Nicholas Manton

* [institute page](http://www.damtp.cam.ac.uk/people/n.s.manton/)

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

## Selected writings

On [[gauge-Higgs unification]]:

* {#Manton79} [[Nicholas Manton]], _A new six-dimensional approach to the Weinberg-Salam model_, Nuclear Physics B Volume 158, Issue 1, 8 October 1979, Pages 141-153 (doi:10.1016/0550-3213(79)90192-5)

On the [[Schwinger effect]]:

* {#AffleckManton82} [[Ian Affleck]], [[Nicholas Manton]], _Monopole pair production in a magnetic field_, Nuclear Physics B Volume 194, Issue 1, (1982) Pages 38-64 (doi:10.1016/0550-3213(82)90511-9

* {#AffleckAlvarezManton82} [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas Manton]], _Pair production at strong coupling in weak external fields_, Nuclear Physics B Volume 197, Issue 3 (1982) Pages 509-519 (doi:10.1016/0550-3213(82)90455-2)

  > (via [[worldline formalism]])

On [[Yang-Mills monopoles]]:

* {#GibbonsManton86} [[Gary Gibbons]], [[Nicholas Manton]], _Classical and Quantum Dynamics of BPS Monopoles_, Nucl. Phys. B274 (1986) 183-224 ([spire:18322](http://inspirehep.net/record/18322), doi:10.1016/0550-3213(86)90624-3)

On [[Skyrmions]] via [[rational maps]] from the [[complex plane]], hence [[holomorphic functions]] from the [[Riemann sphere]], to itself, as for [[Yang-Mills monopoles]]:

* [[Conor J. Houghton]], [[Nicholas Manton]], [[Paul Sutcliffe]], *Rational Maps, Monopoles and Skyrmions*, Nucl. Phys. B510 (1998) 507-537 ([arXiv:hep-th/9705151](https://arxiv.org/abs/hep-th/9705151), doi:10.1016/S0550-3213(97)00619-6)

* [[Nicholas S. Manton]], [[Bernard M.A.G. Piette]], *Understanding Skyrmions using Rational Maps*, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 ([doi:10.1007/978-3-0348-8268-2_27](https://doi.org/10.1007/978-3-0348-8268-2_27), [arXiv:hep-th/0008110](https://arxiv.org/abs/hep-th/0008110))

On constructing [[Skyrmions]] from [[instantons]] in [[D=5 Yang-Mills theory]] ([[hadron Kaluza-Klein theory]]):

* {#AtiyahManton89} [[Michael Atiyah]], [[Nicholas Manton]], _Skyrmions from instantons_, Phys.  Lett.  B, 222(3):438–442, 1989 (doi:10.1016/0370-2693(89)90340-7)

On [[Skyrmion]] [[model (in theoretical physics)|model]] for [[carbon]] [[nucleus]]:

* D. T. J. Feist, P. H. C. Lau, [[Nicholas Manton]], _Skyrmions up to Baryon Number 108_ ([arXiv:1210.1712](https://arxiv.org/abs/1210.1712))

* {#LauMaonton14} P.H.C. Lau, [[Nicholas Manton]], _States of Carbon-12 in the Skyrme Model_,  Phys. Rev. Lett. 113, 232503 (2014) ([arXiv:1408.6680](https://arxiv.org/abs/1408.6680))

On [[skyrmion]] [[model (in theoretical physics)|models]] of [[atomic nuclei]]:

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

* [[Nicholas Manton]], _Lightly Bound Skyrmions, Tetrahedra and Magic Numbers_ ([arXiv:1707.04073](https://arxiv.org/abs/1707.04073))

* [[Nicholas Manton]], *Skyrmions – A Theory of Nuclei*, World Scientific (2022) [[doi:10.1142/q0368 ](https://doi.org/10.1142/q0368 )]

* [[Nicholas Manton]], *Skyrmions as Models for Nuclei*, Ischia (2022) [[pdf](https://agenda.infn.it/event/19753/contributions/170544/attachments/90671/122280/SkyIschia.pdf)]

On patterns in the [[characteristic classes]] ([[Chern classes]]) of [[complex surfaces]], together with some speculations about an anlogy with [[atomic number]] of [[atomic nuclei]]:

* {#AtiyahManton16} [[Michael Atiyah]], [[Nicholas Manton]], _Complex Geometry of Nuclei and Atoms_ ([arXiv:1609.02816](https://arxiv.org/abs/1609.02816))

## Related $n$Lab entries

* [[Skyrmion]]

* [[instanton]]

* [[baryon]]

  [[meson]], [[hadron]]

* [[atomic nucleus]]

* [[gauge-Higgs unification]]

category: people

[[!redirects Nicholas S. Manton]]


Nicholas Meadows

* [webpage](http://www.math.uwo.ca/index.php/profile/148/)

## related $n$Lab entries

* [[(infinity,2)-sheaf]]

category: people


Nicholas Mercator

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

> (not to be confused with [[Gerardus Mercator]])

## Selected writings

On [[logarithms]] (and the [[Mercator series]]):

* [[Nicholas Mercator]], *Logarithmotechnia: Sive Methodus constuendi Logrithmos*, London 1667  ([GoogleBooks](https://books.google.de/books?hl=en&lr=&id=HFl1JqyXpSoC&oi=fnd&pg=PP6&dq=Logarithmotechnia.&ots=2xDsxWNWwO&sig=PfSwMZjoMnW7EAtH79uxX3ATsqc&redir_esc=y#v=onepage&q=Logarithmotechnia.&f=false))

category: people


Nicholas Proudfoot

* [webpage](http://pages.uoregon.edu/njp/)

## Selected writings

On [[intersection cohomology]] of [[matroids]]:

* Tom Braden, [[June Huh]], Jacob P. Matherne, [[Nicholas Proudfoot]], Botong Wang, *Singular Hodge theory for combinatorial geometries* [[arXiv:2010.06088](https://arxiv.org/abs/2010.06088)]

category: people

[[!redirects Nick Proudfoot]]


Nicholas Read

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

## Selected writings

On [[quantum spin liquids]] (then called "[[resonating-valence-bond states]]"):

* [[Nicholas Read]], B. Chakraborty, *Statistics of the excitations of the resonating-valence-bond state*, Phys. Rev. B **40** (1989) 7133 [[doi:10.1103/PhysRevB.40.7133](https://doi.org/10.1103/PhysRevB.40.7133)]

On [[anyons]] satisfying [[non-abelian group|non-abelian]] [[braid group statistics]] in the [[fractional quantum Hall effect]]:

* [[Gregory Moore]], [[Nicholas Read]], _Nonabelions in the fractional quantum Hall effect_, Nucl. Phys. 360B(1991)362 ([pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf), doi:10.1016/0550-3213(91)90407-O)

and generalization to [[su(2)-anyons]] via [[parafermions]]:

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

On [[anyon|anyonic]] [[Majorana zero modes]] in [[topological superconductors]]:

* [[Nicholas Read]], Dmitry Green, _Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect_,  Phys. Rev. B **61** 10267 (2000) $[$[arXiv:cond-mat/9906453](https://arxiv.org/abs/cond-mat/9906453)$]$

category: people


Nicholas Warner

* [webpage](https://dornsife.usc.edu/cf/faculty-and-staff/faculty.cfm?pid=1003810)

## Selected writings

On [[gauged supergravity|gauged]] [[D=7 supergravity]]:

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], [[Nicholas Warner]], _Noncompact Gauging and Critical Points of Maximal Supergravity in Seven Dimensions_, Nucl. Phys. B249 (1985) 381 (doi:10.1016/0550-3213(85)90046-X>doi:10.1016/0550-3213(85)90046-X)

On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", a little later called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

On [[supersymmetry]] and [[G-structure]] (notably [[Spin(7)-structure]] in [[M-theory on 8-manifolds]]):

* {#IshamPope88} [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006))

* {#IshamPopeWarner88} [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_,  Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([cds:185144](http://cds.cern.ch/record/185144), [doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009))

On [[Landau-Ginzburg models]]:

* {#VafaWarner89} [[Cumrun Vafa]], [[Nicholas Warner]], _Catastrophes and the Classification of Conformal Theories_, Phys.Lett. B218 (1989) 51 (doi:10.1016/0370-2693(89)90473-5)

On [[gauged supergravity|gauged]] [[D=5 supergravity]]:

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Gauged $N = 8$ Supergravity in Five Dimensions_, Phys. Lett. 154B, (1985) 268 ([spire:207663](http://inspirehep.net/record/207663), doi:10.1016/0370-2693(85)90361-2)

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Compact and Non–Compact Gauged Supergravity Theories in Five Dimensions_, Nucl. Phys. B272 (1986) 598 ([spire:219727](http://inspirehep.net/record/219727), doi:10.1016/0550-3213(86)90237-3)

On [[E-strings]] as [[M5-branes]] [[wrapped brane|wrapped]] on a [[half K3]]:

* J. A. Minahan, D. Nemeschansky, [[Nicholas Warner]], _Partition Functions for BPS States of the Non-Critical $E_8$ String_, Adv. Theor. Math. Phys.1:167-183, 1998 ([arXiv:hep-th/9707149](https://arxiv.org/abs/hep-th/9707149))

* J. A. Minahan, D. Nemeschansky, [[Cumrun Vafa]], [[Nicholas Warner]], _E-Strings and $N=4$ Topological Yang-Mills Theories_, Nucl. Phys. B527 (1998) 581-623 ([arXiv:hep-th/9802168](https://arxiv.org/abs/hep-th/9802168))

On the [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of [[fuzzy 3-sphere|fuzzy]] [[3-sphere]]-shape, with emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:

* [[Iosif Bena]], [[Nicholas Warner]], _A harmonic family of dielectric flow solutions with maximal supersymmetry_, JHEP 0412:021, 2004 ([arXiv:hep-th/0406145](https://arxiv.org/abs/hep-th/0406145))

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

On non-uniqueness of [[black rings]]:

* [[Iosif Bena]], [[Simon Ross]], [[Nicholas Warner]], _Coiffured Black Rings_, Class.Quant.Grav. 31 (2014) 165015 ([arXiv:1405.5217](https://arxiv.org/abs/1405.5217))

On [[AdS-CFT duality|AdS-CFT dual]] [[type II supergravity]]-solutions corresponding to [[D1-D5-P bound states]] ("[[superstrata]]"):

* [[Iosif Bena]], [[Stefano Giusto]], [[Rodolfo Russo]], [[Masaki Shigemori]], [[Nicholas P. Warner]], *Habemus Superstratum! A constructive proof of the existence of superstrata*, JHEP 1505 (2015) 110 [[arXiv:1503.01463](https://arxiv.org/abs/1503.01463)]

and the non-supersymmetric generalization via [[AdS3-CFT2 duality]] ("[[microstrata]]"):

* [[Bogdan Ganchev]], [[Anthony Houppe]], [[Nicholas P. Warner]], *Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries*, J. High Energ. Phys. **2021** 28 (2021)
[[arXiv:2107.09677](https://arxiv.org/abs/2107.09677)]

