.birmingham.ac.uk/staff/profiles/philosophy/wilson-alastair.aspx)
## Selected writings
Relating the "[[possible worlds]]" of [[modal logic]] to the "[[many-worlds interpretation of quantum mechanics|many worlds]]" of [[interpretations of quantum mechanics|quantum philosophy]]:
* [[Alastair Wilson]], *Modal Metaphysics and the Everett Interpretation* (2006) [[philsci:2635](http://philsci-archive.pitt.edu/2635), [pdf](http://philsci-archive.pitt.edu/2635/1/modalmetaphysicsandeverett.pdf)]
* [[Alastair Wilson]], *The Nature of Contingency: Quantum Physics as Modal Realism*, Oxford University Press (2020) [[ISBN:9780198846215](https://global.oup.com/academic/product/the-nature-of-contingency-9780198846215)]
category: people |
Albanese variety | https://ncatlab.org/nlab/source/Albanese+variety |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The Albanese variety $Alb(X)$ of a [[projective variety|projective]] [[algebraic variety]] $X$ with a chosen basepoint is the universal way of turning this pointed variety into an [[abelian variety]]. Moreover, the Albanese variety of the Albanese variety is the Albanese variety. Thus, taking the Albanese variety defines an [[idempotent monad]] on the category of pointed projective algebraic varieties.
## Definition
By 'variety' let us mean a connected complete algebraic variety over an algebraically closed field. Given any variety $X$ with a chosen basepoint there is an abelian variety called the **Albanese variety** $Alb(X)$. This is defined by the following universal property: there is a map of pointed varieties called the **Albanese map**
$$i_X \colon X \to A(X)$$
such that any map of pointed varieties $f: X \to A$ where $A$ is abelian factors uniquely as $i_X$ followed by a map of abelian varieties (in particular, a group homomorphism):
$$\overline{f} \colon Alb(X) \to A.$$
That is:
$$ f = \overline{f} \circ i_X $$
This process defines a functor
$$ Alb: Var_* \to AbVar $$
from pointed varieties to abelian varieties which has a right adjoint
$$ U: AbVar \to Var_* $$
sending any abelian variety to its underlying pointed variety. The right adjoint $U$ is [[faithful functor|faithful]], but more remarkably it is also [[full functor|full]]: any basepoint-preserving map of varieties between abelian varieties is automatically a group homomorphism. (A proof of this fact is outlined in the article [[abelian variety]].) Moreover, $U$ is [[monadic functor|monadic]]. As a consequence the composite functor
$$ T = U \circ Alb $$
is an [[idempotent monad]] on $Var_*$, and its algebras are the abelian varieties.
It follows that the Albanese map $i_X \colon X \to A(X)$ is the unit of the monad $T$, and $Alb(Alb(X)) \cong Alb(X)$. For more details, see the nCafé discussion [Two miracles in algebraic geometry](https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html).
## Properties
The Albanese variety of $X$ is dual, as an abelian variety, to its [[Picard scheme| Picard variety]].
For $X$ a suitably well behaved ([[smooth variety|smooth]] [[complex variety|complex]], [[projective variety|projective]]) [[algebraic variety]] of [[dimension]] $dim(X)$, its Albanese variety is the [[intermediate Jacobian]] in degree $2 dim(X)-1$:
$$
Alb(X) \coloneqq J^{2 dim(X)-1}(X)
\,.
$$
## Related concepts
* [[Picard scheme]], [[Jacobian variety]]
## References
* [Is forming the Albanese variety a monad?](http://mathoverflow.net/questions/247104/is-forming-the-albanese-variety-a-monad), MathOverflow.
* [[Patrick Walls]], _Intermediate Jacobians and Abel-Jacobi maps_, 2012 ([[WallsJacobian.pdf:file]])
[[!redirects Albanese varieties]]
---
<http://mathoverflow.net/questions/2548/albanese-schemes-when-does-an-initial-abelian-scheme-exist-under-a-given-sch>
nLab page on [[nlab:Albanese variety]]
|
Albert algebra | https://ncatlab.org/nlab/source/Albert+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
The _octonionic Albert algebra_ is the [[Jordan algebra]] of $3$-by-$3$ [[hermitian matrices]] over the [[octonions]] $\mathbb{O}$
$$
\label{3Times3HermitianMatrix}
\mathfrak{h}_3(\mathbb{O})
\;\coloneqq\;
\left\{
\left(
\array{
(x_0 + x_1) & y & \psi_1
\\
y^\ast & (x_0 - x_1) & \psi_2
\\
\psi_1^\ast & \psi_2^\ast & \phi
}
\right)
\;|\;
\array{
x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
\\
y, \psi_1, \psi_2 \in \mathbb{O}
}
\right\}
$$
Similarly the _split-octonionic Albert algebra_ is the algebra of $3$-by-$3$ [[hermitian matrices]] over the [[split-octonions]].
The construction is due to ([Albert 1934](#Albert)), originating in an algebraic approach to [[quantum mechanics]].
## Properties
### Uniqueness
The octonionic and split-octonionic Albert algebras are (up to [[isomorphism]]) the only [[simple algebra|simple]] [[finite-dimensional space|finite-dimensional]] [[formally real algebra|formally real]] [[Jordan algebras]] over the [[real numbers]] that are not [[special Jordan algebra|special]], together comprising the _real Albert algebras_.
Their [[complexifications]] are [[isomorphism|isomorphic]], the _complex-octonionic Albert algebra_, or simply the _complex Albert algebra_. Analogues exist over any [[field]].
An _exceptional Jordan algebra_ (over any [[field]]) is any [[Jordan algebra]] in which an Albert algebra appears as a [[direct summand]]. Every formally real Jordan algebra over the real numbers is either special or exceptional (so they all have excellent self-esteem). The exceptional Jordan algebras are related to the [exceptional Lie algebras](exceptional+Lie+group#lie_algebras_2).
### Relation to 10d super-Spacetime
{#RelationTo10dSuperSpacetime}
The form of the $3 \times 3$-hermitian matrix in (eq:3Times3HermitianMatrix) makes it manifest that the exceptional Jordan algebra is naturally a [[linear map|linear]] [[direct sum]] of the form
$$
\mathfrak{h}_3(\mathbb{O})
\;\simeq_{\mathbb{R}}\;
\mathfrak{h}_2(\mathbb{O})
\oplus
\mathbb{O}^2
\oplus
\mathbb{R}
$$
via
$$
\underset{
\mathfrak{h}_3(\mathbb{O})
}{
\underbrace{
\left\{
\left(
\array{
(x_0 + x_1) & y & \psi_1
\\
y^\ast & (x_0 - x_1) & \psi_2
\\
\psi_1^\ast & \psi_2^\ast & \phi
}
\right)
\right\}
}}
\;\simeq\;
\underset{
\mathfrak{h}_2(\mathbb{O})
}{
\underbrace{
\left\{
\left(
\array{
(x_0 + x_1) & y & 0
\\
y^\ast & (x_0 - x_1) & 0
\\
0 & 0 & 0
}
\right)
\right\}
}
}
\oplus
\underset{
\mathbb{O}^2
}{
\underbrace{
\left\{
\left(
\array{
0 & 0 & \psi_1
\\
0 & 0 & \psi_2
\\
\psi_1^\ast & \psi_2^\ast & 0
}
\right)
\right\}
}
}
\oplus
\underset{
\mathbb{R}
}{
\underbrace{
\left\{
\left(
\array{
0 & 0 & 0
\\
0 & 0 & 0
\\
0 & 0 & \phi
}
\right)
\right\}
}
}
$$
with
$$
\array{
x_0, x_1, \phi \in \mathbb{R} \hookrightarrow \mathbb{O}
\\
y, \psi_1, \psi_2 \in \mathbb{O}
}$$
By the discussion at _[[geometry of physics -- supersymmetry]]_ in the section _[Real spinors in dimension 3,4,6,10](geometry+of+physics+--+supersymmetry#InTermsOfNormedDivisionAlgebraInDimension3To10)_ these summands may be further identified as follows:
* $\mathfrak{h}_2(\mathbb{O}) \simeq \mathbb{R}^{9,1}$ is the incarnation of 10-dimensional [[super-Minkowski spacetime]] via octonionic [[Pauli matrices]];
* under this identification $\mathbb{O}^2 \simeq \mathbf{16}$ is the [[Majorana-Weyl spinor]] [[real spin representation]] of the [[spin group]] $Spin(9,1)$.
$$
\mathfrak{h}_3(\mathbb{O})
\;
\simeq_{\mathbb{R}}
\;
\underset{
dim_{\mathbb{R}} = 26
}{
\underbrace{
\mathbb{R}^{9,1}
\oplus
\mathbf{16}
}}
\oplus
\mathbb{R}
\,.
$$
Under these identifications, $\phi \in \mathbb{R}$ looks like the size of $S^1/(\mathbb{Z}_2)$ in [[Horava-Witten theory]].
This decomposition hence induces an [[action]] of the [[spin group]] $Spin(9,1)$ on the exceptional Jordan algebra. While only the subgroup $Spin(9) \hookrightarrow Spin(9,1)$ of that is an [[isomorphism]] of the [[Jordan algebra]]-[[structure]] itself, the full $Spin(9,1)$-[[action]] does preserve the [[determinant]] on $\mathfrak{h}_3(\mathbb{O})$.
### Automorphisms and exceptional Lie groups
{#Automorphisms}
+-- {: .num_prop}
###### Proposition
**([[general linear group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ preserving [[determinant]] is [[E6]])**
The [[group]] of [[determinant]]-preserving [[linear map|linear]] [[isomorphisms]] of the vector space underlying the octonionic Albert algebra is the [[exceptional Lie group]] [[E6]]${}_{(-26)}$.
=--
(see e.g. ([Manogue-Dray 09](#ManogueDray09))).
+-- {: .num_prop #JordanAutomorphisms}
###### Proposition
**([[Jordan algebra]] [[automorphism group]] of $Mat_{3\times 3}^{herm}(\mathbb{O})$ is [[F4]])**
The [[group]] of [[automorphism]] with respect to the [[Jordan algebra]] structure $\circ$ on the octonionic Albert algebra is the [[exceptional Lie group]] [[F4]]:
$$
Aut\left(
Mat_{3\times 3}^{herm}(\mathbb{O}), \circ
\right)
\;\simeq\;
F_4
\,.
$$
=--
(e.g. [Yokota 09, section 2.2](#Yokota09))
+-- {: .num_prop #JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}
###### Proposition
**([[Jordan algebra]] [[automorphism group]] of $Mat_{3 \times 3}^{herm}(\mathbb{O})$ fixing an [[imaginary number|imaginary]] [[octonion]])**
Fix an [[imaginary number|imaginary]] [[octonion]] $i \in \mathbb{O}$, hence a $\mathbb{R}$-[[linear map|linear]] [[direct sum]] decomposition
$$
\mathbb{O}
\;\simeq_{\mathbb{R}}\;
\mathbb{C} \oplus V
\phantom{AA}\text{with}\phantom{AA}
V \simeq_{\mathbb{R}} \mathbb{C}^3
\,,
$$
and let
\[
\label{ComponentwiseiFixingAutomorphism}
\array{
Mat_{3 \times 3}^{herm}(\mathbb{O})
&\overset{w}{\longrightarrow}&
Mat_{3 \times 3}^{herm}(\mathbb{O})
}
\]
be given componentwise by the identity on $\mathbb{C}$ and by multiplication with some fixed non-vanishing number on $V$.
Then the [[subgroup]] of the [[Jordan algebra]] [[automorphism]] group $Aut\left(Mat_{3\times 3}^{herm}(\mathbb{O}), \circ \right) \;\simeq\; F_4$ (Prop. \ref{JordanAutomorphisms}) of elements that commute with $w$ (eq:ComponentwiseiFixingAutomorphism)
$$
F_4^w
\;\coloneqq\;
\left\{
\alpha \in F_4 \;\vert\; w \alpha = \alpha w
\right\}
$$
is
$$
F_4^w
\;\simeq\;
\big(
SU(3) \times SU(3)
\big)/ \mathbb{Z}_3
\,,
$$
where every element in the [[direct product group]] of [[SU(3)]] with itself
$$
(A, B) \in SU(3) \times SU(3)
$$
[[action|acts]] on an element
$$
\underset{
\in Mat_{3\times 3}^{herm}(\mathbb{O})
}{\underbrace{\;X\;}}
\;\simeq\;
\underset{
\in Mat_{3\times 3}^{herm}(\mathbb{C})
}{\underbrace{\;X_{\mathbb{C}}\;}}
\;+\;
\underset{
\in Mat_{3 \times 3}(\mathbb{C})
}{\underbrace{X_{V}}}
$$
via [[matrix multiplication]] as
\[
\label{MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism}
X_{\mathbb{C}}
+
X_{\mathbb{V}}
\;\mapsto\;
A X_{\mathbb{C}} A^\dagger
\;+\;
B X_{V} A^\dagger
\]
(with $(-)^\dagger$ being the [[conjugate transpose matrix]], hence the [[inverse matrix]] for the [[unitary matrices]] under consideration) and where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]]
$$
\mathbb{Z}_3
\;\subset\;
SU(3) \times SU(3)
$$
which is [[generators and relations|generated]] by the [[pair]] of [[diagonal matrices]]
\[
\label{Z3Generator}
\left(
e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
\;,\;
e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
\right)/
\;\in \;
SU(3) \times SU(3)
\,.
