Daniel Freed

Daniel Freed is a mathematician at University of Texas, Austin.

* [website](http://www.ma.utexas.edu/users/dafr/)

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

Freed's work revolves around the mathematical ingredients and foundations of modern [[quantum field theory]] and of [[string theory]], notably in its more subtle aspects related to [[quantum anomaly cancellation]] (which he was maybe the first to write a clean mathematical account of). In the article _[[Higher Algebraic Structures and Quantization]]_ (1992) he envisioned much of the use of [[higher category theory]] and [[higher algebra]] in [[quantum field theory]] and specifically in the problem of [[quantization]], which has -- and still is -- becoming more widely recognized only much later. He recognized and emphasized the role of [[differential cohomology]] in physics for the description of [[higher gauge field|higher]] [[gauge fields]] and their anomaly cancellation. Much of his work focuses on the nature of the [[Freed-Witten anomaly]] in the quantization of the [[superstring]] and the development of the relevant tools in [[supergeometry]], and notably in [[K-theory]] and [[differential K-theory]]. More recently Freed aims to mathematically capture the [[6d (2,0)-superconformal QFT]].

## Selected writings

Dedicated entries:

* _[[Quantum Fields and Strings]]_

* _[[Higher Algebraic Structures and Quantization]]_

* _[[Dirac charge quantization and generalized differential cohomology]]_

* _[[Loop Groups and Twisted K-Theory]]_

* _[Orientifold précis](http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionDistlerFreedMoore)_

* _[[Topological Quantum Field Theories from Compact Lie Groups]]_

* _[[4-3-2 8-7-6]]_

On [[spin geometry]], [[Dirac operators]] and [[index theory]]:

* {#Freed87} [[Dan Freed]], _Geometry of Dirac operators_, 1987 ([pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]])

On [[quantum anomalies]] via [[index theory]]:

* [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. I. Metrics and connections on determinant bundles_ , Comm. Math. Phys. 106 (1986), no. 1, 159–176 ([doi:10.1007/BF01210930](https://doi.org/10.1007/BF01210930), [euclid:1104115586](https://projecteuclid.org/euclid.cmp/1104115586))

* [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem_ , Comm. Math. Phys. 107 (1986), no. 1, 103–163 ([doi:10.1007/BF01206955](https://doi.org/10.1007/BF01206955), [euclid:1104115934](https://projecteuclid.org/euclid.cmp/1104115934))

On [[instantons]] and [[4-manifolds]]:

* [[Daniel Freed]], [[Karen Uhlenbeck]], _Instantons and Four-Manifolds_, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8](https://link.springer.com/book/10.1007/978-1-4613-9703-8))

On [[topological quantum field theory]]:

* [[Daniel Freed]], *Lectures on topological quantum field theory*, in: *Integrable Systems, Quantum Groups, and Quantum Field Theories*, NATO ASI Series **409** (1992) [[doi:10.1007/978-94-011-1980-1_5](https://doi.org/10.1007/978-94-011-1980-1_5), [pdf](http://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf), [[Freed-TQFTLectures.pdf:file]]]

On [[twisted equivariant K-theory]] with an eye towards [[twisted ad-equivariant K-theory]]:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

On [[quantization]] of the [[electromagnetic field]] in view of [[Dirac charge quantization]]:

* [[Daniel S. Freed]], [[Gregory W. Moore]], [[Graeme Segal]], p. 7 of: *The Uncertainty of Fluxes*, Commun. Math. Phys. 271:247-274, 2007 ([arXiv:hep-th/0605198](https://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3))

* [[Daniel Freed]], [[Gregory Moore]], [[Graeme Segal]], _Heisenberg Groups and Noncommutative Fluxes_, Annals Phys. 322:236-285 (2007) ([arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200))

On [[twisted ad-equivariant K-theory]] of [[compact Lie groups]] and the identification with the [[Verlinde ring]] of [[positive energy representations]] of their [[loop group]]:

* [[Daniel S. Freed]], [[Michael Hopkins]], [[Constantin Teleman]],

  1. _[[Loop Groups and Twisted K-Theory]] I_,

     J. Topology, 4 (2011), 737-789

     ([arXiv:0711.1906](http://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019))

  1. _[[Loop Groups and Twisted K-Theory]] II_,

     J. Amer. Math. Soc. 26 (2013), 595-644

     ([arXiv:0511232](http://arxiv.org/abs/math/0511232), [doi:10.1090/S0894-0347-2013-00761-4](https://doi.org/10.1090/S0894-0347-2013-00761-4))

  1. _[[Loop Groups and Twisted K-Theory]] III_,

     Annals of Mathematics, Volume 174 (2011) 947-1007

     ([arXiv:math/0312155](http://arxiv.org/abs/math/0312155), [doi:10.4007/annals.2011.174.2.5](http://dx.doi.org/10.4007/annals.2011.174.2.5))

* [[Daniel S. Freed]], [[Constantin Teleman]],

  _Dirac families for loop groups as matrix factorizations_,

  Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

  ([arxiv:1409.6051](http://arxiv.org/abs/1409.6051), [doi:10.1016/j.crma.2015.02.011](https://doi.org/10.1016/j.crma.2015.02.011))

On the [[cobordism hypothesis]]:

* [[Daniel Freed]], _The cobordism hypothesis_, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, ([arXiv:1210.5100](http://arxiv.org/abs/1210.5100), [doi:10.1090/S0273-0979-2012-01393-9](https://doi.org/10.1090/S0273-0979-2012-01393-9))

On formalizing [[short-range entanglement]] in [[topological phases of matter]] via [[invertible topological field theories]]:

* {#Freed14} [[Daniel S. Freed]], *Short-range entanglement and invertible field theories* $[$[arXiv:1406.7278](https://arxiv.org/abs/1406.7278) $]$

On classification of [[invertible TQFTs]] via [[reflection positivity]]:

* {#FreedHopkins21} [[Daniel Freed]], [[Michael Hopkins]], *Reflection positivity and invertible topological phases*, Geometry & Topology **25** (2021) 1165–1330  [[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)]

On [[spectral networks]] and [[Chern-Simons theory with complex gauge group]]:

* [[Daniel S. Freed]], [[Andrew Neitzke]], *3d spectral networks and classical Chern-Simons theory* [[arXiv:2208.07420](https://arxiv.org/abs/2208.07420)]

## Related entries

* [[differential cohomology]]

* [[quantum anomaly]]

* [[Freed-Witten anomaly]]

* [[orientifold]]

* [[Chern-Simons theory]]

category: people

[[!redirects D. Freed]]
[[!redirects Daniel Freed]]
[[!redirects Daniel S. Freed]]

[[!redirects Freed]]
[[!redirects Dan Freed]]


Daniel Freedman

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

One of the co-discoverers of [[supergravity]] ([[4d supergravity]]).

## Selected writings

Introducing [[4d supergravity]]:

* {#FreedmanNieuwenhuizenFerrara76} [[Daniel Freedman]], [[Peter van Nieuwenhuizen]], [[Sergio Ferrara]], _Progress toward a theory of supergravity_, Phys. Rev. D13 (1976) 3214 ([doi.org/10.1103/PhysRevD.13.3214](https://doi.org/10.1103/PhysRevD.13.3214))

On [[supergravity]]:

* [[Antoine Van Proeyen]], [[Daniel Freedman]], _Supergravity_, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]

On [[multi-trace operators]] in the [[AdS/CFT correspondence]]:

* [[Eric D'Hoker]], [[Daniel Freedman]], [[Samir Mathur]], A. Matusis, [[Leonardo Rastelli]], _Extremal Correlators in the AdS/CFT Correspondence_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9908160](https://arxiv.org/abs/hep-th/9908160))

category: people


Daniel Friedan

* [webage](http://en.wikipedia.org/wiki/Daniel_Friedan)

## Selected writings

On the [[NSR superstring]] as [[2d CFT]]:

* {#FriedanMartinecShenker86} [[Daniel Friedan]], [[Emil Martinec]], [[Stephen Shenker]], _Conformal invariance, supersymmetry and string theory_, Nucl. Phys., B271 (1986) 93 (doi:10.1016/S0550-3213(86)80006-2)

## Related $n$Lab entries

* [[NSR superstring]], [[2d SCFT]]

* [[picture number]]

category: people

[[!redirects Dan Friedan]]


Daniel Gottesman

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

## Selected writings

Introducing [[stabilizer codes]] for [[quantum error correction]]:

* [[Daniel Gottesman]], *Stabilizer Codes and Quantum Error Correction* ([arXiv:quant-ph/9705052](https://arxiv.org/abs/quant-ph/9705052))

Further on [[quantum error correcting codes]]:

* [[Daniel Gottesman]], *Theory of fault-tolerant quantum computation*, Phys. Rev. A **57** (1998) 127-137 $[$[doi:10.1103/PhysRevA.57.127](https://doi.org/10.1103/PhysRevA.57.127)$]$

* [[Daniel Gottesman]], *Fault-Tolerant Quantum Computation*,  Physics in Canada **63** 4  (2007) 183-189 $[$[arXiv:quant-ph/0701112](https://arxiv.org/abs/quant-ph/0701112)$]$

* Panos Aliferis, [[Daniel Gottesman]], [[John Preskill]], *Quantum accuracy threshold for concatenated distance-3 codes*, Quant. Inf. Comput. 6 (2006) 97-165 ([arXiv:quant-ph/0504218](https://arxiv.org/abs/quant-ph/0504218))

Introducing [[measurement-based quantum computation]]:

* [[Daniel Gottesman]], [[Isaac L. Chuang]], *Quantum Teleportation is a Universal Computational Primitive*, Nature **402** (1999) 390-393 [[doi:10.1038/46503](https://doi.org/10.1038/46503), [arXiv:quant-ph/9908010](https://arxiv.org/abs/quant-ph/9908010)]

category: people


Daniel Gottlieb

* [webpage](http://www.math.purdue.edu/~gottlieb/)

## Related pages

* [[dualizable object]]

* [[Becker-Gottlieb transfer]]

* [[associated infinity-bundle]]

category: people

[[!redirects Daniel Henry Gottlieb]]
[[!redirects Daniel H. Gottlieb]]


Daniel Grady

* [webpage](https://www.gradydaniel.com/)

## Selected writings

On [[differential cohomology]]:

* {#DG18} [[Daniel Grady]], _Differential cohomology and Applications_, talk at _Geometry, Topology & Physics_, NYU Abu Dhabi, April 2018 ([[DiffCohomologyAndApplications18.pdf:file]])

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]]:

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], _Ramond-Ramond fields and twisted differential K-theory_ ([arXiv:1903.08843](https://arxiv.org/abs/1903.08843))

and for [[KO]]:

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

On [[differential cohomotopy|differential]] [[Cohomotopy]] and [[Hypothesis H]]:

* [[Daniel Grady]], [[Hisham Sati]], *Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers*, Journal of Geometry and Physics **165** (2021) 104203 [[arXiv:2001.07640](https://arxiv.org/abs/2001.07640), [doi:10.1016/j.geomphys.2021.104203](https://doi.org/10.1016/j.geomphys.2021.104203)]

On [[local field theory|locality]] of [[extended functorial field theory|extended]] [[functorial field theory]]:

* [[Daniel Grady]], [[Dmitri Pavlov]], _Extended field theories are local_ ([arXiv:2011.01208](https://arxiv.org/abs/2011.01208))

On the [[cobordism hypothesis]] for [[extended functorial field theory|extended]] [[functorial field theory]] equipped with geometric structure (such as [[conformal field theory]] etc.):