* [[Bogdan Ganchev]], [[Stefano Giusto]], [[Anthony Houppe]], [[Rodolfo Russo]], *$AdS_3$ holography for non-BPS geometries*, European Physical Journal C **82** 217 (2022)
 [[https://arxiv.org/abs/2112.03287](https://arxiv.org/abs/2112.03287), [doi:10.1140/epjc/s10052-022-10133-2](https://doi.org/10.1140/epjc/s10052-022-10133-2)]

On (in-)stability of non-[[supersymmetry|supersymmetric]] [[AdS spacetime|AdS]] [[string theory vacua|vacua in string theory]]:

* [[Iosif Bena]], [[Krzysztof Pilch]], [[Nicholas Warner]], _Brane-Jet Instabilities_,  J. High Energ. Phys. 2020, 91 (2020) ([arXiv:2003.02851](https://arxiv.org/abs/2003.02851))

On [[M2-M5-brane bound states]]:

* [[Iosif Bena]], [[Anthony Houppe]], [[Dimitrios Toulikas]], [[Nicholas P. Warner]], *Maze Topiary in Supergravity* [[arXiv:2312.02286](https://arxiv.org/abs/2312.02286)]

[[!redirects Nicholas P. Warner]]

category: people


Nicholas Wheeler

* [personal page](https://www.reed.edu/physics/faculty/wheeler/)

## Selected writings

On [[generalized quantum measurement channels]]:

* {#Wheeler12} [[Nicholas Wheeler]], *Generalized Quantum Measurement* (2012) [[pdf](https://www.reed.edu/physics/faculty/wheeler/documents/Quantum%20Mechanics/Miscellaneous%20Essays/Generalized%20Quantum%20Measurement/Generalized%20Quantum%20Measurements.pdf), [[Wheeler-GeneralizedQuantumMeasurement.pdf:file]]]

category: people


Nicholas Woodhouse

* [webpage](http://www.wadham.ox.ac.uk/fellows-staff/academics/175)

category: people


Nick Benton

* [webpage](http://research.microsoft.com/en-us/um/people/nick/)

## Selected writings

On [[categorical semantics]] for the [[exponential modality]] in [[linear logic]]:

* {#BBP92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Term assignment for intuitionistic linear logic*, Technical report 262, Computer Laboratory, University of Cambridge (August 1992) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-262.pdf), [[BentonBiermanDePaiva-TermAssignment.pdf:file]]]

  > (published abridged as [BBPH92](#BBPH92))

* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]

and specifically as the [[comonad]] induced from an [[adjoint functors|adjunction]] between ordinary [[types]] and [[linear types]]:

* {#Benton95} [[Nick Benton]], *A mixed linear and non-linear logic: Proofs, terms and models*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** [[doi:10.1007/BFb0022251](https://doi.org/10.1007/BFb0022251), [[BentonLinearLogic.pdf:file]]]

On [[modal operator]]-like [[monad in computer science|monads in computer science]]:

* {#BentonBiermanPaiva98} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Computational Types from a Logical Perspective*, Journal of Functional Programming **8** 2  (1998) 177-193 [[doi:10.1017/S0956796898002998](https://doi.org/10.1017/S0956796898002998), [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5778)]

On [[monads in computer science]]:

* [[Nick Benton]], [[John Hughes]], [[Eugenio Moggi]], *Monads and Effects*, in: *Applied Semantics*, Lecture Notes in Computer Science **2395**, Springer (2002) 42-122 [[doi:10.1007/3-540-45699-6_2](https://doi.org/10.1007/3-540-45699-6_2)]

* {#Benton15} [[Nick Benton]], _Categorical Monads and Computer Programming_, in: *[Impact150: stories of the impact of mathematics](https://www.lms.ac.uk/2015/impact150-stories-impact-mathematics)*. London Mathematical Society (2015)  [[pdf](https://www.lms.ac.uk/sites/lms.ac.uk/files/2.%20Benton%20-%20Categorical%20Monads%20and%20Computer%20Programming.pdf), [[Benton-CategoricalMonads.pdf:file]], [doi:10.1112/i150lms/t.0002](https://www.lms.ac.uk/2015/impact150-stories-impact-mathematics)]

On [[dependent linear type theory]]:

* {#KPB15} [[Neelakantan Krishnaswami]], [[Pierre Pradic]], [[Nick Benton]], *Integrating Dependent and Linear Types*,  ACM SIGPLAN Notices **50** 1 (2015)  17–30  [[pdf](https://www.cl.cam.ac.uk/~nk480/dlnl-paper.pdf),  [doi:10.1145/2775051.2676969](https://doi.org/10.1145/2775051.2676969)]

## Related entries

* [[linear type theory]]

* [[dependent linear type theory]]

category: people


Nick E. Bonesteel

* [webpage](http://web2.physics.fsu.edu/~bonesteel/)

## Selected writings

On [[topological quantum compilation]] and approximating (cf. the [[Solovay-Kitaev theorem]]) given [[quantum gates]] by (i.e. [[quantum compilation|compiling]] them to) [[quantum circuit|cicuits]] of [[anyon]] [[braid representation|braid]] [[quantum gates|gates]] (generally considered for [[su(2)-anyons]] and here mostly for universal [[Fibonacci anyons]], to some extent also for non-universal [[Majorana anyons]]):

* [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) $[$[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)$]$

* [[Layla Hormozi]], [[Georgios Zikos]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Compiling*, Phys. Rev. B **75** 165310 (2007) $[$[arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111), [doi:10.1103/PhysRevB.75.165310](https://doi.org/10.1103/PhysRevB.75.165310)$]$

* [[Layla Hormozi]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Computing with Read-Rezayi States*, Phys. Rev. Lett. **103** 160501 (2009) $[$[doi:10.1103/PhysRevLett.103.160501](https://doi.org/10.1103/PhysRevLett.103.160501), [arXiv:0903.2239](https://arxiv.org/abs/0903.2239)$]$

* M. Baraban, [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Resources required for topological quantum factoring*, Phys. Rev. A **81** 062317 (2010) $[$[doi:10.1103/PhysRevA.81.062317](https://doi.org/10.1103/PhysRevA.81.062317), [arXiv:1002.0537](https://arxiv.org/abs/1002.0537)$]$

  > (focus on compiling [[Shor's algorithm]])

* Caitlin Carnahan, Daniel Zeuch, [[Nicholas E. Bonesteel]], *Systematically generated two-qubit anyon braids*, Phys. Rev. A **93** 052328 (2016) [[arXiv:1511.00719](https://arxiv.org/abs/1511.00719), [doi:10.1103/PhysRevA.93.052328](https://doi.org/10.1103/PhysRevA.93.052328)]

and particularly by "weaves":

* [[Steven H. Simon]], [[Nick E. Bonesteel]], [[Michael H. Freedman]], N. Petrovic, [[Layla Hormozi]], *Topological Quantum Computing with Only One Mobile Quasiparticle*, Phys. Rev. Lett. 96 (2006) 070503 ([arXiv:quant-ph/0509175](https://arxiv.org/abs/quant-ph/0509175), [doi:10.1103/PhysRevLett.96.070503](https://doi.org/10.1103/PhysRevLett.96.070503))

category: people

[[!redirects Nick Bonesteel]]
[[!redirects Nicholas E. Bonesteel]]


Nick Evans

* [personal page](https://www.southampton.ac.uk/~evans/)

## Selected writings

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

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

On the [[Higgs field]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), doi:10.1007/JHEP02(2021)058]

On [[colour superconductivity]] in [[holographic QCD]]:

* Kazem Bitaghsir Fadafan, Jesus Cruz Rojas, [[Nick Evans]], _A Holographic Description of Colour Superconductivity_, Phys. Rev. D **98** (2018) 066010  [[arXiv:1803.03107](https://arxiv.org/abs/1803.03107), [doi:10.1103/PhysRevD.98.066010](https://doi.org/10.1103/PhysRevD.98.066010)]

On [[domain walls]] and [[holographic QCD]]:

* [[Nick Evans]], [[Jack Mitchell]], *Domain Wall AdS/QCD*, Phys. Rev. D **104** (2021) 094018 [[arXiv:2108.12152](https://arxiv.org/abs/2108.12152), [doi:10.1103/PhysRevD.104.094018](https://doi.org/10.1103/PhysRevD.104.094018)]

* [[Nick Evans]], [[Jack Mitchell]], *Thermal Transitions in Domain Wall AdS/QCD* [[arXiv:2207.10374](https://arxiv.org/abs/2207.10374)]

On [[holographic QCD]], related non-conformal [[holographic principle|holography]] and [[DBI action]]-effects

* [[Nick Evans]], *Holography for Multi-Representation and Chiral Matter*, talk at *[[Strings 2022]]* [[indico:4940848](https://indico.cern.ch/event/1085701/contributions/4940848), [slides](https://indico.cern.ch/event/1085701/contributions/4940848/attachments/2482358/4261644/Nick%20Evans%20Tuesday-Research-Evans.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=aecca42b-c952-42f0-84c6-5b23230534b7)]

category: people


Nick Gilbert

Nick Gilbert is a British mathematician formerly based in Heriot Watt university in Edinburgh, now retired.

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

* [Researchgate page](https://www.researchgate.net/profile/Nick-Gilbert-3)

His recent research work has benn in the study of [[ordered groupoids]] and the use of [[groupoids]]  in the algebra of rewriting for [[group presentations]] and related structures.

## Selected writings

On [[knot theory]]:

* [[Nick Gilbert]], [[Tim Porter]], *Knots and surfaces*, Oxford University Press (1994) [[ISBN:9780198514909](https://global.oup.com/academic/product/knots-and-surfaces-9780198514909?cc=de&lang=en&)]

On [[rewriting]] in [[group presentations]]:

* [[Nick Gilbert]], Emma Antonia Mcdougall, _Groupoids and the algebra of rewriting in group presentations_, Beiträge zur Algebra und Geometrie **62** 3 (2021) 1-17 [[doi:10.1007/s13366-020-00531-6](https://doi.org/10.1007/s13366-020-00531-6), [arXiv:1901.04348](https://arxiv.org/abs/1901.04348)]

category:people

[[!redirects N. D. Gilbert]]


Nick Gurski

Nick Gurski is an assistant professor at Case Western Reserve University. Previously he was at the University of Sheffield, U.K.