\]
=--
([Yokota 09, theorem 2.12.2](#Yokota09))
\begin{prpn} \label{StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra} The further [[subgroup]] of
$
F_4^w
\simeq
\big(
SU(3) \times SU(3)
\big) / \mathbb{Z}_3
\;\subset\;
F_4
$
(Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion}) which fixes a subspace
$$
Mat_{2 \times 2}^{herm}(\mathbb{C})
\;\subset\;
\underset{
Mat_{3 \times 3}^{herm}( \mathbb{O} )
}{
\underbrace{
Mat_{3 \times 3}^{herm}(\mathbb{C})
\;\oplus\;
Mat_{3 \times 3}(V)
}}
$$
(hence, by the [above](#RelationTo10dSuperSpacetime), a 4d [[Minkowski spacetime]] (incarnated via its [[Pauli matrices]]) inside the 10d [[Minkowski spacetime]] inside the octonionic Albert algebra)
is
$$
\big(
U(1)
\times
SU(2)
\times
SU(3)
\big) / \mathbb{Z}_6
\,,
$$
where the [[quotient group|quotient]] is by the [[cyclic group|cyclic]] [[subgroup]] which is [[generators and relations|generated]] by the element
\[
\left(
\exp\left(2 \pi i \tfrac{1}{6}\right)\;,\;
\exp\left(2 \pi i \tfrac{1}{2}\right) \mathbf{1}_2\;,\;
\exp\left(2 \pi i \tfrac{1}{3}\right) \mathbf{1}_3
\right)
\;\in\;
U(1) \times SU(2) \times SU(3)
\,.
\]
(Hence this group happens to coincide with the _exact_ [[gauge group]] of the [[standard model of particle physics]], see [there](standard+model+of+particle+physics#GaugeGroup)).
\end{prpn}
This was claimed without proof in [Dubois-Violette & Todorov 18](#DuboisVioletteTodorov18). See also [Krasnov 19](#Krasnov19).
\begin{proof} By Prop. \ref{JordanAlgebraAutomorphismsFixingAnImaginaryOctonion} (eq:MatrixMultiplicationRepresentationOfiFixingJordanAutomorphism) it is clear that the subgroup in question is that represented by those [[pairs]] $(A,B) \in SU(3) \times SU(3)$ for which $A$ is $(1 + 2)$-block diagonal. Such matrices $A$ form the [[subgroup]] of [[SU(3)]] of [[matrices]] that may be written in the form
$$
diag\left(
c^2, c^{-1} \mathbf{\sigma}
\right)
$$
for $c \in U(1)$ and $\mathbf{\sigma} \in $ [[SU(2)]]. The [[kernel]] of the [[group homomorphism]]
\[
\label{IdentifyingU1SU2inSU3}
\array{
U(1) \times SU(2) &\longrightarrow& SU(3)
\\
(c,\mathbf{\sigma}) &\mapsto& diag\left(
c^{2}, c^{-1} \mathbf{\sigma}
\right)
}
\]
is clearly the [[cyclic group]]
\[
\label{Z2Generator}
\left\{
(1,\mathbf{1}_2)\;,\;
\left(
e^{2\pi i \tfrac{1}{2}},e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2
\right)
\right\}
\;\simeq\;
\mathbb{Z}_2
\,.
\]
Hence the subgroup in question is
$$
\begin{aligned}
\Big(
\big( U(1) \times SU(2) \big)/ \mathbb{Z}_2
\;\times\;
SU(3)
\Big)/ \mathbb{Z}_3
& \simeq
\Big(
\Big(
\big( U(1) \times SU(2) \big)
\;\times\;
SU(3)
\Big) / \mathbb{Z}_2
\Big) / \mathbb{Z}_3
\\
&\simeq
\Big(
\big( U(1) \times SU(2) \big)
\;\times\;
SU(3)
\Big) / \mathbb{Z}_6
\,,
\end{aligned}
$$
where in the first step we extended the $\mathbb{Z}_2$-[[action]] as the [[trivial action]] on the $SU(3)$-factor, and in the second step we used the evident [[isomorphism]] $\mathbb{Z}_2 \times \mathbb{Z}_3 \simeq \mathbb{Z}_6$ (an application of the "[[fundamental theorem of cyclic groups]]", if you wish).
It remains to see that the [[action]] of $\mathbb{Z}_6$ is as claimed. By the above identification $\mathbb{Z}_6 \simeq \mathbb{Z}_2 \times \mathbb{Z}_3$, it is generated by the _joint_ action of that of the generators of $\mathbb{Z}_3$ and of $\mathbb{Z}_2$, which, by (eq:Z3Generator) and (eq:Z2Generator), is
$$
\underset{
\text{generator of}\, \mathbb{Z}_3
}{
\underbrace{
\Big(
e^{2\pi i \tfrac{1}{3}} \;,\;
e^{2\pi i \tfrac{1}{3}} \mathbf{1}_2\;,\;
e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
\Big)
}
}
\underset{
\text{generator of} \, \mathbb{Z}_2
}{
\underbrace{
\Big(
1 , (e^{2 \pi i \tfrac{1}{2}}) (e^{2 \pi i \tfrac{1}{2}}\mathbf{1}_2), \mathbf{1}_3
\Big)
}
}
\;=\;
\left(
e^{2\pi i \tfrac{1}{3}} \;,\;
\underset{
= e^{2\pi i \tfrac{-1}{6}}
}{
\underbrace{
e^{2\pi i \tfrac{1}{2}} e^{2\pi i \tfrac{1}{3}}
}}
\;
( e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2)
\;,\;
e^{2\pi i \tfrac{1}{3}} \mathbf{1}_3
\right)
$$
as an element in $SU(3) \times SU(3)$, hence is
$$
\Big(
e^{2\pi i \tfrac{1}{6}} \;,\;
e^{2\pi i \tfrac{1}{2}}\mathbf{1}_2 \;,\;
e^{2\pi i \tfrac{1}{3}}\mathbf{1}_3 \;,\;
\Big)
\;\in\;
U(1) \times SU(2) \times SU(3)
$$
under the lift through (eq:IdentifyingU1SU2inSU3).
\end{proof}
## Related concepts
* [[Kantor-Koecher-Tits construction]] $\rightarrow$ [[E7]]
## References
### General
The original article is
* {#Albert} [[Abraham Adrian Albert]], _On a Certain Algebra of Quantum Mechanics_, Annals of Mathematics, Second Series 35 (1): 65–73, (1934)(doi:[10.2307/1968118](http://dx.doi.org/10.2307/1968118), [JSTOR](https://www.jstor.org/stable/1968118)).
A textbook account is in
* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 5 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000
Further discussion:
* {#Baez02} [[John Baez]], [section 3.4](http://math.ucr.edu/home/baez/octonions/node12.html) _$\mathbb{O}P^2$ and the Exceptional Jordan Algebra_ of _The Octonions_, Bull. Amer. Math. Soc. 39 (2002), 145-205. ([web](http://math.ucr.edu/home/baez/octonions/octonions.html))
* {#Yokota09} Ichiro Yokota, _Exceptional Lie groups_ ([arXiv:0902.0431](https://arxiv.org/abs/0902.0431))
See also
* Wikipedia, _[Albert algebra](https://en.wikipedia.org/wiki/Albert_algebra)_
### Possible relation to color gauge structure
{#ReferencesRelationStandardModel}
Attempts to identify aspects of the [[color charge|color]] [gauge group](standard+model+of+particle+physics#GaugeGroup) of the [[standard model of particle physics]] within the exceptional Jordan algebra:
* {#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))
* {#DuboisVioletteTodorov18} Michel Dubois-Violette, Ivan Todorov, _Exceptional quantum geometry and particle physics II_ ([arXiv:1808.08110](https://arxiv.org/abs/1808.08110))
* Ivan Todorov, _Exceptional quantum algebra for the standard model of particle physics_ ([arXiv:1911.13124](https://arxiv.org/abs/1911.13124))
* {#Krasnov19} [[Kirill Krasnov]], _$SO(9)$ characterisation of the Standard Model gauge group_ ([arXiv:1912.11282](https://arxiv.org/abs/1912.11282))
* Latham Boyle, _The Standard Model, The Exceptional Jordan Algebra, and Triality_ ([arXiv:2006.16265](https://arxiv.org/abs/2006.16265))
[[!redirects exceptional Jordan algebra]]
[[!redirects exceptional Jordan algebras]]
[[!redirects Albert algebra]]
[[!redirects Albert algebras]]
[[!redirects real Albert algebra]]
[[!redirects real Albert algebras]]
[[!redirects octonionic Albert algebra]]
[[!redirects octonionic Albert algebras]]
[[!redirects split-octonionic Albert algebra]]
[[!redirects split-octonionic Albert algebras]]
[[!redirects split octonionic Albert algebra]]
[[!redirects split octonionic Albert algebras]]
[[!redirects complex Albert algebra]]
[[!redirects complex Albert algebras]]
[[!redirects complexified Albert algebra]]
[[!redirects complexified Albert algebras]]
[[!redirects complex-octonionic Albert algebra]]
[[!redirects complex-octonionic Albert algebras]]
[[!redirects complex octonionic Albert algebra]]
[[!redirects complex octonionic Albert algebras]]
|
Albert Blakers | https://ncatlab.org/nlab/source/Albert+Blakers |
* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=24490)
## related $n$Lab entries
* [[Blakers-Massey theorem]]
category: people
[[!redirects Albert L. Blakers]]
|
Albert Burroni | https://ncatlab.org/nlab/source/Albert+Burroni | Albert Burroni is a French mathematician based in the University of Paris 7. He has been very influential in applying categorical methods to certain classes of logical structures.
* [Home page](http://www.pps.univ-paris-diderot.fr/~burroni/)
* Article by [[Pierre Ageron]] on [Albert Burroni dans l'école d'Ehresmann. Constructivisme et structuralisme](http://www.math.unicaen.fr/~ageron/histoire-philo.html)in : Actes de la journée mathématique en l'honnneur d'Albert Burroni : catégories, théories algébriques et informatique, Institut de mathématiques de Jussieu (2004) 11-24.
## Selected writings
{#SelectedWritings}
On (among other things) [[virtual double categories]]:
* [[Albert Burroni]], _$T$-catégories (catégories dans un triple)_, Cahiers de topologie et géométrie différentielle catégoriques **12**.3 (1971) 215-321 [[dml:91097](https://eudml.org/doc/91097), [pdf](http://www.numdam.org/article/CTGDC_1971__12_3_215_0.pdf)]
category:people
[[!redirects Burroni]] |
Albert Einstein | https://ncatlab.org/nlab/source/Albert+Einstein |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Albert_Einstein)
* Jeroen van Dongen, *Einstein's Unification*, Cambridge University Press (2013) [[doi:10.1017/CBO9780511781377](https://doi.org/10.1017/CBO9780511781377)]
## Writings and References
{#References}
### The 1919 eclipse
On the (eventual) [[experiment|experimental]] confirmation of [[general relativity]]:
* {#Coles01} Peter Coles, _Einstein, Eddington, and the 1919 Eclipse_ ([arXiv:astro-ph/0102462](https://arxiv.org/abs/astro-ph/0102462))
* Gerard Gilmore, Gudrun Tausch-Pebody, _The 1919 eclipse results which verified General Relativity and their later detractors: a story re-told_ ([arXiv:2010.13744](https://arxiv.org/abs/2010.13744))
### Gravitational waves
The first article that _correctly_ derived [[gravitational waves]] from the [[Einstein equations]] is
* {#Einstein18} [[Albert Einstein]], _Über Gravitationswellen_, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften Berlin (1918), 154-167
In particular this correctly stated that gravitational waves require a [[quadrupole moment]] as a [[source]] (e.g. a rotating binary star system) and not just a [[dipole moment]] (e.g. an oscillating charge) as for [[electromagnetic waves]] (the [[graviton]] has [[spin]] 2, the [[photon]] has spin 1...), thereby correcting a mistake to this effect in the earlier article
* {#Einstein16} [[Albert Einstein]], _Näherungsweise Integration der Feldgleichungen der Gravitation_, Sitzung der physikalisch-mathematischen Klasse vom 22. Juni 1916 ([web](http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/BGG54UCY/index.meta&ww=0.7142&wh=0.7142&wx=0.2315))
The reality of gravitational wave solutions however kept being a cause of concern for many years (Einstein himself was concerned that the linearization approximation used in their derivation might have been too coarse), for a brief account of the early history see
* {#Steinicke05} [[Wolfgang Steinicke]], _Einstein and the Gravitational waves_, Astron. Nachr. / AN 326 (2005), No. 7 – Short Contributions AG 2005 Köln ([pdf](http://www.hs.uni-hamburg.de/DE/GNT/events/pdf/steinicke05.pdf))
* [[Sean Carroll]], _[Einstein vs. Physical Review](http://www.preposterousuniverse.com/blog/2005/09/16/einstein-vs-physical-review/)_, 2005
### Wormholes
Introducing the idea of [[spacetime]] [[wormholes]] in [[gravity]] ([[Einstein-Rosen bridges]]):
* [[Albert Einstein]], [[Nathan Rosen]], *The Particle Problem in the General Theory of Relativity*, Phys. Rev. **48** 73 (1935) ([doi:10.1103/PhysRev.48.73](https://doi.org/10.1103/PhysRev.48.73))
### Entanglement
Introducing the [[Einstein-Podolsky-Rosen paradox]] concerning [[quantum entanglement]]:
* {#EPR} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review 47 (10): 777–780. (1935) ([doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777))
### Philosophy of science
* [[Albert Einstein]], *Physik und Realität*, Journal of The Franklin Institute **221** 3 (1936) [[pdf](https://informationphilosopher.com/solutions/scientists/einstein/Physik_und_Realitat.pdf), [[Einstein-PhysikRealitate.pdf:file]]]
On the assumption of a [[spacetime]] [[continuum]] in view of [[quantum physics]] ([[quantum gravity]]):
> Es ist allerdings darauf hingewiesen worden, dass bereits
die Einführung eines raum-zeitlichen Kontinuums angesichts der molekularen Struktur allen Geschehens im Kleinen möglicherweise als naturwidrig anzusehen sei. Vielleicht weise der Erfolg von Heisenbergs Methode auf eine rein algebraische Methode der Naturbeschreibung, auf die Ausschaltung kontinuierlicher Funktionen aus der Physik hin. Dann aber muss auch auf die Verwendung des Raum-Zeit Kontinuums prinzipiell verzichtet werden. Es ist nicht undenkbar, dass der menschliche Scharfsinn einst Methoden finden wird, welche die Beschreitung dieses Weges möglich machen. Einstweilen aber erscheint dieses Projekt ähnlich wie der Versuch, in einem luftleeren Raum zu atmen.
