virtually fibered conjecture

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## Idea

The _virtually fibered conjecture_ says that every [[closed manifold|closed]], [[irreducible manifold|irreducible]], [[atoroidal 3-manifold|atoroidal]] [[3-manifold]] with infinite [[fundamental group]] has a [[finite cover]] which is a [[surface]] [[fiber bundle]] over the [[circle]].

A proof was announced by [[Ian Agol]] in 2012, based on work by [[Daniel Wise]].

## Related concepts

* [[arithmetic topology]]

* [[Alexander polynomial]]

## References

* Wikipedia, _[Virtually fibered conjecture](https://en.m.wikipedia.org/wiki/Virtually_fibered_conjecture)_

The final step of the proof after the work of Wise is contained in

* [[Ian Agol]], Daniel Groves, Jason Manning, _The virtual Haken conjecture_ [arXiv:1204.2810](http://arxiv.org/abs/1204.2810)

See also

* [[Ian Agol]], _The virtual Haken conjecture_, Talk at ICM 2014 ([video](https://www.youtube.com/watch?v=Y6ACJYvfs5U))


viscosity

[[!redirects shear viscosity]]

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#### Physics
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## Examples

* The discussion of shear viscosity of [[quark-gluon plasmas]] turns out to be subtle. See there.

## Related concepts

* [[elasticity]]

* [[solidity]]

* [[cohesion]]

## References

See also

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


Vishal Lama

Hi, I co-author the mathematical blog [Topological Musings](http://topologicalmusings.wordpress.com) with [Todd Trimble](http://ncatlab.org/nlab/show/Todd+Trimble).

I hope to fill up more of this page as time progresses.

**Books**

There are at least three books that I would highly recommend to an undergraduate student who may have the requisite "mathematical maturity" but perhaps has not tasted the elixir of [Category Theory](http://ncatlab.org/nlab/show/category+theory) yet.

* [Conceptual Mathematics: A First Introduction to Categories](http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&s=books&qid=1255426142&sr=8-1) (Paperback) - F. William Lawvere and Stephen H. Schanuel.

* [Sets for Mathematics](http://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521804442/ref=sr_1_3?ie=UTF8&s=books&qid=1255426295&sr=1-3) - F. William Lawvere and Robert Rosebrugh

* [Topoi: The Categorial Analysis of Logic](http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&s=books&qid=1255426408&sr=1-1) - Robert Goldblatt.

I intend to write reviews (containing details on each chapter, hopefully) for all the aforementioned books some time in the future. It will be a work in progress.

category: people


Vitalik Buterin

__Vitalik Buterin__ is the principal founder of [[zoranskoda:Ethereum]] [[blockchain]]. He was considering the limitations of [[zoranskoda:bitcoin|Bitcoin]] blockchain, where new functionalities were added one by one and came up with the idea of developing a new blockchain with Turing complete runtime for [[smart contract]]s on blockchain, enabling also the development of distributed applications, turning the blockchain into a computer-like platform where new applications could easily be added. Buterin is considered as a prodigy who founded Ethereum at age of 19 and now considered as one of main gurus of blockchain technology. He is one of the authors of the Plasma algorithm for layer 2 network over Ethereum-like blockchains and promotes the Casper proof of stake protocol for Ethereum 2.0.

Other leading figures behind early Ethereum network include Charles Hoskinson and Gavin Wood. Hoskinson is now behind the [[zoranskoda:Cardano]] project, leading company IOHK, and also contributing to [polymath](https://polymath.network) blockchain for financial securities. Gavin Wood introduced Solidity, the major higher level language for smart contracts which compiles to Ethereum [[virtual machine]] bytecode, in August 2014. He is now leading the web 3.0 project and Parity Technologies which is behind Parity client for Ethereum, Polkadot and Substrate projects.

* site [vitalik.ca](https://vitalik.ca)
*  V. Buterin, _Ethereum in 25 minutes_, [youtube](https://www.youtube.com/watch?v=66SaEDzlmP4); _Blockchain and Ethereum security on the higher level_, [youtube](https://www.youtube.com/watch?v=UFDAtStVXbc) 1:21h
* V. Buterin, Ethereum: A next-generation smart contract and decentralized application platform, [pdf](https://github.com/ethereum/wiki/wiki/White-Paper.pdf)
* V. Buterin, _Blockchain and Ethereum security on the higher level_, [youtube](https://www.youtube.com/watch?v=UFDAtStVXbc) 1:21h
* Joseph Poon, Vitalik Buterin, _Plasma: scalable autonomous smart contracts_
[pdf](https://www.plasma.io/plasma.pdf) draft whitepaper
* V. Buterin, _Chain interoperability_, R3 reports, Sep 2016 [pdf](https://www.r3.com/wp-content/uploads/2018/04/Chain_Interoperability_R3.pdf)
* _Sidechains vs Plasma vs Sharding_, [html](https://vitalik.ca/general/2019/06/12/plasma_vs_sharding.html)

Vitalik recently proposed a privacy mixer for Ethereum to enhance anonymity via groupings

* _Minimal mixer design_, proposal at [hackmd](https://hackmd.io/s/rJj9hEJTN); [news at coindesk](https://www.coindesk.com/vitalik-roposes-mixer-to-anonymize-one-off-transactions-on-ethereum)

category: people, computer science, applications


Vitaly Ginzburg

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

## Selected writings

Introducing [[Landau-Ginzburg models]] in [[superconductivity]]:

* [[Vitaly Ginzburg]], [[Lev Landau]], _On the Theory of Superconductivity_,  reprinted In: _On Superconductivity and Superfluidity_ Springer (2009) ([doi:1007/978-3-540-68008-6_4](https://doi.org/10.1007/978-3-540-68008-6_4))

category: people

[[!redirects В. Л. Гинзбург]]


Vitaly Lorman

* [webpage](http://www.math.jhu.edu/~vlorman/)

## Related entries

* [[image of J]]

category: people


VitalyR

## VitalyR ##
I'm a student in China. This is [my home page](https://vitalyr.com).


Vittoria Bussi

* [webpage](https://www.maths.ox.ac.uk/people/profiles/vittoria.bussi)

category: people


Vittorio Giovannetti

* [personal page](https://sites.google.com/site/giovannettivittorio/home)

* [Institute page](https://www.sns.it/en/persona/vittorio-giovannetti)

## Selected writings

Introducing the notion of [[quantum random access memory]]:

* {#GiovanettiyLloydMaccone08} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Quantum random access memory*, Phys. Rev. Lett. **100** 160501 (2008) [[doi:10.1103/PhysRevLett.100.160501](https://doi.org/10.1103/PhysRevLett.100.160501), [arXiv:0708.1879](https://arxiv.org/abs/0708.1879)]

* {#GiovanettiyLloydMaccone08b} [[Vittorio Giovannetti]], [[Seth Lloyd]], [[Lorenzo Maccone]], *Architectures for a quantum random access memory*, Phys. Rev. A **78** 052310 (2008) [[doi:10.1103/PhysRevA.78.052310](https://doi.org/10.1103/PhysRevA.78.052310)]

category: people


Vivek Kumar Singh

* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/researchers/vivek-kumar-singh.html) at NYU Abu Dhabi

* [GoogleScholar page](https://scholar.google.com/citations?user=lL8i4AsAAAAJ&hl=en&oi=sra)

* [inspire page](https://inspirehep.net/authors/1418741)

Member of [[CQTS]] at NYU Abu Dhabi.

## Selected writings
On [[topological quantum computation ]]:

* [[Vivek Kumar Singh]], Akash Sinha, Pramod Padmanabhan, Indrajit Jana, *Dyck Paths and Topological Quantum Computation*, [[arXiv:2306.16062v1](https://arxiv.org/abs/2306.16062)]

On [[topological entanglement entropy]] in [[Chern-Simons theory]]:

* Siddharth Dwivedi, [[Vivek Kumar Singh]], Saswati Dhara, P. Ramadevi, Yang Zhou, Lata Kh Joshi, *Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups*, J. High Energy Phys.02 (2018) 163  [[arXiv:1711.06474](https://arxiv.org/abs/1711.06474), doi:10.1007/JHEP02(2018)163]

* Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, [[Vivek Kumar Singh]], *Topological entanglement and hyperbolic volume*, J. High Energy Phys. **21021** 172 (2021) 172 [[arXiv:2106.03396](https://arxiv.org/abs/2106.03396), doi:10.1007/JHEP10(2021)172]

On the [[knots-quivers correspondence]]:

* [[Vivek Kumar Singh]], S. Chauhan, A. Dwivedi, P. Ramadevi, B.P. Mandal, S. Dwivedi, *Knot-Quiver correspondence for double twist knots*, Phys. Rev. D 108 (2023) 10, **106023** [[arXiv:2303.07036](https://arxiv.org/abs/2303.07036), doi:10.1103/PhysRevD.108.106023]

On [[knot theory]]:

* Satoshi Nawata, P. Ramadevi,[[Vivek Kumar Singh]], *Colored HOMFLY polynomials that distinguish mutant knots
*, J.Knot Theor.Ramifications 26 (2017) 14, **1750096** [[arXiv:1504.00364](https://arxiv.org/abs/1504.00364), doi:10.1142/S0218216517500961]

* Saswati Dhara, A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, [[Vivek Kumar Singh]], A. Sleptsov, *Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations*, Annales Henri Poincare 20 (2019) 12, **4033-4054** [[arXiv:1805.03916](https://arxiv.org/abs/1805.03916), doi:10.1007/s00023-019-00841-z]

On [[topological string]]:

A. Mironov, A. Morozov, An. Morozov, P. Ramadevi,[[Vivek Kumar Singh]], A. Sleptsov, *Checks of integrality properties in topological strings
*, JHEP 08 (2017) 139 [[arXiv:1702.06316](https://arxiv.org/abs/1702.06316), doi:10.1007/JHEP08%282017%29139]

category: people

[[!redirects Vivek Singh]]

[[!redirects Vivek Kr. Singh]]


Vivien M. Kendon

* [institute page](https://www.durham.ac.uk/staff/viv-kendon/)

* [GoogleScholar page](https://scholar.google.com/citations?user=hF8meRwAAAAJ&hl=en)

## Selected writings

On "continuous variable" or *analog* [[quantum computation]]:

* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]

category: people

[[!redirects Vivien Kendon]]
[[!redirects Viv Kendon]]


Vlad Patryshev

Vlad Patryshev vpatryshev@gmail.com
programmer, amateur categorist, organizer of [Bay Area Categories and Types meetup](http://www.meetup.com/Bay-Area-Categories-And-Types/).

category: people

[[!redirects vpatryshev]]
[[!redirects Vlad Patryshev]]


Vladimir A. Fateev

* [institute page](https://www.itp.ac.ru/en/persons/fateev-vladimir-aleksandrovich/)

## Selected writings

A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:

* [[Vladimir S. Dotsenko]], [[Vladimir A. Fateev]], *Conformal algebra and multipoint correlation functions in 2D statistical models*, Nuclear Physics B **240** 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

Introducing [[parafermion]] [[2d CFT]]:

* [[Alexander B. Zamolodchikov]], [[Vladimir A. Fateev]] *Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems*, Sov. Phys. JETP **62** 2 (1985) 215-225 $[$[pdf](http://www.jetp.ras.ru/cgi-bin/dn/e_062_02_0215.pdf), [osti:5929972](https://www.osti.gov/biblio/5929972)$]$

category: people


Vladimir Abramovich Rokhlin

* [Wikipedia page](https://en.m.wikipedia.org/wiki/Vladimir_Abramovich_Rokhlin)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=42580&fChrono=1)

* V I Arnol'd et al., _Vladimir Abramovich Rokhlin (obituary)_,  Russ. Math. Surv. 41 189 ([doi:10.1070/RM1986v041n03ABEH003331](https://iopscience.iop.org/article/10.1070/RM1986v041n03ABEH003331))

## Selected writings

Computing the [[third stable homotopy group of spheres]]:

* [[Vladimir Abramovich Rokhlin]], _On a mapping of the $(n+3)$-dimensional sphere into the $n$-dimensional sphere_,  (Russian) Doklady Akad. Nauk SSSR (N.S.) 80, (1951). 541–544

with a mistake (in the unstable range) corrected in

* [[Vladimir Abramovich Rokhlin]], _New results in the theory of four-dimensional manifolds_, (Russian) Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224.