* [[Daniel Grady]], [[Dmitri Pavlov]], *The geometric cobordism hypothesis* ([arXiv:2111.01095](https://arxiv.org/abs/2111.01095))

category: people

[[!redirects Dan Grady]]


Daniel Gratzer

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

## Selected works

On [[categorical models of dependent types]] via [[locally Cartesian closed categories]]:

* [[Daniel Gratzer]], [[Jonathan Sterling]], *Syntactic categories for dependent type theory: sketching and adequacy* [[arXiv:2012.10783](https://arxiv.org/abs/2012.10783)]

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), doi:10.46298/lmcs-17(3:11)2021]

* Daniel Gratzer, _Implementing Modal Dependent Type Theory_, talk at [ICFP 19](https://icfp19.sigplan.org/) ([slides pdf](https://jozefg.github.io/papers/2019-icfp-modal-slides.pdf))

* [[Daniel Gratzer]], *Syntax and semantics of modal type theory*, PhD Aarhus (2023) [[pdf](https://jozefg.github.io/papers/phd-thesis.pdf)]

On [[XTT]]:

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In [[Herman Geuvers]] (ed.), _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FCSD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).

category: people


Daniel Grayson

* [webpage](http://www.math.uiuc.edu/~dan/index.xhtml)

## Selected writings

On [[symmetry]] and introducing the language of [[homotopy type theory]] for [[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

## Related entries

* [[univalent foundations of mathematics]]

* [[UniMath project]]

category: people

[[!redirects Dan Grayson]]
[[!redirects Daniel R. Grayson]]


Daniel Greenberger

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

## Selected writings

On [[experiment|experimental]] [[quantum physics]] and its [[epistemology|epistemological]] [[interpretation of quantum mechanics|interpretation]]:

* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]

category: people


Daniel Grumiller

* [Google Scholar page](https://scholar.google.com/citations?user=CNAtukIAAAAJ&hl=en)

## Selected writings

On [[flat space holography]]:

* Arjun Bagchi, [[Daniel Grumiller]], *Holograms of Flat Space*, International Journal of Modern Physics DVol. 22, No. 12, 1342003 (2013) ([doi:10.1142/S0218271813420030](https://doi.org/10.1142/S0218271813420030))

* [[Daniel Grumiller]], *Flat Space Holograohy*, 2015 ([pdf](http://quark.itp.tuwien.ac.at/~grumil/pdf/schwarzschild2015.pdf), [[Grumiller_FlatSpaceHolography.pdf:file]])

for [[JT-gravity]]/[[SYK model]]-duality (analogous to [[AdS2/CFT1 duality]]):

* Hamid Afshar, Hernan Gonzalez, [[Daniel Grumiller]], Dmitri Vassilevich, *Flat space holography and complex SYK*, Phys. Rev. D 101, 086024 ([arXiv:1911.05739](https://arxiv.org/abs/1911.05739), [doi:10.1103/PhysRevD.101.086024](http://dx.doi.org/10.1103/PhysRevD.101.086024))

category: people


Daniel Halpern-Leistner

* [personal page](http://pi.math.cornell.edu/~danielhl/)

## Selected writings

On [[algebraic stacks]] (with focus on [[moduli stacks]]):

* [[Daniel Halpern-Leistner]], *Moduli theory*, Lecture notes (2020) [[pdf](http://pi.math.cornell.edu/~danielhl/moduli_theory_notes.pdf), [[Halpern-Leistner_ModuliTheory.pdf:file]]]

category: people


Daniel Harlow

* [webpage](http://www.pctp.princeton.edu/pctp/people/harlow_daniel.html)

## Selected writings

On [[tensor networks]] in [[holographic entanglement entropy]] as [[quantum error correcting codes]]:

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

Introducing the [[HaPPY code]] [[tensor network]] for discussion of [[holographic entanglement entropy]] and [[quantum error correction]]:

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

and more general relation of the [[Ryu-Takayanagi formula]] to [[quantum error correction]]:

* [[Daniel Harlow]], *The Ryu–Takayanagi Formula from Quantum Error Correction*, Comm. Math. Phys. **354** (2017) 865–912 [[doi:10.1007/s00220-017-2904-z](https://doi.org/10.1007/s00220-017-2904-z), [arXiv:1607.03901](https://arxiv.org/abs/1607.03901)]

Review:

* {#Harlow18} [[Daniel Harlow]], _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_ ([arXiv:1802.01040](https://arxiv.org/abs/1802.01040))

* {#Harlow20} [[Daniel Harlow]], *Computation and Holography*, talk at [Snowmass Computational Frontier Workshop 2020](https://indico.fnal.gov/event/43829/) ([pdf](https://indico.fnal.gov/event/43829/contributions/193566/attachments/132763/163346/snowmasscomp.pdf), [[HarlowComputationHolography.pdf:file]])

On the [[black hole firewall problem]] -- points out the significance of the so-called "overlap criterion," and suggests a way around the firewall paradox by arguing that the criterion should be relaxed:

* [[Daniel Harlow]], [[Patrick Hayden]], _Quantum Computation vs. Firewalls_ 2013 ([arXiv:1301.4504](http://arxiv.org/abs/1301.4504))

On the argument that after embedding into consistent [[quantum gravity]], all [[global symmetries]] must become [[local symmetries]], substantiated via [[AdS/CFT]]"

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Symmetries in quantum field theory and quantum gravity_ ([arXiv:1810.05338](https://arxiv.org/abs/1810.05338))

* [[Daniel Harlow]], [[Hirosi Ooguri]], _Constraints on symmetry from holography_, Phys. Rev. Lett. 122, 191601 (2019) ([arXiv:1810.05337](https://arxiv.org/abs/1810.05337))

On the [[covariant phase space]] for [[boundary field theory]]:

* [[Daniel Harlow]], [[Jie-qiang Wu]], *Covariant phase space with boundaries*,  J. High Energ. Phys. **2020** 146 (2020). [[arXiv:1906.08616](https://arxiv.org/abs/1906.08616), doi:10.1007/JHEP10(2020)146]

On [[quantum error correcting codes]] and [[black hole entropy]]:

* [[Chris Akers]], [[Netta Engelhardt]], [[Daniel Harlow]], [[Geoff Penington]], [[Shreya Vardhan]], *The black hole interior from non-isometric codes and complexity* [[arXiv:2207.06536](https://arxiv.org/abs/2207.06536)]

## Related $n$Lab pages

* [[AdS-CFT]]

* [[holographic entanglement entropy]]

* [[holographic quantum error correcting code]]

* [[black hole firewall problem]]

category: people


Daniel Hooper

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

## related $n$Lab entries

* [[dark matter]]

category: people

[[!redirects Dan Hooper]]


Daniel Huybrechts

_Daniel Huybrechts_ is an algebraic geometer at the University of Bonn.

* [homepage](http://www.math.uni-bonn.de/~huybrech)

## Selected writings

On [[complex geometry]]:

* {#Huybrechts04} [[Daniel Huybrechts]],  _Complex geometry: An introduction_, Universitext, Springer (2004) [[doi:10.1007/b137952](https://doi.org/10.1007/b137952), [pdf](http://www.im.ufrj.br/~andrew/ensino/VC20-1/Huybrechts-Complex_Geometry.pdf)]

On [[K3-surfaces]]:

* {#Huybrechts16} [[Daniel Huybrechts]], _Lectures on K3-surfaces_,  Cambridge University Press  2016 ([pdf](http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf), [[HuybrechtsLecturesOnK3.pdf:file]], [doi:10.1017/CBO9781316594193]( https://doi.org/10.1017/CBO9781316594193))

See also:

* Description of a seminar on stability conditions, [[Hall algebras]], [[Donaldson-Thomas invariants]] and [[Stokes phenomenon|Stokes factors]] in Bonn 2009, [Hall.pdf](http://www.math.uni-bonn.de/people/compgeo/Hall.pdf)
* Fourier-Mukai transforms in Algebraic Geometry. Oxford Mathematical Monographs. 2006. 307 pages.

* (with [[Dominic Joyce]], M. Gross) Calabi-Yau manifolds and related geometries. Springer (2002), 244 pages.

* (with M. Lehn) _The Geometry of Moduli Spaces of Sheaves_ Aspects of Mathematics E 31, Vieweg (1997), 260 pages ([[HuybrechtsLehn.pdf:file]])

  A new edition will appear in the Cambridge Mathematical Library of Cambridge Unversity Press.

category: people


Daniel Isaksen

* [website](http://www.math.wayne.edu/~isaksen/)

## Selected writings

On [[hypercovers]] in [[simplicial presheaves]]:

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. ([arXiv:0205027](http://arxiv.org/abs/math/0205027), [K-theory archive](http://www.math.uiuc.edu/K-theory/0563/))

On [[Cayley-Dickson algebras]]:

* Daniel K. Biss, [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras_,
 Communications in Algebra 36 (2), 632-664, 2008 ([arxiv:math/0511691](https://arxiv.org/abs/math/0511691))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Large annihilators in Cayley-Dickson algebras II_, 	Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 ([arxiv:math/0702075](https://arxiv.org/abs/math/0702075))

* Daniel K. Biss, [[Daniel Christensen]], [[Daniel Dugger]], [[Daniel Isaksen]], _Eigentheory of Cayley-Dickson algebras_,
Forum Mathematicum 21(5) (2009), 833-851 ([arxiv:0905.2987](https://arxiv.org/abs/0905.2987))

On [[pro-homotopy theory]] via [[model categories]] of [[pro-objects]]:

* {#FauskIsaksen07} [[Halvard Fausk]], [[D. C. Isaksen]], _Model structures on pro–categories_, Homology, Homotopy and Applications, Vol. 9 (2007), 367–398 ([arXiv:math/0511057](https://arxiv.org/abs/math/0511057))

On computation of [[stable homotopy groups of spheres]] via the [[Adams spectral sequence]]:

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _Stable homotopy groups of spheres_, PNAS October 6, 2020 117 (40) 24757-2476 ([arXiv:2001.04247](https://arxiv.org/abs/2001.04247), [doi:10.1073/pnas.2012335117](https://doi.org/10.1073/pnas.2012335117))

* [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _More stable stems_ ([arXiv:2001.04511](https://arxiv.org/abs/2001.04511))

# Related $n$Lab entries

* [[model structure on simplicial presheaves]]

* [[pro-homotopy theory]]

category: people

[[!redirects D. C. Isaksen]]
[[!redirects Dan Isaksen]]
[[!redirects Daniel Isaksen]]
[[!redirects D. Isaksen]]
[[!redirects Daniel C. Isaksen]]


Daniel Jafferis

* [webpage](https://www.physics.harvard.edu/people/facpages/jafferis)

## Selected writings

On [[D=5 super Yang-Mills theory]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity Backgrounds_ ([arXiv:1305.2886](https://arxiv.org/abs/1305.2886))

On the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Complex Chern-Simons from M5-branes on the Squashed Three-Sphere_ ([arXiv:1305.2891](https://arxiv.org/abs/1305.2891))

Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]] and the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))

Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), doi:10.1007/JHEP12(2017)151]

## Related $n$Lab entries

* [[wall crossing]]

category: people

[[!redirects Daniel Louis Jafferis]]
[[!redirects Daniel L. Jafferis]]


Daniel Kahn

> not to be confuse with [[Daniel Kan]]

***

#Daniel Kahn#

* [MathGenealogy page](http://www.genealogy.ams.org/id.php?id=12751)

## related $n$Lab entries

* [[Kahn-Priddy theorem]]

* [[stable cohomotopy]]

category: people


Daniel Kan

> not to be confused with [[Daniel Kahn]]

Daniel Marinus Kan was a homotopy theorist working at MIT.
He originated much of modern [[homotopy theory]] and [[category theory]].