> To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an attempt to breathe in empty space.
## Further selected writings
* [[Albert Einstein]], *Why Socialism?*, Monthly Review (1949) [[web version](https://monthlyreview.org/2009/05/01/why-socialism/), [Wikipedia entry](https://en.wikipedia.org/wiki/Why_Socialism%3F)]
## Quotes
In his Spencer lecture, delivered at Oxford in 1933, Einstein stressed the importance to be accorded to formal beauty:
> Experience can of course guide us in our choice of serviceable mathematical concepts; it cannot possibly be the source from which they are derived; experience of course remains the sole criterion of the serviceability of a mathematical construction for physics, but the truly creative principle resides in mathematics.
Concerning what are now called [[Einstein equations]] ([[equations of motion]] for [[Einstein gravity]]):
> I have learned something else from the theory of gravitation: No ever so inclusive collection of empirical facts can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the setting up of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or [at least] almost completely.
(From A. Einstein, *Autobiographical Notes*, as translated in P. A. Schilpp: *Albert Einstein -- Philosopher-Scientist*, MJF books, New York (1949, 1951, 1969, 1970), p. 89, [scan](https://ncatlab.org/nlab/files/Einstein-AutobioNotes-from-Schilpp-p88.jpg))
category: people
[[!redirects Einstein]]
|
Albert Georg Passegger | https://ncatlab.org/nlab/source/Albert+Georg+Passegger | category: people |
Albert Lautman | https://ncatlab.org/nlab/source/Albert+Lautman |
__Albert Lautman__ was a French [[philosophy of mathematics|mathematical philosopher]].
He was interested in the structure of advanced mathematics and its creativity and critiziced the [[analytic philosophy|analytic philosophers]] like [[Bertrand Russell|Russell]] and [[Gottlob Frege|Frege]] from the early 20th century who dealt mainly with the issues of a particular logical [[foundation]] and formal aspects and not much on the nature of doing mathematics and its meaning.
As a member of the Resistance, a former prisoner of war and also of Jewish origin, he was killed by a German squad on August 1, 1944.
He influenced the French philosophers [[Gilles Deleuze]] and [[Alain Badiou]], the mathematician and historian of culture and mathematical philosophy [[Fernando Zalamea]], the mathematician, semiolinguist, and philosopher (of science) [[Jean Petitot]], and the philosopher [[David Corfield]].
According to
* Jean Petitot, _Refaire le « Timée » : Introduction à la philosophie mathématique d'Albert Lautman_, 1987
>Although studied very little, Albert Lautman has already been labelled a neo-platonist. Regarded as too speculative despite his exceptional mathematical erudition and his close relationship to Hilbert's axiomatic structuralism, his philosophy of mathematics has not been a subject of specific attention until now. And yet it is, in our opinion, of noteworthy importance. In separating from mathematical theories an additional level of reality lying above, level made up of dialectic- problematic ideas whose understanding is equivalent to the genesis of real theories in which they are determined and achieved, this philosophy of mathematics allows a (transcendental) doctrine of relationships between mathematics and reality to be developed, which goes beyond the dogmatism of logical empiricism without ending up in post-positivistic skepticisms for all that and which articulates the indefinite evolution of the autonomisation and unification of mathematics toward the indefinite production of scientific ontogenèses.
>...
>We believe Albert Lautman is, unemphatically, one of the most inspired philosophers of the century. His theses are of real importance and if we would devote to him only a fraction of the thoughts which we have devoted to another philosopher, who is comparable in stature and opposed in ideas, namely Wittgenstein, he would undoubtedly become one of the most glorious figures of our modernity.
### Lautman on effective and advanced mathematics
According to
* Fernando Zalamea, _Albert Lautman and the Creative
Dialectic of Modern Mathematics_
> With the term 'effective mathematics', Lautman tackles the theories, structures and constructions conceived in the very activity of the mathematician. The term refers to the structure of mathematical knowledge, and what is effective refers to the concrete _action_ of the mathematician to gradually build the mathematical edifice, that such action is constructivist or existential. The mathematical – beyond its ideal set theoretical reconstruction – develops along a hierarchy of real configurations of rather diverse complexity, in which the concepts and examples are connected through structural processes of liberation and saturation, resulting in mathematical creations like mixes between opposite polarities. Lautman detects some _specific_ features of advanced mathematics that are not given in elementary mathematics:
> a) the _complex hierarchisation_ of various theories, irreducible to systems of _intermediate_ deduction;
> b) the _richness_ of the models, irreducible to linguistic manipulation;
> c) the _unity_ of structural methods and of conceptual polarities, beyond the effective multiplicity of models;
> d) the _dynamics_ of the creative activity, in a permanent back-and-forth between freedom and saturation, open to the Platonic division and the Platonic dialectic;
> e) the _mathematically demonstrable relation_ between what is multiple on a given level and what is singular on another, through a sophisticated lattice of mixed ascents and descents.
###Lautman on the abstract world of mathematics and the sensible world of physics as both participating in the same dialectical structure
According to
* Lautman, _Mathematics, Ideas, and the Physical Real_, p231:
>In seeing the sensible thus defined by a mixture of symmetry and dissymmetry, of identity and difference, it is impossible not to recall Plato's [[Timaeus]] (1997). The existence of bodies is based there on the existence of this receptacle that Plato calls the place and whose function consists, as Rivaud has shown in the preface to his edition of the Timaeus (Plato 1932), in making possible the multiplicity of bodies and their alternation in a sin- gle place in the sensible world, just as the role of the Idea of the Other in the intelligible world is to ensure, by its mixture with the Same, both the connection and the separation of types. This reference to Plato enables the understanding that the materials of which the universe is formed are not so much the atoms and molecules of the physical theory as these great pairs of ideal opposites such as the Same and the Other, the Symmetrical and Dissymmetrical, related to one another according to the laws of a harmonious mixture. Plato also suggests more. The properties of place and matter, according to him, are not purely sensible, they are, as Rivaud goes on to say, the geometric and physical transposition of a dialectical theory. It is also possible that the distinction between left and right, as observed in the sensible world, is only the transposition on the plane of experience of a dissymmetrical symmetry which is equally constitutive of the abstract reality of mathematics. A common participation in the same dialectical structure would thus bring to the fore an analogy between the structure of the sensible world and that of mathematics, and would allow a better understanding of how these two realities accord with one another.
### Literature
* wikipedia: [en](http://en.wikipedia.org/wiki/Albert_Lautman)
His collected works have appeared now also in English:
* Albert Lautman, _Mathematics, ideas and the physical real_, collected works, 2011
Jean Petitot wrote an introduction to his philosophy:
* Jean Petitot, [Refaire le « Timée » : Introduction à la philosophie mathématique d'Albert Lautman](http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1987_num_40_1_4488), 1987
An edition of the French-language journal _Philosophiques_ was dedicated to Lautman ([2010, vol. 37 no. 1](http://www.er.uqam.ca/nobel/philuqam/philosophiques/index.php?section=sommaire_par_numeros&vol=37&no=1)). For an English version of one these articles see
* David Corfield, [Lautman and the Reality of Mathematics](http://philsci-archive.pitt.edu/9210/)
category: people, philosophy
[[!redirects Albert Lautman]]
[[!redirects A. Lautman]] |
Albert Nijenhuis | https://ncatlab.org/nlab/source/Albert+Nijenhuis | Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a mathematician at the University of Pennsylvania.
He got his PhD degree in 1952 from the University of Amsterdam, advised by [[Jan Schouten]].
He introduced the notion of a [[natural bundles]] and [[natural operation]] in [[differential geometry]], which was subsequently developed by others.
## Selected writings
On the [[Schouten-Nijenhuis bracket]]:
* [[Albert Nijenhuis]], _Jacobi-type identities for bilinear differential concomitants of certain tensor fields I_, Indagationes Mathematicae **17** (1955) 390–403 [[doi:10.1016/S1385-7258(55)50054-0](https://doi.org/10.1016/S1385-7258(55)50054-0)]
Introducing the notion of [[natural operations]] in [[differential geometry]]:
* [[Albert Nijenhuis]], _Geometric aspects of formal differential operations on tensor fields_, Proceedings of the International Congress of Mathematicians 1958, 463–469.
* [[Albert Nijenhuis]], _Natural bundles and their general properties_, in: _Differential Geometry (in honor of Kentaro Yano)_, Kinokuniya, Tokyo, 1972, pp. 317–334.
Introducing the [[Frölicher-Nijenhuis bracket]]:
* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) **59** (1956) 338–350 and 351–359 [<a href="https://doi.org/10.1016/s1385-7258(56)50046-7">doi:10.1016/s1385-7258(56)50046-7</a>, <a href="https://doi.org/10.1016/s1385-7258(56)50047-9">doI;10.1016/s1385-7258(56)50047-9</a>]
and its refinement for [[almost complex structures]]:
* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms. Part II. Almost-complex structures_, Indagationes Mathematicae (Proceedings) **61** (1958) 414–421 and 422-429 [<a href="https://doi.org/10.1016/S1385-7258(58)50058-4">doi:10.1016/S1385-7258(58)50058-4</a>, <a href="https://doi.org/10.1016/S1385-7258(58)50057-2">doi:10.1016/S1385-7258(58)50057-2</a>]
Giving an explicit formula for what came to be known as the [[Nijenhuis–Richardson bracket]]:
* {#NijenhuisRichardson64} [[Albert Nijenhuis]], [[Roger W. Richardson]], *Cohomology and deformations of algebraic structures*, Bulletin of the American Mathematical Society **70** 3 (1964) 406–412 [[doi:10.1090/s0002-9904-1964-11117-4](http://dx.doi.org/10.1090/s0002-9904-1964-11117-4)]
* [[Albert Nijenhuis]], [[R. W. Richardson]], *Deformations of Lie Algebra Structures*, Journal of Mathematics and Mechanics **17** 1 (1967) 89-105 [[jstor:24902154](https://www.jstor.org/stable/24902154)]
Beware that this other article has a similar title to [Nijenhuis & Richardson 1964](#NijenhuisRichardson64), but does not seem to mention the [[Nijenhuis–Richardson bracket]]:
* [[Albert Nijenhuis]], [[R. W. Richardson]], *Cohomology and deformations in graded Lie algebras*, Bull. Amer. Math. Soc. **72** (1966) 1-29 [[doi:10.1090/S0002-9904-1966-11401-5](https://doi.org/10.1090/S0002-9904-1966-11401-5)]
## Related entries
* [[Nijenhuis tensor]]
* [[Schouten–Nijenhuis bracket]]
* [[Frölicher–Nijenhuis bracket]]
* [[Nijenhuis–Richardson bracket]]
category: people |
Albert R. Hibbs | https://ncatlab.org/nlab/source/Albert+R.+Hibbs |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Albert_Hibbs)
## Selected writings
Introducing [[path integral quantization]]:
* [[Richard Feynman]], [[Albert R. Hibbs]], _Quantum Mechanics and Path Integrals_, New York: McGraw-Hill (1965)
* [[Daniel F. Styer]], [[Richard Feynman]], [[Albert R. Hibbs]], *Quantum Mechanics and Path Integrals: Emended Edition*, Dover (2010) [[ISBN:0486477223](https://store.doverpublications.com/0486477223.html), [pdf](http://www-f1.ijs.si/~ramsak/km1/FeynmanHibbs.pdf)]
category: people
[[!redirects Albert Hibbs]] |
Albert Schwartz > history | https://ncatlab.org/nlab/source/Albert+Schwartz+%3E+history |
> see _[[Albert Schwarz]]_
|
Albert Schwarz | https://ncatlab.org/nlab/source/Albert+Schwarz | __Albert Schwarz__ is a mathematician and a theoretical physicist born in Soviet Union and now Professor at University of California-Davis ([web](http://www.math.ucdavis.edu/~schwarz)). He was one of the pioneers of [[Morse theory]] and brought up a first example of a [[topological quantum field theory]]. Schwarz worked on some examples in [[noncommutative geometry]]. He is "S" of the famous [[AKSZ model]].
* German Wikipedia, _[Albert S. Schwarz](http://de.wikipedia.org/wiki/Albert_S._Schwarz)_
* {#Schwarz04} [[Albert Schwarz]], _My Life In Science_, 2004 ([pdf](https://www.math.ucdavis.edu/~schwarz/bion.pdf), [[AlbertSchwarzLifeInScience.pdf:file]])
## Selected writings
(See also the [list of arXiv articles](http://arxiv.org/find/hep-th/1/au:+Schwarz_A/0/1/0/all/0/1) of A. Schwarz.)