* [website](http://casfaculty.case.edu/nick-gurski/cv/)

## Selected writings

On [[tricategories]]:

* [[Nick Gurski]], _An algebraic theory of tricategories_, PhD thesis (2007) [[pdf](https://dokumen.tips/documents/an-algebraic-theory-of-tricategories.html), [[Gurski-AlgebraicTricategories.pdf:file]]]

On [[iterated loop spaces]] and [[coherence theorems]] for [[monoidal 2-categories]] ([[braided monoidal 2-category|braided]], [[sylleptic monoidal 2-category|sylleptic]], [[symmetric monoidal 2-category|symmetric]]) and, somewhat implicitly: [[sylleptic 3-groups]]:

* [[Nick Gurski]], *Loop spaces, and coherence for monoidal and braided monoidal bicategories*, Advances in Mathematics Volume 226, Issue 5, 20 March 2011, Pages 4225-4265 ([arXiv:1102.0981](http://arxiv.org/abs/1102.0981), [doi:10.1016/j.aim.2010.12.007](https://doi.org/10.1016/j.aim.2010.12.007))

* [[Nick Gurski]], [[Angélica M. Osorno]], *Infinite loop spaces, and coherence for symmetric monoidal bicategories*, Adv. Math. 246 (2013) 1-32 ([arXiv:1210.1174](https://arxiv.org/abs/1210.1174))

## Related $n$Lab page

* [[tricategory]]

* [[Trimble n-category]]

category: people


Nick Huggett

Nick Huggett is  a Professor in the Philosophy Department at the University of Illinois at Chicago.

* [webpage](https://philpeople.org/profiles/nick-huggett)

category:people


Nick Rozenblyum

* [webpage](http://math.uchicago.edu/~nick/)

* [[Notes on geometric Langlands]]

category: people

[[!redirects N. Rozenblyum]]


Nico Courts

Nico Courts ([personal webpage](https://nicocourts.com/)) is a graduate student at University of Washington, Seattle studying [[representation theory]].

category: people


Nicola Di Vittorio

* [webpage](https://divittorio.github.io)

## Selected writings

A formal approach to [[pasting diagram|pasting]] in [[Gray categories]]:

* {#DiVittorio2023} [[Nicola Di Vittorio]], _A Gray-categorical pasting theorem_, Theory and Applications of Categories **39** 5 (2023) 150-171 [[tac:39-05](http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html)]

On lifting [[derivators]] from the [[homotopy categories of (infinity,1)-categories]] to the [[homotopy 2-categories]] of [[(infinity,2)-categories]]:

* [[Nicola Di Vittorio]], *2-derivators* (2023). [[arXiv:2309.05216](https://arxiv.org/abs/2309.05216)]

category: people


Nicola Gambino

Nicola Gambino is associate professor of [[mathematics]] at Manchester.

* [website](https://personalpages.manchester.ac.uk/staff/nicola.gambino/)

## Selected writings

On [[inductive types]] in [[homotopy type theory]] via homotopy-[[initial algebras over an endofunctor]]:

* {#AwodeyGambinoSojakova12} [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Inductive types in homotopy type theory*, LICS'12: (2012) 95–104 [[arXiv:1201.3898](http://arxiv.org/abs/1201.3898), [doi:10.1109/LICS.2012.21](https://doi.org/10.1109/LICS.2012.21), [Coq code](https://github.com/HoTT/HoTT/tree/master/Coq/IT)]

  \linebreak

  Exposition:

  [[Steve Awodey]], *Inductive types in HoTT* (Jan 2012) [[blog post](http://homotopytypetheory.org/2012/01/19/inductive-types-in-hott/)]

and analogously on [[higher inductive types]]:

* [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Homotopy-initial algebras in type theory*, Journal of the ACM **63** 6 (2017) 1–45 [[arXiv:1504.05531](http://arxiv.org/abs/1504.05531), [doi:10.1145/3006383](https://doi.org/10.1145/3006383)]

On the [[constructive model structure on simplicial sets]]:

* {#GambinoSattlerSzumilo19} [[Nicola Gambino]], [[Christian Sattler]], [[Karol Szumiło]], _The constructive Kan-Quillen model structure: two new proofs_ ([arXiv:1907.05394](https://arxiv.org/abs/1907.05394))

On [[Kripke-Joyal semantics]] and [[forcing]] in [[homotopy type theory]]:

* {#AGH21} [[Steve Awodey]], [[Nicola Gambino]], [[Sina Hazratpour]], *Kripke-Joyal forcing for type theory and uniform fibrations* (2021) [[arXiv:2110.14576](https://arxiv.org/abs/2110.14576)]

Including discussion of [[enriched profunctors]]\ bimodules, in the context of operads and [[analytic functors]]:

* [[Nicola Gambino]], [[André Joyal]], _On operads, bimodules and analytic functors_,
Memoirs of the American Mathematical Society, 249 (1184), 2017, (v) + 110pp.
 [Arxiv preprint: 1405.7270](https://arxiv.org/pdf/1405.7270.pdf)  [DOI:10.1090/memo/1184](https://doi.org/10.1090/memo/1184)

From the AMS website: _The authors develop further the theory of operads and also analytic functors. In particular, they introduce the bicategory OpdBim of operad bimodules, that has operads as 0-cells, operad bimodules as 1-cells and operad bimodule maps as 2-cells, and prove that it is cartesian closed. In order to obtain this result, the authors extend the theory of distributors and the formal theory of monads._

## Related entries

* [[homotopy limit]]

* [[monad]]

* [[type theory]], [[homotopy type theory]]

* [[higher inductive type]], [[initial algebra over an endofunctor]], [[W-types]]

* [[software verification]]

category: people
[[!redirects N. Gambino]]


Nicola Pinamonti

* [webpage](http://www.pinamonti.it/index.php?PG=prima)

## Related $n$Lab entries

* [[cosmology]]

* [[cosmological constant]]

* [[quantum gravity]]

category: people


Nicolaas de Bruijn

Nicolaas Govert de Bruijn (known as Dick) was a Dutch mathematician, famous for the [[Automath]] project.

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

## Selected writings

Introducing [[de Bruijn indices]]:

* [[Nicolaas de Bruijn]], *Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem*, Indagationes Mathematicae (Proceedings) **75** 5 (1972) 381-392 [doi:10.1016/1385-7258(72)90034-0, [pdf](http://alexandria.tue.nl/repository/freearticles/597619.pdf)]

Introducing the notion of [[type telescopes]]:

* [[Nicolaas de Bruijn]], §3.1 *Telescopic mappings in typed lambda calculus*, Information and Computation **91** 2 (1991) 189-204 [doi:10.1016/0890-5401(91)90066-B]

## Related entries

* [[Automath]]

* [[de Bruijn indices]]

* [[type telescope]]

[[!redirects Dick de Bruijn]]
[[!redirects N.G. de Bruijn]]


Nicolaas Kuiper

[[!redirects N. H. Kuiper]]

* [wikipedia page](https://en.wikipedia.org/wiki/Nicolaas_Kuiper)

## Selected writings:

Proof of [[Kuiper's theorem]] (that [[U(ℋ)]] is [[contractible]] in the [norm topology](UH#NormTopology)):

* {#Kuiper64} [[Nicolaas Kuiper]], *Contractibility of the unitary group in Hilbert space*, Topology, 3, 19-30 (1964)

On [[microbundles]]:

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles I. Elementary Theory_, Invent. Math., 1, (1966), 1 – 17.

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles II. Semisimplicial Theory_, Invent. Math., 1, (1966), 243 – 259.

The [[Arnold-Kuiper-Massey theorem]]:

* [[Nicolaas Kuiper]], _The quotient space of $\mathbb{C}P(2)$ by complex conjugation is the 4-sphere_, Mathematische Annalen, 1974 ([doi:10.1007/BF01432386](https://doi.org/10.1007/BF01432386))

category:people

[[!redirects N. H. Kuiper]]


Nicolaas Landsman

* [website](http://www.math.ru.nl/~landsman/)

## Selected writings

On [[strict deformation quantization]] of the [[particle]] coupled to a [[Yang-Mills theory|Yang-Mills]] [[background field]]:

* [[Klaas Landsman]], *Strict deformation quantization of a particle in external gravitational and Yang-Mills fields*, Journal of Geometry and Physics __12__ 2 (1993) 93-132 [doi:10.1016/0393-0440(93)90010-C]

Survey of [[strict deformation quantization]]:

* [[Nicolaas P. Landsman]], *Classical and quantum representation theory* [[arXiv:hep-th/9411172](https://arxiv.org/abs/hep-th/9411172)]

On [[C-star algebra|$C^\ast$-]][[groupoid algebras]] (including [[group algebras]]) as [[strict deformation quantizations]] of [[Lie-Poisson structures]] in the generality of [[Lie algebroids]]:

* [[Nicolaas P. Landsman]], B. Ramazan, Ex. 11.1 in: *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]

On the [[Born rule]] in [[quantum mechanics]] (at the end with an eye towards [[Bohr toposes]]):

* [[Klaas Landsman]], *The Born rule and its interpretation*, in: *Compendium of Quantum Physics*, Springer (2009) 64-70 [[doi:10.1007/978-3-540-70626-7_20](https://doi.org/10.1007/978-3-540-70626-7_20), [pdf](https://www.math.ru.nl/~landsman/Born.pdf), [[Landsman-BornRule.pdf:file]]]

On [[Bohr toposes]]:

* {#HeunenLandsmanSpitters09} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], *Bohrification of operator algebras and quantum logic*, Synthese **186** 3 (2012) 719-752 [[arXiv:0905.2275](http://arxiv.org/abs/0905.2275), [doi;10.1007/s11229-011-9918-4](https://doi.org/10.1007/s11229-011-9918-4)]

On the foundations of [[quantum theory]] ([[quantum probability]], [[operator algebra]], [[measurement problem]], [[classical limit]], [[spontaneous symmetry breaking]], [[Bohr topos]]):

* {#Landsman17} [[Klaas Landsman]], _Foundations of quantum theory -- From classical concepts to Operator algebras_, Springer Open 2017 ([doi:10.1007/978-3-319-51777-3](https://link.springer.com/book/10.1007/978-3-319-51777-3), [pdf](https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf))

On the [[Penrose singularity theorem]] and the [[cosmic censorship hypothesis]]:

* [[Klaas Landsman]], _Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose_ ([arXiv:2101.02687](https://arxiv.org/abs/2101.02687))

## Related entries

* [[Bohr topos]]

[[!redirects Nicolaas P. Landsman]]

[[!redirects Klaas Landsman]]

category: people


Nicolae Popescu

__Nicolae Popescu__ was a Romanian mathematician with focus on algebra, especially study of rings and modules, [[abelian categories]] and localization theory. Cf. also [[Gabriel-Popescu theorem]].