French translation in:

* Lucien Guillou, Alexis Marin (eds.), _A la Recherche de la Topologie Perdue: I. Du côté de chez Rohlin. II. Le côté de Casson_, Progress in Mathematics 62, Birkhäuser Boston 1985 (ISBN:0817633294, 9780817633295)

category: people

[[!redirects V. A. Rokhlin]]
[[!redirects V. A. Rohlin]]


Vladimir Aleksandrovich Fock

> disambiguation: [[Vladimir Fock]], mathematician at Strasbourg

***

Soviet Rusiian physicist. His name is attached to _[[Fock space]]_, _[[Hartree-Fock method]]_ etc.

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

## Selected writings

Introducing the [[adiabatic theorem]]:

* [[Max Born]], [[Vladimir A. Fock]],  *Beweis des Adiabatensatzes*, Zeitschrift für Physik **51**  (1928) 165–180 ([doi:10.1007/BF01343193](https://doi.org/10.1007/BF01343193))

[[!redirects Vladimir A. Fock]]

category: people


Vladimir Arnold

Владимир Игоревич Арнольд (Vladimir Igorevich Arnol'd) was a Russian mathematician and mathematical physicist, working in Moscow and Paris.

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

* B. Khesin & S. Tabachnikov: *Vladimir Igorevich Arnold. 12 June 1937 - 3 June 2010*, Biogr. Mems Fell. R. Soc. **64** (2018) 7–26  [[doi:10.1098/rsbm.2017.0016](https://doi.org/10.1098/rsbm.2017.0016), [pdf](https://royalsocietypublishing.org/doi/pdf/10.1098/rsbm.2017.0016)]

* biography at MacTutor [history of math archive](http://www-history.mcs.st-andrews.ac.uk/Biographies/Arnold.html)

## Selected writings

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

* {#Arnold69} [[Vladimir Arnold]], _The cohomology ring of the colored braid group_, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 ([mathnet:mz6827](http://mi.mathnet.ru/eng/mz6827))

On [[classical mechanics|classical]] [[Hamiltonian mechanics]] formulated via [[symplectic geometry]] (ie. with [[phase space]] understood as a [[symplectic manifold]]):

* [[Vladimir Arnol'd]], _[[Mathematical methods of classical mechanics]]_, Graduate Texts in Mathematics **60**, Springer (1978) [[doi:10.1007/978-1-4757-1693-1](https://doi.org/10.1007/978-1-4757-1693-1)]

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

* [[Vladimir Arnold]], _Ramified covering $\mathbb{C}P^2  \to S^4$, hyperbolicity and projective topology_, Siberian Math. Journal 1988, V. 29, N 5, P.36-47

* [[Vladimir Arnold]], _On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms_, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.

On methods of [[topology]] and [[homotopy theory]] in [[hydrodynamics]]:

* [[Vladimir Arnold]], [[Boris Khesin]], _Topological methods in hydrodynamics_, Applied Mathematical Sciences __125__, Springer 1998 ([doi:10.1007/b97593](https://link.springer.com/book/10.1007/b97593))

On [[mathematics]] and its relation to [[physics]]:

* {#Arnold95} [[Vladimir I. Arnold]], *Will mathematics survive? Report on the Zurich Congress*, The Mathematical Intelligencer **17** (1995) 6–10 [[doi:10.1007/BF03024363](https://doi.org/10.1007/BF03024363)]

  > "At the beginning of this century a self-destructive democratic principle was advanced in mathematics [which] led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste of a dying religion, like Druids, parasitic on science and technology, recruiting acolytes in the mathematical schools by Zombie-like mental subjection. Bizarre questions like [[Fermat's last theorem|Fermat's problem]] or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."

* {#Arnold98} [[Vladimir I. Arnold]], *On teaching mathematics*, Russ. Math. Surv. **53** 1 (1998) 229 [[doi:10.1070/RM1998v053n01ABEH000005](https://iopscience.iop.org/article/10.1070/RM1998v053n01ABEH000005), [web](https://www.math.fsu.edu/~wxm/Arnold.htm)]

  > "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

  > The Jacobi identity (which forces the altitudes of a triangle to meet in a point) is an experimental fact in the same way as the fact that the earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.

  > In the middle of the twentieth century an attempt was made to separate physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy’s warning that ugly mathematics has no permanent place under the sun).

  > Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was a universal hatred of mathematicians, both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users.

  > [...]

  > A teacher of mathematics who has not got to grips with at least some of the volumes of the [[Course of Theoretical Physics|course]] by Landau and Lifshitz will then become a relic like the person nowadays who does not know the difference between an open and a closed set."

* {#Arnold00} [[Vladimir I. Arnold]], *Polymathematics: Is mathematics a single science  or a set of arts?*, in: [[Vladimir Arnold|V. Arnold]], [[Peter Lax|P. Lax]], [[Barry Mazur|B. Mazur]] (eds.), *Mathematics: Frontiers and Perspectives*, AMS (2000) 403-416 [[ISBN: 	978-0-8218-2697-3](https://bookstore.ams.org/mfp), [pdf](https://math.ucr.edu/home/baez/Polymath.pdf)]

  > "All mathematics is divided into three parts: [[cryptography]] (paid for by CIA, KGB and the like), [[hydrodynamics]] (supported by manufacturers of atomic submarines) and [[classical mechanics|celestial mechanics]] (financed by military and other institutions dealing with missiles, such as NASA).

  > Cryptography has generated [[number theory]], [[algebraic geometry]] over [[finite fields]], [[algebra]]", [[combinatorics]] and [[computers]].

  > Hydrodynamics procreated [[complex analysis]], [[partial differential equations]], [[Lie groups]] and algebra theory, [[cohomology theory]] and scientific [[computing]].

  > Celestial mechanics is the origin of [[dynamical systems]], [[linear algebra]], [[topology]], [[variational calculus]] and [[symplectic geometry]].

  > The existence and mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation)."

  > [...]

  > "[[David Hilbert|Hilbert]] tried to predict the future development of mathematics and to influence it by [[Hilbert's problems|his Problems]]. The development of mathematics in the 20th century has followed a different path. The most important achievements -- the flourishing of [[homotopy theory]] and of [[differential topology]], the [[geometry|geometrisation]] of all branches of mathematics, its fusion with [[theoretical physics]], the discovery of the algorithmically undecidable problems and the appearance of [[computers]] -- all this went in a different (if not opposite) direction.

  > The influence of [[Henri Poincaré|H. Poincaré]] and of [[Hermann Weyl|H. Weyl]] on the science of the 20th century was much deeper. To Poincaré, who created modern mathematics, [[topology]] and [[dynamical systems]] theory, the future of mathematics lay in the development of [[mathematical physics]], oriented to the description of the [[relativity|relativistic]] and [[quantum physics|quantum]] phenomena."

  > [...]

## Related entries

* [[symplectic geometry]]

* [[Hamiltonian mechanics]]

category: people

[[!redirects Владимир Игоревич Арнольд]]
[[!redirects Владимир Арнольд]]
[[!redirects Vladimir Arnold]]
[[!redirects Vladimir Arnol'd]]
[[!redirects Vladimir Arnol'd]]
[[!redirects Vladimir I. Arnold]]
[[!redirects Vladimir I. Arnol'd]]
[[!redirects Vladimir I. Arnol'd]]
[[!redirects V. I. Arnold]]
[[!redirects V. I. Arnol'd]]
[[!redirects V. I. Arnol'd]]


Vladimir Baranovsky

__Vladimir Baranovsky__ is a mathematician at the University of California at Irvine. Main area: algebraic geometry.

* [web](http://www.math.uci.edu/~vbaranov)
* _Orbifold cohomology as periodic cyclic homology_, Internat. Journal of Math, 14 (2003), no 8,  791-812, [pdf](http://math.uci.edu/%7Evbaranov/math/08orbicyclic.pdf)
* _A universal enveloping for L-infinity algebras_, Math Res Lett __15__ (2008) [pdf](http://math.uci.edu/%7Evbaranov/math/12uofl.pdf)

category: people


Vladimir Berkovich

* [personal webpage](http://www.wisdom.weizmann.ac.il/~vova/), [institute webpage](http://www.wisdom.weizmann.ac.il/math/profile/scientists/berkovich-profile.html)

## related entries

* [[non-archimedean analytic geometry]], [[Berkovich analytic space]]

category: people


Vladimir Bogachev

Vladimir Bogachev (Владимир Игоревич Богачев) is a Russian mathematician.

## Selected writings

## References

[Math-Net.Ru page](http://www.mathnet.ru/eng/person8592).


Vladimir Danilov

* [webpage](http://mathecon.cemi.rssi.ru/en/danilov/)

category: people


Vladimir Dobrev

* [webpage](http://theo.inrne.bas.bg/~dobrev/)

category: people


Vladimir Drinfel'd

**Vladimir Gershonovich Drinfel'd** (Russian: Владимир Гершонович Дринфельд, Ukrainian: Володимир Гершонович Дрінфельд) is an Ukrainian-born (Feb 4, 1954) mathematician, recipient of Fields Medal (1990), now a professor at the University of Chicago. He was student of [[Yuri Manin]] at Moscow; his main specialty is algebraic geometry, including applications to arithmetic, [[automorphic form]]s, [[representation theory]] and mathematical physics. His famous discoveries and new concepts Drinfel'd modular varieties, Drinfel'd shtukas (Drinfel'd elliptic modules), ADHM-construction of instantons, [[Drinfel'd-Sokolov reduction]] in the theory of [[integrable systems]], algebraic approach to quantum inverse scattering method in [[integrable systems]], and related subject of [[quantum group]]s including Yangians and quasi-Hopf algebras, geometric Langlands program, chiral algebras (with A. Beilinson), certain category of infinite-dimensional bundles in algebraic geometry (now sometimes called Drinfel'd bundles), and the general construction of quotient [[dg-categories]] in [[homological algebra]]. His ideas also influenced the beginnings of the new subject of [[derived algebraic geometry]].

* [wikipedia](http://en.wikipedia.org/wiki/Vladimir_Drinfel%27d)

## 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]

Introducing [[quantum groups]] and the [[Drinfeld double]]-construction:

* {#Drinfeld87} [[Vladimir Drinfeld]], _Quantum groups_, in: A. Gleason (ed.) *[Proceedings of the](https://archive.org/details/proceedingsofint0002inte_v5c3/mode/2up) [1986 International Congress of Mathematics](https://inspirehep.net/conferences/966215?ui-citation-summary=true)* **1** (1987) 798-820 [[pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/ICM1986.1.ocr.pdf)]

  expanded version:

  Journal of Soviet Mathematics **41** (1988) 898–915 [[doi:10.1007/BF01247086](https://doi.org/10.1007/BF01247086)]

On [[chiral algebras]]:

* A. A. Beilinson, V. Drinfeld, _[[Chiral Algebras]]_, AMS 2004 (the final form of a preprint in various forms since around 1995, cf. [here](http://www.math.uchicago.edu/~mitya/langlands.html)).