On [[noncommutative tori]]:
* [[Marc Rieffel]], [[Albert Schwarz]], _Morita equivalence of multidimensional noncommutative tori_, Int. J. Math. 10 (1999) 289-299 ([arXiv:math/9803057](https://arxiv.org/abs/math/9803057))
On [[Morita equivalence]] and [[duality in physics]]/[[duality in string theory]]:
* {#Schwarz98} [[Albert Schwarz]], _Morita equivalence and duality_ ([arXiv:hep-th/9805034](http://arxiv.org/abs/hep-th/9805034))
and specifically in relation to [[T-duality]]:
* {#Pioline99} B. Pioline, [[Albert Schwarz]], _Morita equivalence and T-duality (or $B$ versus $\Theta$)_ ([arXiv:hep-th/9908019](http://arxiv.org/abs/hep-th/9908019))
On [[partition functions]]
* [[Albert Schwarz]] _The partition function of a degenerate functional_, Commun. Math. Phys. __67__, 1 (1979)
Introducing the [[AKSZ sigma-model]]:
* {#AKSZ} M. Alexandrov, [[Maxim Kontsevich]], [[Albert Schwarz]], [[Oleg Zaboronsky]], _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))
On [[supergeometry]] as taking place over the [[base topos]] on the [[site]] of [[super points]]
* [[Albert Schwarz]], _On the definition of superspace_, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, ([russian original pdf](http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf))
* Anatoly Konechny and [[Albert Schwarz]],
_On $(k \oplus l|q)$-dimensional supermanifolds_, in: [[Julius Wess]], V. Akulov (eds.) _Supersymmetry and Quantum Field Theory_ (D. Volkov memorial volume) Springer-Verlag, 1998 ,
Lecture Notes in Physics, 509 ([arXiv:hep-th/9706003](http://arxiv.org/abs/hep-th/9706003))
_Theory of $(k \oplus l|q)$-dimensional supermanifolds_ Sel. math., New ser. 6 (2000) 471 - 486
{#KonechnySchwarz98}
* [[Albert Schwarz]], I- Shapiro, _Supergeometry and Arithmetic Geometry_ ([arXiv:hep-th/0605119](http://arxiv.org/abs/hep-th/0605119))
On the [[Lie algebra cohomology]] of the [[super Poincaré Lie algebra]] ([[brane scan]] of [[Green-Schwarz sigma-models]]):
* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries_ ([arXiv:1011.4731](http://arxiv.org/abs/1011.4731))
* [[Mikhail Movshev]], [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra_, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 ([arXiv:1106.0335](http://arxiv.org/abs/1106.0335))
On [[D=10 super Yang-Mills theory]]:
* [[Mikhail Movshev]], [[Albert Schwarz]], _On maximally supersymmetric Yang-Mills theories_, Nucl.Phys. B681 (2004) 324-350 ([arXiv:hep-th/0311132](https://arxiv.org/abs/hep-th/0311132))
On [[supersymmetry]] and [[equivariant localization]]:
* [[Albert Schwarz]], [[Oleg Zaboronsky]], _Supersymmetry and localization_, Comm. Math. Phys. __183__, 2 (1997), 463-476 ([euclid:cmp/1158328185](http://projecteuclid.org/euclid.cmp/1158328185))
On [[BV-formalism]]
* _Geometry of [[Batalin-Vilkovisky quantization]]_, Commun. Math. Phys. __155__, 249 (1993), [euclid](http://projecteuclid.org/euclid.cmp/1104253279)
On the [[semiclassical approximation]] in [[BV-formalism]]:
* _Semiclassical approximation in Batalin-Vilkovisky formalism_, Comm. Math. Phys. __158__ (1993), no. 2, 373--396, [euclid](http://projecteuclid.org/euclid.cmp/1104254246)
On [[quantum field theory]] and [[topology]]:
* monograph: *Quantum field theory and topology*, Grundlehren der Math. Wissen. __307__, Springer 1993. (translated from Russian original Kvantovaja teorija polja i topologija, *Nauka*, Moscow, 1989. 400 pp.)
* scientific reminiscences, [pdf](http://www.math.ucdavis.edu/~schwarz/bion.pdf)
* V. Kac, A. Schwarz, _Geometric interpretation of the partition function of $2$D gravity_, Phys. Lett. B __257__ (1991), no. 3-4, 329--334, [doi](http://dx.doi.org/10.1142/S0217732391000634)
* A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, _Pseudoparticle solutions of the Yang-Mills equations_, Phys. Lett. B __59__ (1975), no. 1, 85--87, http://dx.doi.org/10.1016/0370-2693(75)90163-X
* S. N. Dolgikh, A. A. Rosly, A. S. Schwarz, _Supermoduli spaces_, Comm. Math. Phys. 135 (1990), no. 1, 91--100, [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1104201921)
* V. N. Romanov, A. S. Švarc, _Anomalies and elliptic operators_, (Russian) Teoret. Mat. Fiz. __41__ (1979), no. 2, 190--204.
* Математические основы квантовой теории поля, Atomizdat, Moscow, 1975. 368 pp.
On 1-[[twisted de Rham cohomology]]:
* [[Albert Schwarz]], [[Ilya Shapiro]], *Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"*, Nucl. Phys. B **809** (2009) 547-560 [[arXiv:0809.0086](https://arxiv.org/abs/0809.0086), [[doi:10.1016/j.nuclphysb.2008.10.005](https://doi.org/10.1016/j.nuclphysb.2008.10.005)]
On [[quantum mechanics]] and [[quantum field theory]]:
* Igor Frolov, [[Albert Schwarz]], *Quantum mechanics and quantum field theory. Algebraic and geometric approaches*, lecture notes [[arXiv:2301.03804](https://arxiv.org/abs/2301.03804)]
On [[super Riemann surfaces]] and [[fat graphs]]:
* [[Albert S. Schwarz]], [[Anton M. Zeitlin]], *Super Riemann surfaces and fatgraphs* [[arXiv:2307.02706](https://arxiv.org/abs/2307.02706)]
## Related entries
* [[supergeometry]]
* [[10d super Yang-Mills theory]], [[IKKT matrix model]]
category: people
[[!redirects Albert S. Schwarz]]
|
Alberto Cattaneo | https://ncatlab.org/nlab/source/Alberto+Cattaneo | * [homepage](http://user.math.uzh.ch/cattaneo)
* publication list: [pdf](http://user.math.uzh.ch/cattaneo/allpub.pdf)
## Selected writings
On [[3-manifolds]]:
* {#BottCattaneo98} [[Raoul Bott]], [[Alberto Cattaneo]], _Integral Invariants of 3-Manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))
On [[deformation quantization]]:
* [[Alberto Cattaneo]], _From topological field theory to deformation quantization and reduction_, ICM 2006. ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/icm.pdf))
On higher order [[Vassiliev invariants]] as [[Chern-Simons theory]]-[[correlators]], hence as [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Chern-Simons propagators]] assigned to [[edges]] of [[Feynman diagrams]] in the [[graph complex]]:
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))
* [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218))
On the [[Poisson sigma-model]]:
* {#CattaneoFelder} [[Alberto Cattaneo]], [[Giovanni Felder]], _A path integral approach to the Kontsevich quantization formula_, Commun. Math. Phys. 212, 591--611 (2000) [doi](http://dx.doi.org/10.1007/s002200000229), [math.QA/9902090](http://arxiv.org/abs/math/9902090).
* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and deformation quantization_, Mod. Phys. Lett. A 16, 179--190 (2001) [hep-th/0102208](http://arxiv.org/abs/hep-th/0102208)
* [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and symplectic groupoids_ , (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61--93 (Birkhäuser, 2001)
[math.SG/0003023](http://arxiv.org/abs/math/0003023).
* [[Alberto Cattaneo]], [[Giovanni Felder]], _On the AKSZ formulation of the Poisson sigma model_, Lett. Math. Phys. 56, 163--179 (2001) [math.QA/0102108](http://arxiv.org/abs/math/0102108).
On [[BV-BRST formalism]] compatible with decomposing [[spacetime]] into pieces ([[BV-BFV formalism]]):
* {#CattaneoMnevReshetikhin12} [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical BV theories on manifolds with boundary_, [arXiv:1201.0290](http://arxiv.org/abs/1201.0290); _Classical and quantum Lagrangian field theories with boundary_, [arXiv:1207.0239](http://arxiv.org/abs/1207.0239); _Perturbative quantum gauge theories on manifolds with boundary_, [arxiv/1507.01221](http://arxiv.org/abs/1507.01221)
* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]
On the local formulation (allowing decomposition and re-gluing of [[spacetime]]) of the [[phase space]] of [[Einstein gravity]] in [[first-order formulation of gravity|first-order formulation]] via [[BV-BFV formalism]]:
* {#CattaneoSchiavina15} [[Alberto Cattaneo]], [[Michele Schiavina]], _BV-BFV approach to General Relativity, Einstein-Hilbert action_, J. Math. Phys. **57** 023515 (2016) [[arXiv:1509.05762](https://arxiv.org/abs/1509.05762), [doi:10.1063/1.4941410](https://doi.org/10.1063/1.4941410)]
* {#CattaneoSchiavina17a} [[Alberto Cattaneo]], [[Michele Schiavina]], _The reduced phase space of Palatini-Cartan-Holst theory_, Ann. Henri Poincaré **20** (2019) 445 [[arXiv:1707.05351](https://arxiv.org/abs/1707.05351), [doi:10.1007/s00023-018-0733-z](https://doi.org/10.1007/s00023-018-0733-z)]
* {#CattaneoSchiavina17b} [[Alberto Cattaneo]], [[Michele Schiavina]], *BV-BFV approach to General Relativity: Palatini-Cartan-Holst action*, Adv. Theor. Math. Phys. **23** (2019) 1801-1835 [[arXiv:1707.06328](https://arxiv.org/abs/1707.06328), [doi:10.4310/ATMP.2019.v23.n8.a3](https://doi.org/10.4310/ATMP.2019.v23.n8.a3)]
* [[Alberto S. Cattaneo]], *Phase space for gravity with boundaries*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* (2023) [[arXiv:2307.04666](https://arxiv.org/abs/2307.04666)]
and for flux observables:
* [[Alberto S. Cattaneo]], Alejandro Perez, *A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity*, Class. Quant. Grav. **34** (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), [doi:10.1088/1361-6382/aa69b4](https://doi.org/10.1088/1361-6382/aa69b4)]
and relation to [[BF-theory]]:
* [[Alberto S. Cattaneo]], [[Leon Menger]], [[Michele Schiavina]], *Gravity with torsion as deformed BF theory* [[arXiv:2310.01877](https://arxiv.org/abs/2310.01877)]
On the [[relativistic particle]], and related systems, in [[BV-BRST formalism]]:
* [[Alberto Cattaneo]], [[Michele Schiavina]], _On time_, Lett. Math. Phys. **107** (2017) 375-408 [[doi:10.1007/s11005-016-0907-x](https://doi.org/10.1007/s11005-016-0907-x), [arXiv:1607.02412](http://arxiv.org/abs/1607.02412)]
## Related entries
* [[Poisson sigma-model]]
* [[deformation quantization]]
* [[BV-quantization]]
* [[first-order formulation of gravity]]
[[!redirects A. Cattaneo]]
[[!redirects A. S. Cattaneo]]
[[!redirects Alberto S. Cattaneo]]
[[!redirects Cattaneo]]
category: people |
Alberto Elduque | https://ncatlab.org/nlab/source/Alberto+Elduque |
* [personal page](https://personal.unizar.es/elduque/)
* [GoogleScholar page](https://scholar.google.com/citations?user=zrIPjEwAAAAJ&hl=en)
## Selected writings
On [[composition algebras]]:
* {#Elduque21} [[Alberto Elduque]], *Composition algebras*, Chapter 2 in: Abdenacer Makhlouf (ed.), *Algebra and Applications I: Non-associative Algebras and Categories*, Sciences-Mathematics, ISTEWiley (2021) 27-57 [[arXiv:1810.09979](https://arxiv.org/abs/1810.09979), [ISBN:978-1-789-45017-0](https://www.wiley.com/en-ae/Algebra+and+Applications+1:+Non+associative+Algebras+and+Categories-p-9781789450170)]
category: people |
Alberto Facchini | https://ncatlab.org/nlab/source/Alberto+Facchini |
* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=153516)
editor of *[Journal of Algebra and its Applications](https://www.worldscientific.com/worldscinet/jaa)*
## Related entries
* [[list of journals publishing homotopy theory and category theory]]
category: people
|
Alberto Gandolfi | https://ncatlab.org/nlab/source/Alberto+Gandolfi |
* [institute page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/alberto-gandolfi.html)
## Selected writings
Relating the [[Brownian loop soup]] to [[conformal field theory]]:
* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]
* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), <a href="https://doi.org/10.1007/JHEP11(2022)009">doi:10.1007/JHEP11(2022)009</a>]
category: people |
Alberto García Raboso | https://ncatlab.org/nlab/source/Alberto+Garc%C3%ADa+Raboso |
* [webpage](http://www.math.toronto.edu/agraboso/)
category: people
[[!redirects Alberto Garcia Raboso]]
[[!redirects Alberto Raboso]] |
Alberto Marchisio | https://ncatlab.org/nlab/source/Alberto+Marchisio |
* [institute page](https://ti.tuwien.ac.at/ecs/people/marchisio)
* [GoogleScholar page](https://scholar.google.com/citations?user=6QPlLnAAAAAJ&hl=en)
## Selected writings
On [[quantum computation|quantum]] [[quantum machine learning|machine learning]]:
* Kamila Zaman, [[Alberto Marchisio]], Muhammad Abdullah Hanif, [[Muhammad Shafique]], *A Survey on Quantum Machine Learning: Current Trends, Challenges, Opportunities, and the Road Ahead* [[arXiv:2310.10315](https://arxiv.org/abs/2310.10315)]
category: people |
Alberto Verjovsky | https://ncatlab.org/nlab/source/Alberto+Verjovsky |
* [Institute page](https://www.matem.unam.mx/fsd/alberto)
## Selected writings
On [[Hopf fibrations]] and [[twistor fibrations]]:
* [[Bonaventure Loo]] and [[Alberto Verjovsky]], _On quotients of Hopf fibrations_, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 ([hdl:10077/4637]( http://hdl.handle.net/10077/4637), [pdf](https://www.openstarts.units.it/bitstream/10077/4637/1/LooVerjovskyRendMat26.pdf))
On [[twistor fibrations]]:
* {#SeadeVerjovsky02} [[José Seade]], [[Alberto Verjovsky]], _Higher dimensional complex Kleinian groups_, Math Ann 322, 279–300 (2002) ([doi:10.1007/s002080100247](https://doi.org/10.1007/s002080100247))
and also on the equivariant [[Arnold-Kuiper-Massey theorem]]:
* Le, [[José Seade]], [[Alberto Verjovsky]], Section 4 of: _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t. 49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))
category: people
|
Alberto Vezzani | https://ncatlab.org/nlab/source/Alberto+Vezzani |
* [personal page](http://users.mat.unimi.it/users/vezzani/)
* [GoogleScholar page](https://scholar.google.it/citations?user=9jKNgDMAAAAJ&hl=en)
## Selected writings
On [[Grothendieck's yoga of six operations]] for [[motives]] in [[rigid analytic geometry]]:
* [[Joseph Ayoub]], [[Martin Gallauer]], [[Alberto Vezzani]], *The six-functor formalism for rigid analytic motives*, Forum of Mathematics, Sigma **10** (2022) E61 [[doi:10.1017/fms.2022.55](https://doi.org/10.1017/fms.2022.55), [arXiv:2010.15004](https://arxiv.org/abs/2010.15004)]
category: people |
Alberto Zaffaroni | https://ncatlab.org/nlab/source/Alberto+Zaffaroni |
* [webpage](https://virgilio.mib.infn.it/~zaffaron/)
## Selected writings
On embedding [[D=3 N=4 super Yang-Mills theory]] in [[M-theory]]:
* M. Porrati, [[Alberto Zaffaroni]], _M-Theory Origin of Mirror Symmetry in Three Dimensional Gauge Theories_, Nucl. Phys. B490 (1997) 107-120 ([arXiv:hep-th/9611201](https://arxiv.org/abs/hep-th/9611201))
[[(p,q)5-brane webs]] intersected with [[orientifolds]]:
* [[Amihay Hanany]], [[Alberto Zaffaroni]], _Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry_, JHEP 9907 (1999) 009 ([arXiv:hep-th/9903242](https://arxiv.org/abs/hep-th/9903242))
On [[M-theory on S1/G_HW times H/G_ADE]]
* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))
* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))
On [[D=3 N=4 super Yang-Mills theory]]:
* [[Stefano Cremonesi]], [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopole operators and Hilbert series of Coulomb branches of 3d $\mathcal{N} = 4$ gauge theories_, JHEP 01 (2014) 005 ([arXiv:1309.2657](https://arxiv.org/abs/1309.2657))
category: people |
Albrecht Bertram | https://ncatlab.org/nlab/source/Albrecht+Bertram |
* [personal page](https://www.math.utah.edu/~bertram/)
## Selected writings
On [[Gromov-Witten theory]]:
* {#Bertram02} [[Albrecht Bertram]], *Stable Maps and Gromov-Witten Invariants*, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 ([pdf](http://users.ictp.it/~pub_off/lectures/lns019/Bertram/Bertram.pdf), [[Bertram_StableMaps.pdf:file]])
category: people
|
Albrecht Dold | https://ncatlab.org/nlab/source/Albrecht+Dold |
* [website](http://www.mathunion.org/Publications/Bulletins/38/IMU.EC/DoldA.html)
* [Wikipedia entry](http://de.wikipedia.org/wiki/Albrecht_Dold)
**Albrecht Dold** (1928-2011) was a German mathematician who made important contributions in [[algebraic topology]]. He worked at Princeton, Columbia University, ETH Zürich and held a chair at Heidelberg University from 1963 to his retirement in 1996. Besides numerous research articles he published the textbook _Lectures on Algebraic Topology_ (1972) that became a widely used classic in the field.