* planetmath: [Nicolae Popescu, mathematician](http://planetmath.org/nicolaepopescumathematician)

* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)
* Nicolae Popescu, Liliana Popescu, _Theory of categories_. Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1979.

category: people


Nicolai Kraus

* [webpage](http://www.cs.nott.ac.uk/~psznk/)

## Selected writings

On the [[universal property]] of [[propositional truncation]]:

* [[Nicolai Kraus]], *The General Universal Property of the Propositional Truncation*, in *TYPES 2014* Leibniz International Proceedings in Informatics (LIPIcs) **39** (2015) [[arXiv:1411.2682](https://arxiv.org/abs/1411.2682), [doi:10.4230/LIPIcs.TYPES.2014.111](https://doi.org/10.4230/LIPIcs.TYPES.2014.111)]

On [[homotopy n-types]] ("[[homotopy levels]]") in [[homotopy type theory]]:

* [[Nicolai Kraus]], *Truncation levels in Homotopy Type Theory*, Nottingham (2015) [[pdf](https://eprints.nottingham.ac.uk/28986/1/thesis.pdf), [eprints:28986](https://eprints.nottingham.ac.uk/28986)]

On [[weakly constant functions]] and [[propositional truncation]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*,  Logical Methods in Computer Science **13** 1 [doi:10.23638/LMCS-13(1:15)2017, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]

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

* [[Jakob von Raumer]], [[Nicolai Kraus]], *Path Spaces of Higher Inductive Types in Homotopy Type Theory*,  34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Vancouver, Canada, 2019 ([arXiv:1901.06022](https://arxiv.org/abs/1901.06022))

On [[quotient inductive types]]:

* {#ADK2017} [[Thorsten Altenkirch]], [[Nils A. Danielsson]], [[Nicolai Kraus]], _Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type_, FoSSaCS 2017, Lecture Notes in Computer Science **10203** Springer (2017) [[arXiv:1610.09254](https://arxiv.org/abs/1610.09254)]

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)]

On [[free groups]] and [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Thorsten Altenkirch]], *Free Higher Groups in Homotopy Type Theory*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, (2018) [[arXiv;1805.02069](https://arxiv.org/abs/1805.02069), [doi:10.1145/3209108.3209183](https://doi.org/10.1145/3209108.3209183)]

On [[quotient types]] and [[higher inductive types]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

On [[ordinals]] in [[type theory]] and [[marked extensional well-founded orders]]:

* {#deJongEtAl23} [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], [[Chuangjie Xu]], *Set-Theoretic and Type-Theoretic Ordinals Coincide* [[arXiv:2301.10696](https://arxiv.org/abs/2301.10696)]

## Related entries

* [[univalent category]]

* [[category object in an (infinity,1)-category]]

category: people


Nicolai map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The *Nicolai map* is a (non-[[local quantum field theory|local]] and non-linear) transformation of [[boson|bosonic]] [[quantum field theories|quantum fields]] to [[free quantum field|free fields]] whose existence characterizes the original bosonic [[quantum field theory]] as arising by [[fermionic path integral|integrating out the fermions]] of a [[global symmetry|globally]] [[supersymmetry|supersymmetric]] field theory.

## References

The original discussion:

* [[Hermann Nicolai]], *On a new characterization of scalar supersymmetric theories*, Physics Letters B **89** 3–4 (1980) Pages 341-346 [doi:10.1016/0370-2693(80)90138-0]

* [[Hermann Nicolai]], *Supersymmetry and functional integration measures*, Nuclear Physics B
**176** 2 (1980) 419-428 [doi:10.1016/0550-3213(80)90460-5]

* [[Hermann Nicolai]], *Supersymmetric functional integration measures*, NATO Advanced Study Institute on Supersymmetry, Bonn (Aug 1984) 393-420 [[cds:155731](https://cds.cern.ch/record/155731)]

The infinitesimal version:

* [[Olaf Lechtenfeld]], *Konstruktion der Nicolai-Abbildung in supersymmetrischen Feldtheorien*, PhD thesis, Bonn (1984) [[pdf](https://lib-extopc.kek.jp/preprints/PDF/2000/0030/0030157.pdf), [[Lechtenfeld-NicolaiAbbildung.pdf:file]]]

* K. Dietz, [[Olaf Lechtenfeld]], *Nicolai maps and stochastic observables from a coupling constant flow*, Nuclear Physics B **255** (1985) 149-173 [doi:10.1016/0550-3213(85)90132-4]

* K. Dietz, [[Olaf Lechtenfeld]], *Ghost-free quantisation of non-abelian gauge theories via the Nicolai transformation of their supersymmetric extensions*, Nuclear Physics B **259** 2–3 (1985) 397-411 [doi:10.1016/0550-3213(85)90642-X]

* [[Olaf Lechtenfeld]], *Stochastic variables in ten dimensions?*, Nuclear Physics B **274** 3–4 (1986) 633-652 [doi:10.1016/0550-3213(86)90531-6]

* [[Rainald Flume]], [[Olaf Lechtenfeld]], *On the stochastic structure of globally supersymmetric field theories*, Physics Letters B **135** 1–3 (1984) 91-95 [doi:10.1016/0370-2693(84)90459-3]

Construction in terms of a [[Dyson formula|path-ordered integral]]:

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Universal form of the Nicolai map*, Phys. Rev. D **104** 021701 (2021) [[arXiv:2104.00012](https://arxiv.org/abs/2104.00012), [doi:10.1103/PhysRevD.104.L021701](https://doi.org/10.1103/PhysRevD.104.L021701)]

* [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Is the Nicolai map unique?*,  J. High Energ. Phys. **2022** 139 (2022) [[arXiv:2207.09471](https://arxiv.org/abs/2207.09471), doi:10.1007/JHEP09(2022)139]

On its [[non-perturbative effects|non-perturbative nature]]:

* [[Olaf Lechtenfeld]], *Supersymmetric large-order perturbation with the Nicolai map*, Physics Letters B
**835** (2022) 137507 [[arXiv:2208.06420](https://arxiv.org/abs/2208.06420), [doi:10.1016/j.physletb.2022.137507](https://doi.org/10.1016/j.physletb.2022.137507)]

Review:

* [[Olaf Lechtenfeld]], *The Nicolai-map approach to supersymmetry* [[arXiv:2309.00481](https://arxiv.org/abs/2309.00481)]

On the Nicolai map for the [[CP^N sigma-model|$\mathbb{C}P^n$ sigma-model]]:

* [[Lorenzo Casarin]], [[Olaf Lechtenfeld]], [[Maximilian Rupprecht]], *Nicolai maps with four-fermion interactions* [[arXiv:2310.19946](https://arxiv.org/abs/2310.19946)]

[[!redirects Nicolai maps]]


Nicolai Reshetikhin

[[!redirects Nicolai Reshitikhin]]

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

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

## Selected writings

Introducing the [[Reshetikhin-Turaev construction]]:

* {#ReshetikhinTuraev91} [[Nikolai Reshetikhin]], [[Vladimir Turaev]], _Invariants of 3-manifolds via link polynomials and quantum groups_. Invent. Math. 103 (1991), no. 3, 547–597. ([doi:10.1007/BF01239527](https://doi.org/10.1007/BF01239527), [pdf](http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf))

On [[Lie groups]] and [[quantum groups]]:

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

On [[conformal blocks]] for [[Liouville theory]]:

* [[Babak Haghighat]], [[Yihua Liu]], [[Nicolai Reshetikhin]], *Flat Connections from Irregular Conformal Blocks* [[arXiv:2311.07960](https://arxiv.org/abs/2311.07960)]

## Related entries

* [[3d TQFT]]

* [[Chern-Simons theory]]

* [[quantization of 3d Chern-Simons theory]]

category: people

[[!redirects Nikolai Reshetikhin]]
[[!redirects Reshetikhin]]


Nicolaos Toumbas

* [Institute page](https://ucy.ac.cy/dir/en/component/comprofiler/userprofile/nick)

## Selected writings

Introducing the notion of [[giant gravitons]]:

* [[John McGreevy]], [[Leonard Susskind]], [[Nicolaos Toumbas]], _Invasion of the Giant Gravitons from Anti-de Sitter Space_, JHEP 0006:008, 2000 ([arXiv:hep-th/0003075](https://arxiv.org/abs/hep-th/0003075))

category: people


Nicolas Berkouk

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

## Selected writings

On [[level set]] [[persistent homology]] and [[sheaf theory]]:

* {#BerkoukGinotOudot19} [[Nicolas Berkouk]], [[Grégory Ginot]], [[Steve Oudot]], *Level-sets persistence and sheaf theory* $[$[arXiv:1907.09759](https://arxiv.org/abs/1907.09759), [talk](Center+for+Quantum+and+Topological+Systems#GinotAtHomotopicalPerspectivesOnTDAJune2022)$]$

category: people


Nicolas Ginoux

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=93592)

## related $n$Lab entries

* [[wave equation]]

* [[Green's function]]

category: people


Nicolas Moeller

* [webpage](http://www.theorie.physik.uni-muenchen.de/17ls_th_cosmology_en/members/former_mem/mueller1/index.html)

## Related entries

* [[string field theory]]

category: people


Nicolas Orantin

* [webpage](https://www.math.tecnico.ulisboa.pt/~norantin/research.html)

## related $n$Lab entries

* [[topological recursion]]

* [[random matrix]]

category: people


Nicolas Ricka

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

## related $n$Lab entries

* [[Anderson duality]]

category: people


Nicolas Robles

* [webpage](http://www.math.uzh.ch/index.php?id=assistenten&L=&key1=7073&key2=&key3=&keySemId=)

## related $n$Lab entries

* [[zeta function regularization]]

category: people


Nicolas Spaltenstein

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12935)

> (image from [here](https://pnp.mathematik.uni-stuttgart.de/idsr/idsr1/geckmf/spaltenstein.html))

## Selected writings

Influential precusor discussion eventually leading to the [[model structure on unbounded chain complexes]]:

* [[Nicolas Spaltenstein]], *Resolutions of unbounded complexes*, Compositio Mathematica **65** 2 (1988) 121-154 [[numdam:CM_1988__65_2_121_0](http://www.numdam.org/item?id=CM_1988__65_2_121_0), [eudml:89885](https://eudml.org/doc/89885)]

category: people


Nicolas Tabareau

[[!redirects Nicholas Tabareau]]

Nicolas Tabareau is a French computer scientist. He is an INRIA researcher based in the École des Mines de Nantes.

* [home page](http://www.emn.fr/z-info/ntabareau/)

category:people


Nicole Berline

* [webpage](http://www.math.polytechnique.fr/~berline/)

## Selected writings

On [[heat kernels]], [[Dirac operators]] and [[index theory]]:

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], Section 7.1 of: *Heat Kernels and Dirac Operators*, Springer 2004 ([ISBN:9783540200628](https://www.springer.com/gp/book/9783540200628))

category: people


Nicolò Petri

* webpage ?