On [[braided monoidal category|braided]] [[fusion categories]]:

* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

On [[prismatic cohomology]]

* [[Vladimir Drinfeld]], _Prismatization_ ([arXiv:2005.04746](https://arxiv.org/abs/2005.04746))

category: people

[[!redirects Vladimir Drinfeld]]
[[!redirects Vladimir Drinfel'd]]
[[!redirects V. Drinfeld]]
[[!redirects V. Drinfel'd]]
[[!redirects V. G. Drinfel'd]]
[[!redirects Drinfeld]]
[[!redirects Drinfel'd]]
[[!redirects Владимир Дринфельд]]
[[!redirects Володимир Дрінфельд]]


Vladimir Fock

Professor of mathematics at Institut de Recherche Mathématique Avancée in Strasbourg. Collaborating with [[Alexander Goncharov]].

* [webpage](http://www-irma.u-strasbg.fr/~fock/)

> disambiguation: [[Vladimir Aleksandrovich Fock]], soviet russian physicist

category: people


Vladimir Glaser

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

## writings

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

## related $n$Lab entries

* [[renormalization]]

* [[causal perturbation theory]],

* [[lcoally covariant perturbative quantum field theory]]

category: people


Vladimir Gribov

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

* Ya.I. Azimov, _Vladimir Naumovich Gribov: Pieces of biography_ ([arXiv:1608.05727](http://arxiv.org/abs/1608.05727))

## related $n$Lab entries

* [[Gribov ambiguity]]

* [[S-matrix]]

category: people


Vladimir Guletskii

* [web](http://pcwww.liv.ac.uk/~guletski/)

* grant [Lambda-structures in stable categories](http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/I034017/1)

## Selected writings

On [[symmetric powers]] in [[higher algebra|higher]] [[homotopical algebra]] (and on [[monoidal localization|monoidal]] [[Bousfield localization of model categories]]):

* {#GorchinskiyGuletskii16} [[Sergey Gorchinskiy]], [[Vladimir Guletskii]], *Symmetric powers in abstract homotopy categories*, Adv. Math., **292** (2016) 707-754 [[arXiv:0907.0730](https://arxiv.org/abs/0907.0730), [doi:10.1016/j.aim.2016.01.011](https://doi.org/10.1016/j.aim.2016.01.011)]

[[zeta function|Zeta function]]s in triangulated context

* Vladimir Guletskii, _Zeta functions in triangulated categories_, Mathematical Notes __87__, 3 (2010) 369--381, [math/0605040](https://arxiv.org/abs/math/0605040)

category:people


Vladimir Hinich

* [website](http://math.haifa.ac.il/hinich/WEB/hinich.htm)

* [recent publications](http://math.haifa.ac.il/hinich/WEB/publications.htm)

## Selected writings

On [[model category]] [[structures]] on [[algebra over an operad|algebras over operads]] in [[chain complexes]] (and introducing the [[model structure on unbounded chain complexes]]):

* {#Hinich97} [[Vladimir Hinich]],  *Homological algebra of homotopy algebras*, Communications in Algebra **25** 10   (1997) 3291-3323 [[arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), [doi:10.1080/00927879708826055](https://doi.org/10.1080/00927879708826055), Erratum: [arXiv:math/0309453](http://arxiv.org/abs/math/0309453)]

The [[model structure on dg-coalgebras]] (in [[characteristic zero]]) as a [[model structure for L-infinity algebras|model structure for $L_\infty$-algebras]] and the [[Quillen equivalence]] between [[dg-Lie algebras]] as well as the interpretation in terms of formal $\infty$-stacks ([[L-infinity algebras|$L_\infty$-algebras]]):

* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, Journal of Pure and Applied Algebra **162** 2 (2001) 209-250 [[arXiv:9812034](http://arxiv.org/abs/math/9812034), doi:10.1016/S0022-4049(00)00121-3]

On the [[enriched Yoneda lemma]]:

* {#Hinich16} [[Vladimir Hinich]], *Enriched Yoneda lemma*, Theory and Applications of Categories **31** 29 (2016) 833-838 [[tac:31-29](http://www.tac.mta.ca/tac/volumes/31/29/31-29abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/31/29/31-29.pdf)]

On ([[Lie algebra weight system|Lie algebra-]])[[weight systems]] on [[chord diagrams]]:

* [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197))

On [[(infinity,1)-colimits|$\infty$-colimits]] and [[Day convolution]] in the context of [[enriched (infinity,1)-categories|enriched $\infty$-categories]]:

* [[Vladimir Hinich]], *Colimits in enriched ∞-categories and Day convolution*, Theory and Applications of Categories **39** 12 (2023) 365-422 [[tac:39-12](http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html), [arXiv:2101.09538](https://arxiv.org/abs/2101.09538)]

category: people

[[!redirects V. Hinich]]

[[!redirects Hinich]]


Vladimir Lifschitz

* [website](https://www.cs.utexas.edu/~vl/)

## Selected publications

On [[Lifschitz realizability]]:

* [[Vladimir Lifschitz]], *$CT_0$ is stronger than $CT_0!$*. Proceedings of the American Mathematical Society, Volume 73, Number 1, January 1979, pp. 101–106. [[pdf](https://www.ams.org/journals/proc/1979-073-01/S0002-9939-1979-0512067-X/S0002-9939-1979-0512067-X.pdf)]

category: people


Vladimir Retakh

__Vladimir Retakh__ is a professor of mathematics at Rutgers University, formerly in Moscow, specialized in combinatorics, algebra and their applications.

* [web](http://www.math.rutgers.edu/~vretakh)

## Selected writings

On [[quasideterminants]] and [[Cohn localization]]:

* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_ (2007) [[[RetakhWilson-Quasideterminants.pdf:file]]]

[[!redirects V. Retakh]]


Vladimir Rubtsov

__Vladimir Rubtsov__ (also spelled Roubtsov in France) is a mathematician and mathematical physicist in Angers, France ([[integrable system]]s, geometry in physics, Painlevé transcendents, character varieties, [[tau function]]s, [[Lie algebroid]]s, noncommutative algebra, Yang-Baxter equations).

* [webpage](https://www.math.univ-angers.fr/~volodya/index_uk.html)
* [[A. Odesskii]], V. Rubtsov, V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935) [arxiv/1208.2935](https://arxiv.org/abs/1208.2935); _Bi-hamiltonian ordinary differential equations with matrix variables_, Theor. and Math. Phys. 171(1): 442--447 (2012)
* Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, _Painlevé monodromy manifolds, decorated character varieties, and cluster algebras_, IMRN __24__ (2017) 7639–7691 [arxiv/1511.03851](https://arxiv.org/abs/1511.03851) [doi](https://doi.org/10.1093/imrn/rnw219); _Algebras of quantum monodromy data and decorated character varieties_, in: Geometry and physics: A Festschrift in honour of Nigel Hitchin, Oxford Univ. Press 2018 [arxiv/1705.01447](https://arxiv.org/abs/1705.01447); _Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras_, Adv. Math. __376__ (2021) 107442 [arxiv/1905.02772](https://arxiv.org/abs/1905.02772) [doi](https://doi.org/10.1016/j.aim.2020.107442)
* Marta Mazzocco, Vladimir Rubtsov, _Confluence on the Painlevé monodromy manifolds, their Poisson structure and quantisation_, [arXiv/1212.6723](https://arxiv.org/abs/1212.6723)
* U. Bruzzo, I. Mencattini, V. Rubtsov, P. Tortella, _Nonabelian Lie algebroid extensions_, [arXiv:1305.2377](https://arxiv.org/abs/1305.2377)
* D. Gurevich, V. Rubtsov, _Yang-Baxter equation and deformation of associative and Lie algebras_, in: Quantum Groups, Springer Lecture Notes in Math. __1510__ (1992) 47-55 [doi](https://doi.org/10.1007/BFb0101177)
* B. Enriquez, V. Rubtsov, _Hitchin systems, higher Gaudin Hamiltonians and r-matrices_, Math. Res. Letters __3__ (1996) n° 3, 343-357,  [alg-geom/9503010](https://arxiv.org/abs/alg-geom/9503010)
* [[Vladimir Retakh]], Vladimir Rubtsov, _Noncommutative Toda chains, Hankel quasideterminants and [[Painleve transcendent|Painlevé]] II equation_, [arxiv/1007.4168](http://arxiv.org/abs/1007.4168)

category: people

[[!redirects V. Rubtsov]]


Vladimir S. Dotsenko

* [institute page](https://www.itp.ac.ru/en/persons/dotsenko-vladimir-stepanovich/)

## Selected writings

A precursor result to the [[hypergeometric construction of KZ solutions]], specifically of [[conformal blocks]] for [[affine Lie algebra]]/[[WZW-model]]-[[2d CFTs]]:

* [[Vladimir S. Dotsenko]], [[Vladimir A. Fateev]], *Conformal algebra and multipoint correlation functions in 2D statistical models*, Nuclear Physics B **240** 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

category: people


Vladimir S. Retakh

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

category: people

[[!redirects Vladimir Retakh]]


Vladimir Turaev

[[!redirects Turaev]]
Vladimir Turaev is a mathematician of Russian origin, who has made fundamental contributions to [[quantum field theory|Quantum Field Theory]], low dimensional [[topology]], [[knot theory]], and their interactions with [[mathematical physics]] and [[algebra]].

He is currently based at Indiana University ([Home page](https://math.indiana.edu/about/faculty/touraev-vladimir.html) which is not very informative!)

## Selected writings

* _On certain enumeration problems in two-dimensional topology_ ,
Math. Res. Lett.16(3) (2009), 515-529.

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 [[knot theory]] and [[3-manifolds]]:

* [[Vladimir Turaev]], *Quantum invariants of knots and 3-manifolds*, de Gruyter Studies in Mathematics **18** Walter de Gruyter & Co. (1994) [[doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221)]

On  [[HQFT|homotopy quantum field theory]]:

*  _Homotopy Quantum Field Theory_ , EMS Tracts in Math.10, European Math. Soc. Publ. House, Zurich 2010.

On [[Dijkgraaf-Witten theory]]:

* Dijkgraaf-Witten invariants of surfaces and projective representations of groups,
J. Geom. Phys.  57(11)  (2007), 2419-2430.

On [[braid groups]]:

* [[Christian Kassel]], [[Vladimir Turaev]], _Braid Groups_, GTM **247** Springer Heidelberg 2008 ([doi:10.1007/978-0-387-68548-9](https://link.springer.com/book/10.1007/978-0-387-68548-9), [webpage](http://irma.math.unistra.fr/~kassel/Braids-bk.html))

## Related entries

* [[Reshetikhin-Turaev construction]]

* [[Reshetikhin-Turaev invariant]]

* [[Turaev-Viro model]]

* [[unitary functorial field theory]]

[[!redirects Turaev]]

category: people


Vladimir Vapnik

* [website](https://datascience.columbia.edu/people/vladimir-vapnik/)

### Selected writings

On [[statistical learning theory]]:

* [[Vladimir Vapnik]], _Statistical Learning Theory_, Wiley (1998) [[ISBN:978-0-471-03003-4](https://www.wiley.com/en-us/Statistical+Learning+Theory-p-9780471030034)]

* [[David Corfield]], [[Bernhard Schölkopf]], [[Vladimir Vapnik]], _Falsificationism and statistical learning theory: Comparing the Popper and Vapnik-Chervonenkis dimensions_, Journal for General Philosophy of Science **40** 1 (2009) 51-58 [[doi:10.1007/s10838-009-9091-3](https://doi.org/10.1007/s10838-009-9091-3)]

category: people


Vladimir Vershinin

* [MathNet page](http://www.mathnet.ru/eng/person8972)

## Selected writings

On the [[group homology]] and [[group cohomology]] of [[braid groups]]:

* [[Vladimir Vershinin]], *Homology of Braid Groups and their Generalizations*, Banach Center Publications (1998) **42** 1 421-446 ([pdf](https://hopf.math.purdue.edu/Vershinin/hobr.pdf), [dml:208821](https://eudml.org/doc/208821))

category: people


Vladimir Voevodsky

Владимир Воеводский (who published in English as Vladimir Voevodsky) was a Russian mathematician working in the Institute for Advanced Study.