He got his PhD degree in 1955 from the Hebrew University of Jerusalem, advised by [[Samuel Eilenberg]].

His PhD students include [[David L. Rector]], [[Aldridge K. Bousfield]], [[Stewart B. Priddy]], [[Emmanuel Dror Farjoun]], [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Jeffrey H. Smith]], [[David A. Blanc]].

***

* [English wikipedia entry](http://en.wikipedia.org/wiki/Daniel_Kan)

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

*  [[Clark Barwick]], [[Michael Hopkins]], [[Haynes Miller]], and [[Ieke Moerdijk]], _Daniel M. Kan (1927—2013)_, Notices of the American Mathematical Society, **62-09** (2015) ([pdf](https://www.ams.org/notices/201509/rnoti-p1042.pdf), [doi:10.1090/noti1282](https://doi.org/10.1090/noti1282)).

[[Clark Barwick]] has posted (on 7 August 2013) the following:

> On Sunday, 4 August, 2013, Daniel M. Kan died peacefully at his home in Newton, MA, surrounded by his family. It was his 86th birthday. There was a small burial service Monday afternoon.

> Dan received his Ph.D in 1955, and after short-term positions at Columbia, Princeton, and Hebrew University, he joined the Department of Mathematics at MIT in 1959, where he remained until his retirement in 1993. Dan continued to do mathematics until the last week of his life.

> In his long career, Dan published more than 70 papers with 15 coauthors. His lifelong mathematical pursuit was abstract [[homotopy theory]], and many of his ideas were so natural and flexible that they quickly became incorporated into the very fabric of [[algebraic topology]]. He supervised 15 Ph.D students (all of them at MIT), and he influenced many more through his unique seminar in algebraic topology, which today is known as the _Kan Seminar_.

## Selected writings

On [[simplicial groups]] and introducing the [[simplicial loop group]]-functor:

* {#Kan58a} [[Daniel M. Kan]], *A combinatorial definition of homotopy groups*, Annals of Mathematics **67** 2 (1958) 282–312 [[doi:10.2307/1970006](https://doi.org/10.2307/1970006)]

* {#Kan58b} [[Daniel M. Kan]], _On homotopy theory and c.s.s. groups_, Ann. of Math. 68 (1958) 38-53 [[jstor:1970042](https://www.jstor.org/stable/1970042)]

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)]

* [[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 (doi:10.1016/0022-4049(72)90023-0)

On [[derived hom-spaces]] ([[function complexes]]) in [[projective model structures on simplicial presheaves]]:

* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [doi:10.1016/1385-7258(83)90051-3, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]

On the $G$-[[Borel model structure]] and its [[Quillen equivalence]] with the [[slice model structure]] over the [[simplicial classifying space]] $\overline W G$:

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

On [[simplicial localization]]:

* {#DwyerKanLocalizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3, [pdf](http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf)]

* {#DwyerKanCalculating} [[William Dwyer]], [[Daniel Kan]], _Calculating simplicial localizations_, J. Pure Appl. Algebra 18 (1980), 17-35 [doi:10.1016/0022-4049(80)90113-9, [pdf](http://www.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf)]

* {#DwyerKan80FunctionComplexes} [[William Dwyer]], [[Daniel Kan]], _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)]

* {#DwyerKanEquivalences} [[William Dwyer]], [[Daniel Kan]], *Equivalences between homotopy theories of diagrams*, in: *Algebraic topology and algebraic K-theory*, Ann. of Math. Stud. **113**, Princeton University Press (1988) [[doi:10.1515/9781400882113-009](https://doi.org/10.1515/9781400882113-009)]

On (enhancement and generalization of) [[Elmendorf's theorem]] in [[equivariant homotopy theory]]:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

Introducing the [[model structure on simplicial groupoids]]:

* [[William Dwyer]], [[Daniel Kan]],  *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

On [[homotopy commutative diagrams]]:

* [[William Dwyer]], [[Dan Kan]], [[Jeff Smith]], _Homotopy commutative diagrams and their realizations_, Journal of Pure and Applied Algebra 57 (1989) 5-24 (doi:10.1016/0022-4049(89)90023-6)

On [[homotopy limits]]:

* *[[Homotopy Limits, Completions and Localizations]]*

On [[derived functors]] such as [[homotopy limit]]-functors on [[model categories]] and more general [[homotopical categories]]:

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], **[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]**, Mathematical Surveys and Monographs **113**, AMS 2004 ([ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf))

## Related entries

* [[homotopy theory]]

* [[Kan extension]]

* [[Kan lift]]

* [[Dold-Kan correspondence]]

* [[Kan fibration]]

* [[Kan complex]]

* [[relative category]]

category: people

[[!redirects Dan Kan]]
[[!redirects D. M. Kan]]
[[!redirects Kan]]
[[!redirects Daniel M. Kan]]


Daniel Kastler

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

## related $n$Lab entries

* [[Haag-Kastler axioms]]

* [[quantum group]]

category: people


Daniel Lazard

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

## Selected writings

Establishing what came to be known as *[[Lazard's criterion]]* (for characterizing [[flat modules]]):

* {#Lazard} [[Daniel Lazard]], _Sur les modules plats_ C. R. Acad. Sci. Paris **258** (1964) 6313-6316

category: people


Daniel Leivant

* [website](https://www.cs.indiana.edu/~leivant/)

category: people


Daniel Lenz

* [webpage](http://www.analysis-lenz.uni-jena.de)

## related $n$Lab entries

* [[pseudogroup]]

category: people


Daniel Licata

* [website](http://www.cs.cmu.edu/~drl/)

## Selected writings

On [[verified programming|programming]] in [[homotopy type theory]]:

* *Programming in Homotopy Type Theory*, Talk at IFIP Wroking Group 2.8 meeting, November, 2012 ([pdf](https://dlicata.wescreates.wesleyan.edu/pubs/lh122tttalks/lh12wg2.8.pdf), [[LicataProgrammingInHoTT.pdf:file]])

Introduction to basics of [[synthetic homotopy theory|synthetic]] [[homotopy theory]] in terms of [[homotopy type theory]]:

* [[Guillaume Brunerie]],  [[Dan Licata]], [[Peter LeFanu Lumsdaine]]: _Homotopy theory in type theory_, 2013 ([pdf slides](http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf), [[Licata-HomotopyInTypeTheory.pdf:file]], [blog entry 1](https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction), [blog entry 2](https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/))

Formalization of (the proof of) the [[Blakers-Massey theorem]] in [[homotopy type theory]]:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

On [[adjoint logic]]:

* [[Daniel Licata]], [[Mike Shulman]], and [[Mitchell Riley]], _A Fibrational Framework for Substructural and Modal Logics (extended version)_, in Proceedings of 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) ([doi: 10.4230/LIPIcs.FSCD.2017.25](http://drops.dagstuhl.de/opus/volltexte/2017/7740/), [pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf))

On [[synthetic (∞,1)-category theory]] in [[cubical type theory]] with [[bicubical sets]]:

* {#WeaverLicata20} [[Matthew Weaver]], [[Daniel Licata]], *A Constructive Model of Directed Univalence in Bicubical Sets*, in: *Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20*, Association for Computing Machinery (2020) 915–928 [[doi:10.1145/3373718.3394794](https://doi.org/10.1145/3373718.3394794)]

On [[homotopy type theory|homotopy]] [[dependent linear type theory]] of dependent [[stable homotopy types]] with [[categorical semantics]] in [[parametrized spectra]]:

* {#RileyFinsterLicata21} [[Mitchell Riley]], [[Eric Finster]], [[Daniel Licata]], *Synthetic Spectra via a Monadic and Comonadic Modality* ([arXiv:2102.04099](https://arxiv.org/abs/2102.04099))

category: people

[[!redirects Dan Licata]]

[[!redirects Daniel R. Licata]]


Daniel Litt

* [webpage](http://www.daniellitt.com)

## related $n$Lab entries

* [[modular form]], [[automorphic form]]

category: people


Daniel Loss

* [institute page 1](https://www.quantumtheory.unibas.ch/people/loss/)

* [institute page 2](https://physik.unibas.ch/en/persons/daniel-loss/)

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

## Selected writings

The idea of [[qbits]] realized on [[quantum mechanics|quantum mechanical]] [[spinors]] (e.g. [[electron]]-spin or [[nucleus]]-spin, cf. [spin resonance qbits](nuclear+magnetic+resonance#SpinResonanceQBitsReferences)):

* [[Daniel Loss]], [[David P. DiVincenzo]], *Quantum Computation with Quantum Dots*, Phys. Rev. A **57** 120 (1998) $[$[arXiv:cond-mat/9701055](https://arxiv.org/abs/cond-mat/9701055), [doi:10.1103/PhysRevA.57.120](https://doi.org/10.1103/PhysRevA.57.120)$]$

category:people


Daniel Luckhardt

Daniel Luckhardt is currently a postdoc at Ben-Gurion University of the Negev ([page](https://www.math.bgu.ac.il/~luckhard/)). He did his PhD in Göttingen.


Daniel Miller

Grad student at Cornell interested in Galois representations. ([webpage](http://www.math.cornell.edu/~dkmiller))


Daniel Moskovich

* [institute page](https://in.bgu.ac.il/en/QI/Pages/Daniel-Moskovich.aspx)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=105199)

## Selected writings

Introducing the notion of "tangle machines" (such as for describing [[adiabatic quantum computation]]):

* Avishy Y. Carmi, [[Daniel Moskovich]],*Tangle Machines*, Proc. R. Soc. A **471** (2015) 20150111 [[arXiv:1404.2862](https://arxiv.org/abs/1404.2862), [doi:10.1098/rspa.2015.0111](https://doi.org/10.1098/rspa.2015.0111)]

category: people


Daniel Murfet

__Daniel Murfet__ is an algebraic geometer; his PhD advisor was [[Amnon Neeman]]. Daniel keeps a blog [The Rising Sea](http://therisingsea.org) which is not very active, but hosts lots of (nice and reliable) Daniel's expositional notes (pdf files) on algebraic geometry, homological algebra and some category theory.