Some biographical notes:
* Klaus Volkert: _Vier Heidelberger Topologen 1935-1996_, Jahresbericht der Deutschen Mathematiker-Vereinigung **124**:215–238 (2022). ([doi](https://doi.org/10.1365/s); [pdf](https://link.springer.com/content/pdf/10.1365/s13291-022-00254-8.pdf))
## Selected writings
Introducing the [[Chern-Dold character]]:
* [[Albrecht Dold]], _Relations between ordinary and extraordinary homology_, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 ([mathnet:mat350](http://mi.mathnet.ru/eng/mat350)), reprinted in: J. Adams & G. Shepherd (Authors), _Algebraic Topology: A Student's Guide_ (London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press 1972 ([doi:10.1017/CBO9780511662584.015](https://doi.org/10.1017/CBO9780511662584.015))
On [[dualizable objects]], [[traces]] and the [[Becker-Gottlieb transfer]]:
* [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]
On [[algebraic topology]]:
* {#Dold95} [[Albrecht Dold]], *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf))
category: people |
Albrecht Fröhlich | https://ncatlab.org/nlab/source/Albrecht+Fr%C3%B6hlich |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Albrecht_Fr%C3%B6hlich)
## Selected writings
On [[algebraic number theory]]:
* {#FroehlichCassels67} [[J. W. S. Cassels]], [[Albrecht Fröhlich]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])
category: people
[[!redirects Albrecht Froehlich]]
|
Albrecht Klemm | https://ncatlab.org/nlab/source/Albrecht+Klemm |
* [webpage](http://www.th.physik.uni-bonn.de/people/aklemm/)
## Selected writings
On [[geometric engineering of quantum field theory]]:
* {#KatzKlemmVafa97} [[Sheldon Katz]], [[Albrecht Klemm]], [[Cumrun Vafa]], _Geometric Engineering of Quantum Field Theories_, Nucl.Phys. B497 (1997) 173-195 ([arXiv:hep-th/9609239](http://arxiv.org/abs/hep-th/9609239))
On [[toroidal orbifolds]] in [[string theory]]:
* Jens Erler, [[Albrecht Klemm]], _Comment on the Generation Number in Orbifold Compactifications_, Commun. Math. Phys. 153:579-604, 1993 ([arXiv:hep-th/9207111](https://arxiv.org/abs/hep-th/9207111))
On the [[M5-brane elliptic genus]]:
* Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 [[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)]
On [[E-strings]] in [[F-theory]]:
* Jie Gu, [[Babak Haghighat]], [[Albrecht Klemm]], Kaiwen Sun, Xin Wang, _Elliptic Blowup Equations for 6d SCFTs. III: E--strings, M--strings and Chains_ ([arXiv:1911.11724](https://arxiv.org/abs/1911.11724))
On the [[mirror map]]:
* S. Hosono, [[Albrecht Klemm]], [[Stefan Theisen]], [[Shing-Tung Yau]], _Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces_, Commun. Math. Phys. __167__ (1995) 301-350 [[doi:10.1007/BF02100589](https://doi.org/10.1007/BF02100589), [arXiv:hep-th/9308122](https://arxiv.org/abs/hep-th/9308122)]
On [[topological string theory]] and [[integrable systems]]:
* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]
## Related entries
* [[topological recursion]]
* [[topological string]]
category: people
|
Alcides Buss | https://ncatlab.org/nlab/source/Alcides+Buss |
* [webpage](http://mtm.ufsc.br/~alcides/)
category: people
|
Aldridge Bousfield | https://ncatlab.org/nlab/source/Aldridge+Bousfield |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Aldridge_Bousfield)
## Selected writings
On [[homotopy limits]], [[completion of a space|completions]] and [[localization of a space|localizations]] (such as [[p-completion]] and [[rationalization]]):
* {#BousfieldKan71} [[Aldridge Bousfield]], [[Daniel Kan]], *Localization and completion in homotopy theory*, Bull. Amer. Math. Soc. **77** 6 (1971) 1006-1010 [[doi:10.1090/S0002-9904-1971-12837-9](https://doi.org/10.1090/S0002-9904-1971-12837-9), [pdf](https://www.ams.org/journals/bull/1971-77-06/S0002-9904-1971-12837-9/S0002-9904-1971-12837-9.pdf)]
On [[orthogonal factorization systems]]:
* [[Aldridge Bousfield]], *Constructions of factorization systems in categories*, Journal of Pure and Applied Algebra **9** 2-3 (1977) 207-220 [<a href="https://doi.org/10.1016/0022-4049(77)90067-6">doi:10.1016/0022-4049(77)90067-6</a>]
Introducing what came to be known as *[[Bousfield localization]]* ([[Bousfield localization of spectra|of spectra]] via that of [[Bousfield localization of model categories|model categories]]):
* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_, Topology Volume 18 Issue 4 (1979) (<a href="https://doi.org/10.1016/0040-9383(79)90018-1">doi:10.1016/0040-9383(79)90018-1</a>, [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))
* [[Daniel M. Kan]] and [[Aldridge K. Bousfield]], *[[Homotopy Limits, Completions and Localizations]]*, Lecture Notes in Mathematics 304 (1972; 1987), Springer ([doi:10.1007/978-3-540-38117-4](https://link.springer.com/book/10.1007/978-3-540-38117-4))
On the [[core of a ring]]:
* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _The core of a ring_, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (<a href="https://doi.org/10.1016/0022-4049(72)90023-0">doi:10.1016/0022-4049(72)90023-0</a>)
On [[rational homotopy theory]] and the [[fundamental theorem of dg-algebraic rational homotopy theory]]:
* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS, vol. 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))
Establishing the [[Bousfield-Friedlander model structure]], in particular the [[stable model category|stable]] [[model structure on topological sequential spectra]]:
* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])
On the [[universal Kan fibration]] (origin of the [[univalence axiom]]):
* {#Bousfield06} [[Aldridge Bousfield]], [email to VV from 01 May 2006 10:10:30 CDT](https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/YSQz-jJ_AAAJ) $[$[[BousfieldOnUnivalence.jpg:file]]$]$
## Related $n$Lab entries
* [[rational homotopy theory]]
* [[Bousfield-Friedlander model structure]]
* [[Bousfield localization of spectra]]
* [[Bousfield localization of model categories]]
* [[Bousfield-Friedlander theorem]]
* [[proper model category]]
* [[semi-left-exact left Bousfield localization]]
category: people
[[!redirects A. K. Bousfield]]
[[!redirects Aldridge K. Bousfield]]
[[!redirects Aldridge Knight Bousfield]]
[[!redirects Bousfield]]
|
ALE space | https://ncatlab.org/nlab/source/ALE+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _asymptotically locally Euclidean space_ or _ALE space_ for short is a solution to the Euclidean [[Einstein equations]] which is a [[blow-up]] of an [[ADE-orbifold]] singularity $\mathbb{C}^2/\Gamma$ for [[finite group|finite]] [[subgroup]] $\Gamma \hookrightarrow SU(2)$.
[[!include KK-monopole geometries -- table]]
## In string theory
* In [[M-theory]]: [[KK-monopole]]
In [[F-theory]]: [[D7-brane]]
## References
An [[ADE classification]] of 4d ALE-spaces is due to
* {#Kronheimer89} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))
In
* {#IntriligatorSeiberg96} [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_ ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))
this result is interpreted physically as describing the [[moduli space]] of [[vacua]] of [[gauge theories]] with [[spontaneously broken symmetry]] ("Higgs branches"). See at _[[3d mirror symmetry]]_ for more on this.
For application in [[string theory]] see at _[[KK-monopole]]_ and see
* {#JohnsonMyers96} [[Clifford Johnson]], [[Robert Myers]], _Aspects of Type IIB Theory on ALE Spaces_, Phys.Rev. D55 (1997) 6382-6393 ([arXiv:hep-th/9610140](https://arxiv.org/abs/hep-th/9610140))
[[!redirects ALE spaces]]
[[!redirects asymptotically locally Euclidean space]]
[[!redirects asymptotically locally Euclidean spaces]]
|
Alejandro Adem | https://ncatlab.org/nlab/source/Alejandro+Adem |
* [webpage](http://www.math.ubc.ca/~adem/)
## Selected writings
On [[G-spaces]], [[equivariant homotopy theory]] and [[equivariant cohomology]]:
* [[Alejandro Adem]], [[James Davis]], *Topics in Transformation Groups*, Chapter 1 in: *Handbook of Geometric Topology*, 2001, Pages 1-54 ([doi:10.1016/B978-044482432-5/50002-0](https://doi.org/10.1016/B978-044482432-5/50002-0), [pdf](https://personal.math.ubc.ca/~adem/topics.pdf))
On [[braid group representations]] seen in the [[topological K-theory]] of the [[classifying space]] of the [[braid group]]:
* [[Alejandro Adem]], [[Daniel C. Cohen]], [[Frederick R. Cohen]], *On representations and K-theory of the braid groups*, Math. Ann. **326** (2003) 515-542 ([arXiv:math/0110138](https://arxiv.org/abs/math/0110138), [doi:10.1007/s00208-003-0435-8](https://doi.org/10.1007/s00208-003-0435-8))
On [[free action|free]] [[group actions on n-spheres]] (or rather on [[product topological space|product spaces]] of [[n-spheres]]):
* [[Alejandro Adem]], _Constructing and deconstructing group actions_, in [[Paul Goerss]], [[Stewart Priddy]], *Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory*, Contemporary Mathematics **346** AMS 2004 ([arXiv:0212280](http://arxiv.org/abs/math/0212280), [doi:10.1090/conm/346](http://dx.doi.org/10.1090/conm/346))
On [[group cohomology]] of [[finite groups]]:
* [[Alejandro Adem]], _Lectures on the cohomology of finite groups_ ([pdf](http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf))
* [[Alejandro Adem]], [[R. James Milgram]], _Cohomology of Finite Groups_, Springer 2004
On [[orbifolds]]:
* {#AdemKlaus} [[Alejandro Adem]], Michele Klaus, _Lectures on orbifolds and group cohomology_ ([pdf](http://www.math.ubc.ca/~adem/hangzhou.pdf), [[AdemKlausOrbifolds.pdf:file]])
On [[orbifolds]], [[orbifold cohomology]] and specifically on [[Chen-Ruan cohomology]] and [[equivariant K-theory]]:
* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))
On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:
* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))
* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64. ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))
On [[orbifolds]] in [[mathematical physics]] and in particular in [[string theory]]:
* [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310, American Mathematical Society, 2002
## Related $n$Lab entries
* [[orbifold]], [[Riemannian orbifold]]
* [[group actions on spheres]]
* [[equivariant K-theory]]
* [[Bredon cohomology]]
* [[string topology]]
* [[sphere]]
* [[Borel construction]]
* [[finite group]]
* [[transgression]]
category: people
|
Alejandro Corichi | https://ncatlab.org/nlab/source/Alejandro+Corichi |
* [InSpire page](https://inspirehep.net/authors/1013037)
* [GoogleScholar page](https://scholar.google.com/citations?user=OO8beHMAAAAJ)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alejandro_Corichi)
## Selected writings
On [[Fock space]]-[[quantization]] of the [[electromagnetic field]] based on careful analysis of its [[covariant phase space]]:
* {#Corichi98} [[Alejandro Corichi]], *Introduction to the Fock Quantization of the Maxwell Field*, Rev. Mex. Fis. **44** 4 (1998) 402-412 [[arXiv:physics/9804018](https://arxiv.org/abs/physics/9804018)]
and generalizing to the case of [[spacetimes]] [[manifold with boundary|with boundaries]]:
* [[Alejandro Corichi]], Juan D. Reyes, Tatjana Vukasinac, *On covariant and canonical Hamiltonian formalisms for gauge theories* [[arXiv:2312.10229]](https://arxiv.org/abs/2312.10229)]
category: people
|
Alejandro Uribe | https://ncatlab.org/nlab/source/Alejandro+Uribe |
* [webpage](http://www.math.lsa.umich.edu/~uribe/)
## related $n$Lab pages
* [[theta function]], [[quantization of 3d Chern-Simons theory]]
category: people
|
Aleks Kissinger | https://ncatlab.org/nlab/source/Aleks+Kissinger |
* [website](http://web.comlab.ox.ac.uk/people/Aleks.Kissinger/)
## Selected writings
On [[classical structures]] in the [[quantum information theory via dagger-compact categories]] and their graphical formalization in the [[ZX-calculus]]:
* [[Aleks Kissinger]], *Graph Rewrite Systems for Classical Structures in $\dagger$-Symmetric Monoidal Categories*, MSc thesis, Oxford (2008) [[pdf](https://www.cs.ox.ac.uk/people/bob.coecke/Aleks.pdf), [[Kissinger-CLassicalStructures.pdf:file]]]
* [[Aleks Kissinger]], *Exploring a Quantum Theory with Graph Rewriting and Computer Algebra*, in: *Intelligent Computer Mathematics. CICM 2009*, Lecture Notes in Computer Science **5625** (2009) 90-105 [[doi:10.1007/978-3-642-02614-0_12](https://doi.org/10.1007/978-3-642-02614-0_12)]
On [[quantum information theory via dagger-compact categories]]:
* [[Bob Coecke]], [[Aleks Kissinger]], *Picturing Quantum Processes -- A First Course in Quantum Theory and Diagrammatic Reasoning*, Cambridge University Press (2017) [[ISBN:9781107104228](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB&isbn=9781107104228)]
Relating the [[ZX-calculus]] to [[braided fusion categories]] for [[anyon]] [[braiding]]:
* [[Fatimah Rita Ahmadi]], [[Aleks Kissinger]], *Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics* $[$[arXiv:2211.03855](https://arxiv.org/abs/2211.03855)$]$
On the online proof assistant *[[Globular]]* for higher dimensional [[rewriting]] via [[semistrict]] [[globular set|globular]] [[higher categories]]:
* [[Krzysztof Bar]], [[Aleks Kissinger]], [[Jamie Vicary]], *Globular: an online proof assistant for higher-dimensional rewriting*, Logical Methods in Computer Science **14** 1 (2018) [[doi:10.23638/LMCS-14(1:8)2018](https://doi.org/10.23638/LMCS-14(1:8)2018), [arXiv:1612.01093](https://arxiv.org/abs/1612.01093)]
## Related entries
* [[quantum information]]
* [[rewriting]]
category: people |
Aleksandar Mikovic | https://ncatlab.org/nlab/source/Aleksandar+Mikovic | Aleksandar Mikovic is Associate Professor at the Department of Mathematics, Lusófona University, Lisbon. He is a member of the Grupo de Física Matemática of the Universidade de Lisboa.