* IAS [obituary](https://www.ias.edu/news/2017/vladimir-voevodsky-obituary)

* former [web site](http://www.math.ias.edu/~vladimir/Site3/home.html)

* [Wikipedia article](http://secure.wikimedia.org/wikipedia/en/wiki/Vladimir_Voevodsky) .

Voevodsky received a [[Fields medal]] in 2002 for a [[proof]] of the [[Milnor conjecture]]. The proof crucially uses [[A1-homotopy theory]] and [[motivic cohomology]] developed by Voevodsky for this purpose. In further development of this in 2009 Voevodsky announced a proof of the [[Bloch-Kato conjecture]].

After this work in [[algebraic geometry]], [[cohomology]] and [[homotopy theory]] Voevodsky turned to the [[foundations of mathematics]] and worked on [[homotopy type theory]] which he described as a new "[[univalence|univalent]] foundations" for modern mathematics with its emphasis on [[homotopy theory]] and [[higher category theory]].

## Selected writings

Introducing the modern notion of [[equivalence in type theory]] (namely via [[contractible type|contractible]] [[fiber type|fibers]]) and thereby fixing the [[univalence axiom]] of [Hofmann & Streicher (1998),  §5.4](univalence+axiom#HofmannStreicher98) (due to the [subtlety with quasi-inverses](equivalence+in+type+theory#TheIssueWithQuasiInverses)):

* {#UnivalentFoundationsProject} [[Vladimir Voevodsky]],  *Univalent Foundations Project* (2010) [[pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/univalent_foundations_project.pdf), [[Voevodsky-UFP2010.pdf:file]]]

Proposal for a "[[Homotopy Type System]]" (cf. *[[semisimplicial type]]*):

* [[Vladimir Voevodsky]], _A type system with two kinds of identity types_ (Feb. 2013) [[[Voevodsky-HTS.pdf:file]]]

* [[Vladimir Voevodsky]], _A simple type system_ (Jan 2013) [[[Voevodsky-TTS.pdf:file]], [Implementation](https://github.com/DanGrayson/checker)]

## Selected Talks

*  {#Talk14} _The origins and motivations for univalent foundations_, IAS 2014 ([adapted transcript](https://www.ias.edu/ideas/2014/voevodsky-origins), [video]( https://video.ias.edu/voevodsky14/))

* list of video-recorded talks on [[homotopy type theory]]: [here](http://video.ias.edu/taxonomy/term/42).

## Selected Interviews

* An interview is [here](http://www.youtube.com/watch?v=vcDaQTPH-Rc).

* {#InterviewWithVoevodskyByMikhailov} [[Roman Mikhailov]], *Интервью Владимира Воеводского*, Princeton 2012 ([Part 1](https://baaltii1.livejournal.com/198675.html), [Part 2](https://baaltii1.livejournal.com/200269.html))

* _Le bifurcation de Vladimir Voevodsky_, interview conducted by Fondation Sciences Mathématiques de Paris, 2014, [video](https://vimeo.com/99586217), [transcript in French](http://smf4.emath.fr/Publications/Gazette/2014/142/smf_gazette_142_87-94.pdf).

## Related $n$Lab entries

*  [[homotopy theory]]
*  [[algebraic geometry]]
*  [[algebraic topology]]
*  [[K-theory]]
*  [[A1-homotopy theory]]
*  [[Voevodsky motive]]
*  [[Kapranov–Voevodsky 2-vector space]]
*  [[homotopy type theory]]
*  etc ...

category: people

[[!redirects Владимир Воеводский]]
[[!redirects Vladimir Voevodsky]]
[[!redirects Vladimir Voevodskij]]
[[!redirects Vladimir Voevodskiĭ]]
[[!redirects V. Voevodsky]]
[[!redirects Voevodsky]]


Vladimiro Sassone

* [Home Page](http://www.ecs.soton.ac.uk/people/vs)

category: people


Vladimír Souček

* [webpage](http://www.karlin.mff.cuni.cz/~soucek/)

## related $n$Lab entries

* [[BGG sequence]]

* [[parabolic geometry]], [[Cartan geometry]]

category: people

[[!redirects Vladimir Soucek]]
[[!redirects Vladimir Souček]]
[[!redirects Vladimír Soucek]]


Vladislav Kupriyanov

* [MPI page](https://www.mpp.mpg.de/en/news/news/detail/max-planck-institut-fuer-physik-begruesst-vladislav-kupriyanov-als-capes-humboldt-stipendiaten/)

## Selected writings

* [[Vladislav Kupriyanov]]

  **$L_\infty$-Bootstrap Approach to Non‐Commutative Gauge Theories**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [doi:1903.02867](https://arxiv.org/abs/1903.02867) [doi:10.1002/prop.201910010](https://doi.org/10.1002/prop.201910010)

on [[L-infinity algebras]] in [[noncommutative geometry|noncommutative]] [[gauge theory]]

category: people


Voevodsky motive

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

#Contents#
* table of contents
{:toc}

## Idea

[[A. Suslin]] and [[Vladimir Voevodsky|V. Voevodsky]] have realized a [[triangulated category]] $D^b(\mathcal{MM}_k)$ which is supposed to be the bounded [[derived category]] of the hypothetical [[abelian category]] of [[mixed motives]] over $k$, predicted by Grothendieck--Beilinson--Deligne.
There are variants developed by Hanamura and M. Levine. There is a different "derived" approach to mixed motives, namely the $A^1$-homotopy theory of [[F. Morel]] and [[V. Voevodsky]].

See at _[[motive]]_ the section _[Contructions of the derived category of mixed motives](http://ncatlab.org/nlab/show/motive#DerivedMotives)_.

## Applications

Voevodsky used the derived category of mixed motives to solve [[Milnor's conjecture]] in [[algebraic K-theory]].

## Related concepts

See also

* [[motivic homotopy theory]]

* [[motive]]

  * [[pure motive]]

    * [[Chow motive]], [[numerical motive]]

  * [[mixed motive]]

  * [[noncommutative motive]]

## References

* Wikipedia article on motives and references therein.

* A. Beilinson, V. Vologodsky, _A DG guide to Voevodsky's motives_, [math.AG/0604004](http://de.arxiv.org/abs/math/0604004)

* M.V. Bondarko, _Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura_, [math.AG/0601713](http://de.arxiv.org/abs/math/0601713)

* M.V. Bondarko, _Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey_, [arxiv:0903.0091](http://de.arxiv.org/abs/0903.0091)

* [[V.  Voevodsky]], _Motives over [[simplicial scheme|simplicial schemes]]_, Journal of K-Theory, Volume 5, Issue 01 , pp 1 - 38,
(preliminary version in K-theory preprint archive: [here](http://www.math.uiuc.edu/K-theory/0638/).

Models for Voevodsky motives:

* Peter Bonart, *Triangulated Categories of Big Motives via Enriched Functors* (2023) [[arXiv:2310.17349](https://arxiv.org/abs/2310.17349)]

[[!redirects Voevodsky motives]]


Vojta's conjecture

#Contents#
* table of contents
{:toc}

## Idea

Vojta introduced a dictionary between [[value distribution theory]]
of Nevanlinna and [[Diophantine approximation theory]] of Roth and suggested that this dictionary should continue to hold in higher dimensions.
This leads to qualitative conjectures in
[[arithmetic geometry]] which cover almost every important conjecture in the field; notably the [[abc conjecture]],
[[Mordell's conjecture]], some of [[Lang's conjecture]]s.  The perspective given by [[Arakelov geometry|Arakelov theory]] is central in Vojta's conjectures.

## Related concepts

* [[abc conjecture]]

## References

The original article is

* [[Paul Vojta]], _Diophantine approximations and value distribution theory_, Lecture Notes in Mathematics, 1239, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0072989, ISBN 978-3-540-17551-3, MR883451

See also

* David McKinnon, _Vojta's main conjecture for blowup surfaces_ ([pdf](http://www.math.uwaterloo.ca/~dmckinno/Papers/Vojta.Main.pdf))

[[!redirects Vojta conjecture]]


Vojtěch Pravda

* [webpage](http://users.math.cas.cz/~pravda/)

## related $n$Lab entries

* [[universal spacetime]]

category: people

[[!redirects Vojtech Pravda]]


Volker Braun

* [webpage](http://www.sas.upenn.edu/~vbraun/)

## Selected writings

On [[torsion subgroup|torsion]] in the [[D-brane charge quantization in topological K-theory]]:

* [[Volker Braun]], *K-Theory Torsion* [arXiv:hep-th/0005103]

Computation of the [[twisted ad-equivariant K-theory]]-groups for all [[simply connected topological space|simply connected]] [[compact Lie group|compact]] [[simple Lie groups]]:

* [[Volker Braun]], _Twisted K-Theory of Lie Groups_, JHEP 0403 (2004) 029 ([arXiv:hep-th/0305178](https://arxiv.org/abs/hep-th/0305178))

On [[single trace operators]]/[[BMN operators]] in [[super Yang-Mills theory]] regarded as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation operator]] and application to [[AdS-CFT duality]]:

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

On [[D-branes]] in [[Gepner models]] via [[boundary conformal field theory]]:

* {#BraunSchaeferNameki05} [[Volker Braun]], [[Sakura Schafer-Nameki]], _D-Brane Charges in Gepner Models_, J. Math. Phys. 47 (2006) 092304 ([arXiv:hep-th/0511100](https://arxiv.org/abs/hep-th/0511100))

On [[duality in string theory|duality]] of [[M-theory on G2-manifolds]] with [[heterotic string theory on CY3-manifolds]]:

* {#BraunSchaeferNameki17} [[Andreas Braun]], [[Sakura Schafer-Nameki]], _Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality_, JHEP04(2018)126 ([arXiv:1708.07215](https://arxiv.org/abs/1708.07215))

## Related pages

* [[string phenomenology]]

category: people


Volker Puppe

* [webpage](http://www.math.uni-konstanz.de/~puppe/)

category: people


Volker Schomerus

* [webpage](http://wwwiexp.desy.de/sfb676/researchers/schomerus/)

category: people


Volodin model

#Volodin model for K-theory

This is the simplicial set /  complex constructed by Volodin, using a construction similar to that of the [[Vietoris complex]].  It is the [[Volodin space]] of the family of subgroups of the stable general linear group described as follows:

We let $T_n^\sigma(R)$ be the subgroup of $G\ell_n(R)$ formed by the $\sigma$-triangular matrices, (discussed at [[higher generation by subgroups]]), and then look at all such subgroups for all $n$, considering the stable general linear group $G\ell(R)$ as the colimit of the nested sequence of all the $G\ell_n(R)$, take $G = G\ell(R)$. Considering the family,  $\mathcal{H}$, of all the $T_n^\sigma(R)$, form the corresponding [[Volodin space]].

##References

*  A. A. [[Suslin]] and M. Wodzicki, _Excision in algebraic K-theory_, The Annals of Mathematics,
136, (1992), 51 – 122.

* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative
rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887)


Volodin space

[[!redirects Volodin spaces]]
#Volodin space
* table of contents
{: toc}

##Idea

_Volodin spaces_ are the [[Vietoris complex]] analogues of the nerve of a family of subgroups discussed in the entry, [[higher generation by subgroups]].  They provide a way of building a geometric object that provides a means of comparing the information on the 'big group' that is 'stored' by subgroups within the family.