* [gScholar](https://scholar.google.com/citations?user=ilWvLq0AAAAJ) profile

## Selected writings

On [[Grothendieck universes]] in [[category theory]]:

* [[Daniel Murfet]], _Foundations for category theory_ (2006) [[pdf](http://therisingsea.org/notes/FoundationsForCategoryTheory.pdf), [[Murfet-GrothendieckUniverses.pdf:file]]]

On [[machine learning]]:

* S. Wei, Daniel Murfet, M. Gong, H. Li , J. Gell-Redman, T. Quella, _Deep learning is singular, and that’s good_, IEEE Transactions on neural networks and learning systems [pdf](https://www.suswei.com/publication/wei-2022-singular/wei-2022-singular.pdf)

On a category of [[matrix factorizations]] in the context of [[Landau-Ginzburg models]], described in terms of [[linear logic]] and the [[geometry of interactions]]:

* [[Daniel Murfet]], _The cut operation on matrix factorisations_, Journal of Pure and Applied Algebra
**222** 7 (2018) 1911-1955 [[arXiv:1402.4541](https://arxiv.org/abs/1402.4541), [doi:10.1016/j.jpaa.2017.08.014](https://doi.org/10.1016/j.jpaa.2017.08.014)]

Introduction to [[linear logic]] with amplification of its [[categorical semantics]] via [[linear algebra]] (in [[Vect]]/[[FinDimVect]]):

* {#Murfet14} [[Daniel Murfet]], *Logic and linear algebra: an introduction* [[arXiv:1407.2650](http://arxiv.org/abs/1407.2650)]

On a [[bicategory]] of [[Landau-Ginzburg models]] with [[matrix factorizations]] as [[1-morphisms]]:

* [[Nils Carqueville]], [[Daniel Murfet]], *Adjunctions and defects in Landau-Ginzburg models*, Advances in Mathematics **289** (2016) 480-566 [[doi:10.1016/j.aim.2015.03.033](https://doi.org/10.1016/j.aim.2015.03.033), [arXiv:1208.1481](https://arxiv.org/abs/1208.1481)]

A bicategory of [[Landau-Ginzburg model]]s where matrix factorizations supply morphisms is studied in

* [[Nils Carqueville]], Daniel Murfet, _Adjunctions and defects in Landau-Ginzburg models_, Advances in Mathematics 289 (2016) 480--566 [doi](https://doi.org/10.1016/j.aim.2015.03.033)

[[!redirects D. Murfet]]


Daniel N. Blaschke

* [InSpire page](https://inspirehep.net/authors/1032120)

* [GoogleScholar page](https://scholar.google.com/citations?user=lpba2OwAAAAJ&hl=en)

## Selected writings

On the [[phase space]] and canonical [[quantization]] of [[Yang-Mills theory]]:

* {#BlaschkeGieres21} [[Daniel N. Blaschke]], [[François Gieres]], *On the canonical formulation of gauge field theories and Poincaré transformations*, Nuclear Physics B **965** (2021) 115366 [[arXiv:2004.14406](https://arxiv.org/abs/2004.14406), [doi:10.1016/j.nuclphysb.2021.115366](https://doi.org/10.1016/j.nuclphysb.2021.115366)]

category: people

[[!redirects Daniel Blaschke]]


Daniel P. Arovas

* [institute page](https://physics.ucsd.edu/Directory/Person/5)

* [GoogleScholar page](https://scholar.google.com/citations?user=Ehx8cp8AAAAJ&hl=en)

## Selected writings

On [[anyon statistics|anyon phases]] (specifically in the [[quantum Hall effect]]) as [[Berry phases]] of a [[quantum adiabatic theorem|adiabatic]] transport of anyon positions:

* {#AvrosSchriefferWilczek84} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], _Fractional Statistics and the Quantum Hall Effect_, Phys. Rev. Lett. 53, 722 (1984) $[$[doi:10.1103/PhysRevLett.53.722](https://doi.org/10.1103/PhysRevLett.53.722)$]$

The model of [[anyon statistics]] as an [[Aharonov-Bohm effect]] of a "[[fictitious gauge field]]" sourced by and coupled to each anyon:

* {#ArovasScriefferWilczekZee85} [[Daniel P. Arovas]], [[John Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](Frank+Wilczek#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

category: people

[[!redirects Daniel Arovas]]


Daniel Quillen

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

* Scanned images of Quillen's research notebooks, Clay Mathematical Institute, [web](http://www.claymath.org/publications/quillen-notebooks).

## Related $n$Lab entries

* [[model category]], [[model structure on topological spaces]]

* [[Quillen's theorems]]

* [[algebraic K-theory]]

* [[quasi-free algebra]]

* [[cotangent complex]]

* [[Andre-Quillen cohomology]]

* [[Quillen-Lichtenbaum conjecture]]

* [[complex cobordism]]

* [[formal group law]]

* and much much more....

## Selected writings

Introducing [[model category]]-theory for [[homotopy theory]]:

* {#Quillen67} [[Daniel Quillen]], _Homotopical algebra_, Lecture Notes in Mathematics __43__, Springer-Verlag 1967, iv+156 pp ([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)).

On [[rational homotopy theory]]:

* {#Quillen69} [[Dan Quillen]], _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([JSTOR](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

Proof that [[topological realization]] takes [[Kan fibrations]] to [[Serre fibrations]]:

* {#Quillen68} [[Daniel Quillen]], *The geometric realization of a Kan fibration is a Serre fibration*, Proc. AMS 19 (1968) 1499-1500 [[doi:1968-019-06/S0002-9939-1968-0238322-1](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1), [pdf](https://www.ams.org/journals/proc/1968-019-06/S0002-9939-1968-0238322-1/S0002-9939-1968-0238322-1.pdf)]

On the (co)homology of [[commutative algebras]]:

* [[D. G. Quillen]], 1970, _On the (co-)homology of commutative rings_, in _Proc. Symp. on Categorical Algebra_, 65–87, American Math. Soc.

On [[cobordism cohomology theory]]:

* {#Quillen71} [[Daniel Quillen]], _Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations_, Advances in Mathematics 7 (1971), 29–56 (doi:10.1016/0001-8708(71)90041-7)

On [[equivariant de Rham cohomology]]:

* {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], _Superconnections, Thom classes and equivariant differential forms_, Topology 25, 85 (1986)  (doi:10.1016/0040-9383(86)90007-8)

Also

* [[J. Cuntz]], [[Daniel Quillen]], _Algebra extensions and nonsingularity_, J. Amer. Math. Soc. __8__  (1995),  no. 2, 251–289, [MR96c:19002](http://www.ams.org/mathscinet-getitem?mr=1303029), [doi](http://ams.mpim-bonn.mpg.de/leavingmsn?url=http://dx.doi.org/10.2307/2152819); _Cyclic homology and nonsingularity_,  J. Amer. Math. Soc. __8__  (1995),  no. 2, 373–442, [MR96e:19004](http://www.ams.org/mathscinet-getitem?mr=1303030), [doi](http://dx.doi.org/10.2307/2152822)

On the [[group completion theorem]]:

* [[Daniel Quillen]], _On the group completion of a simplicial monoid_, Appendix Q in: [[Eric Friedlander]], [[Barry Mazur]], _Filtrations on the homology of algebraic varieties_, Memoir of the A.M.S., Vol. 110, no. 529 (1994) ([doi:10.1090/memo/0529](http://dx.doi.org/10.1090/memo/0529), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/quillencomp.pdf))

[[!redirects Dan Quillen]]
[[!redirects D. Quillen]]
[[!redirects Daniel G. Quillen]]
[[!redirects D. G. Quillen]]
[[!redirects Quillen]]
category: people


Daniel R. Simon

* webpage?

## Selected writings

Early highlighting of expected properties of [[quantum computation]] ("[[quantum supremacy]]") in view of algorithms such as [[Simon's algorithm]]/[[Shor's algorithm]]:

* [[Daniel R. Simon]], *On the power of quantum computation*, SIAM Journal on Computing **26** 5 (1997) [[doi:10.1137/S0097539796298637](https://doi.org/10.1137/S0097539796298637), [pdf](https://courses.cs.washington.edu/courses/cse599/01wi/papers/simon_qc.pdf)]

On [[Simon's algorithm]]:

* *[Exploring Simon’s Algorithm with Daniel Simon](https://aws.amazon.com/blogs/quantum-computing/simons-algorithm/)* (Oct 2021)

category: people

[[!redirects Daniel Simon]]


Daniel Roggenkamp

* [website](http://www.physics.rutgers.edu/people/pips/Roggenkamp.html)

## Selected writings

Discussion of [[Rozansky-Witten theory|Rozansky-Witten]] [[defect QFT]] as an [[extended TQFT]] via the [[cobordism hypothesis]]:

* [[Ilka Brunner]], [[Nils Carqueville]], [[Daniel Roggenkamp]], *Truncated affine Rozansky--Witten models as extended TQFTs*,  Commun. Math. Phys. (2023) [[arXiv:2201.03284](https://arxiv.org/abs/2201.03284), [doi:10.1007/s00220-022-04614-4](https://doi.org/10.1007/s00220-022-04614-4)]

category: people


Daniel Schroeder

* [webpage](http://physics.weber.edu/schroeder/)

## Selected writings

On [[perturbative quantum field theory]]:

* [[Michael Peskin]], [[Daniel Schroeder]], _An Introduction to Quantum Field Theory_, 1995 ([spire:407703](https://inspirehep.net/literature/407703), [ISBN 9780201503975](https://www.routledge.com/An-Introduction-To-Quantum-Field-Theory/Peskin-Schroeder/p/book/9780201503975))

category: people


Daniel Schäppi

* [webpage](https://homepages.uni-regensburg.de/~scd37703/index.html)
* [MO website](http://mathoverflow.net/users/1649/daniel-schappi)

## related $n$Lab entries

* [[Tannaka duality]]

category: people

[[!redirects Daniel Schaeppi]]


Daniel Spivak

* [website](https://web.archive.org/web/20230203063320/http://www.math.utoronto.ca/cms/people/students/graduate/spivakdaniel/)

## Selected writings

* [[Matthew Kennedy]], [[Astra Kolomatskaia]], [[Daniel Spivak]] *An infinite quantum Ramsey theorem*, Journal of Operator Theory, Volume 84, Issue 1, Summer 2020  pp. 49-65. ([doi:10.7900/jot.2018dec11.2259](http://dx.doi.org/10.7900/jot.2018dec11.2259), [arXiv:1711.09526](https://arxiv.org/abs/1711.09526))

category:people


Daniel Sternheimer

* [webpage](http://monge.u-bourgogne.fr/dsternh/)

## Selected writings

Introducing [[formal deformation quantization]]:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* {#BayenFlatoFronsdalLichnerowiczSternheimer78b} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. II. Physical applications_, Annals of Physics **111** 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]

## Related entries

* [[deformation quantization]]

* [[singleton representation]]

category: people


Daniel Tanré

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=56161)

## Selected writings

On [[rational homotopy theory]]:

* [[Yves Félix]], [[John Oprea]], [[Daniel Tanré]], _Algebraic models in geometry_, Oxford University Press 2008 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tanre.pdf), [ISBN:9780199206520](https://global.oup.com/academic/product/algebraic-models-in-geometry-9780199206520))

On [[rational homotopy theory]] of non-[[nilpotent spaces]] via [[deck transformation|deck]]-[[Borel-equivariant rational homotopy theory]] of their [[universal cover|universal]] [[covering spaces]]:

* [[Antonio Gómez-Tato]], [[Stephen Halperin]], [[Daniel Tanré]], *Rational homotopy theory for non-simply connected spaces*, Trans. Amer. Math. Soc. 352 (2000), 1493-1525 ([doi:10.1090/S0002-9947-99-02463-0](https://doi.org/10.1090/S0002-9947-99-02463-0), [jstor:118074](https://www.jstor.org/stable/118074))

* [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], [[Daniel Tanré]], _Homotopy theory of complete Lie algebras and Lie models of simplicial sets_, Journal of Topology (2018) 799-825 ([arXiv:1601.05331](https://arxiv.org/abs/1601.05331), [doi:10.1112/topo.12073](https://doi.org/10.1112/topo.12073))

On the [[ordinary cohomology]] of [[configuration spaces of points]]:

* [[Yves Félix]], [[Daniel Tanré]], _The cohomology algebra of unordered configuration spaces_, Journal of the LMS, Vol 72, Issue 2 ([arxiv:math/0311323](https://arxiv.org/abs/math/0311323), [doi:10.1112/S0024610705006794](https://doi.org/10.1112/S0024610705006794))

On [[rational parametrized stable homotopy theory]]:

* {#FelixMurilloTanre10} [[Yves Félix]], Aniceto Murillo [[Daniel Tanré]], _Fibrewise stable rational homotopy_, Journal of Topology, Volume 3, Issue 4, 2010, Pages 743–758 ([doi:10.1112/jtopol/jtq023](https://doi.org/10.1112/jtopol/jtq023))

On rational models for [[spherical fibrations]] and [[Thom spaces]]:

* {#FelixOpreaTanre16} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Lie-model for Thom spaces of tangent bundles_, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 ([pdf](http://www.ams.org/journals/proc/2016-144-04/S0002-9939-2015-12829-8/S0002-9939-2015-12829-8.pdf), [doi:10.1090/proc/12829](https://doi.org/10.1090/proc/12829))

[[!redirects Daniel Tanre]]

category: people


Daniel Thomas

* [personal page](https://danbthomas1.wixsite.com/research)

* [institute page](https://www.qmul.ac.uk/spcs/staff/research-staff/profiles/danielthomas.html)

## Selected writings

On [[cosmic structure formation]] in view of [[MOND]]:

* [[Daniel Thomas]], [[Ali Mozaffari]], [[Tom Zlosnik]], *Consistent cosmological structure formation on all scales in relativistic extensions of MOND* ([arXiv:2303.00038](https://arxiv.org/abs/2303.00038))

category: people


Daniel Waldram

Daniel Waldram is professor for theoretical physics at Imperial College London.