## Selected writings

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))

* [[Nicolò Petri]], section 6.5 of: _Supersymmetric objects in gauged supergravities_ ([arxiv:1802.04733](https://arxiv.org/abs/1802.04733))

* {#Petri18} [[Nicolò Petri]], _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

On [[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

Similarly for [[D2-D6-NS5 brane bound states]]:

* Federico Faedo, [[Yolanda Lozano]], [[Nicolo Petri]], _Searching for surface defect CFTs within $AdS_3$_, JHEP 11 (2020) 052 ([arXiv:2007.16167](https://arxiv.org/abs/2007.16167))

On [[AdS2/CFT1]] in [[massive type IIA string theory]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_2$ solutions and their massive IIA origin_, JHEP 05 (2019) 107 ([arXiv:1811.11572](https://arxiv.org/abs/1811.11572))

On [[AdS2/CFT1]] on [[D1-D3 brane intersections]]:

* [[Giuseppe Dibitetto]], [[Yolanda Lozano]], [[Nicolò Petri]], Anayeli Ramirez, _Holographic Description of M-branes via $AdS_2$_ ([arXiv:1912.09932](https://arxiv.org/abs/1912.09932))

[[!redirects Nicolo Petri]]


Nicolò Sibilla

* [website](http://guests.mpim-bonn.mpg.de/sibilla/)

category: people


Niels A. Obers

* [personal page](https://www.nbi.dk/~obers/Welcome.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=a5crI5YAAAAJ&hl=en)

## Selected writings

On [[U-duality]], [[M-theory]] and [[M-branes]]:

* {#ObersPioline99} [[Niels Obers]], [[Boris Pioline]], *U-duality and M-Theory*, Phys. Rept. **318** (1999) 113-225 [[arXiv:hep-th/9809039](https://arxiv.org/abs/hep-th/9809039), doi:10.1016/S0370-1573(99)00004-6]

On the *[[blackfold]]*-approach to describing [[black branes]]:

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Essentials of Blackfold Dynamics*, JHEP 1003 063 (2010) [[arXiv:0910.1601](https://arxiv.org/abs/0910.1601), doi:10.1007/JHEP03(2010)063]

* [[Roberto Emparan]], [[Troels Harmark]], [[Vasilis Niarchos]], [[Niels A. Obers]], *Blackfolds in Supergravity and String Theory*, JHEP 1108 (2011) 154 [[arXiv:1106.4428](https://arxiv.org/abs/1106.4428)doi:10.1007/JHEP08(2011)154]

On [[black brane|black]] [[M2-M5 brane bound states]]:

* Jay Armas, [[Vasilis Niarchos]], [[Niels A. Obers]], Section 2 of: _Thermal transitions of metastable M-branes_, J. High Energ. Phys. (2019) 2019: 128 [[arXiv:1904.13283](https://arxiv.org/abs/1904.13283)]

category: people

[[!redirects Niels Obers]]


Niels Bohr

One of the founding fathers of [[quantum mechanics]].

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

## Selected writings

On the [[interpretation of quantum mechanics]] (see also at *[[Bohr topos]]*):

* {#Bohr49} [[Nils Bohr]], *Discussion with Einstein on Epistemological Problems in Atomic Physics*, in: P. A. Schilpp (ed.), *Albert Einstein, Philosopher-Scientist*, The Library of Living Philosophers **VII** Evanston (1949) 201-241, and *Niels Bohr Collected Works* **7** (1996) 339-381 [doi:10.1016/S1876-0503(08)70379-7]

> [pp. 209]: "I advocated a point of view conveniently termed 'complementarity", suited to embrace the characteristic features of individuality of quantum phenomena, and at the same time to clarify the peculiar aspects of the observational problem in this field of experience. For this purpose, it is decisive to recognize that *however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms*. The argument is simply that by the word 'experiment' we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.

> This crucial point, which was to become a main theme of the discussions reported in the following, implies the *impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear.*"

For commentary see:

* {#Scheibe73} [[Erhard Scheibe]], p. 24 of: *Bohr's interpretation of quantum mechanics*, Chapter I in: _The logical analysis of quantum mechanics_, Oxford: Pergamon Press (1973)

and see at _[[interpretation of quantum mechanics]]_ the section _[Bohr's standpoint](interpretation+of+quantum+mechanics#BohrStandpoint)_.

> when it comes to atoms, language can be used only as in poetry.

([[Niels Bohr]], [1920](https://en.wikiquote.org/wiki/Niels_Bohr))

## Related $n$Lab entries

* [[Bohr-Sommerfeld quantization]]

* [[Bohr topos]], [[interpretation of quantum mechanics]]

category: people

[[!redirects Niels Bohr]]
[[!redirects Nils Bohr]]

[[!redirects Bohr]]


Niels Henrik Abel

[[!redirects Nils Abel]]
[[!redirects Niels Abel]]

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

There are many theorems which are called Abel's theorem.

* Springer [eom]: [abel theorem](http://www.encyclopediaofmath.org/index.php/Abel_theorem)

Abel-Ruffini's theorem -- generic algebraic equation of order 5 or higher is insoluble in radicals.

Abel-Jacobi's theorem is treated under [[Jacobian variety]].

## Related entries

* [[abelian group]]

category: people

[[!redirects Nils Henrik Abel]]
[[!redirects Abel's theorem]]


Niels Tuning

* [webpage](https://www.nikhef.nl/~h71/)

## Selected writings

On [[flavour physics]] and [[flavour anomalies]] seen at the [[LHCb experiment]]:

* [[Niels Tuning]], _Recent results from LHCb_, Nikhef 2020 ([nikhef event:2253](https://indico.nikhef.nl/event/2253), [pdf](https://indico.nikhef.nl/event/2253/material/slides/0.pdf), [[TuningFlavourAnomaly20.pdf:file]])

  > In recent years a number of results in flavour physics have drawn some attention due to tensions with respect to Standard Model predictions. An overview of these results will be shown, together with a few recent results from this Spring, both on the flavour anomalies and 'classical' flavour physics

category: people


Niels van der Weide

* [website](https://nmvdw.github.io/)

## Publications

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

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

* [[Niccolò Veltri]], [[Niels van der Weide]], *Constructing Higher Inductive Types as Groupoid Quotients*, Logical Methods in Computer Science **17** 2 (2021)  [[arXiv:2002.08150](https://arxiv.org/abs/2002.08150), doi:10.23638/LMCS-17(2:8)2021]

* [[Niels van der Weide]], *Constructing 1-Truncated Finitary Higher Inductive Types as Groupoid Quotients*, [[Homotopy Type Theory Electronic Seminar Talks]], 6 February 2020 ([video](https://www.youtube.com/watch?v=sQhqQTMcMKw), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weide-2020-02-06-HoTTEST.pdf))

On [[finite sets]] in [[homotopy type theory]]/[[univalent foundations of mathematics]]:

* [[Dan Frumin]], [[Herman Geuvers]], [[Léon Gondelman]], [[Niels van der Weide]], _Finite Sets in Homotopy Type Theory_, in *CPP 2018: Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs* (2018) 201–214 [[doi:10.1145/3167085](https://doi.org/10.1145/3167085), [pdf](http://cs.ru.nl/~nweide/FiniteSetsInHoTT.pdf)]

On [[2-dimensional type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 12 (2022) 1–14, [[doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334)]

On [[univalent bicategory|univalent]] [[bicategories]] in [[homotopy type theory]]:

* [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], _Bicategories in Univalent Foundations_, Mathematical Structures in Computer Science **31** 10 (2022) 1232-1269 [[arXiv:1903.01152](https://arxiv.org/abs/1903.01152), [doi:10.1017/S0960129522000032](https://doi.org/10.1017/S0960129522000032)&rbrack

## Related entries

* [[univalent foundations of mathematics]]

* [[UniMath project]]

category: people


Nielsen invariant

Nielsen invariant is introduced in

* J. Nielsen, _Über die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ringflächen, die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ringflächen_, Math. Ann. __82__(1-2):83–93, 1920.

Nielsen invariant and Reidemeister torsion are rather related. There are many article now on this relation, see

* Alexander Fel'shtyn, _New directions in Nielsen-Reidemeister theory_, [arxiv/0712.2601](http://arxiv.org/abs/0712.2601); _Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem_, [arxiv/0807.0045](http://arxiv.org/abs/0807.0045); _The growth rate of Floer homology and symplectic zeta function_, [arxiv/1101.4495](http://arxiv.org/abs/1101.4495)

There are also some precursors like [[dilogarithm]] functions from the end of 19th century, closely related to the subject.


Nielsen-Schreier theorem

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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_theorem #NielsenSchreierTheorem}
###### Theorem
**(Nielsen--Schreier theorem)**

Every [[subgroup]] $H$ of a [[discrete group|discrete]] [[free group]] $G$ is itself a [[free group]]. Moreover, if $G$ is free on $k$ generators and $H$ has index $n$ in $G$, then $H$ is free on $n k - n + 1$ generators.

=--
The second part of this yields what is called the _Schreier index formula_.

This has a number of different proofs. The first proof, perhaps nowadays the most familiar proof, is based on covering space theory (in particular, covering spaces of topological graphs). The second proof is quite similar in spirit but is based on groupoids (another way of viewing homotopy 1-types), and has the advantage that it circumvents the point-set topology considerations inherent to covering space theory.

+-- {: .proof}
###### Topological Proof (sketch)
**of theorem \ref{NielsenSchreierTheorem}**

A free group $G = F(S)$ is, by the [[van Kampen theorem]], a [[fundamental group]] of a bouquet[^fine] of $S$ many circles, which is in particular a [[connected space|connected]] 1-dimensional [[CW-complex]] $X$ (in simpler language, a connected graph). By general [[covering space]] theory, given a (pointed) connected space $X$ with $\pi_1(X, x) = G$, subgroups $H \subseteq G$ are in bijective correspondence with isomorphism classes of connected covering spaces $p: (E, e) \to (X, x)$, with $\pi_1(E, e) \cong H$. Now, a covering space $E$ of a connected graph $X$ is also a connected graph. But any connected graph is [[homotopy equivalence|homotopy equivalent]] to a bouquet of circles, whose fundamental group is a free group. Thus $\pi_1(E, e)$ is a free group.

The second statement is proved by observing that the Euler characteristic of $X$ is $\chi(X) = 1 - \rho(G)$, where $\rho(G)$ is the number of generators of the free group $G$, and $\chi(E) = n\chi(X)$ if $p \colon E \to X$ is a covering space with $n$ points in each fiber.

Full details may be found in [May](#Maybook). Key technical ingredients include: (1) each connected graph $E$ contains a maximal [[tree]] $T$ (using [[Zorn's lemma]]), (2) the quotient map $E \to E/T$ is a homotopy equivalence, and $E/T$ is a bouquet of circles.
=--

This topological proof can be reformulated in more algebraic language, using a little groupoid theory (groupoids being [[homotopy n-type|homotopy 1-types]]). A key construction here is the [[action groupoid]] $X \sslash G$ formed from a $G$-set $X$, also called a homotopy quotient or homotopy orbit space.

+-- {: .proof}
###### Groupoidal Proof
**of theorem \ref{NielsenSchreierTheorem}**

By the discussion at _[free groupoid -- fundamental group](http://ncatlab.org/nlab/show/free+groupoid#FundamentalGroup)_, we may think of the [[free group]] $F(S)$ as the [[fundamental group]] of a homotopy 1-type which is freely built from a single vertex and one loop from that vertex to itself for each element in $S$. This is the _[[free groupoid]]_ $*\sslash F(S)$ on this [[directed graph]] (a bouquet of circles). It is a classical fact (see at _[[universal principal bundle]]_) that the [[universal cover]] of this is the [[contractible]] groupoid $\mathbf{E}(F(S))$, the [[action groupoid]] $F(S)\sslash F(S)$ of $F(S)$ acting on itself from the right and that its [[quotient]] by the $F(S)$-action from the left recovers the original groupoid with fundamental group $F(S)$. If we instead quotient only by the given subgroup $H \hookrightarrow F(S)$, we obtain a connected groupoid with fundamental group $H$ (meaning simply that at any point or object $x$ of the groupoid, the group $\hom(x, x)$ is isomorphic to $H$).