They were essentially introduced by [[Volodin]] as part of his approach to higher [[algebraic K-theory]]. We will discuss them via another approach that is explicit in work by [[Suslin]], on the equivalence of the Volodin K-theory with that of [[Quillen]].

##Preliminaries

Let $X$ be a non-empty set, and denote by $E(X)$, the [[simplicial set]] having $E(X)_p = X^{p+1}$, so a $p$-simplex is a $p+1$ tuple, $\underline{x}= (x_0,\ldots, x_p)$, each $x_i \in X$, and in which
$$d_i(\underline{x}) = (x_0,\ldots, \hat{x_i}, \ldots x_p),$$and
$$s_j(\underline{x}) = (x_0,\ldots, x_j, x_j, \ldots x_p),$$
so $d_i$ omits $x_i$, whilst $s_j$ repeats $x_j$.
+--{: .un_lemma}
######Lemma
The simplicial set, $E(X)$, is contractible.
=--
The proof is fairly easy to construct and is 'well known'.

The case we are really interested in is when we replace the general set, $X$, by the underlying set of a group, $G$. (As is often done, we will not introduce a special notation for the underlying set of $G$, just writing $G$ for it.) In this case, we have the simplicial set $E(G)$  and the group, $G$, acts freely on $E(G)$ by

$$g\cdot(g_0,\ldots , g_p) = (gg_0,\ldots, gg_p).$$

(Here we have used a left action of $G$, and leave you to check that the evident right action could equally well be used.)

The quotient simplicial set of orbits, will be denoted $G\backslash E(G)$. It is often useful to write $[g_1,\ldots,g_p]$ for the orbit of the $p$-simplex $(1,g_1,g_1g_2,\ldots, g_1g_2\ldots g_p)\in E(G)_p$.

It is 'instructive' to calculate the faces and degeneracy maps in this notation.  We will only look at $[g_1,g_2]$ in detail. This element has representative $(1,g_1,g_1g_2)$. We thus have:

* $d_0(1,g_1,g_1g_2) = (g_1,g_1g_2) \equiv (1,g_2)$,
so $d_0[g_1,g_2] = [g_2]$;
*  $d_1(1,g_1,g_1g_2) = (1,g_1g_2)$,
so $d_1[g_1,g_2] = [g_1g_2]$;
*  $d_2(1,g_1,g_1g_2) = (1,g_1)$,

so $d_2[g_1,g_2] = [g_1]$.

(That looks familiar!)

For the degeneracies,

*  $s_0(1,g_1,g_1g_2) = (1,1,g_1,g_1g_2)$, so $s_0 [g_1,g_2] =[1,g_1,g_2] $;
*  $s_1(1,g_1,g_1g_2) = (1,g_1,g_1,g_1g_2)$, so $s_1 [g_1,g_2] =  [g_1,1,g_2] ;$

and similarly $s_2 [g_1,g_2] =  [g_1,g_2,1]$.

The general formulae are now easy to guess and to prove - so they will be left to you, and then the following should be obvious.

+--{: .un_lemma}
######Lemma
There is a natural simplicial isomorphism,
$$G\backslash E(G)\xrightarrow{\cong}Ner(G[1])= BG.$$
=--
We thus have that $G\backslash E(G)$ is a [[classifying space]] for $G$.

This construction of $E(G)$ is exactly that of the nerve of the [[action groupoid]] of the action of $G$ on itself by left multiplication.

##Volodin spaces

We put ourselves in the context of a group, $G$, and a family, $\mathcal{H}$, of subgroups of $G$ as in the context of [[higher generation by subgroups]].  We suppose that $\mathcal{H}= \{H_i\mid i\in I\}$ for some indexing set, $I$.
+--{.un_defn}
(cf. Suslin-Wodzicki, (ref. below) p. 65.)
We denote by $V(G,\mathcal{H})$, or $V(\mathfrak{H})$, the simplicial subset of $E(G)$ formed by simplices, $(g_0,\ldots,g_p)$, that satisfy the condition that there is some $i\in I$ such that, for all $0\leq j,k\leq p$, $g_j g_k^{-1}\in H_i$.

The simplicial set, $V(G,\mathcal{H})$, will be called the _Volodin space_ of $(G,\mathcal{H})$.
=--

##References

*  A. A. [[Suslin]] and M. Wodzicki, _Excision in algebraic K-theory_, The Annals of Mathematics,
136, (1992), 51 – 122.

* [[I. Volodin]], _Algebraic K-theory as extraordinary homology theory on the category of associative
rings with unity_, Izv. Akad. Nauk. SSSR, 35, (Translation: Math. USSR Izvestija Vol. 5 (1971)
No. 4, 859-887)


Volodymyr Lyubashenko

**Volodymyr Lyubashenko** (Володимир Любашенко) is a [[mathematical physics|mathematical physicist]] and [[mathematics|mathematician]] from Kiev. He has been working in [[potential theory]], conformal field theory, [[Hopf algebras]], quantum groups, low-dimensional topology, category theory, operads and higher category theory, and [[homological algebra|homological]] (and most recently homotopical) [[algebra]].

* a list of papers [html](http://www.math.ksu.edu/~lub/papers.html)

* webpages at [Kiev](http://www.imath.kiev.ua/~lub), [Kansas State U.](http://www.math.ksu.edu/~lub)

## selected writings

* Yu. Bespalov, V. Lyubashenko, O. Manzyuk, _Pretriangulated $A_\infty$-categories_, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 ([ps.gz](http://www.math.ksu.edu/~lub/cmcMonad.gz))

  (on [[pretriangulated A-∞ categories]])

  * notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are [here](http://www.math.ksu.edu/~lub/cmcMoCor.pdf).

## related $n$Lab entries

* [[hypersimplex]]

* [[Serre functor]]

* [[Tannaka duality]]

* [[A-∞ category]]

* [[pretriangulated A-∞ category]]

* [[(∞,1)-bimodule]]

* [[Atiyah 2-framing]]

category: people

[[!redirects Володимир Любашенко]]


volume

#Contents#
* table of contents
{:toc}

## Idea

For $(X, \mu)$ a [[measure space]], the [[integral]] $\int_X \mu$ is, if it exists, the _volume_ of $X$ as seen by the [[measure]] $\mu$.

## Related concepts

* [[length]]

* [[area]]

* [[density]], [[volume form]]

* [[complex volume]]

[[!redirects volumes]]


volume conjecture

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _volume conjectures_ are a class of [[conjectures]] (slightly differing in generality and assumptions), saying that on a suitable (in particular: [[hyperbolic 3-manifold|hyperbolic]]) [[3-manifold]] $X$ a [[large N limit]] of [[SU(n)]]-[[Chern-Simons theory]] [[quantum observables]] ($N$-[[colored Jones polynomials]] or more generally [[RT-invariants]], [[TV-invariants]]) is [[equality|equal]] to the [[volume]] or [[complex volume]] of $X$.

For a few special cases of 3-manifolds there are explicit [[proofs]] of the volume conjecture(s). Besides this there is an abundance of [[computer experiment|numerical evidence]] for the volume conjectures, using computer algebra such as [[SnapPy]] (see also [Zickert 07](#Zickert07)). In fact experimentation with these numerics is what has been driving the formulation of further variants of the volume conjecture.

Hence [[experimental mathematics]] strongly suggests that the volume conjectures are true. But a conceptual explanation (let alone [[proof]]) in terms of [[quantum field theory]] has remained open ([Witten 14, bottom of p. 4](#Witten14)).

But an explanation via [[duality in string theory]], combining [[AdS/CFT duality]] with the [[3d/3d correspondence]] for [[wrapped brane|wrapped]] [[M5-branes]], is argued for in [Gang-Kim-Lee 14, 3.2](#GangKimLee14), [Gang-Kim 18 (21)](#GangKim18), see [below](#AsAdSCFTPlus3d3dDuality).

### For $SU(2)$ on knot complements

The original _volume conjecture_ (also "Kashaev's conjecture", due to [Kashaev 95](#Kashaev95), and understood in terms of the $N$-[[colored Jones polynomial]] by [Murakami-Murakami 01](#MurakamiMurakami01)) states that the [[large N limit]] of the $N$-[[colored Jones polynomial]] (for [[gauge group]] [[SU(2)]]) of a [[knot]] $K$ gives the simplicial [[volume]] of its [[complement]] in the [[3-sphere]] (for [[hyperbolic knots]] this is the volume of the complementary [[hyperbolic 3-manifold]])

\[
  \label{KashaevConjecture}
  lim_{N \to \infty}
  \left(
    \frac{
       2 \pi log
    }
    {N}
      \left\vert
        V_N(K; q = e^{\frac{2 \pi i}{N}})
      \right\vert
  \right)
  \;=\;
  vol(K).
\]

Here $V_N(K; q)$ is the ratio of the values of the $N$-[[colored Jones polynomial]] of $K$ and of the [[unknot]]

$$
  V_N(K; q) = \frac{J_N(K; q)}{J_N(\bigcirc; q)}.
$$

The simplicial volume of a knot complement can be found via its unique **torus** decomposition  into hyperbolic pieces and Seifert fibered pieces by a system of tori. The simplicial volume is then the sum of the hyperbolic volumes of the hyperbolic pieces of the decomposition.

If one omits the [[absolute value]] in (eq:KashaevConjecture) then the volume conjecture instead involves the [[complex volume]] ([MMOTY 02, Conjecture 1.2](#MMOTY02)).

### For $SU(2)$ on general 3-manifolds
 {#IdeaGenerally}

More generally, volume conjectures state [[convergence of a sequence|convergence]] of the [[Turaev-Viro invariants]] or [[Reshetikhin-Turaev invariants]] on general [[hyperbolic 3-manifolds]] to the [[volume]] or [[complex volume]], respectively.

See ([Chen-Yang 15](#ChenYang15))

### For $SU(n)$

Generalization from [[gauge group]] [[SU(2)]] to [[SU(n)]]: [Chen-Liu-Zhu 15](#ChenLiuZhu15)

## Proof strategies

### As combined AdS/CFT + 3d/3d duality for wrapped M5-branes
 {#AsAdSCFTPlus3d3dDuality}

In [Gang-Kim-Lee 14b, 3.2](#GangKimLee14b), [Gang-Kim 18 (21)](#GangKim18) it is argued that the [[volume conjecture]] for [[Chern-Simons theory]] on [[hyperbolic 3-manifolds]] $\Sigma^3$ is the combined statement of two [[dualities in string theory]]:

1. [[AdS/CFT duality]]

1. [[3d-3d correspondence]]

for the situation of [[M5-branes]] [[wrapped brane|wrapped on]] $\Sigma^3$ ([DGKV 10](3d-3d+correspondence#DGKV10)):

For review of the literature see also [Dimofte 16, Section 4.1](#Dimofte16).

## Related concepts

* [[hyperbolic manifold]]

* [[analytically continued Chern-Simons theory]]

* [[Borel regulator]]

* [[membrane instanton]]

## References

### General

Original articles include

* {#Kashaev95} [[Rinat Kashaev]], _A Link Invariant from Quantum Dilogarithm_,  Modern Physics Letters AVol. 10, No. 19, pp. 1409-1418 (1995) ([arXiv:q-alg/9504020](https://arxiv.org/abs/q-alg/9504020))

* {#Kashaev95} [[Rinat Kashaev]], _The Hyperbolic Volume Of Knots From The Quantum Dilogarithm_ Lett. Math. Phys. 39 (1997) 269-275 ([arXiv:q-alg/9601025](https://arxiv.org/abs/q-alg/9601025))

* {#KashaevTirkkonen99} [[Rinat Kashaev]], O. Tirkkonen, _Proof of the volume conjecture for torus knots_, Journal of Mathematical Sciences (2003) 115: 2033 ([arXiv:math/9912210](http://arxiv.org/abs/math/9912210))

* {#MurakamiMurakami01} [[Hitoshi Murakami]], [[Jun Murakami]], _The Colored Jones Polynomial And The Simplicial Volume Of A Knot_, Acta Math. 186 (2001) 85-104 ([euclid.acta/1485891370](https://projecteuclid.org/euclid.acta/1485891370))

* {#MMOTY02} [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, _Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links_, Experiment. Math. 11 (2002) 427-435 ([arXiv:math/0203119](https://arxiv.org/abs/math/0203119))

* {#Murakami04} [[Hitoshi Murakami]], _Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot_, Internat. J. Math. 15 (2004) 547-555.