* [webpage](http://www3.imperial.ac.uk/people/d.waldram)

## Selected writings:

On [[gaugino condensation]] in [[heterotic string theory]] and [[Horava-Witten theory]]:

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Gaugino condensation in M theory on $S^1/Z_2$_, Phys. Rev. D 57, 7529 (1998) ([arXiv:hep-th/9711197](https://arxiv.org/abs/hep-th/9711197), [doi:10.1103/PhysRevD.57.7529](https://doi.org/10.1103/PhysRevD.57.7529))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Five-Branes and Supersymmetry Breaking in M-Theory_, JHEP 9904:009, 1999 ([arXiv:hep-th/9901017](https://arxiv.org/abs/hep-th/9901017))

On [[string phenomenology]] in [[heterotic M-theory]]:

* {#DOPW99} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Models from Heterotic M-theory_, Adv. Theor. Math. Phys. 5 (2002) 93-137 ([arXiv:hep-th/9912208](https://arxiv.org/abs/hep-th/9912208))

* {#DOPW00} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Model Vacua in Heterotic M-Theory_, talk at [Strings '99](http://strings99.aei.mpg.de/), Potsdam, Germany, 19 - 24 Jul 1999 ([arXiv:hep-th/0001101](https://arxiv.org/abs/hep-th/0001101))

On the [[F-theory]]-perspective on [[heterotic M-theory on ADE-orbifolds]]:

* [[Monika Marquart]], [[Daniel Waldram]], _F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$_ ([arXiv:hep-th/0204228](https://arxiv.org/abs/hep-th/0204228))

On [[D=11 N=1 supergravity]] solutions for the [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[supersymmetric cycles]]:

* [[Jerome Gauntlett]], [[Nakwoo Kim]], [[Daniel Waldram]], _M-Fivebranes Wrapped on Supersymmetric Cycles_, Phys. Rev. D63 (2001) 126001 ([arXiv:hep-th/0012195](https://arxiv.org/abs/hep-th/0012195))

On [[flux compactifications]] in terms of [[G-structure]]:

* [[Jerome Gauntlett]], Dario Martelli, Stathis Pakis, [[Daniel Waldram]], _G-Structures and Wrapped NS5-branes_, Commun.Math.Phys. 247 (2004) 421-445 ([arxiv:hep-th/0205050](https://arxiv.org/abs/hep-th/0205050))

On generalized [[G-structures]] and [[exceptional generalized geometry]] of [[flux compactifications]] and in [[AdS/CFT]]:

* [[Daniel Waldram]], *Fluxes, holography and uses of Exceptional Generalized Geometry*, talk at *[[Strings 2022]]* [[indico:4940880](https://indico.cern.ch/event/1085701/contributions/4940880), [slides](https://indico.cern.ch/event/1085701/contributions/4940880/attachments/2482886/4262576/Daniel%20Waldram%20Strings2022-waldram.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e222a3a9-ebf4-445c-ada0-f7b1d592d908) ]

## Related $n$Lab entries

* [[exceptional generalized geometry]]

category: people

[[!redirects Dan Waldram]]


Daniel Wise

* [webpage](http://www.math.mcgill.ca/wise/)

## related $n$Lab entries

* [[virtually fibered conjecture]]

* [[cubulation]]

category: people


Daniel Zwanziger

* [institute page](https://as.nyu.edu/faculty/daniel-zwanziger.html)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Daniel_Zwanziger)

## Selected writings

On [[Dirac charge quantization]]:

* [[Daniel Zwanziger]], *Quantum Field Theory of Particles with Both Electric and Magnetic Charges*, Phys. Rev. **176** (1968) 1489 [[doi:10.1103/PhysRev.176.1489](https://doi.org/10.1103/PhysRev.176.1489)]

category: people


Daniela Petrisan

* [webpage](https://www.irif.fr/~petrisan/)

## Selected writings

On [[nominal sets]]:

* Daniela Petrisan, _Investigations into Algebra and Topology over Nominal Sets_, PhD thesis, University of Leicester, 2010. [pdf](https://www.irif.fr/~petrisan/thesis.pdf)

On [[semialgebras]] and [[weak distributive laws]]:

* [[Daniela Petrisan]] and Ralph Sarkis. _Semialgebras and Weak Distributive Laws_, Proceedings 37th Conference on
Mathematical Foundations of Programming Semantics, EPTCS 351, 2021, pp. 218-241. ([doi:10.4204/EPTCS.351.14](https://doi.org/10.4204/EPTCS.351.14))

category: people


Daniele Amati

* [webpage](http://www.sissa.it/tpp/members.php?ID=12)

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

## Quotes

> {#Witten21stCenturyQuote} Back in the early '70s, the Italian physicist, Daniele Amati reportedly said that [[string theory]] was part of 21st-century physics that fell by chance into the 20th century. I think it was a very wise remark. ([[Edward Witten]], [Nova interview 2003](http://www.pbs.org/wgbh/nova/elegant/view-witten.html), also [American Scientist Astronomy Issue 2002](http://www.sns.ias.edu/~witten/papers/string.pdf))

## Selected writings

On examples of [[universal spacetimes]]:

* [[Daniele Amati]], [[Ctirad Klimčík]], _Nonperturbative computation of the Weyl anomaly for a class of nontrivial backgrounds_, Phys. Lett. B, 219:443–447, 1989 ([spire:269390](inspirehep.net/record/269390), 10.1016/0370-2693(89)91092-7)

## Related $n$Lab entries

* [[universal spacetime]]

category: people


Daniele Angella

* [webpage](https://sites.google.com/site/danieleangella/home)

## Selected writings

On [[Bott-Chern cohomology]]:

* [[Daniele Angella]], Adriano Tomassini, _On Bott-Chern cohomology and formality_, J. Geom. Phys. 93 (2015), 52-61 ([arXiv:1411.6037](https://arxiv.org/abs/1411.6037))

For [[orbifolds]]:

* [[Daniele Angella]], _Cohomologies of certain orbifolds_, J. Geom. Phys. 71 (2013), 117--126 ([arXiv:1211.2561](https://arxiv.org/abs/1211.2561))

category: people


Daniele Guido

* [webpage](http://www.mat.uniroma2.it/~guido/)

## related $n$Lab entries

* [[spin-statistics theorem]]

category: people


Daniele Turi

* [GoogleScholar page](https://scholar.google.com/citations?user=02OVN_4AAAAJ&hl=en)

## Selected writings

On [[higher-order abstract syntax]]:

* [[Marcelo Fiore]], [[Gordon Plotkin]], [[Daniele Turi]]. _Abstract syntax and variable binding_, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158). IEEE (1999) [[doi:10.1109/LICS.1999.782615](https://doi.org/10.1109/LICS.1999.782615), [pdf](https://homepages.inf.ed.ac.uk/gdp/publications/Abstract_Syn.pdf), [webpage](https://www.dcs.ed.ac.uk/home/dt/abstractsyn.html)]

category: people


Daniell integral

[[Lebesgue integral]]s are defined as a construction in [[measure theory]], that is the measure precedes the integral; alternatively one can introduce integrals as linear functionals satisfying some axioms, in that approach measure can be defined using integral, hence the integral precedes measure. A [[Daniell integral]] is a particular axiomatically defined type of such functionals.

* V. I. Sobolev, [Daniell integral](https://www.encyclopediaofmath.org/index.php/Daniell_integral), [[eom]] entry
* wikipedia: [Daniell integral](https://en.wikipedia.org/wiki/Daniell_integral)
* P. J. Daniell, _A general form of integral_, Annals of Mathematics, Second Series, 19 (4): 279–294 (1918) [doi](http://dx.doi.org/10.2307/1967495)

category: analysis


Daniil Frumin

* Contact email: dfrumin at cs dot ru dot nl


Danil Annenkov

* [personal page](https://dannenkov.me/)

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

## Selected writings

On a version of the [[Homotopy Type System]] with application to [[semi-simplicial types in homotopy type theory]]:

* [[Danil Annenkov]], [[Paolo Capriotti]], [[Nicolai Kraus]], [[Christian Sattler]]: *Two-Level Type Theory and Applications* [[arXiv:1705.03307](https://arxiv.org/abs/1705.03307)]

category: people


Daniël Otten

* [personal page](https://www.otten.co/)

* [institute page](https://www.illc.uva.nl/People/person/5117/Dani%C3%ABl-Otten)

## Selected writings

On [[categorical semantics]] for [[weak type theories]]:

* [[Daniël Otten]], *Models for Propositional Type Theory*, talk at *[[DutchCATS]]* (11 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_otten.pdf)]

category: people

[[!redirects Daniel Otten]]


Daniël Prins

## Selected writings

On [[flux compactifications]] and ([[generalized Calabi-Yau structure|generalized]]) [[Calabi-Yau structure]] via [[G-structures]] and [[generalised complex geometry]]:

* [[Daniël Prins]], [[Dimitrios Tsimpis]], _IIA supergravity and M-theory on manifolds with $SU(4)$ structure_, Phys. Rev. D 89.064030 ([arXiv:1312.1692](https://arxiv.org/abs/1312.1692))

* {#Prins16} [[Daniël Prins]], _On flux vacua, $SU(n)$-structures and generalised complex geometry_, Université Claude Bernard -- Lyon I, 2015.  ([arXiv:1602.05415](https://arxiv.org/abs/1602.05415), [tel:01280717](https://tel.archives-ouvertes.fr/tel-01280717))

* [[Ruben Minasian]], [[Daniël Prins]], [[Hagen Triendl]], _Supersymmetric branes and instantons on curved spaces_, JHEP 10 (2017) 159 ([arXiv:1707.07002](https://arxiv.org/abs/1707.07002))

[[!redirects Daniel Prins]]


Danny Stevenson

* [website](https://researchers.adelaide.edu.au/profile/daniel.stevenson)

## Selected writings

Towards higher [[Atiyah Lie algebroids]] for [[bundle gerbes]]:

* {#Stevenson06} [[Danny Stevenson]], *Lie 2-algebras and the geometry of gerbes*, Unni Namboodiri Lectures 2006 [[pdf](http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf), [[Stevenson-Lie2AlgebrasAndGerbes.pdf:file]]]

On [[décalage]] and the [[simplicial loop group]] construction:

* [[Danny Stevenson]], _Décalage and Kan's simplicial loop group functor_, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 ([arXiv:1112.0474](https://arxiv.org/abs/1112.0474), [tac:26-28](http://www.tac.mta.ca/tac/volumes/26/28/26-28abs.html))

On [[non-abelian cohomology]] and [[principal ∞-bundles]]:

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](https://arxiv.org/abs/1203.2460))

* [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](https://arxiv.org/abs/1203.2461))

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 ([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

category: people

[[!redirects Daniel Stevenson]]


Darboux integration

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## Idea

A method of [[integration]].