Thus, consider the [[quotient]] $H \backslash \mathbf{E}F(S) \simeq H \backslash \backslash *$, [[generalized the|the]] connected groupoid whose [[fundamental group]] is $\pi_1 = H$. By the properties of [[free groupoids]] discussed at _[free groupoid -- fundamental group](http://ncatlab.org/nlab/show/free+groupoid#FundamentalGroup)_ it is sufficient to show that $H \backslash \mathbf{E}F(S)$ is isomorphic to the [[free groupoid]] on a connected [[directed graph]]. But $\mathbf{E}F(S)$ itself is the free groupoid on a directed graph, namely on the _action graph_ $F(S)\sslash S$ (which is the same as the [[Cayley graph]] given by a set $S$ of generators and no relations), and we may consider the quotient graph $H \backslash (F(S) \sslash S)$. The free groupoid functor $F$ preserves this quotient. Thus we have

$$
  H \backslash \mathbf{E}F\left(S\right) = H  \backslash F\left( F\left(S\right)\sslash S \right) =  F \left(\left(H \backslash F\left(S\right)\right)\sslash S\right)
  \,.
$$
as desired.
=--

+-- {: .num_remark}
###### Remark

The original algebraic proof of theorem \ref{NielsenSchreierTheorem} was rather long and complicated, based on Nielsen's method of short cancellations in [[combinatorial group theory]] based on words (Nielsen's transformations of words, word length functions, ...). The class of _'projective groups'_ ( that is, in this context, [[projective object]]s in [[Grp|the category of groups]]) coincides with the class of free groups.

The above simple [[homotopy theory|homotopy-theoretic]] proof was indicated in ([Higgins](#Higgins)), also ([Gilbert-Porter](#GilbrtPorter)).

=--
##Related entries

* [[Cayley graph]]

## Related concepts

* [submodules of a free module](free+module#SubmodulesOfFreeModules)

+-- {: .num_theorem #DedekindTheorem}
###### Theorem
**(Dedekind's theorem)**

Every subgroup of a [[free abelian group]] is itself a free abelian group.

=--

See at _[pid - Structure theory of modules](http://ncatlab.org/nlab/show/principal+ideal+domain#StructureTheoryOfModules)_ for details.

## References

The restricted statement that every subgroup of a free _[[abelian group]]_ is itself free was originally given by [[Richard Dedekind]].

[[Jakob Nielsen]] proved the statement for finitely-generated subgroups in 1921. The full theorem was proven in

* [[Otto Schreier]], *Die Untergruppen der freien Gruppen*, Abh. Math. Semin. Univ. Hambg. **5** (1927) 161–183 [[doi:10.1007/BF02952517](https://doi.org/10.1007/BF02952517)]

The topological proof is due to

* [[Reinhold Baer]], [[Friedrich Levi]], _Freie Produkte und ihre Untergruppen_, Compositio Math. 3: 391–398.

and another one due to [[Jean-Pierre Serre]].

Versions of the topological proof are given in many places. One is

* [[Peter May]], _A Concise Course in Algebraic Topology_, Chicago Lectures in Mathematics, U. Chicago Press (1999), pp. 34-35. Revised version available online ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf)), chapter 4.
{#Maybook}

Another can be found reviewed towards the end of

* [[N. D. Gilbert]], [[T. Porter]], _Knots and Surfaces_, Oxford U.P., 1994.
 {#GilbertPorter}

Other related texts include

* H. Zieschang, _Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam_, Invent. Math. __10__, 4--37, 1970.

* W. Magnus, A. Karras, D. Solitar, _Combinatorial group theory_

* R. Lyndon, P. Schupp, _Combinatorial group theory_, Springer 1977 (Russian transl. Mir, Moskva 1980)

A groupoid-based proof of the Nielsen-Schreier theorem appears as theorem 9 in chapter 14 in

* [[Philip Higgins|P. J. Higgins]], _Categories and groupoids_, [Repr. Theory Appl. Categ., 1 – 178](http://www.emis.de/journals/TAC/reprints/articles/7/tr7abs.html) , ISSN
1201-561X, reprint of the 1971 original _Notes on categories and groupoids_, (Van Nostrand
Reinhold, London; MR0327946) with a new preface by the author.
 {#Higgins}

Similar material can also be found in

* [[R. Brown]], _Topology and groupoids_, Booksurge, 2006.

[^fine]: A bouquet of circles is the [[coproduct]] of a collection of circles, each with a basepoint, in the category of [[pointed spaces]]. In other words, a [[pushout|wide pushout]] in [[Top]] of inclusions of a 1-point space into circles, with the evident pointing.

Discussion in [[homotopy type theory]]/[[univalent foundations]] (see also [[mathematics presented in HoTT]]):

* {#Swan20} [[Andrew Swan]], _On the Nielsen-Schreier Theorem in Homotopy Type Theory_ ([arXiv:2010.01187](https://arxiv.org/abs/2010.01187))

[[!redirects Nielsen's theorem]]
[[!redirects Nielsen theorem]]
[[!redirects Nielsen–Schreier theorem]]
[[!redirects Nielsen--Schreier theorem]]

[[!redirects Dedekind-Nielsen-Schreier theorem]]


Niemeier lattice

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[lattice (discrete subgroup)]]

* [[Leech lattice]], [[E8-lattice]]

* [[Mathieu moonshine]], [[umbral moonshine]]

## References

The original article:

* [[Hans-Volker Niemeier]], *Definite quadratische Formen der Dimension 24 und Diskriminante 1*, Journal of Number Theory **5** 2 (1973) 142-178 [doi:10.1016/0022-314X(73)90068-1]

See also:

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

Relation to [[heterotic string theory]]:

* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, [inspire:233203](https://inspirehep.net/literature/233203)]

[[!redirects Niemeier lattices]]


Nigel Higson

* [webpage](http://www.math.psu.edu/higson/math/Home.html)

category: people


Nigel Hitchin

Nigel Hitchin is professor of pure [[mathematics]] at Oxford.

* [website](http://people.maths.ox.ac.uk/~hitchin/)

## Selected writings

Introducing the [[ADHM construction]] for [[Yang-Mills instantons]]:

* [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65**  3 (1978) 185-187  [doi:10.1016/0375-9601(78)90141-X]

On the [[moduli spaces of monopoles]]:

* [[Michael Atiyah]], [[Nigel Hitchin]], _The geometry and dynamics of magnetic monopoles_  M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 ([jstor:j.ctt7zv206](https://www.jstor.org/stable/j.ctt7zv206))

* [[Michael Atiyah]], [[Nigel Hitchin]], J. T. Stuart and M. Tabor, _Low-Energy Scattering of Non-Abelian Magnetic Monopoles_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 315, No. 1533, New Developments in the Theory and Application of Solitons (Aug. 13, 1985), pp. 459-469 ([jstor:37546](https://www.jstor.org/stable/37546))

On [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]:

* {#AtiyahDijkgraafHitchin10} [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]], _Geometry and physics_, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 ([pdf](http://rsta.royalsocietypublishing.org/content/368/1914/913.full.pdf), [doi;10.1098/rsta.2009.0227](https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227))

On [[SL(2,O)]]:

* {#Hitchin18} [[Nigel Hitchin]], _$SL(2)$ over the octonions_, Mathematical Proceedings of the Royal Irish Academy. Vol. 118. No. 1. Royal Irish Academy, 2018 ([arXiv:1805.02224](https://arxiv.org/abs/1805.02224))

## Related entries

* [[moduli space of monopoles]]

* [[Hitchin connection]]

* [[Hitchin fibration]]

* [[Hitchin functional]]

* [[generalized complex geometry]]

* [[Kobayashi-Hitchin correspondence]]

* [[SL(2,O)]]

category: people

[[!redirects Hitchin]]


Nigel Ray

* [webpage](https://personalpages.manchester.ac.uk/staff/nigel.ray/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=24511)

## Selected writings

On [[E-infinity spaces|$E_\infty$-spaces]] and [[E-infinity ring spectra|$E_\infty$-ring sectra]]:

* [[Peter May]] with contributions by [[Frank Quinn]], [[Nigel Ray]] and [[Jørgen Tornehave]], _$E_\infty$-Ring spaces and $E_\infty$ ring spectra_, Lecture Notes in Mathematics 577, Springer 1977 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf), [cds:1690879](https://cds.cern.ch/record/1690879))

On the [[quaternionic bordism ring]]:

* [[Nigel Ray]], _The symplectic bordism ring_, Volume 71, Issue 2 March 1972, pp. 271-282 ([doi:10.1017/S0305004100050519](https://doi.org/10.1017/S0305004100050519))

* {#GorbunovRay92} [[Vassily Gorbunov]], [[Nigel Ray]], _Orientations of $Spin$ Bundles and Symplectic Cobordism_, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]], [doi: 10.2977/prims/1195168855](https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4))

* [[Vassily Gorbunov]], [[Nigel Ray]], [[Paul Turner]], _On the Hopf Ring for a Symplectic Oriented Spectrum_, American Journal of Mathematics Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 ([doi:10.2307/2374960](https://doi.org/10.2307/2374960))

category: people


Nijenhuis tensor

## Related concepts

* [[complex structure]]

* [[Newlander-Nirenberg theorem]]

## References

* _Dozen definitions of the Nijenhuis tensor_ ([pdf](http://www.math.uit.no/ansatte/boris/Images/1/12DEF-NJ.pdf))

[[!redirects Nijenhuis tensors]]


Nijenhuis–Richardson bracket

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Suppose $M$ is a [[smooth manifold]].
Recall that any differential $(k+1)$-form $K\in\Omega^{k+1}(M,T M)$ valued in the [[tangent bundle]] of $M$ gives rise to a [[graded derivation]] $\iota_K$ of degree $k$ on the algebra of [[differential forms]] on $M$: on 1-forms we have $\iota_K \omega=\omega\circ K$ and on higher forms we extend using the Leibniz identity.

Concretely,
$$\iota_K \omega(X_1,\ldots,X_{k+l})=1/((k+1)!(l-1)!)\sum_\sigma (-1)^\sigma \omega(K(Y_1,\ldots,Y_{k+1}),Y_{k+2},\ldots),$$
where $Y_i=X_{\sigma(i)}$.

The map $\iota$ defined an injective homomorphism of [[graded vector spaces]] from $\Omega^{\bullet+1}(M,T M)$ to graded derivations of $\Omega(M)$.  Its image comprises precisely those derivations that vanish on 0-forms and is closed under the commutator operation.  Transferring the bracket to its source yields the __Nijenhuis–Richardson bracket__:
$$[K,L]^\wedge = \iota_K L-(-1)^{k l}\iota_L K,$$
where $\iota_K(\omega\otimes X)=\iota_K \omega\otimes X$.

## Classification of graded derivations of differential forms

The Nijenhuis–Richardson bracket is an important ingredient in the classification of graded derivations of differential forms.

See the article [[Frölicher–Nijenhuis bracket]] for more information.