Generalization to [[Reshetikhin-Turaev construction]] on closed manifold, to the  [[Turaev-Viro construction]] on [[manifolds with boundary]], and to more general [[roots of unity]] than considered before is in

* {#ChenYang15} [[Qingtao Chen]], [[Tian Yang]], _A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary_ ([arXiv:1503.02547](http://arxiv.org/abs/1503.02547))

* Dongmin Gang, Mauricio Romo, Masahito Yamazaki, _All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory_,     Commun. Math. Phys. (2018) 359: 915. ([arXiv:1704.00918](https://arxiv.org/abs/1704.00918), [doi:10.1007/s00220-018-3115-y](https://doi.org/10.1007/s00220-018-3115-y))

Generalization to [[SU(n)]]:

* {#ChenLiuZhu15} [[Qingtao Chen]], [[Kefeng Liu]], Shengmao Zhu, _Volume conjecture for $SU(n)$-invariants_ ([arXiv:1511.00658](https://arxiv.org/abs/1511.00658))

Review includes

* {#Murakami11} [[Hitoshi Murakami]], _An Introduction to the Volume Conjecture_ ([arXiv:1002.0126](https://arxiv.org/abs/1002.0126)) In: _Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory_, Contemporary Mathematics Volume 541, AMS 2011  ([doi:10.1090/conm/541](http://dx.doi.org/10.1090/conm/541))

* {#Witten14} [[Edward Witten]], pp. 4 of _Two Lectures On The Jones Polynomial And Khovanov Homology_ ([arXiv:1401.6996](http://arxiv.org/abs/1401.6996))

* Wikipedia, _[Volume conjecture](http://en.wikipedia.org/wiki/Volume_conjecture)

See also

* {#Neumann03} [[Walter Neumann]], _Extended Bloch group and the Cheeger-Chern-Simons class_, Geom. Topol. 8 (2004) 413-474 ([arXiv:math/0307092](http://arxiv.org/abs/math/0307092))

* {#Zickert07} [[Christian Zickert]], _The volume and Chern-Simons invariant of a representation_, Duke Math. J., 150 (3):489-532, 2009 ([arXiv:0710.2049](http://arxiv.org/abs/0710.2049))

* {#Neumann11} [[Walter Neumann]], _Realizing arithmetic invariants of hyperbolic 3-manifolds_, Contemporary Math 541 (Amer. Math. Soc. 2011), 233--246 ([arXiv:1108.0062](http://arxiv.org/abs/1108.0062))

* {#GaroufalidisThurstonZickert11} [[Stavros Garoufalidis]], [[Dylan Thurston]], [[Christian Zickert]], _The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds_ ([arXiv:1111.2828](http://arxiv.org/abs/1111.2828), [Euclid](http://projecteuclid.org/euclid.dmj/1259332507))

### Via string theory

#### General

Speculative discussion in terms of [[quantum field theory]] or [[string theory]] includes

* {#Gukov03} [[Sergei Gukov]], _Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial_, Commun. Math. Phys. 255 (2005) 577-627 ([arXiv:hep-th/0306165](https://arxiv.org/abs/hep-th/0306165))

* {#DijkgraafFuji09} [[Robbert Dijkgraaf]], Hiroyuki Fuji, _The Volume Conjecture and Topological Strings_ ([arXiv:0903.2084](http://arxiv.org/abs/0903.2084))

* {#DimofteGukov10} [[Tudor Dimofte]], [[Sergei Gukov]], _Quantum Field Theory and the Volume Conjecture_, Contemporary Mathematics 541 (2011), p.41-67 ([arxiv:1003.4808](http://arxiv.org/abs/1003.4808))

* {#Dimofte16} [[Tudor Dimofte]], Section 4.1 of:  _Perturbative and nonperturbative aspects of complex Chern-Simons Theory_, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 ([arXiv:1608.02961](https://arxiv.org/abs/1608.02961))

A conceptual explanation of the volume conjecture via [[analytically continued Chern-Simons theory]] was proposed in

* {#Witten10} [[Edward Witten]], _Analytic Continuation Of Chern-Simons Theory_, AMS/IP Stud. Adv. Math 50 (2011): 347 ([arXiv:1001.2933](https://arxiv.org/abs/1001.2933))

(but it seems that as a sketch or strategy for a rigorous proof, it didn't catch on).

#### As AdS/CFT + 3d/3d duality for wrapped M5-branes

Suggestion that the statement of the [[volume conjecture]] is really [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, Section 3.2 of: _Holography of 3d-3d correspondence at Large $N$_, JHEP 04 (2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], around (21) of: _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))

[[!redirects volume conjectures]]


volume form

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

# Contents
* table of contents
{: toc}

## Idea

The _volume form_ on a finite-[[dimension|dimensional]] [[orientation|oriented]] (pseudo)-[[Riemannian manifold]] $(X,g)$ is the [[differential form]] whose [[integral]] over pieces of $X$ computes the [[volume]] of $X$ as measured by the [[metric]] $g$.

If the manifold is unoriented, then we get a _volume [[pseudoform]]_ instead, or equivalently a volume [[density]] (of weight $1$).  We can also consider volume (pseudo)-forms in the absence of a metric, in which case we have a choice of volume forms.

## Definition

### General

For $X$ a general [[smooth manifold]] of finite [[dimension]] $n$, a __volume form__ on $X$ is a nondegenerate (nowhere vanishing) differential $n$-[[differential form|form]] on $X$, equivalently a nondegenerate [[section]] of the [[canonical line bundle]] on $X$.  A __volume pseudoform__ or __volume element__ on $X$ is a positive definite [[density]] (of rank $1$) on $X$, or equivalently a positive definite differential $n$-[[pseudoform]] on $X$.

A volume form defines an [[orientation]] on $X$, the one relative to which it is positive definite.  If $X$ is already oriented, then we require the orientations to agree (to have a volume form on $X$ qua oriented manifold); that is, a volume form on an oriented manifold must be positive definite (just as a volume pseudoform on any manifold must be).  In this situation, there is essentially no difference between a form and a pseudoform, hence no difference between a volume form and a volume pseudoform or volume element.

### For Riemannian manifolds

More specifically, for $(X,g)$ a (pseudo)-[[Riemannian manifold]] of [[dimension]] $n$, *the* __volume pseudoform__ or __volume element__ $vol_g$ is a specific differential $n$-[[pseudoform]] that measures the volume as seen by the metric $g$.  If $X$ is [[orientation|oriented]], then we may interpret $vol_g \in \Omega^n(X)$ as a differential $n$-form, also denoted $vol_g$.

This $vol_g$ is characterized by any of the following equivalent statements:

* The symmetric square $vol_g \cdot vol_g$ is equal to the $n$-fold [[wedge product]] $g \wedge \cdots \wedge g$, as elements of $\Omega^n(X) \otimes \Omega^n(X)$, and $vol_g$ is [[positive n-form|positive]] (meaning that its [[integral]] on any [[open submanifold]] is nonnegative).

* The volume form is the image under the [[Hodge star]] operator $\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X)$ of the [[smooth function]] $1 \in \Omega^0(X)$

  $$
    vol_g = \star_g 1
    \,.
  $$

* In local oriented coordinates, $vol_g = \sqrt{|det(g)|}$, where $det(g)$ is the [[determinant]] of the [[matrix]] of the coordinates of $g$.  In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to $vol_g = \sqrt{\det(g)}$.  (Note that local coordinates for a pseudoform include a local orientation, so this makes sense regardless of whether $X$ is oriented.)

* For $(E, \Omega)\colon T X \to iso(n)$ the [[Lie algebra valued differential form]] on $X$ with values in the [[Poincare Lie algebra]] $iso(n)$ that encodes the metric and orientation (the _[[spin connection]]_ $\Omega$ with the _[[vielbein]]_ $E$), the volume form is the image of $(E,\Omega)$ under the canonical volume [[Lie algebra cohomology|Lie algebra cocycle]] $vol \in CE(iso(n))$:

  $$
    vol_g = vol(E)
    \,.
  $$

  See [[Poincare Lie algebra]] for more on this.

### Degenerate cases

If we allow a volume (pseudo)form to be degenerate, then most of this goes through unchanged.  In particular, a degenerate (pseudo)-[[Riemannian metric]] defines a degenerate volume pseudoform (and hence a degenerate volume form on an oriented manifold).

However, a degenerate $n$-form $\omega$ does not specify an orientation in general, so there is not necessarily a good notion of volume form on an unoriented manifold.  On the other hand, if the open submanifold on which $\omega \ne 0$ is [[dense subspace|dense]], then there is at most one compatible orientation, although there still may be none.  Of course, on an oriented manifold, forms are equivalent to pseudoforms, so we still know what a degenerate volume form is there.

To remove the requirement of positivity is much more drastic; an arbitrary $n$-pseudoform is simply a $1$-[[density]] (and an arbitrary $n$-form is a $1$-pseudodensity).  This is at best a notion of *signed* volume, rather than volume.

## Properties

Since an $n$-(pseudo)form is [[positive n-form|positive]] iff its [[integration of differential forms|integral]] on any [[open submanifold]] is nonnegative and nondegenerate iff its integral on sufficiently small [[inhabited subspace|inhabited]] open submanifolds is nonzero, a volume (pseudo)form may be defined as one whose integral on any inhabited open submanifold is (strictly) positive.

A volume (pseudo)form is also equivalent to an [[absolutely continuous measure|absolutely continuous]] positive [[Radon measure]] on $X$.  Here, nondegeneracy corresponds precisely to absolute continuity.

If I remember correctly, every volume (pseudo)form comes from a metric, which is unique iff $n \leq 1$.

## Related concepts

* [[density]]

* [[divergence]]

[[!redirects volume form]]
[[!redirects volume forms]]
[[!redirects volume pseudoform]]
[[!redirects volume pseudoforms]]
[[!redirects volume element]]
[[!redirects volume elements]]


volume of a Lie groupoid

#Idea#

One expects a notion of volume measure on [[Lie groupoid]]s (also known as [[differentiable stack]]s), which generalizes the notion of [[groupoid cardinality]] of finite groupoids in that it reduces the volume of the object space by the degree to which [[automorphism]]s encode [[homological resolution|weak quotient]]s.

A solution to this was proposed by Alan Weinstein, and re-interpreted in terms of 2-vector bundles by Richard Hepworth.

One would expect some relation of this to the [[BV theory|Lagrangian BV formalism]], which is also a formalism for integration over $L_\infty$-[[Lie infinity-algebroid|algebroid]]s.