## Related concepts

* [[Lebesgue integration]]

* [[Riemann integration]]

* [[Jordan content]]

## References

* Wikipedia, _[Darboux integral](http://en.wikipedia.org/wiki/Darboux_integral)_

[[!redirects Darboux integral]]
[[!redirects Darboux integrals]]


Darboux's theorem

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## Idea

Darboux's theorem states that a [[smooth manifold]] $X$ equipped with a [[differential 1-form]] $\theta$ which is sufficiently non-degenerate admits local [[coordinate charts]] $\phi_i \colon\mathbb{R}^{2n} \to X$ on which $\theta$ takes the canonical form $\phi_i^\ast \theta = \sum_{k} x^{2k} \mathbf{d} x^{2k+1}$.

In particular

* for $(X, \omega)$ a [[symplectic manifold]] there are local charts in which the [[symplectic form]] $\omega$ takes the form $\phi_i^\ast \omega = \sum_{k} \mathbf{d} x^{2k} \wedge \mathbf{d} x^{2k+1}$;

* similarly, for [[contact manifolds]] there are adapated local charts.

(e.g. [Arnold 78, p. 362](#Arnold78))

For the case of symplectic manifolds the Darboux theorem may also be read as saying that a [[G-structure]] for $G = Sp(2n)$ the [[symplectic group]] (hence an [[almost symplectic structure]]) is an [[integrable G-structure]] already when it is first-order integrable, i.e. [[torsion of a G-structure|torsion]]-free, i.e. symplectic. See at _[integrability of G-structures -- Examples -- Symplectic structure](integrability+of+G-structures#ExampleSymplecticStructure)_.

## Related concepts

* [[canonical coordinates]], [[canonical momenta]]

* [[integrable G-structure]]

## References

Lecture notes include

* [[Andreas Čap]], section 1.8 of _Differential Geometry 2_, 2011/2012 ([pdf](http:///www.mat.univie.ac.at/~cap/files/Diffgeom2.pdf))

* {#Pasquotto} Federica Pasquotto, _Linear $G$-structures by example_ ([pdf](http://www.few.vu.nl/~pasquott/course16.pdf))

Textbook accounts include

* {#Arnold78} [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate texts in Mathematics 60 (1978)

See also

* Wikipedia, _[Darboux theorem](http://en.wikipedia.org/wiki/Darboux%27s_theorem)_

A version for [[supergeometry]] is due to

* [[Bertram Kostant]], in Lecture Notes in Mathematics vol.570, 177, (Bleuler, K. and Reetz, A.eds), Proc. Conf. on Diff. Geom. Meth. in Math. Phys., Bonn 1975., Springer-Verlag, Berlin, 1977

A version for prequantized symplectic manifolds ([[prequantum bundles]]) is discussed in

* {#FaureTsujii12} Frédéric Faure, Masato Tsujii, around prop. 2.15 of _Prequantum transfer operator for symplectic Anosov diffeomorphism_ ([arXiv:1206.0282](http://arxiv.org/abs/1206.0282))

A version for [[shifted symplectic structures]] ([[BV-BRST complex|BV-brackets]]) on [[stacks]] is discussed in

* [[Christopher Brav]], [[Vittoria Bussi]], [[Dominic Joyce]], _A 'Darboux theorem' for derived schemes with shifted symplectic structure_ ([arXiv:1305.6302](http://arxiv.org/abs/1305.6302))

* [[Oren Ben-Bassat]], [[Christopher Brav]], [[Vittoria Bussi]], [[Dominic Joyce]], _A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications_ ([arXiv:1312.0090](http://arxiv.org/abs/1312.0090))

[[!redirects Darboux theorem]]

[[!redirects Darboux coordinate chart]]
[[!redirects Darboux coordinate charts]]

[[!redirects Darboux chart]]
[[!redirects Darboux charts]]

[[!redirects Darboux coordinate]]
[[!redirects Darboux coordinates]]


Darin Morrison

## Selected writings

A formalization in [[HoTT]]-[[Agda]] of general [[(n,r)-categories]] for $n,r \in \mathbb{N} \sqcup \{\infty\}$, defined as [[coinductive types]] of infinity-graphs, with operations defined by induction-coinduction, is implemented in

* {#Morrison19} Darin Morrison, _[agda-nr-cats](https://github.com/5HT/agda-nr-cats)_,
_[agda-infinity-categories](https://github.com/freebroccolo/agda-infinity-categories/)

See also at _[[internal category in homotopy type theory]]_.

category: people


dark matter

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## Idea

The [[observable universe]] at large scales seems to behave as if it contains large amount of [[matter]] which does have a [[gravity|gravitational effect]] but does not emit [[electromagnetic radiation]] and hence remains "dark" to telescopes. This hypothetical matter is therefore called _dark matter_.

There are various astronomical observations that suggest the existence of dark matter

1. galaxy rotation curves -- the speed of rotation of [[galaxies]] as a function of the distance from their center cannot be explained by the observed visible matter, but an explanation of this dependence by gravitating matter requires the presence of plenty of dark matter;

1. cosmic microwave background fluctuations -- the measured fluctuations in the [[cosmic microwave background]] are very well fitted by the [[standard model of cosmology]] with cold dark matter (CDM) included. (see also [Resonaances, 18 Jan 2013](#Resonaances)).

1. Simulations of [[structure formation]] in the universe seem to require existence of dark matter in order to reproduce results compatible with observation.

Any further details about the nature of this hypothetical dark matter remain elusive to date. Possible classes of candidates go by various names. For instance "[[weakly interacting massivle particle|weakly interacting massive particles]]" ("[[WIMP]]"s), hence massive particles that interact via [[gravity]] and the [[weak nuclear force]], but not via [[electromagnetism]].
Certain [[neutrino]]-dark matter scenarios are being discussed ([Neutrino White 16](#NeutrinoWhite16))
 In [[supersymmetry|supersymmetric]] [[theory (physics)|theories]] with [[R-parity]] imposed, the _lightest supersmmetric particle_ (LSP, such as the [[gravitino]] or the [[neutralino]]) is traditionally regarded as a natural candidate for dark matter ([EHNOS 84](#EHNOS84), reviewed in [Ellis-Olive 10](#EllisOlive10)).

## On galactic scales
 {#Properties}

The cold dark matter paradigm always worked excellently on cosmic scales, while seemingly facing problems with reproducing various observations on the scale of [[galaxies]]. But various recent computer simulations (e.g. with [FIRE](structure+formation#ReferencesFIRE), see the references [below](#ReferencesComputerSimulation)) point to these problems all going away when the complex dynamics of star-formation feedback/back-reaction on galactic [[structure formation]] (see there for more) is taken into account. We discuss this for:

1. _[The cusp-core behaviour](#CuspCoreProblem)_

1. _[Galactic rotation profiles](#GalacticRotationCurves)_

1. _[Missing satellites](#MissingSatellites)_

### Cusp-core behaviour
 {#CuspCoreProblem}

Early numerical simulation of dark matter exhibited dark matter halos around galaxies with a sharp density maximum at the center of the galaxy -- called a "cusp" in the density profile -- while observation on actual galaxies tends to favour a noticeably smoother density distribution at the center -- called a "core" of the density profile.

This discrepancy between [[theory (physics)|theory]] and [[experiment]] is (or was) known as the _cusp-core problem_ for dark matter [[model (physics)|models]] (see the references [below](#ReferencesCoreCuspProblem)). It is (or was) one of several issues that the dark matter model has (or had) on the scale of galaxies and smaller.

There are two potential resolutions to this problem: a) Either dark matter is more "exotic" than assumed in these simulations, such as being [[fuzzy dark matter]] or similar. Or, b), more mundanely, the existing simulations do not properly resolve a subtle effect shown already by more ordinary dark matter.

Recent investigations support possibility b), indicating that the density cusps seen in previous simulations are but an artifact of overly simplified numerical models, and that would-be cusps are diluted and hence smoothed out by bursty star formation leading to dark matter heating ([Pontzen-Governato 14](#PontzenGovernato14), [Read-Walker-Steger-18](#ReadWalkerSteger18)) and/or by dynamical friction in the interstallar medium ([ESH 01](#ESH01)). See [Read 19](#Read19) for review.

From [[Justin Read]] [22 Aug 2018](https://twitter.com/ReadDark/status/1032176578808168448) on [Read-Walker-Steger 18](#ReadWalkerSteger18):

> On large scales, the [[standard model of cosmology|standard cosmological model]] (LCDM) gives an excellent description of the [[cosmic microwave background]] [[radiation]] and the [[structure formation|growth of structure]]. However, on small scales, in our cosmic backyard known as the Local Group, there have been long-standing puzzles.  The oldest of these is the “cusp-core” problem. Gas rich dwarf galaxies have much less dark matter (DM) in their inner regions than early numerical models predicted. This could point to more exotic DM models, or to a failure of the cosmological model.

> However, there is a simpler solution. The early numerical models, above, considered a universe that contains only DM. Recent models that include gas cooling, star formation, and gas blow-out due to exploding stars find that star formation in dwarfs is bursty.

> This bursty star formation occurs due to repeated cycles of gas inflow and outflow. These cause the inner gravitational potential of the dwarf galaxy to fluctuate, kinematically "heating up" the dark matter and lowering its inner density ([Pontzen-Governato 14](#PontzenGovernato14)).

>  A key prediction of such models is that, at a fixed dark matter halo mass, more star formation leads to more dark matter "heating" and, therefore, a lower inner dark matter density, at least in dwarf galaxies. This is the prediction we set out to test in [Read-Walker-Steger 18](#ReadWalkerSteger18).

\linebreak

### Galactic rotation profile
 {#GalacticRotationCurves}

Computer simulation ([FIRE](https://fire.northwestern.edu)) of [[galaxy|galactic]] [[structure formation]] using the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]] qualitatively reproduces the peculiar [[galactic rotation curves]] that motivated dark matter (or [[MOND]], for that matter) in the first place ([Hopkins et al. 17, Figure 4, Figure 5](#Hopkins17)):

\begin{center}
\begin{imagefromfile}
  "file_name": "FIRE2GalacticRotationCurves.jpg",
  "width": 800
\end{imagefromfile}
\end{center}

> graphics grabbed from ([Hopkins et al. 17](#Hopkins17))

and also reproduces well the baryonic [[Tully-Fisher relation]] ([El-Badry et al. 18, Figure 4](#ElBadry18)) which used to be an issue in the [[standard model of cosmology|standard]] [[cold dark matter]] [[model (physics)|model]]:

\begin{center}
\begin{imagefromfile}
  "file_name": "FIRE2TullyFisher.jpg",
  "width": 500
\end{imagefromfile}
\end{center}

> graphics grabbed from ([El-Badry et al. 18](#ElBadry18))

More in detail, [[galaxy rotation curves]] exhibit a close correlation between [[angular velocity]] and total _visible_ enclosed math at any given [[radius]], called the _[[radial acceleration relation]]_ (RAR) or _[[mass-discrepancy acceleration relation]]_. This, too, is reproduced in [[cold dark matter]]-[[model (in theoretical physics)]]  computer simulation ([Santos-Santos et al. 15](#SantosSantosEtAl15), [Cintio-Lelli 15](#CintioLelli15), [Keller-Wadsley 16](#KellerWadsley16), [Ludlow et al 16](#LudlowEtAl16)). These early simulations were not found conclusive in [Lelli et al 16, section 8.2](#LelliEtAl16). But more detailed analysis ([PSF 18](#PSF18)) and more refined simulation ([Dutton-Maccio-Obreja-Buck 19](#DuttonMaccioObrejaBuck19)) has then been claimed to confirm the statement.