## Related concepts

* [[Frölicher–Nijenhuis bracket]]

* [[Schouten–Nijenhuis bracket]]

## References

The formula for the NR bracket is made explicit in and thus derives its name from:

* [[Albert Nijenhuis]], [[Roger W. Richardson]], Section 2 of: _Cohomology and deformations of algebraic structures_,  Bulletin of the American Mathematical Society 70:3 (1964), 406–412.  [doi](http://dx.doi.org/10.1090/s0002-9904-1964-11117-4).

* [[Albert Nijenhuis]], [[Roger W. Richardson]], Section 5 of: *Deformations of Lie Algebra Structures*, Journal of Mathematics and Mechanics **17** 1 (1967) 89-105 [[jstor:24902154](https://www.jstor.org/stable/24902154)]

but the existence of the bracket is already implicit in:

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part I.  Derivations in the graded ring of differential forms_, Indagationes Mathematicae (Proceedings) 59 (1956), 338–350 and 351–359.  [doi:10.1016/s1385-7258(56)50046-7][1] and [doi:10.1016/s1385-7258(56)50047-9][2].

[1]: https://doi.org/10.1016/s1385-7258(56)50046-7
[2]: https://doi.org/10.1016/s1385-7258(56)50047-9

and refined to [[almost complex structures]] in

* [[Alfred Frölicher]], [[Albert Nijenhuis]], _Theory of vector-valued differential forms.  Part II.  Almost-complex structures_, Indagationes Mathematicae (Proceedings) 61 (1958), 414–421 and 422-429.  [doi:10.1016/s1385-7258(58)50057-2][3] and [doi:10.1016/s1385-7258(58)50058-4][4].

[3]: https://doi.org/10.1016/s1385-7258(58)50057-2
[4]: https://doi.org/10.1016/s1385-7258(58)50058-4

See also:

* Wikipedia, *[Nijenhuis–Richardson bracket](https://en.wikipedia.org/wiki/Nijenhuis%E2%80%93Richardson_bracket)*

A textbook account: Chapter 16 of

* [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008).  [PDF](https://www.mat.univie.ac.at/~michor/dgbook.pdf).


Nik Weaver

* [personal page](https://www.math.wustl.edu/~nweaver/)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=51888)

## Selected writings

On [[Hilbert bimodules]]:

* [[Nik Weaver]], _Hilbert bimodules with involution_, Canad. Math. Bull. **44** 3 (2001) 355–369 [[arXiv:math/9908119](http://arxiv.org/abs/math/9908119), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/DC7055E821BEA9438350CCB325EA64CA/S0008439500014466a.pdf/hilbert-bimodules-with-involution.pdf)]

On the [[Calkin algebra]]:

* {#PhillipsWeaver} N. Christopher Phillips, [[Nik Weaver]], _The Calkin algebra has outer automorphisms_, Duke Math. J. **139** 1 (2007) 185-202 [[arXiv:math/0606594](https://arxiv.org/abs/math/0606594), [doi:10.1215/S0012-7094-07-13915-2](https://projecteuclid.org/journals/duke-mathematical-journal/volume-139/issue-1/The-Calkin-algebra-has-outer-automorphisms/10.1215/S0012-7094-07-13915-2.short)]

On [[quantum mechanics]] and [[quantization]]:

* [[Nik Weaver]], *Mathematical Quantization*, Routledge (2011) [[ISBN 9781584880011](https://www.routledge.com/Mathematical-Quantization/Weaver/p/book/9781584880011)]

On [[von Neumann algebras]], [[noncommutative measure theory]] and [[quantum relations]]:

* {#Weaver10} [[Nik Weaver]], *Quantum relations* [[arXiv:1005.0354](https://arxiv.org/abs/1005.0354)]

* {#WeaverKuperberg11} [[Nik Weaver]], [[Greg Kuperberg]], *A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations*, Memoirs of the AMS **215** 1010 (2011) [[ams:memo-215-1010](https://bookstore.ams.org/memo-215-1010)]

On [[predicative mathematics]]:

*  [[Nik Weaver]], [papers on conceptualism](https://web.archive.org/web/20100616093000/https://www.math.wustl.edu/~nweaver/conceptualism.html).

category: people


Nikita Karpenko

* [personal page](https://sites.ualberta.ca/~karpenko/)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=96576)

## Selected writings

On [[quadratic forms]]:

* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]

category: people


Nikita Kolganov

* [personal page](https://itmp.msu.ru/en/institute/phd-students/kolganov-n-m)

## Selected writings

On universality of [[braid representation|braid]] [[quantum gates]] of [[su(2)-anyons]] at higher [[Chern-Simons level|level]] $k$ (and also for $SU(N)$-anyons with $N \gt 2$):

* [[Nikita Kolganov]], [[Sergey Mironov]], [[Andrey Morozov]], *Large $k$ topological quantum computer*, Nuclear Physics B
**987** (2023) 116072 [[arXiv:2105.03980](https://arxiv.org/abs/2105.03980), [doi:10.1016/j.nuclphysb.2023.116072](https://doi.org/10.1016/j.nuclphysb.2023.116072)]

category: people


Nikita Markarian

* [webpage](https://nikitamarkarian.wordpress.com)

## related $n$Lab entries

* [[Weyl n-algebra]]
* [[Atiyah class]]

## Selected writing

* Nikita Markarian, _The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem_, J. Lond. Math. Soc. (2) __79__:1 (2009) 129--143 [doi](https://doi.org/10.1112/jlms/jdn064)
* Nikita Markarian, _Weyl $n$-algebras_, [arXiv:1504.01931](http://arxiv.org/abs/1504.01931)

category: people


Nikita Nekrasov

* [webpage](http://www.ihes.fr/~nikita/)

## Selected writings

On [[Lorentzian orbifolds]] as [[target spacetimes]] in [[string theory]]:

* [[Nikita Nekrasov]], _Milne Universe, Tachyons, and Quantum Group_, Surveys High Energ. Phys. 17 (2002) 115-124 ([arXiv:hep-th/0203112](https://arxiv.org/abs/hep-th/0203112))

On the [[Gromov-Witten/Donaldson-Thomas correspondence]]:

* [[Davesh Maulik]], [[Nikita Nekrasov]], [[Andrei Okounkov]], [[Rahul Pandharipande]], _Gromov–Witten theory and Donaldson–Thomas theory, I_. Compositio Mathematica. 142 (5): 1263–1285. (2003) ([arXiv:math/0312059](https://arxiv.org/abs/math/0312059). [doi:10.1112/S0010437X06002302](http://www.journals.cambridge.org/abstract_S0010437X06002302))

On [[Knizhnik-Zamolodchikov equations]] controlling [[codimension]]$=2$ [[defect QFT|defects]] in [[D=4 super Yang-Mills theory]]:

* [[Nikita Nekrasov]], *BPS/CFT correspondence V: BPZ and KZ equations from $q q$-characters* [[arXiv:1711.11582](https://arxiv.org/abs/1711.11582)]

* [[Nikita Nekrasov]], Alexander Tsymbaliuk, *Surface defects in gauge theory and KZ equation*, Letters in Mathematical Physics **112** 28 (2022) [[arXiv:2103.12611](https://arxiv.org/abs/2103.12611), [doi:10.1007/s11005-022-01511-8](https://doi.org/10.1007/s11005-022-01511-8)]

* Saebyeok Jeong, Norton Lee, [[Nikita Nekrasov]], *Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations*,  J. High Energ. Phys. **2021** 120 (2021) [[arXiv:2103.17186](https://arxiv.org/abs/2103.17186), doi:10.1007/JHEP10(2021)120]

category: people


Niklas Beisert

## Selected writings

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation opetator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertStaudacher03} [[Niklas Beisert]], [[Matthias Staudacher]], _The $\mathcal{N}=4$ SYM Integrable Super Spin Chain_,
Nucl. Phys. B670:439-463, 2003 ([arXiv:hep-th/0307042](https://arxiv.org/abs/hep-th/0307042))

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

* {#BeisertEtAl10} [[Niklas Beisert]], [[Luis Alday]], [[Radu Roiban]], [[Sakura Schafer-Nameki]], [[Matthias Staudacher]], [[Alessandro Torrielli]], [[Arkady Tseytlin]], et. al., _Review of AdS/CFT Integrability: An Overview_, Lett. Math. Phys. 99, 3 (2012) ([arXiv:1012.3982](https://arxiv.org/abs/1012.3982))

category: people


Niko Inassaridze

* ([homepage](http://www.rmi.ge/~inas/))

category:people


Niko Naumann

* [webpage](http://homepages.uni-regensburg.de/~nan25776/)

## Selected writings

On [[topological modular forms]] ([[tmf]]):

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_, Int. Math. Res. Not. (2013) ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

On [[secondary characteristic classes]] in [[string cobordism cohomology theory]] and in [[tmf]]:

* [[Ulrich Bunke]], [[Niko Naumann]], _Secondary Invariants for String Bordism and tmf_, Bulletin des Sciences Mathématiques, Volume 138, Issue 8, December 2014, Pages 912-970 ([doi:10.1016/j.bulsci.2014.05.002](https://doi.org/10.1016/j.bulsci.2014.05.002))

## Related $n$Lab entries

* [[moduli stack]] of [[formal groups]]

* [[moduli stack of tori]]

* [[KR cohomology theory]]

category: people


Nikolai Durov

__Nikolai Durov__  (Николай Валерьевич Дуров) is a Russian mathematician from St. Petersburg with main current interests in [[arithmetic geometry]], currently employed at St. Petersburg  Department of the Steklov Institute of Mathematics.

* [homepage](http://www.pdmi.ras.ru/eng/perso/durov.php) at Steklov
* [mathnet.ru](http://www.mathnet.ru/php/person.phtml?&personid=34084&option_lang=eng) entry
* wikipedia: [Nikolai Durov](http://en.wikipedia.org/wiki/Nikolai_Durov)

Durov obtained his Ph.D. in 2007 in Bonn under [Gerd Faltings](http://en.wikipedia.org/wiki/Gerd_Faltings):

* _New approach to Arakelov geometry_, [arxiv/0704.2030](http://arxiv.org/abs/0704.2030)

Durov's mathematical work preceding his study in Bonn includes his work on classical [[Galois theory]] of polynomial equations; it provides essentially the third historically available method to compute algorithmically a Galois group of a given equation. His method is however statistical and some random data are included in input. The algorithm terminates with probability $1$ for all equations iff the [[Riemann hypothesis]] is true. The exposition of these results is in

* N. V. Durov, _Computation of the Galois group of a polynomial with rational coefficients. I._ (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), Vopr.Teor. Predst. Algebr. i Grupp. 11, 117--198, 301; English translation in J. Math. Sci. (N. Y.) 134 (2006), no. 6, 2511--2548 (MR2006b:12006)
* N. V. Durov, _Computation of the Galois group of a polynomial with rational coefficients. II._ (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 321 (2005), Vopr. Teor. Predst. Algebr. i Grupp. 12, 90--135, 298; English translation in J. Math. Sci. (N. Y.) 136 (2006), no. 3, 3880--3907 (MR2006e:12004)

Nikolai Durov is also an experienced computer programmer. He was a member of a St Petersburg State University student team winning a student [world tournament](http://www.cs.northwestern.edu/~agupta/_projects/acm_ibm/acm2001.htm) in programming. His high school education was in Italy. His younger brother [Pavel V. Durov](http://durov.vkontakte.ru) is a professional programmer and main constructor behind [one of the most popular internet sites](http://Vkontakte.ru) in Russia. The company is not any more in their control. Nikolai and Pavel together established communication platform and company [Telegram](https://telegram.org) (see [FAQ](https://telegram.org/faq) and [wikipedia](https://en.wikipedia.org/wiki/Telegram_(messaging_service)) where they worked on Nikolai's envisioned [[zoranskoda:Telegram Open Network]] (TON) which he classifies as a 5th generation [[blockchain]] project enhanced with additional DNS, proxy and (torrent-like) storage infrastructure.