#References#

* [[Alan Weinstein]], _The volume of a differentiable stack_ ([arXiv](http://arxiv.org/abs/0809.2130))

* Richard Hepworth, _2-Vector Bundles and the Volume of a Differentiable Stack ([pdf](http://www.hepworth.staff.shef.ac.uk/files/2VB.pdf))

[[!redirects volume of Lie groupoids]]


von Neumann algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### AQFT
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

\tableofcontents

## Idea

A _von Neumann algebra_ or _$W^*$-algebra_ is an important and special kind of [[operator algebra]], relevant in particular to [[measure theory]] and [[quantum mechanics]]/[[quantum field theory]] in its algebraic formulation as [[AQFT]]. Specifically, (non-commutative) von Neumann algebras can be understood as the [[Isbell duality|formal duals]] of ([[non-commutative geometry|non-commutative]]) [[localizable measurable spaces]] (or [[measurable locales]]); see the section _[Relation to measurable spaces](#RelationToMeasureSpaces)_ below.

## History and terminology

Since terminology varies in the literature, we will say something about this first.  There are no precise definitions here; see below for those.

(Of course, _$W^*$-algebras_ should not be confused with _[[W-algebras]]_ in (logarithmic) [[conformal field theory]].)

[[John von Neumann]] originally studied certain [[operator algebras]] (back then they were called _[[rings]] of [[linear operator|operators]]_), defined as unital $*$-subalgebras of the algebra $B(H)$ of [[bounded operators]] on some [[Hilbert space]] $H$ that are [[closed subspace|closed]] in any of the several [[operator topology|operator topologies]] on $B(H)$ (except for the norm topology, which gives $C^*$-[[C-star-algebra|algebras]]); the ultraweak topology is most convenient for our purposes.

One disadvantage of such a definition is that it makes it difficult to separate properties of von Neumann algebras from properties of their [[representations]] on Hilbert spaces.  For example, all faithful representations induce the same ultraweak topology on a von Neumann algebra, but different representations induce different weak topologies.  Furthermore, not all von Neumann algebras come automatically
equipped with a representation on a Hilbert space, such as the [[coproduct]] of two von Neumann algebras (although such a representation can always be constructed).  Finally, this definition unnecessarily confuses two very distinct notions: algebras and [[modules]] (or representations).

Therefore, we may use the modern abstract terminology in which a von Neumann algebra is defined as an [[associative algebra|algebra]] with certain [[extra structure|structures]] and [[extra property|properties]].  It then becomes a theorem that every von Neumann algebra has a free representation on a Hilbert space (such as Haagerup\'s standard form), so we may study von Neumann algebras in the historical concrete sense if we wish; but now we think of these as particular *representations* of algebras.

In the old terminology, morphisms of representations
of von Neumann algebras (von Neumann algebras in the historical concrete sense) are sometimes called _spatial morphisms of von Neumann algebras_ (as opposed to the _abstract morphisms_ that we will define below).  Similarly, the concrete von Neumann algebras themselves are sometimes called _von Neumann algebras_, whereas the abstract von Neumann algebras are called _$W^*$-algebras_.  Compare the historic definitions of $C^*$-algebras, as well as other examples of [[concrete and abstract structures]] such as [[manifolds]].

The [[nPOV]] dictates that a clear distinction between
the categories of algebras and modules must be maintained,
in particular, modules should not be confused with algebras.
Hence we stick to the modern terminology, which also seems
to be preferred in new papers on von Neumann algebras,
see for example [arXiv:1110.5671v1](http://arxiv.org/abs/1110.5671).

## Definitions

For completeness, we give both the modern abstract and historical concrete definitions.

### Abstract von Neumann algebras {#abstractdefn}

We build on the concepts of [[Banach space]] and (abstract) $C^*$-[[C-star-algebra|algebra]].  In this definition, a Banach space is a [[complex number|complex]] Banach space and a [[morphism]] of Banach spaces is a [[short linear map]] (a complex-linear map of norm at most $1$); a $C^*$-algebra is a complex unital $C^*$-algebra, and a morphism of $C^*$-algebras is a unital $*$-homomorphism (which is necessarily also a short linear map).  Note in particular that an [[isomorphism]] of either must be an [[isometry]].

Given a Banach space $A$, a __[[predual]]__ of $A$ is a Banach space $V$ whose [[dual Banach space]] $V^*$ is isomorphic to $A$:
$$ V^* \overset{i}\to A $$
(or more properly, equipped with such an isomorphism $i$).  Similarly, given a morphism $f\colon A \to B$ (properly, with $A$ and $B$ so equipped), a __predual__ of $f$ is a morphism $t\colon W \to V$ whose dual is isomorphic to $f$:
$$ \array {
   V^*                      & \overset{i}\to  & A \\
   \mathllap{t^*}\downarrow &                 & \downarrow\mathrlap{f} \\
   W^*                      & \underset{j}\to & B
} .$$

With these preliminaries, a __$W^*$-algebra__ or ("abstract") *von Neumann algebra* is a $C^*$-algebra that admits a predual (or more properly, equipped with one), and a $W^*$-[[homomorphism]] is a $C^*$-homomorphism that admits a predual.  In this way, the [[category]] of $W^*$-algebras becomes a [[subcategory]] of the category of $C^*$-algebras.

It is a theorem (see [below](#preduals)) that the predual of a $W^*$-algebra or $W^*$-homomorphism is essentially unique; we speak of [[generalized the|the]] predual of $A$, write it $A_*$, and identify $A$ with $(A_*)^*$ (and similarly for morphisms).  (So in fact we don\'t need the word 'equipped'; being a $W^*$-algebra is an [[extra property]], not an [[extra structure]], on a $C^*$-algebra.)

### Concrete von Neumann algebras

Fix a [[complex number|complex]] [[Hilbert space]] $H$ and consider the [[associative algebra|algebra]] $B(H)$ of [[bounded operators]] on $H$.  A ("concrete") __von Neumann algebra__ on $H$ is a unital $*$-subalgebra of $B(H)$ that is [[closed subspace|closed]] in the [[weak operator topology]], or equivalently in the [[ultraweak topology]] or in the [[strong topology]].  As such is automatically [[closed subspace|closed]] in the [[norm topology]], the von Neumann algebras form a (particularly nice) class of concrete $C^*$-[[C-star-algebra|algebras]] on $H$, where the latter are defined as unital $*$-subalgebras of $B(H)$ closed under the norm topology.

We equip a von Neumann algebra with the topology induced by its inclusion into $B(H)$ equipped with the ultraweak topology.  An __abstract morphism__ of von Neumann algebras can then be defined as a unital $*$-homomorphism that is continous in the ultraweak topology.  Here we are disregarding the data of the inclusion of a von Neumann algebra into $B(H)$ and treating it as an algebra on its own.

Alternatively, we can define a von Neumann algebra $A$
as a unital $*$-algebra that admits an injective morphism into $B(H)$ for some Hilbert space $H$ such that the image of the inclusion is closed in the ultraweak topology on $B(H)$.  One can then prove that the topology induced on $A$ by the ultraweak topology on $B(H)$ does not depend on the choice of $H$ or the particular inclusion of $A$ into $B(H)$.  Hence one can define an abstract morphism of von Neumann algebras as a unital morphism of $*$-algebras that is continuous in the ultraweak topology.

It is a theorem that the category of (concrete) von Neumann algebras and abstract morphisms is [[equivalence of categories|equivalent]] to the category of (abstract) $W^*$-algebras and $W^*$-homomorphisms.  Similarly, we get the category of [[representations]] of $W^*$-algebras on Hilbert spaces using instead the __spatial morphisms__ of concrete von Neumann algebras.

## Sakai's theorem and properties of preduals
   {#preduals}

Sakai's theorem states that preduals considered in the abstract definition
are necessarily unique.
More precisely, given a von Neumann algebra $A$
we consider the category whose objects are pairs $(V,f)$,
where $V$ is a Banach space and $f\colon V^*\to A$ is an isomorphism
of Banach spaces.  A morphism from $(V,f)$ to $(W,g)$
is a morphism $h\colon V\to W$ of Banach spaces
such that $f h^* = g$.

Sakai's theorem then states that in the above category
there is exactly one morphism
between any pair of objects, which is necessarily an isomorphism.
In particular, the category of preduals is canonically isomorphic
to the [[terminal category]].

Sakai's theorem can be extended to morphisms of von Neumann algebras.
Thus preduals of von Neumann algebras and their morphisms
are unique up to a unique isomorphism, in particular we can talk
about [[generalized the|the]] predual of a von Neumann algebra and [[generalized the|the]] predual
of a morphism of von Neumann algebras.

The weak topology induced on a von Neumann algebra by its predual
is called the **ultraweak topology**.
The role of the ultraweak topology for von Neumann algebras
is analogous to the role of the norm topology for C\*-algebras.
In particular, morphisms of von Neumann algebras are precisely
those morphisms of C\*-algebras that are continuous
in the ultraweak topology.

Consider the [[dual vector space|dual space]] $V$ of a von Neumann algebra $A$ equipped
with the ultraweak topology.
The [[topological vector space]] $V$ canonically embeds
into the dual of $A$ as a Banach space, the embedding map being
induced by the canonical continuous map from $A$
equipped with the norm topology to $A$
equipped with the ultraweak topology.
Thus $V$ is also a Banach space.
There is a canonical morphism of Banach spaces from $A$ to $V^*$
given by evaluating an element of $V$ on the given element of $A$.
This morphism is in fact an isomorphism, hence $V$ is the predual of $A$.
In other words, the predual of a von Neumann algebra is canonically
isomorphic to its dual in the ultraweak topology.
Similarly, the predual of a morphism of von Neumann algebras
is canonically isomorphic to its dual in the ultraweak topology.

## Elementary examples

The easiest example of a von Neumann algebra is given by the $C^*$-algebra $B(H)$
of bounded operators on a complex [[Hilbert space]] $H$.
The predual can be canonically identified with the Banach space
of [[trace class operator|trace class operators]].

Any $C^*$-subalgebra of $B(H)$ closed in the ultraweak topology
is again a von Neumann algebra.

Another example is $L^\infty(X)$ under pointwise almost everywhere multiplication, where $X$ is a σ-finite [[measure space]] or a [[localizable measurable space]]. These are (up to isomorphism) all of the _commutative_ von Neumann algebras, according to a specialized version of the [[Gelfand duality theorem]].

A faithful representation (in fact, the standard from) of $L^\infty(X) \hookrightarrow B(L^2(X))$ is given by considering $L^2(X)$ as an $L^\infty(X)$-module given by pointwise almost everywhere multiplication.

## Properties of morphisms of von Neumann algebras

## The category of von Neumann algebras

The category of von Neumann algebras is not a [[locally presentable category]]
since its [[small objects]] are precisely von Neumann algebras of dimension 0 and 1.
See Theorem 4.2 in [Chirvasitu–Ko](#CK).

The [[forgetful functor]] from von Neumann algebras
to [[sets]] that sends a von Neumann algebra
to its [[unit ball]] is a [[right adjoint functor]].
In fact, it is a [[monadic functor]] and preserves all [[sifted colimits]].

Thus, [[limits]] and [[sifted colimits]] of von Neumann algebras can be computed
on the level of underlying unit balls.

Small [[coproducts]] of von Neumann algebras exist.
There is also a “reduced” version of small [[coproducts]],
known as [[free products]],
which can be defined in a manner analogous to the [[spatial tensor product]].

## Monoidal structures

There are two different [[tensor products]] one can define on von Neumann algebras.

First, one can use the usual universal property of tensor products
and postulate that morphisms $A\otimes B\to M$
are in a natural bijection with pairs of morphisms
$A\to M$ and $B\to M$ whose images commute in $M$.
This yields a [[symmetric monoidal structure]] on von Neumann algebras.
This monoidal structure is not [[closed]].

Secondly, one can also define a “reduced” version, known as the _spatial tensor product_.
Given two von Neumann algebras $A$ and $B$, their spatial tensor product
is the von Neumann algebra generated by $A\otimes 1$ and $1\otimes B$
in the von Neumann algebra $B(L^2 A\otimes L^2 B)$,
where $L^2 A$ and $L^2 B$ are the Haagerup standard form of $A$ and $B$ respectively.
This also results in a [[symmetric monoidal structure]].
Furthermore, passing to the [[opposite category]]
yields a [[closed monoidal structure]].