A conceptual explanation of the mechanism by stellar feedback is discussed in [GBFH 19](#GBFH19).

### Missing satellites?
 {#MissingSatellites}

On the apparent resolution of the [missing satellite problem](https://en.wikipedia.org/wiki/Dwarf_galaxy_problem):

[Garrison-Kimmel et al. 17 ](#GarrisonKimmel17), see [Garrison-Kimmel 18](#GarrisonKimmel18)

\linebreak

## Related concepts

* [[bullet cluster]]

* [[fuzzy dark matter]]

* [[dark energy]], [[dark radiation]]

* [[standard model of cosmology]]

* [[MOND]]

## References

### General

Review includes

* {#Einasto09} Jaan Einasto, _Dark matter_ ([arXiv:0901.0632](https://arxiv.org/abs/0901.0632)) 2009

* [[Syksy Räsänen]], _Dark matter_ cosmo 2015 lecture notes ([pdf](http://www.courses.physics.helsinki.fi/teor/cos1/cosmo2015_07.pdf))

* Wikipedia, _[Dark matter](http://en.wikipedia.org/wiki/Dark_matter)_

* [[Matthew Strassler]], _[Current hints of dark matter](http://profmattstrassler.com/articles-and-posts/relativity-space-astronomy-and-cosmology/dark-matter/current-hints-of-dark-matter-413/)_

* {#Resonaances} Resonaances, _[How many neutrinos in the sky](http://resonaances.blogspot.de/2013/01/how-many-neutrinos-in-sky.html)_

* {#BertoneHooper16} Gianfranco Bertone, Dan Hooper, _A History of Dark Matter_ ([arXiv:1605.04909](https://arxiv.org/abs/1605.04909))

* {#Hooper18} [[Daniel Hooper]], _In Defense of Dark Matter_, 2018 ([web](http://online.kitp.ucsb.edu/online/cdm-c18/hooper/))

Discussion of possible matter representations that could serve as dark matter:

* 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))

Discussion of [[neutrino]]-dark matter:

* {#NeutrinoWhite16} _A White Paper on keV Sterile Neutrino Dark Matter_,  Journal of Cosmology and Astroparticle Physics, Volume 2017, January 2017 ([arXiv:1602.04816](https://arxiv.org/abs/1602.04816))

Issues with defining gravitating mass on non-stationary spacetimes:

* Zhi-Wei Wang, [[Samuel L. Braunstein]], _Could dark matter be a natural consequence of a dynamical universe?_ ([arXiv:2011.09923](https://arxiv.org/abs/2011.09923))

### Evidence
 {#Evidence}

Evidence for dark matter on large cosmic scales is extremely strong, see the above reviews (...). Evidence on smaller scales, starting around the scale of galaxies is problematic, see the above reviews (...). In particular, as yet there is no direct detection of any dark matter particle.

(...)

Observational constraints:

* Torsten Bringmann, [[Maxim Pospelov]], *Novel direct detection constraints on light dark matter*, Phys. Rev. Lett. **122** (2019) 171801 [[arXiv:1810.10543](https://arxiv.org/abs/1810.10543), [doi:10.1103/PhysRevLett.122.171801](https://doi.org/10.1103/PhysRevLett.122.171801)]

Outlook on prospect of direct detection of dark matter:

* Gianfranco Bertone, Tim M. P. Tait, _A New Era in the Quest for Dark Matter_, Nature volume 562, pages 51–56 (2018)  ([arXiv:1810.01668](https://arxiv.org/abs/1810.01668))

* [[Maxim Pospelov]], _Dark matter identification efforts_, talk at *[[Strings 2022]]* [[indico:4940821](https://indico.cern.ch/event/1085701/contributions/4940821), [video](https://ustream.univie.ac.at/media/core.html?id=118664ce-20a5-4b62-a3b3-735ce5deea20)]

Argument that dark matter has already be seen across a range of direct detection experiments, but mis-interpreted as noise, due to negligence of the possibility of [[inelastic scattering|inelastic]] [[plasmon]] excitations in [[crystal|crystalline]] [[solid state physics|solid state]] detector materials:

* {#KBKK20} Noah Kurinsky, Daniel Baxter, Yonatan Kahn, Gordan Krnjaic, _A Dark Matter Interpretation of Excesses in Multiple Direct Detection Experiments_ ([arXiv:2002.06937](https://arxiv.org/abs/2002.06937))

* Yonatan Kahn, from slide 27 on in: _Dark matter review_, talk at Lake Louise Winter Institute, 2020 ([cern:event/846070/contributions/3693217/](https://indico.cern.ch/event/846070/contributions/3693217/))

Claimed rebuttal:

* Alan E. Robinson, Émile Michaud, _Comment on A dark matter interpretation of excesses in multiple direct detection experiments [arXiv:2002.06937]_ ([arXiv:2002.08893](https://arxiv.org/abs/2002.08893))

> The result of [Kurinsky et al.](#KBKK20) should not be taken as evidence for dark matter, although it does highlight the ongoing need to investigate the effect of collective modes how [sic] we detect radiation.

Reply:

* Noah Kurinsky, Daniel Baxter, Yonatan Kahn, Gordan Krnjaic, Peter Abbamonte, _Reply to Robinson and Michaud, arXiv:2002.08893_ ([arXiv:2003.00101](https://arxiv.org/abs/2003.00101))

> the points raised by RM do not invalidate our primary conclusions, as they pertain to a much different energy scale than we discuss in our paper.

### Core-cusp problem
  {#ReferencesCoreCuspProblem}

* {#ESH01} Amr El-Zant, Isaac Shlosman, Yehuda Hoffman, _Dark Halos: The Flattening of the Density Cusp by Dynamical Friction_,  The Astrophysical Journal, Volume 560, Number 2 ([arXiv:astro-ph/0103386](https://arxiv.org/abs/astro-ph/0103386))

* {#PontzenGovernato14} Andrew Pontzen, Fabio Governato, _Cold dark matter heats up_, Nature, 506, 171 - 178 (13 Feb 2014) ([arXiv:1402.1764](https://arxiv.org/abs/1402.1764))

* Jose Oñorbe, Michael Boylan-Kolchin, James S. Bullock, Philip F. Hopkins, Dušan Kerěs, Claude-André Faucher-Giguère, Eliot Quataert, Norman Murray, _Forged in FIRE: cusps, cores, and baryons in low-mass dwarf galaxies_, Monthly Notices of the Royal Astronomical Society, Volume 454, Issue 2, 1 December 2015 ([arXiv:1502.02036](https://arxiv.org/abs/1502.02036))

* [[Justin Read]], O. Agertz, M. L. M. Collins, _Dark matter cores all the way down_, Monthly Notices of the Royal Astronomical Society, Volume 459, Issue 3, 1 July 2016 ([arXiv:1508.04143](https://arxiv.org/abs/1508.04143))

* {#ReadWalkerSteger18} [[Justin Read]], M. G. Walker, P. Steger, _Dark matter heats up in dwarf galaxies_, Monthly Notices of the Royal Astronomical Society  ([arXiv:1808.06634](https://arxiv.org/abs/1808.06634), [doi:10.1093/mnras/sty3404](https://doi.org/10.1093/mnras/sty3404),  [talk recording](http://online.kitp.ucsb.edu/online/cdm-c18/read/), [press release](https://www.surrey.ac.uk/news/dark-matter-move))

* {#Read19} [[Justin Read]], _[Dark matter heats up in dwarf galaxies](https://www.simonsfoundation.org/event/dark-matter-heats-up-in-dwarf-galaxies/)_, Simons Foundation Lecture 2019

* Matthew D. A. Orkney, [[Justin Read]], et al., _EDGE: Two routes to dark matter core formation in ultra-faint dwarfs_ ([arXiv:2101.02688](https://arxiv.org/abs/2101.02688))

See also

* Wikipedia, _[Cuspy halo problem](https://en.wikipedia.org/wiki/Cuspy_halo_problem)_

### Computer simulation
 {#ReferencesComputerSimulation}

Discussion of computer simulation of [[dark matter]] [[structure formation]] on [[galaxy]]-scales:

* {#Hopkins17} Hopkins et al. _FIRE-2 Simulations: Physics versus Numerics in Galaxy Formation_. Monthly Notices of the Royal Astronomical Society, Volume 480, Issue 1, 11 October 2018, Pages 800–863 ([arXiv:1702.06148](https://arxiv.org/abs/1702.06148), [doi:10.1093/mnras/sty1690](https://doi.org/10.1093/mnras/sty1690))

* {#ElBadry18} El-Badry et al. _Gas Kinematics in FIRE Simulated Galaxies Compared to Spatially Unresolved HI Observations_, Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 2, 21 June 2018, Pages 1536–1548 ([arXiv:1801.03933](https://arxiv.org/abs/1801.03933), [doi:10.1093/mnras/sty730](https://doi.org/10.1093/mnras/sty730))

* {#GarrisonKimmel17} Shea Garrison-Kimmel et al. _Not so lumpy after all: modeling the depletion of dark matter subhalos by Milky Way-like galaxies_ ([arXiv:1701.03792](https://arxiv.org/abs/1701.03792))

* {#GarrisonKimmel18} Shea Garrison-Kimmel, _Next-generation Galaxy Formation Simulations with FIRE_, 2018 ([video recording](https://youtu.be/sSkrm66uDvw))

See also

* Jie Wang, Sownak Bose, Carlos S. Frenk, Liang Gao, Adrian Jenkins, Volker Springel, Simon D. M. White, _Universality in the structure of dark matter haloes over twenty orders of magnitude in halo mass_ ([arXiv:1911.09720](https://arxiv.org/abs/1911.09720))

### Galaxy rotation curves

On the [[radial acceleration relation]]:

* {#CintioLelli15} Arianna Di Cintio, Federico Lelli, _The mass discrepancy acceleration relation in a $\Lambda CDM$ context_, Monthly Notices of the Royal Astronomical Society: Letters, Volume 456, Issue 1, 11 February 2016, Pages L127–L131 ([arXiv:1511.06616](https://arxiv.org/abs/1511.06616), [doi:10.1093/mnrasl/slv185](https://doi.org/10.1093/mnrasl/slv185))

* {#SantosSantosEtAl15} Isabel M. Santos-Santos et al. _The distribution of mass components in simulated disc galaxies_ ([arXiv:1510.02474](https://arxiv.org/abs/1510.02474))

* {#KellerWadsley16} B.W. Keller, J.W. Wadsley, _$\Lambda CDM$ is Consistent with SPARC Radial Acceleration Relation_ ([arXiv:1610.06183](https://arxiv.org/abs/1610.06183))

* {#LudlowEtAl16} Aaron D. Ludlow et. al. _The Mass-Discrepancy Acceleration Relation: a Natural Outcome of Galaxy Formation in Cold Dark Matter halos_, Phys. Rev. Lett. 118, 161103 (2017) ([arXiv:1610.07663](https://arxiv.org/abs/1610.07663))

* {#PSF18} Chiara Di Paolo, Paolo Salucci, Jean Philippe Fontaine, _The Radial Acceleration Relation (RAR): the crucial cases of Dwarf Discs and of Low Surface Brightness galaxies_, ApJ 2019 ([arXiv:1810.08472](https://arxiv.org/abs/1810.08472))

* {#PSF18} Chiara Di Paolo, Paolo Salucci, Jean Philippe Fontaine, _The Radial Acceleration Relation (RAR): the crucial cases of Dwarf Discs and of Low Surface Brightness galaxies_, ApJ 2019 ([arXiv:1810.08472](https://arxiv.org/abs/1810.08472))