* [webpage](http://gfm.cii.fc.ul.pt/people/amikovic/)
##Selected writings
On [[DBI-action]] and [[Green-Schwarz action functional]] for [[D-branes]]:
* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_, Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))
On [[spin networks]]:
* [[João Faria Martins]], [[Aleksandar Mikovic]], _Invariants of spin networks embedded in three-manifolds_, Comm. Math. Phys. (2006) 279, pp. 381-399 ([arXiv:gr-qc/0612137](https://arxiv.org/pdf/gr-qc/0612137.pdf), [doi:10.1007/s00220-008-0422-8](https://doi.org/10.1007/s00220-008-0422-8))
On [[piecewise flat spacetimes]] and [[quantum gravity]]:
* [[Aleksandar Mikovic]], _Piecewise Flat Metrics and Quantum Gravity_ ([arXiv:2001.11439](https://arxiv.org/abs/2001.11439))
category:people |
Aleksandar Nanevski | https://ncatlab.org/nlab/source/Aleksandar+Nanevski |
* [webpage](http://software.imdea.org/~aleks/)
category: people
|
Aleksandr Aleksandrov | https://ncatlab.org/nlab/source/Aleksandr+Aleksandrov |
Алекса́ндр Дани́лович Алекса́ндров (Aleksandr Danilovič Aleksandrov) was a Russian [[geometry|geometer]].
* [Wikipedia entry](http://en.wikipedia.org/wiki/Aleksandr_Danilovich_Aleksandrov) (English)
## Related articles
* [[Alexandrov space]]
category: people
[[!redirects Алекса́ндр Дани́лович Алекса́ндров]]
[[!redirects Александр Данилович Александров]]
[[!redirects Александр Д. Александров]]
[[!redirects А. Д. Александров]]
[[!redirects Александр Александров]]
[[!redirects Aleksandr Danilovič Aleksandrov]]
[[!redirects Aleksandr Danilovich Aleksandrov]]
[[!redirects Aleksandr D. Aleksandrov]]
[[!redirects A. D. Aleksandrov]]
[[!redirects Aleksandr Aleksandrov]]
[[!redirects Aleksander Aleksandrov]]
[[!redirects Alexander Alexandrov]]
|
Aleksei Starobinsky | https://ncatlab.org/nlab/source/Aleksei+Starobinsky |
Alexei Starobinsky (1948 - 2023)
* [personal page](http://www.itp.ac.ru/en/persons/starobinsky-aleksei-aleksandrovich/)
* [InSpire page](https://inspirehep.net/authors/987711)
* *[‘A Physicist Has to Be a Romantic’](https://www.hse.ru/en/news/305811947.html)*, HSE University News (Sep 2019)
## Selected writings
Introducing the [[Starobinsky model of cosmic inflation]] (and with it the field of [[cosmic inflation]] in the first place):
* {#Starobisnky80} [[Aleksei Starobinsky]], *A new type of isotropic cosmological models without singularity*, Phys. Lett. B **91** (1980) 99-102 \[<a href="https://doi.org/10.1016/0370-2693(80)90670-X">doi:10.1016/0370-2693(80)90670-X</a>\]
* [[Aleksei Starobinsky]], *Isotropization of arbitrary cosmological expansion given an effective cosmological constant*, JETP Lett. **37** (1983) 66-69 [[spire:187801](https://inspirehep.net/literature/187801)]
* [[Aleksei Starobinsky]], *The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy*, Sov. Astron. Lett. **9** (1983) 302 [[spire:199078](https://inspirehep.net/literature/199078)]
## Related entries
* [[Starobinsky model of cosmic inflation]]
* [[eternal inflation]]
category: people |
aleph | https://ncatlab.org/nlab/source/aleph |
An aleph (ℵ) is the [[cardinality]] of an [[infinite set|infinite]] [[well-ordered set|well-ordered]] (or well-orderable) [[set]]. Assuming the [[axiom of choice]] (in the guise of the [[well-ordering theorem]]), every [[cardinal number]] is either a [[natural number]] or an aleph.
The alephs are themselves well-ordered; for an [[ordinal number]] $\mu$, we denote the $\mu$th aleph as $\aleph_\mu$.
In particular $\aleph_0$ is the the cardinality of the set of [[natural numbers]].
[[!redirects aleph]]
[[!redirects alephs]]
[[!redirects ℵ]]
[[!redirects ℵs]]
|
Alessandra Buonanno | https://ncatlab.org/nlab/source/Alessandra+Buonanno |
* [webpage](http://www.aei.mpg.de/1282509/Homepage_of_Alessandra_Buonanno)
Buonanno made the theoretical prediction ([Buonanno-Damour 98](#BuonannoDamour98), see also [Damour 10](Thibault+Damour#Damour10)) of the [[gravitational wave]]-signal emitted by inspiralling [[relativistic binaries]] which has later been observed by [[LIGO]].
## Selected writings
On the theoretical prediction of the [[gravitational wave]]-signal produced by [[relativistic binaries]] (as later detected by [[LIGO]]):
* {#BuonannoDamour98} [[Alessandra Buonanno]], [[Thibault Damour]], _Effective one-body approach to general relativistic two-body dynamics_, Phys.Rev. D59 (1999) 084006 ([arXiv:gr-qc/9811091](http://arxiv.org/abs/gr-qc/9811091))
* {#BuonannoDamour00} [[Alessandra Buonanno]], [[Thibault Damour]], _Binary black holes coalescence: transition from adiabatic inspiral to plunge_, IX Marcel Grossmann Meeting in Rome, July 2000 ([arXiv:gr-qc/0011052](http://arxiv.org/abs/gr-qc/0011052))
* [[Alessandra Buonanno]], _The making of high-precision gravitational waves_, talk at ICTS 2019 ([home page](https://www.icts.res.in/lectures/gravitationalwaves2019))
## Related $n$Lab entries
* [[gravitational wave]]
* [[relativistic binary]]
category: people |
Alessandra Di Pierro | https://ncatlab.org/nlab/source/Alessandra+Di+Pierro |
* [GoogleScholar page](https://scholar.google.ae/citations?user=t-1lOBUAAAAJ&hl=zh-CN)
## Selected writings
On [[kernel methods]] in [[topological data analysis]] via [[quantum computation]]:
* Massimiliano Incudini, Francesco Martini, [[Alessandra Di Pierro]], *Higher-order topological kernels via quantum computation*, 2023 IEEE International Conference on Quantum Computing and Engineering, QCE **1** (2023) [[arXiv:2307.07383](https://arxiv.org/abs/2307.07383), [doi:10.1109/QCE57702.2023.00076](https://doi.ieeecomputersociety.org/10.1109/QCE57702.2023.00076)]
category: people |
Alessandro Luongo | https://ncatlab.org/nlab/source/Alessandro+Luongo |
* [personal page](https://luongo.pro/)
## Selected writings
On [[quantum computation|quantum]] [[algorithms]] for [[data analysis]]:
* [[Alessandro Luongo]], *[Quantum algorithms for data analysis](https://quantumalgorithms.org)* [[quantumalgorithms.org](https://quantumalgorithms.org)]
category: people
|
Alessandro Nagar | https://ncatlab.org/nlab/source/Alessandro+Nagar |
* [webpage](http://www.ihes.fr/~nagar/Home.html)
## related $n$Lab entries
* [[gravitational wave]]
category: people |
Alessandro Strumia | https://ncatlab.org/nlab/source/Alessandro+Strumia |
* [Institute page Pisa](https://sites.google.com/a/unipi.it/pisa-theory-group/staff/strumia-alessandro)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alessandro_Strumia)
## Selected writings
On [[dark matter]] [[model (in theoretical physics)|models]]:
* Marco Cirelli, Nicolao Fornengo, [[Alessandro Strumia]], _Minimal Dark Matter_, Nucl.Phys.B753:178-194, 2006 ([arXiv:hep-ph/0512090](https://arxiv.org/abs/hep-ph/0512090))
On the [[Higgs field]] [[vacuum stability]]:
* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_, JHEP12(2013)089 ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))
* {#EGMRSST15} Jose R. Espinosa, [[Gian Giudice]], Enrico Morgante, Antonio Riotto, Leonardo Senatore, [[Alessandro Strumia]], Nikolaos Tetradis, _The cosmological Higgstory of the vacuum instability_ ([arXiv:1505.04825](https://arxiv.org/abs/1505.04825))
* {#Strumia17} [[Alessandro Strumia]], _Higgs and Vacuum (In)Stability_, talk at GGI 2017 ([pdf](https://indico.cern.ch/event/660870/contributions/2746200/attachments/1538374/2411246/2017-HiggsDecay.pdf))
and implications for [[split supersymmetry]]:
* [[Gian Giudice]], [[Alessandro Strumia]], _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_, Nuclear Physics B Volume 858, Issue 1, 1 May 2012, Pages 63-83 ([arXiv:1108.6077](https://arxiv.org/abs/1108.6077))
On [[flavour anomalies]]:
* {#StrumiaEtAl17} Guido D'Amico, Marco Nardecchia, Paolo Panci, Francesco Sannino, [[Alessandro Strumia]], Riccardo Torre, Alfredo Urbano,
_Flavour anomalies after the $R_{K^\ast}$ measurement_,
J. High Energ. Phys. (2017) 2017 ([arXiv:1704.05438](https://arxiv.org/abs/1704.05438))
## related $n$Lab entries
* [[Higgs field]]
* [[flavour anomaly]]
* [[asymptotic safety]] (or not)
category: people |
Alessandro Tomasiello | https://ncatlab.org/nlab/source/Alessandro+Tomasiello |
* [webpage](http://moby.mib.infn.it/~atom/)
## Selected writings
On [[D8-branes]], [[D6-branes]], and [[D6-D8-brane bound states]] in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:
* Fabio Apruzzi, [[Marco Fazzi]], Dario Rosa, [[Alessandro Tomasiello]], _All $AdS_7$ solutions of type II supergravity_, JHEP 04 (2014) 064 ([arxiv:1309.2949](https://arxiv.org/abs/1309.2949))
On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]]:
* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))
(and on [[D6-D8 brane intersections]] and their [[fuzzy funnel]] [[noncommutative geometry]])
* {#DHTV14} [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))
* [[Ibrahima Bah]], Achilleas Passias, [[Alessandro Tomasiello]], _$AdS_5$ compactifications with punctures in massive IIA supergravity_, JHEP11 (2017)050 ([arXiv:1704.07389](https://arxiv.org/abs/1704.07389))
category: people |
Alessandro Torrielli | https://ncatlab.org/nlab/source/Alessandro+Torrielli |
Alessandro Torrielli works on [[integrable systems]], especially in the context of [[AdS-CFT]] and [[Yangian]] symmetry, at Surrey University.