* TON whitepaper, 23 page [pdf](https://cdn.crowdfundinsider.com/wp-content/uploads/2018/03/TON-White-Paper-Telegram-ICO.pdf), mirrors [pdf](https://drive.google.com/file/d/1ucUeKg_NiR8RxNAonb8Q55jZha03WC0O/view) [pdf](https://icorating.com/upload/whitepaper/gNQ7e9z3lCGi519Wz8mmC0Kg8aA0goeZKAQ802vo.pdf)
* Nikolai Durov, _Telegram Open Network_, technical overview, Dec 2017, 132 pp. [pdf](https://toncoin.tech/TON-official-whitepaper.pdf), mirror [pdf](https://www.kriptovaluta.hr/wp-content/uploads/2018/03/TON-Technology.pdf); _Telegram Open Network Blockchain_, Sep 2018, 121 pp. [pdf](https://www.docdroid.net/qY4sQEv/telegram-open-network-blockchain-september-5-2018.pdf); _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf); _Fift: a brief introduction_ (about his new Forth-like stack based programming language on TON), May 23, 2019, 87 pp. pdf is at channel [t.me/Tgram/170](https://t.me/Tgram/170); _Catchain consensus: an outline_, [pdf](https://test.ton.org/catchain.pdf)

The technical overview sporadically uses the notation from [[type theory]].

Durov's earlier publications also include

* N. Durov, S. Meljanac, A. Samsarov, [[Z. Škoda]], _A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra_, Journal of Algebra 309, n. 1, 318--359 (2007)
([doi:jalgebra](http://dx.doi.org/10.1016/j.jalgebra.2006.08.025))
([math.RT/0604096](http://front.math.ucdavis.edu/math.RT/0604096)).

where in chapters 7--9 Durov presented a flexible theory of a class of functors which can be viewed as representing generalizations of [formal schemes](http://en.wikipedia.org/wiki/Formal_scheme) but over an arbitrary ring, and with weaker assumptions. This theory is then applied to a problem in Lie theory and deformation theory; an interesting chapter on symplectic Weyl algebras is included. In chapter 10 an alternative method using [[Hopf algebras]] rather than geometry is presented.

Recently he introduced the notion of a [[vectoid]] and the related notion of an [[algebrad]] which is a generalization of the notions of a symmetric and a non-symmetric operad:

* Nikolai Durov, _Classifying vectoids and generalisations of operads_, [arxiv/1105.3114](http://arxiv.org/abs/1105.3114), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273
* _Classifying vectoids and generalizations of operads_, talk at
The International Conference "Contemporary Mathematics"
June 12, 2009, video: [link](http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=345)

Other sources:

* _Computation of derived absolute tensor square of the ring of integers_, talk at 2nd annual conference-meeting MIAN–POMI "Algebra and Algebraic Geometry", St. Petersburg, December 25, 2008, [link](http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=267)
* _Arithmetic intersection theory and homotopical algebra_, seminar 2007
* N. V. Durov, _Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties)_, preprint POMI 13/2012 (in Russian) [abstract](http://www.pdmi.ras.ru/preprint/2012/12-13.html), [pdf.gz](ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2012/13-12_rus.pdf.gz)
* N. V. Durov, _МУЛЬТИПЛИКАТИВНЫЕ МОНОИДЫ ${\mathbb{F}}_p$-АЛГЕБР И АБСОЛЮТНЫЕ ТЕНЗОРНЫЕ ПРОИЗВЕДЕНИЯ КОНЕЧНЫХ ПОЛЕЙ_, (Multiplicative monoids of ${\mathbb{F}}_p$-algebras and absolute tensor products of finite fields), preprint POMI 12/2012 (in Russian) [abstract](http://www.pdmi.ras.ru/preprint/2012/12-12.html) [pdf.gz](ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2012/12-12_rus.pdf.gz)
* N. V. Durov, _Homotopy theory of normed sets I. Basic constructions_, Algebra i Analiz, 29:6 (2017),  35–98 [mathnet.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1562&option_lang=eng); _Homotopy theory of normed sets II. Model categories_, Algebra i Analiz, 30:1 (2018) 32–95 [mathnet.ru](http://mi.mathnet.ru/eng/aa1571)

His paper on normed sets above (part 1 out of 3) is partly extending ideas from

* [[Frederic Paugam]], _Overconvergent global analytic geometry_, 2015, [arXiv:1410.7971v2](https://arxiv.org/abs/1410.7971)

[[!redirects Nikolai Durov]]
[[!redirects Николай Дуров]]
[[!redirects Николай Валерьевич Дуров]]


Nikolai Kudasov

* [github page](https://github.com/fizruk)

* [GoogleScholar page](https://scholar.google.com/citations?user=s6fwk54AAAAJ&hl=en)

## Selected writings

Formalization of the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]] via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

category: people


Nikolaj Kuntner

[[!redirects NikolajKuntner]]
[[!redirects Nikolaj Kuntner]]
[[!redirects Nikolaj+Kuntner]]
[[!redirects Nikolaj-Kuntner]]
[[!redirects NikolajK]]
[[!redirects Nikolaj K]]
[[!redirects Nikolaj+K]]
[[!redirects Nikolaj-K]]

I'm a physicist interested in clear and encompassing foundations.
I create video content on type systems and category theoretical concepts _[on youtube](https://www.youtube.com/channel/UCcrSMnEYhIPX_p127jI23qw/videos)_.

![2014](http://i.imgur.com/mBK4dTV.jpg)

+-- {: .query}
[[Nikolaj Kuntner]]: how to do a query text
=--

![2018](https://i.imgur.com/hcXFEOC.png)


Nikolay Bogolyubov

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

## Selected writings

On ([[AQFT|algebraic]]) [[quantum field theory]]:

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

## Related entries

* [[causal perturbation theory]]

* [[Bogoliubov's formula]]

category: people

[[!redirects Nikolay Bogoliubov]]

[[!redirects Bogoliubov]]
[[!redirects Bogolyubov]]


Niles Johnson

## About

Niles Johnson is an associate professor in [[algebraic topology]] at The Ohio State University.

Website:

* [nilesjohnson.net](http://nilesjohnson.net)

## Selected writings

On [[exotic 7-spheres]]:

* [[Niles Johnson]], _Visualizing 7-manifolds_, 2012 ([nilesjohnson.net/seven-manifolds.html](https://nilesjohnson.net/seven-manifolds.html))

On [[2-categories]] and [[2-category theory]]:

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

On bimonoidal or [[rig categories]]

* [[Niles Johnson]], [[Donald Yau]], _Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory_ ([three volume book](https://nilesjohnson.net/En-monoidal.html); abridged version [arXiv:2107.10526](https://arxiv.org/abs/2107.10526)).

On [[Mackey functors]] [[enriched]] over [[closed category|closed]] [[multicategories]]:

* [[Niles Johnson]], [[Donald Yau]], _Homotopy Theory of Enriched Mackey Functors_ ([arXiv:2212.04276](https://arxiv.org/abs/2212.04276))

## Related pages

* [[Hopf fibration]]

category:people


nilpotence theorem

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

#Contents#
* table of contents
{:toc}

## Idea

A fundamental statement in [[stable homotopy theory]]/[[higher algebra]]/[[chromatic homotopy theory]].

## Statement

For any [[ring spectrum]] $R$ the kernel of the canonical [[morphism]]

$$
  \pi_\bullet R \longrightarrow MU_\bullet(R)
$$

to the [[MU]]-homology of $R$ consists of [[nilpotent elements]].

This is due to [Devinatz-Hopkins-Smith 88](#DevinatzHopkinsSmith88) See [Lurie 10, theorem 3](#Lurie10)

## Special cases

The Nishida nilpotence theorem ([Nishida 73](#Nishida73)) is the special case for the [[sphere spectrum]], saying that all positive-degree elements in the [[stable homotopy groups of spheres]] are nilpotent.

This is indeed a special case. The [[MU]]-homology of the sphere spectrum is the [Lazard ring](https://ncatlab.org/nlab/show/MU#homotopy_groups_cobordism_and_lazard_ring) and hence is torsion-free, whereas all positive-degree elements of the stable homotopy ring are torsion by the [[Serre finiteness theorem]] and therefore
belong to the aforementioned kernel.

## Consequences

### Relation with $\infty$-fields

The nilpotence theorem implies that every [[∞-field]] is an [[∞-module]] over the [[Morava K-theory]] [[spectrum]] $K(n)$, for some $n$. See at _[[∞-field]]_.

## Related concepts

* [[Serre finiteness theorem]]

* [[thick subcategory theorem]]

## References

* {#Nishida73} [[Goro Nishida]], _The nilpotency of elements of the stable homotopy groups of spheres_, Journal of the Mathematical Society of Japan 25 (4): 707–732,  (1973) ([euclid:euclid.jmsj/1240435467](https://projecteuclid.org/euclid.jmsj/1240435467), [euclid:jmsj/1240435467](https://projecteuclid.org/euclid.jmsj/1240435467), ISSN 0025-5645, MR 0341485)

* {#Ravenel84} [[Douglas Ravenel]], Section 10.1 of: _Localization with Respect to Certain Periodic Homology Theories_, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 ([doi:10.2307/2374308](https://doi.org/10.2307/2374308), [jstor:2374308](https://www.jstor.org/stable/2374308))

* {#DevinatzHopkinsSmith88} [[Ethan Devinatz]], [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory I_, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 ([jstor:1971440](https://www.jstor.org/stable/1971440))

* {#HopkinsSmith98} [[Michael Hopkins]], [[Jeffrey Smith]], _Nilpotence and Stable Homotopy Theory II_, Annals of Mathematics Second Series, Vol. 148, No. 1 (Jul., 1998), pp. 1-49 ([jstor:120991](https://www.jstor.org/stable/120991))

* {#Lurie10} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 25 _The Nilpotence lemma_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf))

[[!redirects nilpotence lemma]]

[[!redirects Nishida nilpotence theorem]]
[[!redirects Nishida nilpotence lemma]]

[[!redirects Hopkins-Smith nilpotence theorem]]