* {#DuttonMaccioObrejaBuck19} Aaron A. Dutton, Andrea V. Macciò, Aura Obreja, Tobias Buck, _NIHAO XVIII: Origin of the MOND phenomenology of galactic rotation curves in a LCDM universe_ ([arXiv:1902.06751](https://arxiv.org/abs/1902.06751))

Critical comments in

* {#LelliEtAl16} Federico Lelli, Stacy S. McGaugh, James M. Schombert, Marcel S. Pawlowski, section 8.2 of _One Law To Rule Them All: The Radial Acceleration Relation of Galaxies_ ([arXiv:1610.08981](https://arxiv.org/abs/1610.08981))

Conceptual explanation by stellar feedback:

* {#GBFH19} Michael Y. Grudić, Michael Boylan-Kolchin, Claude-André Faucher-Giguère, Philip Hopkins, _Stellar feedback sets the universal acceleration scale in galaxies_ ([arxiv:1910.06345](https://arxiv.org/abs/1910.06345))

### Axions

The observation that the "invisible [[axion]]" is a candidate for dark matter is due to three groups:

* {#PreskillWiseWilcek83} [[John Preskill]], M. Wise, [[Frank Wilcek]], _Cosmology of the invisible axion_, Phys. Lett. B 120:127-32 (1983)

* {#AbbottSikivie83} L.F. Abbott, P. Sikivie, _A Cosmological Bound on the Invisible Axion_, Phys.Lett. B120:133-36 (1983)

* {#DineFischler83} [[Michael Dine]], [[Willy Fischler]], _The Not So Harmless Axion_, Phys.Lett. B120:137-141 (1983)

A proposal that [[axions]] account for [[fuzzy dark matter]], and thus fix the [[WIMP]] model for dark matter with its problem of reproducing galactic rotation curves, is in

* {#HOTW16} Lam Hui, Jeremiah P. Ostriker, Scott Tremaine, [[Edward Witten]], _On the hypothesis that cosmological dark matter is composed of ultra-light bosons_, Phys. Rev. D 95, 043541 (2017) ([arXiv:1610.08297](https://arxiv.org/abs/1610.08297))

Discussion in the broader context of [[Kalb-Ramond fields]]:

* Christian Capanelli, Leah Jenks, Edward W. Kolb, Evan McDonough, *Cosmological Implications of Kalb-Ramond-Like-Particles* [[arXiv:2309.02485](https://arxiv.org/abs/2309.02485)]

### Lightest super-partner

The observation that the lightest supersymmetric particle would be a natural dark matter candidate goes back to

* {#EHNOS84} [[John Ellis]],  J.S. Hagelin, Dimitri V. Nanopoulos, [[Keith Olive]], M. Srednicki  _Supersymmetric relics from the Big Bang_, Nuclear Physics B 238[2]: 453-76, 1984 ([SPIRE](http://inspirehep.net/record/191839?ln=en))

with review in

* {#EllisOlive10} [[John Ellis]], [[Keith Olive]], _Supersymmetric Dark Matter Candidates_ ([arXiv:1001.3651](http://arxiv.org/abs/1001.3651))

### Gravitinos
 {#ReferencesGravitinos}

On the [[gravitino]] as a dark matter candidate:

* {#EllisOlive10} [[John Ellis]], [[Keith Olive]], _Supersymmetric Dark Matter Candidates_ ([arXiv:1001.3651](http://arxiv.org/abs/1001.3651))

A proposal for super-heavy [[gravitinos]] as [[dark matter]], by embedding [[D=4 N=8 supergravity]] into [[E10]]-[[U-duality]]-invariant [[M-theory]]:

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

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

following the proposal towards the end of

* {#GellMann83} [[Murray Gell-Mann]], introductory talk at _[Shelter Island II](https://en.wikipedia.org/wiki/Shelter_Island_Conference)_, 1983 ([[Gell-Mann_ShelterIslandII_1983.pdf:file]])

  in: _Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics_. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

Further discussion:

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Supermassive gravitinos and giant primordial black holes_ ([arXiv:2007.11889](https://arxiv.org/abs/2007.11889))

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Evidence for a stable supermassive gravitino with charge $2/3$?* [[arXiv:2303.09131](https://arxiv.org/abs/2303.09131)]

[[!redirects cold dark matter]]

[[!redirects core-cusp problem]]
[[!redirects core-cusp problems]]

[[!redirects cusp-core problem]]
[[!redirects cusp-core problems]]

[[!redirects cuspy halo problem]]
[[!redirects cuspy halo problems]]

[[!redirects missing satellite problem]]
[[!redirects missing satellite problems]]

[[!redirects dwarf galaxy problem]]
[[!redirects dwarf galaxy problems]]


dark radiation

#Contents#
* table of contents
{:toc}

## Example

* [[axion]]

## Related concepts

* [[dark matter]], [[dark energy]]

* [[standard model of cosmology]]

## References

* Wikipedia, _[Dark radiation](http://en.wikipedia.org/wiki/Dark_radiation)_


Dasein

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Cohesive toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to [[Hegel]] _[[Science of Logic]]_ ([§ 191](Science+of+Logic#191)) _Dasein_ ("determinate being") is the [[Aufhebung]] of [[becoming]], which in turn is the [[unity of opposites]] of [[nothing]] and [[being]].

| |  | [[Dasein]] | |  |
|--|--|--|--|--|
| [[becoming]] : | [[nothing]] | $\;\;\;\dashv$ | [[being]] | : [[ceasing]] |

According to [[Hegel]] _[[Phenomenology of Spirit]]_ ([Vorrede §55](http://ncatlab.org/nlab/show/Phenomenology+of+Spirit#Vorrede55)) this is the _[[Nous]]_ of [[Anaxagoras]].

Following suggestions by [[William Lawvere]] as discussed in detail at _[Aufhebung --  Examples -- Aufhebung of Becoming -- Determinate being](Aufhebung#AufhebungOfBecoming)_, this is formalized by the [[level of a topos|level]] given by the [[sharp modality]] in its [[unity of opposites|opposition]] to the [[flat modality]]:

$$
  \array{
     \flat \; &\dashv& \;\;\sharp
     \\
     \vee \; &\nearrow_{\mathrlap{Dasein}}& \;\;\vee
     \\
     \emptyset \; &\dashv& \;\;\ast
  }
  \,.
$$

So $\sharp$ is a higher [[level of a topos|level]] version of $\ast$ ([[being]]). Indeed, this is the statement of ([SoL § 194](Science+of+Logic#194)).

## Related concepts

* [[category of being]]

[[!redirects determinate being]]

[[!redirects dasein]]

[[!redirects Daseyn]]
[[!redirects daseyn]]


daseinisation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Given a [[C*-algebra]] $A$ thought of as the [[algebra of observables]] of a [[quantum mechanical system]], write $ComSub(A)$ for its [[poset of commutative subalgebras]]. Then the [[presheaf topos]] over $ComSub(A)$ with its canonical [[spectral presheaf]] as well as the [[presheaf topos]] over the [[opposite category]] $ComSub(A)^{op}$ canonically regarded as a [[ringed topos]] -- the "[[Bohr topos]]", might both be regarded as [[topos theory|topos-theoretic]] incarnations of the [[phase space]] of the given [[quantum mechanical system]]. By standard [[quantum mechanics]] every [[self-adjoint operator]] $a \in A_{sa}$ is to be regarded as an "[[observable]] on phase space", in some sense. Hence one may ask if $a$ induces in a precise sense a function on the phase space [[internalization|internal]] to these toposes.

A construction from each $a \in A_{sa}$ of a [[clopen subset]] $\delta^o(a) \subset \Sigma_A$ of the [[spectral presheaf]] $\Sigma$ of $A$ has been given in ([Isham-Döring 07](#IshamDoering07)) for [[von Neumann algebras]] $A$. There this is called the "daseinisation" of $a$. An analogous construction for the [[Bohr toposes]] of [[C*-algebras]] has been given in ([Heunen-Landsman-Spitters 09](#HeunenLandsmanSpitters09)). A direct identification of [[quantum observables]] with homorphisms of [[ringed toposes]] out of the Bohr topos is discussed at _[Bohr topos -- The observables](Bohr+topos#TheObservables)_.

## References

* [[Andreas Döring]], [[Chris Isham]], _A Topos Foundation for Theories of Physics_  ([arXiv:quant-ph/0703060](http://arxiv.org/abs/quant-ph/0703060), [arXiv:quant-ph/0703062](http://arxiv.org/abs/quant-ph/0703062), [arXiv:quant-ph/0703066](http://arxiv.org/abs/quant-ph/0703066))
  {#IshamDoering07}

* [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], _A Topos for Algebraic Quantum Theory_ Communications in Mathematical Physics 291. 63-110. 2009
  {#HeunenLandsmanSpitters09}

[[!redirects daseinisation]]
[[!redirects daseinization]]


data

#Contents#
* table of contents
{:toc}

## Idea

**Data** are [[values]] of [[qualitative]] or [[quantitative]] [[variables]], belonging to a set of items. Data in [[computing]] (or [[data processing]]) are represented in a [[structure]], often [[tabular]] (represented by [[rows]] and [[columns]]), a [[tree]] (a [[set]] of [[nodes]] with [[parent]]-[[children]] [[relation|relationship]] or a [[graph]] structure (a set of [[interconnected]] nodes). Data are typically the [[result]]s of [[measurement]]s and can be [[visualised]] using [[graph]]s or [[image]]s.

Data as an [[abstract]] [[concept]] can be viewed as the lowest level of [[abstraction]] from which [[information]] and then [[knowledge]] are derived.

* [From the Wikipedia article on data](https://en.wikipedia.org/wiki/Data)

## Related concepts

* [[data type]]

* [[database]]

* [[topological data analysis]]


data analysis

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[topological data analysis]]

* [[statistical inference]]

## References

See also:

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


data type

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of [[type]] of [[data]] in [[computer science]].

## Examples

* [[Bool]], [[bit]]

* [[String (computer science)]]

In [[quantum computation]]:

* [[qbit]], [[qdit]]

## Related concepts

* [[data]]

* [[database]], [[database theory]]

* [[monadic effect]]

* [[function type]]

* [[object-oriented programming]]

* [[topological data analysis]]

## References

* Jim Morris, _Types are not sets_, Proceedings of the 1st annual ACM SIGACT-SIGPLAN symposium on Principles of programming languages (POPL), 1973. ([doi](http://dx.doi.org/10.1145/512927.512938))

* [[John C. Reynolds]], _Towards a Theory of Type Structure_, Colloquium on Programming, Paris, 9-11 April 1974. ([pdf](http://repository.cmu.edu/cgi/viewcontent.cgi?article=2289&context=compsci))

* Barbara Liskov and Stephen Zilles, _Programming with Abstract Data Types_, Proceedings of the ACM SIGPLAN symposium on Very high level languages, pp. 50-59, 1974. ([doi](http://dx.doi.org/10.1145/800233.807045)) ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.136.3043))

* {#Sale77} [[Arthur Sale]], _Primitive data types_, The Australian Computer Journal, Vol. 9, No. 2, July 1977 ([pdf](http://eprints.utas.edu.au/139/1/PrimitiveData.pdf))

* D. J. Lehmann, M. B., Smyth, *Algebraic specification of data types: A synthetic approach*,  Math. Systems Theory **14** (1981) 97–139 [[doi:10.1007/BF01752392](https://doi.org/10.1007/BF01752392)]

  > (via [[CPOs]] in [[domain theory]])

[[!redirects data types]]


database

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[data]], [[data type]]

* [[database theory]]

* [[store comonad]]

* [[lens (in computer science)]]

## References

See also:

* Wikipedia, *[Database](https://en.m.wikipedia.org/wiki/Database)*

[[!redirects databases]]