* [webpage](http://www.surrey.ac.uk/maths/people/torrielli_alessandro/index.htm)
## Selected writings
* Alessandro Torrielli, _Yangians, S-matrices and AdS/CFT_, J. Phys. A44:263001, 2011 ([arXiv:1104.2474](http://arxiv.org/abs/1104.2474))
On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation opetator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:
* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))
On the [[sine-Gordon equation]], the [[Thirring model]] and their [[duality in physics|duality]]:
* [[Alessandro Torrielli]], *LonTI Lectures on Sine-Gordon and Thirring* [[arXiv:2211.01186](https://arxiv.org/abs/2211.01186)]
category: people
|
Alessandro Valentino | https://ncatlab.org/nlab/source/Alessandro+Valentino |
* [website](http://www.uni-math.gwdg.de/sandro/)
## Selected writings
On [[QFTs with defects]] via [[higher algebra|higher]] [[categorical algebra]]:
* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], *Bicategories for boundary conditions and for surface defects in 3-d TFT*, Commun. Math. Phys. **321** (2013) 543–575 [[arXiv:1203.4568](https://arxiv.org/abs/1203.4568), [doi:10.1007/s00220-013-1723-0](https://doi.org/10.1007/s00220-013-1723-0)]
category: people |
Alessandro Vichi | https://ncatlab.org/nlab/source/Alessandro+Vichi |
## Selected writings
On the [[conformal bootstrap]]:
* [[David Poland]], [[Slava Rychkov]], [[Alessandro Vichi]], _The Conformal Bootstrap: Numerical Techniques and Applications_, Rev. Mod. Phys. 91, 15002 (2019) ([arXiv:1805.04405](https://arxiv.org/abs/1805.04405))
Application of [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:
* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))
exposition in:
[[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))
|
Alessio Lomuscio | https://ncatlab.org/nlab/source/Alessio+Lomuscio | [[!redirects A. Lomuscio]]
* [Home page](http://www.doc.ic.ac.uk/~alessio/index.html)
category:people |
Alessio Marrani | https://ncatlab.org/nlab/source/Alessio+Marrani |
* [Inspire page](https://inspirehep.net/authors/1029392)
# Selected writings
On [[magic supergravity]]:
* {#CacciatoriCerchiaiMarrani12} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Squaring the Magic_ [[arXiv:1208.6153](http://arxiv.org/abs/1208.6153)]
* {#CacciatoriCerchiaiMarrani2013} Sergio L. Cacciatori, [[Bianca Cerchiai]], [[Alessio Marrani]], _Magic coset decompositions_, Adv. Theor. Math. Phys. **17** 5 (2013) 1077-1128 [[doi:10.4310/ATMP.2013.v17.n5.a4](https://doi.org/10.4310/ATMP.2013.v17.n5.a4)]
On generalizations of [[exceptional structures]], including [[E8]], [[octonions]] and the [[exceptional Jordan algebra]]:
* [[Piero Truini]], [[Michael Rios]], [[Alessio Marrani]], *The Magic Star of Exceptional Periodicity*, J. Phys.: Conf. Ser. **1194** (2019) 012106 [[arXiv:1711.07881](https://arxiv.org/abs/1711.07881), [doi:10.1088/1742-6596/1194/1/012106](https://iopscience.iop.org/article/10.1088/1742-6596/1194/1/012106)]
On [[D=14 supersymmetry]]:
* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D 99, 046004 (2019) ([arXiv:1811.06101](https://arxiv.org/abs/1811.06101))
On [[12-dimensional supergravity]], [[D=14 supersymmetry]] et al. and further indications that [[M-theory]] in 10+1 dimensions may be understood as the [[KK-compactification]] on Cayley-plane [[fibers]] of some kind of [[bosonic M-theory]] in 26+1 dimensions:
* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _The Geometry of Exceptional Super Yang-Mills Theories_, Phys. Rev. D **99** (2019) 046004 [[arXiv:1811.06101](https://arxiv.org/abs/1811.06101), [doi:10.1103/PhysRevD.99.046004](https://doi.org/10.1103/PhysRevD.99.046004)]
* [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], *Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B **808** (2020) 135674 [[arXiv:1906.10709](https://arxiv.org/abs/1906.10709), [doi:10.1016/j.physletb.2020.135674](https://doi.org/10.1016/j.physletb.2020.135674)]
reviewed in:
* [[Alessio Marrani]], *Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory*, talk at *[M-Theory and mathematics 2023](https://ncatlab.org/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (Jan, 2023) [[web](/nlab/show/M-Theory+and+Mathematics#Marrani2023)]
Relation to the [[Monster group]], [[Moonshine]] and the [[Monster vertex operator algebra]]:
* [[Alessio Marrani]], [[Michael Rios]], [[David Chester]], *Monstrous M-theory*, Symmetry **15** 2 (2023) 490; [[doi:10.3390/sym15020490](https://doi.org/10.3390/sym15020490), [arXiv:2008.06742](https://arxiv.org/abs/2008.06742)]
On the work of [[Mike Duff]]:
* [[Leron Borsten]], [[Alessio Marrani]], [[Christopher N. Pope]], [[Kellogg Stelle]], *Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday*, Proceedings of the Royal Society A,
**478** (2022) 2259 [[doi:10.1098/rspa.2022.0166](https://doi.org/10.1098/rspa.2022.0166)]
category: people |
Alex Arvanitakis | https://ncatlab.org/nlab/source/Alex+Arvanitakis |
* [webpage](http://www.damtp.cam.ac.uk/user/asa49/)
## Selected writings
On [[supersymmetry and division algebras]], the corresponding [twistor space](twistor+space#TwistorSpace) and its [[anti de Sitter spacetime|AdS]] version:
* [[Alex Arvanitakis]], Alec E. Barns-Graham, [[Paul Townsend]], _Twistor description of spinning particles in AdS_, JHEP 01 (2018) 059 ([arXiv:1710.09557](https://arxiv.org/abs/1710.09557))
On [[branes]] in [[exceptional generalized geometry]]:
* [[Alex Arvanitakis]], Chris D. A. Blair, _Type II strings are Exceptional_ ([arXiv:1712.07115](https://arxiv.org/abs/1712.07115))
* [[Alex Arvanitakis]], Chris Blair, _The Exceptional Sigma Model_ ([arXiv:1802.00442](https://arxiv.org/abs/1802.00442))
On [[higher WZW terms]] for [[branes]]:
* [[Alex Arvanitakis]], _Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 ([arXiv:1804.07303](https://arxiv.org/abs/1804.07303), [doi:10.4310/ATMP.2019.v23.n5.a1](https://dx.doi.org/10.4310/ATMP.2019.v23.n5.a1))
On [[L-infinity algebra]]-[[structure]] in [[perturbative quantum field theory]]:
* {#Arvanitakis19} [[Alex Arvanitakis]], _The $L_\infty$-algebra of the S-matrix_ ([arXiv:1903.05643](https://arxiv.org/abs/1903.05643))
On [[Lagrangian correspondence]]s
* Alex S. Arvanitakis, _Topological defects as lagrangian correspondences_, Phys. Rev. D 107, 066016, 2023 [arXiv:2301.04143](https://arxiv.org/abs/2301.04143) [doi](https://doi.org/10.1103/PhysRevD.107.06601)
## Related $n$Lab entries
* [[exceptional generalised geometry]]
* [[higher WZW term]], [[branes]]
* [[L-infinity algebra]], [[perturbative quantum field theory]]
categories: people |
Alex Cole | https://ncatlab.org/nlab/source/Alex+Cole |
* [webpage](https://grad.wisc.edu/2020/05/06/alexander-cole/)
## Selected writings
Application of [[topological data analysis]] ([[persistent homology]]) to analysis of [[cosmological structure formation]]:
* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))
and to analysis of [[phase transitions]]:
* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))
On the [[weak gravity conjecture]]:
* Lars Aalsma, [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]],
_A New Spin on the Weak Gravity Conjecture_ ([arXiv:2011.05337](https://arxiv.org/abs/2011.05337))
category: people |
Alex Eskin | https://ncatlab.org/nlab/source/Alex+Eskin |
* [webpage](https://www.math.uchicago.edu/~eskin/)
## Selected writings
On [[pillowcase orbifolds]]:
* {#EskinOkounkov05} [[Alex Eskin]], [[Andrei Okounkov]], _Pillowcases and quasimodular forms_, In: Ginzburg V. (ed.) _Algebraic Geometry and Number Theory_, Progress in Mathematics, vol 253. Birkhäuser 2006
([arXiv:math/0505545](https://arxiv.org/abs/math/0505545), [doi:10.1007/978-0-8176-4532-8_1](https://doi.org/10.1007/978-0-8176-4532-8_1))
category: people |
Alex Heller | https://ncatlab.org/nlab/source/Alex+Heller | Alex Heller, (born on July 9, 1925, died January 31,
2008), was professor of mathematics at the Graduate School and University Center, CUNY in New York. He was a PhD student of [[Sammy Eilenberg]] and made some very important contributions to [[homotopy theory]] and to [[category theory]].
* [mathematics genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=4773)
* [[Alex-Heller.pdf|Obituary:file]] prepared by [[Noson Yanofsky]].
##References
A complete MathSciNet listing of his publications is given here in [[Alex-Heller-pub.pdf|pdf-format:file]] and in [[Alex-Heller-bibtex.bib|bibtex-format:file]]. Some selected papers of direct relevance to the themes of the nLab are listed below.
* Alex Heller, _Homotopy resolutions of semi-simplicial complexes_,Trans. Amer. Math. Soc. 80 (1955), 299-344, (available [here](http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0075588-0/).
* Alex Heller, _On the representability of homotopy functors_
J. London Math. Soc. (2) 23 (1981), no. 3, 551–562
* Alex Heller, _Homotopy theories_ , Memoirs of the American Mathematical Society, Vol. 71, No 383 (1988).
* Alex Heller, _Stable homotopy theories and stabilization_ , [MR](http://www.ams.org/mathscinet-getitem?mr=1431157)
* Alex Heller, Robert A Di Paola, _Dominical categories: recursion theory without elements_, J. Symbolic Logic 52 (1987), no. 3, 594-635.
* Alex Heller, _Stable homotopy categories_,
Bull. Amer. Math. Soc. 74 (1968) 28-63
* Alex Heller, _On stochastic processes derived from Markov chains
Ann. Math. Statist. 36 (1965) 1286-1291
* Alex Heller, _The loop-space functor in homological algebra_,
Trans. Amer. Math. Soc. 96 1960 382-394
* Alex Heller, _Homological algebra in abelian categories_,
Ann. of Math. (2) 68 1958 484-525
* Alex Heller, _Relative homotopy_,
J. London Math. Soc. (2) 44 (1991), no. 3, 537-552.
category: people
|
Alex Hoffnung | https://ncatlab.org/nlab/source/Alex+Hoffnung | chains. Here we will sketch the idea and eventually should formalize this on the page for Borel-Moore homology. In the meantime the Wikipedia article plus its references and the Chriss-Ginzburg book `Representation Theory and Complex Geometry` provide plenty of information.
$$ H_\bullet(Y) = (possibly\; unbounded)\; i-chains\; in\; Y\;(with\; coefficients\; in \;\mathbb{C})$$
$$ H_i(Y) = i-dim\; subspace\;of\; Y \;without\; boundary \;(possibly \;singular\; or\; unbounded)$$
The construction starts by triangulating the space and considering the vector space of formal sums of $i$-simplices.
Of course, one wants independence of triangulation so we need to take a direct limit of these vectors spaces over all refinements of the triangulation. The boundary map then comes from simplicial homology and we have a chain complex. The homology of this complex is the Borel-Moore homology.
+--{.query}
To anyone reading this: These are notes from the University of Ottawa/Fields Institute summer school which took place last week (if you are reading this before Friday July 3). Adam Katz and I are trying to make our way through the notes from several of the classes. Anyone is welcome to add comments, corrections, insights or just join our discussion. Actually it would be great if we had some more people involved. I think the plan is something like: 1) Understand Kamnitzer's lectures on Borel-Weil, Ginzburg construction and geometric Satake, then move to Savage's lectures relating the geometry from Kamnitzer and the combinatorics from Kang's lectures. So Kang's lectures will hopefully serve as background reading to understand Savage's lectures.
Alex
=--
+--{.query}
Other projects:
1) Understand q-Schur algebras and relationship to representation theory of Hecke algebras
2) Learn lots of things in Ginzburg-Chriss.
3) Write my thesis!
4) Write everything else I should be writing.
5) Get a job! :)
I guess 3) and 5) are taken care of now.
If anyone wants to talk about the first item that would be great. I guess this is my new recruiting station to get people to talk to me about math.
=--
category: people
[[!redirects Alexander Hoffnung]] |
Alex Kavvos | https://ncatlab.org/nlab/source/Alex+Kavvos |
* [personal page](https://seis.bristol.ac.uk/~tz20861/)
## Selected writings
On [[directed homotopy type theory]]:
* [[Alex Kavvos]], *A quantum of direction* (2019) [[pdf](https://seis.bristol.ac.uk/~tz20861/papers/meio.pdf)]
On [[modal type theory]]:
* [[Daniel Gratzer]], [[G. Alex Kavvos]], [[Andreas Nuyts]], [[Lars Birkedal]]: _Multimodal Dependent Type Theory_, Logical Methods in Computer Science **17** 3 (2021) lmcs:7713 [[arXiv:2011.15021](https://arxiv.org/abs/2011.15021), <a href="https://doi.org/10.46298/lmcs-17(3:11)2021">doi:10.46298/lmcs-17(3:11)2021</a>]
category: people
[[!redirects G. Alex Kavvos]]
[[!redirects G. A. Kavvos]]
|
Alex Kehagias | https://ncatlab.org/nlab/source/Alex+Kehagias |
* [webpage](http://www.physics.ntua.gr/~kehagias/)
## Selected writings
Discussion of [[pp-wave spacetimes]] as [[Penrose limits]] of [[anti de Sitter spacetime|AdSp]] [[product manifold|x]] [[n-sphere|S^q]] spacetimes:
* E. Floratos, [[Alex Kehagias]],
_Penrose Limits of Orbifolds and Orientifolds_, JHEP 0207 (2002) 031 ([arXiv:hep-th/0203134](https://arxiv.org/abs/hep-th/0203134))
On [[Horava-Witten theory]] fopr [[7d supergravity]]:
* {#GherghettaKehagias02} Tony Gherghetta, [[Alex Kehagias]], _Anomaly Cancellation in Seven-Dimensional Supergravity with a Boundary_, Phys.Rev. **D68** (2003), 065019, ([arXiv:hep-th/0212060](http://arxiv.org/abs/hep-th/0212060))
On the [[Starobinsky model of cosmic inflation]]:
* {#KehagiasDizgahRiotto13} [[Alex Kehagias]], Azadeh Moradinezhad Dizgah, Antonio Riotto, _Comments on the Starobinsky Model of Inflation and its Descendants_, Phys. Rev. D 89, 043527 (2014) ([arXiv:1312.1155](http://arxiv.org/abs/1312.1155))
* {#FKR13} [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, _On the Starobinsky Model of Inflation from Supergravity_, Nucl. Phys. B 876, 187 (2013) ([arXiv:1307.1137](http://arxiv.org/abs/1307.1137))
* [[Sergio Ferrara]], [[Alex Kehagias]], Antonio Riotto, _The Imaginary Starobinsky Model and Higher Curvature Corrections_ ([arXiv:1405.2353](http://arxiv.org/abs/1405.2353))
* {#FerrarKehagias14} [[Sergio Ferrara]], [[Alex Kehagias]], _Higher Curvature Supergravity, Supersymmetry Breaking and Inflation_ ([arXiv:1407.5187](http://arxiv.org/abs/1407.5187))
* {#DFKRU14} [[Ioannis Dalianis]], [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, [[Rikard von Unge]], _Supersymmetry Breaking and Inflation from Higher Curvature Supergravity_ ([arXiv:1409.8299](http://arxiv.org/abs/1409.8299))
* [[Luis Alvarez-Gaume]], [[Alex Kehagias]], [[Costas Kounnas]], [[Dieter Luest]], Antonio Riotto, _Aspects of Quadratic Gravity_ ([arXiv:1505.07657](http://arxiv.org/abs/1505.07657))
## Related $n$Lab entries
* [[7d supergravity]]
* [[Starobinsky model of cosmic inflation]]
category: people
|
Alex Simpson | https://ncatlab.org/nlab/source/Alex+Simpson | * [webpage](https://www.fmf.uni-lj.si/si/imenik/32646/)
* [old webpage](http://homepages.inf.ed.ac.uk/als/)
## Selected writings
On [[exact real computer arithmetic]]:
* Dave Plume (supervised by [[Martín Escardó]], [[Alex Simpson]]): *A Calculator for Exact Real Number Computation*, University of Edinburgh (1998) [[web](https://www.dcs.ed.ac.uk/home/mhe/plume/)]
On [[measure theory]], [[probability theory]], and [[locales]]:
* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))
On [[probability theory]] in [[topos theory]]:
* _Probability sheaves_, talk at Topos à l'IHÉS, November 2015 ([YouTube](https://www.youtube.com/watch?v=IMGoluu1mgc))
On [[cartesian closed category|cartesian closed]] [[convenient categories of topological spaces]], such as [[compactly generated topological spaces]]:
* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))
On [[first-order set theory]] and [[categorical logic]]:
* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], *Relating first-order set theories, toposes and categories of classes*, Annals of Pure and Applied Logic **165** 2 (2014) 428-502 [[doi:10.1016/j.apal.2013.06.004](https://doi.org/10.1016/j.apal.2013.06.004)]
## Related $n$Lab entries
* [[Banach-Tarski paradox]]
category: people
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Alexander A. Migdal | https://ncatlab.org/nlab/source/Alexander+A.+Migdal |
* [Institute page](https://as.nyu.edu/faculty/alexander-migdal.html)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexander_Arkadyevich_Migdal)
## Selected writings
Early discussion of [[flux tubes]]/[[Wilson lines]] as effective [[strings]] in [[Yang-Mills theory]] ([Gauge/String duality](AdS-CFT+correspondence#PolyakovGaugeStringDualityReferences)):
* {#MakeenkoMigdal81} [[Yuri Makeenko]], [[Alexander A. Migdal]], *Quantum chromodynamics as dynamics of loops*, Nuclear Physics B **188** 2 (1981) 269-316 [<a href="https://doi.org/10.1016/0550-3213(81)90258-3">doi:10.1016/0550-3213(81)90258-3</a>]
> "So the [[world sheet]] of [[string]] should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric [[flux tube]] in the [[monopole]] plasma."
On the [[large-N limit]]:
* [[Alexander A. Migdal]], _Loop equations and $1/N$ expansion_, Physics Reports **102** (4) 199-290 (1983) [<a href="https://doi.org/10.1016%2F0370-1573%2883%2990076-5">doi</a>]
category: people
[[!redirects Alexander Migdal]]
[[!redirects Alexander Arkadyevich Migdal]]
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Alexander B. Atanasov | https://ncatlab.org/nlab/source/Alexander+B.+Atanasov | [[!redirects Alexander B. Atanosov]]
* [personal page](http://abatanasov.com/)
## Selected writings
On the [[geometric Langlands correspondence]] with emphasis on [[Yang-Mills monopoles]]:
* [[Alexander B. Atanasov]], *Magnetic Monopoles, 't Hooft Lines, and the Geometric Langlands Correspondence*, 2018 ([pdf](http://abatanasov.com/Files/Thesis.pdf), [slides](http://abatanasov.com/Files/Thesis%20Presentation.pdf))
category: people
[[!redirects Alexander Atanasov]]
[[!redirects Alex Atanasov]]
|
Alexander Beilinson | https://ncatlab.org/nlab/source/Alexander+Beilinson | **Alexander (or Sasha) Beĭlinson** is currently a professor at University of Chicago. He was student of [[Yuri Manin]] at Moscow State University, with main works in algebraic geometry. He has made visionary contributions to the study of algebraic cycles, automorphic forms and L-functions, [[algebraic K-theory]], Hodge theory, [[motive]]s and [[motivic cohomology]]. He conjectured the category of motivic sheaves with remarkable cohomological properties, what is often called the Beĭlinson dream. Some of his works, especially those in collaboration with [[Vladimir Drinfeld|Vladimir Drinfel'd]] are of large importance to mathematical physics, especially on their concept of **[[chiral algebra]]s** which are an approach to a chiral part of the [[conformal field theory]] on a curve, which is a geometric counterpart of the theory of [[vertex operator algebra]]s. In late 1980s Beĭlinson proposed a geometric analogue of the Langlands program, now called [[geometric Langlands]] program, which has been continued in his collaboration with Drinfel'd and also by Ed Frenkel, Dennis Gaitsgory, [[Ivan Mirković]], Kari Vilonen, and more recently taken up by mathematical physics community led by [[Edward Witten|Witten]].
Beĭlinson has substantial contributions to [[geometric representation theory]], which has been revolutionized after two discoveries: his proof (with input from Bernstein) of Kazhdan-Lusztig's connjectures in 1980, and his paper with Bernstein on what is now called [[Beĭlinson-Bernstein localization]] theorem, which lead to influx of the algebraic methods involving algebraic D-modules to representation theory. With Kazhdan, Beĭlinson has used D-modules in the proof of Jantzen's conjecture, where he introduced the notion of [[D-affinity]] and the geometric viewpoint via D-schemes. In the work on Hitchin fibration and Hitchin integrable system (with Drinfel'd) much of technique of algebraic geometry on ind-schemes, including study of D-modules is developed and used. In similar spirit to D-modules, he was also using perverse modules; with [[Ofer Gabber]], Bernstein and Deligne he developed their basic theory including deep and extremely powerful theorem for usage in representation theory, the **decomposition theorem** (see a survey: [pdf](http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf)). On technical side, he also described appropriate gluing procedure for the derived categories of perverse sheaves, involving t-structures.
Beĭlinson has shown a remarkable structure of the bounded derived category of coherent sheaves on projective spaces, and its connections to quivers. This work, together with subsequent work with [[Joseph Bernstein]] and also later works of [[Mikhail Kapranov]] and [[Alexei Bondal]] marked the birth of the [[derived noncommutative algebraic geometry]]. With [[Victor Ginzburg]], [[Yuri Manin|Manin]], [[Wolfgang Soergel]] and others, Beĭlinson introduced a wide picture of "Koszul duality patterns" in [[representation theory]].
His other works concentrated on [[motives]], [[higher regulators]], [[epsilon-factors]], and so on.
* [geometric Langlands homepage](http://www.math.uchicago.edu/~mitya/langlands.html)
* wikipedia: [Alexander Beilinson](http://en.wikipedia.org/wiki/Alexander_Beilinson)
## Selected writings
Introducing the notion of [[perverse sheaves]] (and of [[t-structures]] on [[triangulated categories]]):
* [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]
See also:
* {#Beilinson85} [[Alexander Beilinson]] _Higher regulators and values of L-functions_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))
* {#Beilinson80} [[Alexander Beilinson]], _Higher regulators of curves_, Funct. Anal. Appl. 14 (1980), 116-118, [mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng).
* {#Beilinson87} [[Alexander Beilinson]], _Height pairing between algebraic cycles_, in _K-Theory, Arithmetic and Geometry_, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, [DOI](http://dx.doi.org/10.1007/BFb0078364).
* A. Beilinson, J. Bernstein, _Localisations de $\mathfrak{g}$–modules_, C. R. Acad. Sci. Paris __292__ (1981), 15–18.
* A. A. Beilinson, [[V. Drinfeld]], _[[Chiral Algebras]]_, AMS 2004 (a preprint in various forms since around 1995, cf. [here](http://www.math.uchicago.edu/~mitya/langlands.html)).
* A. A. Beilinson, V. Ginzburg, W. Soergel, _Koszul duality patterns in representation theory_, J. Amer. Math. Soc. __9__ (2): 473–527 (1996).
* A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, _Koszul duality_, J. Geom. Phys. __5__ (1988), no. 3, 317--350.
* A. Beilinson, [[Victor Ginzburg|V. Ginzburg]], _[[wall crossing|Wall-crossing]] functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999)
* A. Beĭlinson, [[J. Bernstein]], _A proof of Jantzen conjectures_, I. M. Gelʹfand Seminar, 1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf)
## Related entries
* [[Deligne-Beilinson cohomology]]
* [[Beilinson regulator]]
* [[Beilinson conjecture]]
* [[derived noncommutative algebraic geometry]]
* [[p-adic Hodge theory]]
[[!redirects Beilinson]]
[[!redirects A. A. Beilinson]]
[[!redirects A.A. Beilinson]]
[[!redirects A. Beilinson]]
[[!redirects Sasha Beilinson]]
[[!redirects Beĭlinson]]
[[!redirects A. A. Beĭlinson]]
[[!redirects A. Beĭlinson]]
category: people |
Alexander Belavin | https://ncatlab.org/nlab/source/Alexander+Belavin |
* [webpage](http://www.itp.ac.ru/en/persons/belavin-aleksander-abramovich/)
* [Wikipedia entry](https://en.wikipedia.org/wiki/Alexander_Belavin)
## Selected writings
Introducing [[conformal field theory]]:
* [[Alexander Belavin]], [[Alexander Polyakov]], [[Alexander Zamolodchikov]], _Infinite conformal symmetry in two–dimensional quantum field theory_, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (<a href="https://doi.org/10.1016/0550-3213(84)90052-X">doi:10.1016/0550-3213(84)90052-X</a>)
On [[gravitational instantons]]:
* {#BelavinBurlankov76} [[Alexander Belavin]], D. Burlankov, _The renormalisable theory of gravitation and the Einstein equations_, Physics Letters A Volume 58, Issue 1, 26 July 1976, Pages 7-8 (<a href="https://doi.org/10.1016/0375-9601(76)90530-2">doi:10.1016/0375-9601(76)90530-2</a>)
## Related $n$Lab entries
* [[conformal bootstrap]], [[operator product expansion]]
category: people |
Alexander Belopolsky | https://ncatlab.org/nlab/source/Alexander+Belopolsky |
* [mathematics genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=112023)
## related entries
* [[picture changing operator]]
* [[differential form on a supermanifold]]
* [[superstring field theory]]
category: people
|
Alexander Berenbeim | https://ncatlab.org/nlab/source/Alexander+Berenbeim |
Blog on [[cohesive homotopy type theory]]:
* Alexander Berenbeim, _Explorations of the interesection of Cohesive Homotopy Type Theory, Allegory Theory, and Model Categories_ ([web](http://topostheorist.wordpress.com))
category: people
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Alexander Berglund | https://ncatlab.org/nlab/source/Alexander+Berglund |
* [webpage](http://staff.math.su.se/alexb/)
## Selected writings
On [[rational models of mapping spaces]] via [[L-∞ algebra]]:
* [[Alexander Berglund]], _Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras_, Homology, Homotopy and Applications, Volume 17 (2015) Number 2 ([arXiv:1110.6145](https://arxiv.org/abs/1110.6145), [doi:10.4310/HHA.2015.v17.n2.a16]( http://dx.doi.org/10.4310/HHA.2015.v17.n2.a16))
## Related $n$Lab entries
* [[nilpotent L-∞ algebra]]
category: people |
Alexander Bobenko | https://ncatlab.org/nlab/source/Alexander+Bobenko |
* [webpage](http://page.math.tu-berlin.de/~bobenko/)
category: people
|
Alexander Braverman | https://ncatlab.org/nlab/source/Alexander+Braverman |
* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Alexander_Braverman)
## Selected writings
On a mathematical definition of [[Coulomb branches]] of [[D=3 N=4 super Yang-Mills theory]]:
* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal{N} = 4$ gauge theories, II_, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 ([arXiv:1601.03586](http://arxiv.org/abs/1601.03586))
* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], _Line bundles over Coulomb branches_ ([arXiv:1805.11826](https://arxiv.org/abs/1805.11826))
(relation to [[Hilbert schemes]])
category: people |