## $W^*$-categories

## Modules over von Neumann algebras

## Bimodules over von Neumann algebras and Connes fusion

## Modular algebra and Tomita--Takesaki theory

* [[modular theory]]

## Gelfand duality for commutative von Neumann algebras

See the article [[commutative von Neumann algebra]].

## Relevance

The [[Gelfand duality theorem]] states that there is a contravariant [[equivalence of categories|equivalence]] between the [[category]] of commutative von Neumann algebras and the category of compact strictly localizable enhanced [[measurable space|measurable spaces]]; that is, the [[opposite category]] of one is equivalent to the other. See [Relation to Measurable Spaces](#RelationToMeasureSpaces) below. General von Neumann algebras are seen then as a 'noncommutative' measurable spaces in a sense analogous to [[noncommutative geometry]].

The importance of von Neumann algebras for ([[higher category theory|higher]]) [[category theory]] and topology lays in the evidence that von Neumann algebras are deeply connected with the low dimensional [[quantum field theory]] (2d [[CFT]], [[TQFT]] in low dimensions, inclusions of [[von Neumann algebra factor|factor]]s, [[quantum group]]s and [[knot theory]]; [[elliptic cohomology]]: works of Wenzl, Vaughan Jones, Anthony Wasserman, Kerler, Kawahigashi, Ocneanu, Szlachanyi etc.).

The highlights of their structure theory include the results on classification of [[von Neumann algebra factor|factors]] ([[Alain Connes]], 1970s) and theory of inclusions of subfactors (V. Jones). (Hilbert) [[bimodule]]s over von Neumann algebras have a remarkable tensor product due Connes ([[Connes fusion]]). Following Segal's manifesto

* Graeme Segal, _Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)_. Séminaire Bourbaki, Vol. 1987/88.  Astérisque  No. 161-162  (1988), Exp. No. 695, 4, 187--201 (1989).

and its update

* [[Graeme Segal]], _What is an elliptic object?_  Elliptic cohomology,  306--317, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007.

on hypothetical connections between [[CFT]] and [[elliptic cohomology]], Stolz and Teichner have made a case for a role von Neumann algebras seem to play in a partial realization of that program:

* [[Stefan Stolz]] and [[Peter Teichner]], _What is an elliptic object?_ ([pdf](http://math.berkeley.edu/~teichner/Papers/Oxford.pdf))

See also the [Wikipedia entry](http://en.wikipedia.org/wiki/Von_Neumann_algebra) entry for more on von Neumann algebras and a list of references and links.

## General

The _[[bicommutant theorem]]_ (as known as the _double commutant theorem_ , or _von Neumann's double commutant theorem_ ) is the following result.

Let $A \subseteq B(H)$ be a sub-[[star-algebra]] of the [[C-star algebra]] of [[bounded linear operator|bounded linear operators]] on a [[Hilbert space]] $H$. Then $A$ is a [[von Neumann algebra]] on $H$ if and only if $A = A''$, where $A'$ denotes the [[commutant]] of $A$.

Notice that the condition of $A$ being a von Neumann algebra (being closed in the [[weak operator topology]]; "weak" here can be replaced by "strong", "ultrastrong", or "ultraweak" as described in [[operator topology]]), which is a [[topology|topological]] condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).

## Relation to measurable spaces
 {#RelationToMeasureSpaces}

The [[Gelfand-Naimark theorem|Gel'fand–Naimark theorem]] states that the category of [[localizable measurable spaces]] is contravariantly [[equivalence of categories|equivalent]] to (that is equivalent to the [[opposite category|opposite]] of) the category of commutative von Neumann algebras.  As such, arbitrary von Neumann algebras may be interpreted as 'noncommutative' measurable spaces in a sense analogous to [[noncommutative geometry]]. See at _[[noncommutative probability space]]_.

## Topics of interest for the understanding of AQFT

This paragraph will collect some facts of interest for the aspects of [[AQFT]].

In this paragraph $\mathcal{M}$ will always be a von Neumann algebra acting on a [[Hilbert space]] $\mathcal{H}$ with [[commutant]] $\mathcal{M}'$.

### Vectors

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

A [[vector]] $x \in \mathcal{H}$ is a **[[cyclic vector]]** if $\mathcal{M}x$ is dense in $\mathcal{H}$.

=--

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

A [[vector]] $x \in \mathcal{H}$ is a **[[separating vector]]** if $M(x) = 0$ implies $M = 0$ for all $M \in \mathcal{M}$.

=--

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

The notions of cyclic and separating are dual with respect to the commutant, that is a vector is cyclic for $\mathcal{M}$ iff it is separating for $\mathcal{M}'$.

=--

### Projections in von Neumann algebras

One crucial feature of von Neumann algebras is that they contain
"every projection one could wish for": there are three points that make this statement precise:

* the linear combinations of projections are norm dense in a von Neumann algebra

* [[Gleason's theorem]]

* Murray--von Neumann classification of [[von Neumann algebra factor|factors]]

#### Projections are norm dense

First let us note that every element $A$ of a von Neumann algebra can trivially be written as a linear combination of two selfadjoint elements:
$$
A = \frac{1}{2} (A + A^*) + i\frac{1}{2i} (A - A^*)
$$

Then, by the [[spectral theorem]] every selfadjoint element A is represented by it's spectral measure E via
$$
A = \integral_{-\|A\|}^{\|A\|} \lambda E(d\lambda)
$$
The integral converges in norm to A and all spectral projections are elements of the von Neumann algebra. It immediatly follows that the set of finite sums of multiples of projections is norm dense in every von Neumann algebra.

#### Gleason's theorem {#GleasonsTheorem}

See [[Gleason's theorem]].

#### Murray--von Neumann classification of factors
To be done...

### Miscellaneous

* [[split inclusion of von Neumann algebras]]

## Related concepts

* [[enveloping von Neumann algebra]]

* [[von Neumann algebra factor]]

* [[W-star category]]

* [[quantum relation]]

## References
 {#References}

The original definition is due to [[John von Neumann]]:

* [[J. v. Neumann]], _Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren_, Mathematische Annalen 102:1 (1930), 370-427, [doi](http://dx.doi.org/10.1007/bf01782352).

Monographs:

* [[Masamichi Takesaki]]_, _Theory of Operator Algebras_, Encyclopaedia of Mathematical Sciences **124**, **125**, **127** (1979, 2002, 2003)

* {#Blackadar06} [[Bruce Blackadar]], *Operator Algebras -- Theory of $C^\ast$-Algebras and von Neumann Algebras*, Encyclopaedia of Mathematical Sciences **122**, Springer (2006) [[doi:10.1007/3-540-28517-2](https://doi.org/10.1007/3-540-28517-2)]

On [[von Neumann algebras]] in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *From quantum fields to local von Neumann algebras*, Rev. Math. Phys. **4** spec01 (1992) 15-47 [[doi:10.1142/S0129055X92000145](https://doi.org/10.1142/S0129055X92000145)]

See also:

* [[Jacob Lurie]], _von Neumann algebras_, lecture series (2011) ([web](http://www.math.harvard.edu/~lurie/261y.html))

* Abraham Westerbaan, _The Category of von Neumann Algebras_, [1804.02203](https://arxiv.org/abs/1804.02203) 2018

* {#CK} [[Alexandru Chirvasitu]], Joanna Ko, _Monadic forgetful functors and (non-)presentability for C∗- and W*-algebras_, [arXiv:2203.12087](https://arxiv.org/abs/2203.12087).

In [[quantum physics]]/[[quantum computing]]:

as [[categorical semantics]]

for the [[quantum lambda-calculus]]:

* [[Kenta Cho]], [[Abraham Westerbaan]], *Von Neumann Algebras form a Model for the Quantum Lambda Calculus*, QPL 2016 [[arXiv:1603.02133](https://arxiv.org/abs/1603.02133), [pdf](https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf), slides:[pdf](http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf), [[Cho-vNQuantumCalculus.pdf:file]]]

and for [[QPL]]:

* [[Kenta Cho]], *Semantics for a Quantum Programming Language by Operator Algebras*, EPTCS **172** (2014) 165-190 [[arXiv:1412.8545](https://arxiv.org/abs/1412.8545), [doi:10.4204/EPTCS.172.12](https://doi.org/10.4204/EPTCS.172.12)]

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von Neumann algebra factor

+-- {: .rightHandSide}
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#### Algebra
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#### Functional analysis
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#Contents#
* table of contents
{:toc}

## Definition

For $A$ a [[von Neumann algebra]] write $A'$ for its [[commutant]] in the ambient algebra $B(\mathcal{H})$ of [[bounded operator]]s.

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

A von Neumann algebra $A$ is called a **factor** if its [[center]] is trivial
$$
  Z(A) := A \cap A' = \mathbb{C}1
  \,.
$$

Equivalently: if $A$ and its [[commutant]] $A'$ generate the full algebra of [[bounded operators]] $B(\mathcal{H})$.

=--

## Properties

Every von Neumann algebra may be written as a [[direct integral]] over factors. ([von Neumann 49](#vNeumann49))

## Classification

Factors are classified in terms of the [[K-theory]] of their categories of finite [[W\*-modules]].
A [[W\*-module]] over a factor $A$ is _finite_ if it is not isomorphic to its proper submodule.

### Type I

Type I factors are characterized by the condition that the K-theory of finite modules
is isomorphic to $\mathbf{Z}$, the group of integers.
The only factors of this type are of the from $B(H)$, bounded operators on a Hilbert space $H$.

### Type II

Type II factors are characterized by the condition that the K-theory of finite modules
is isomorphic to $\mathbf{R}$, the group of real numbers.

Type II factors are subdivided into two classes: type II$_1$ factors are characterized
by the condition that $A$ is a finite $A$-module,
whereas for a type II$_\infty$ factor $A$ is not a finite $A$-module.

### Type III

Type III factors are characterized by the condition that the K-theory of finite modules
is trivial, i.e., only the zero module is finite.

Type III factors are further subdivided into three classes,
according to the structure of the center of their [[modular algebra]],
which is a commutative von Neumann algebra graded by purely imaginary numbers,
whose graded components are [[noncommutative L^p-spaces]].

By the [[von Neumann duality]] for commutative von Neumann algebras,
the spectrum of this center is a [[measurable space]] equipped with a σ-ideal
of negligible sets and the grading yields an action of $\mathbf{R}$, the group of real numbers.
This object is known as the [[noncommutative flow of weights]].

If the center is trivial (so the spectrum is a point), the factor has type III$_1$.
If the action of $\mathbf{R}$ is not periodic, then the factor has type III$_0$.
If the action is periodic with period $\lambda$, a positive real number,
then the factor has type III$_{\exp(-\lambda)}$.

## Related concepts

* [[C-star algebra]]

* [[uniformly hyperfinite algebra]]

## References

### General
 {#ReferencesGeneral}

The original articles:

The classification of factors into types I, II, III and the construction of examples not of type I:

* [[Francis J. Murray]], [[John von Neumann]], *On Rings of Operators*, Annals of Mathematics, Second Series, **37** 1 (1936) 116-229 [[doi:10.2307/1968693](https://doi.org/10.2307/1968693), [jstor:1968693](https://www.jstor.org/stable/1968693)]

Discussion of [[traces]] on these factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. II*, Trans. Amer. Math. Soc. **41** (1937) 208-248  [[doi:10.2307/1989620](https://doi.org/10.2307/1989620), [jstor:1989620](https://www.jstor.org/stable/1989620)]

On [[isomorphism]] of factors and proof of a single isomorphism class of approximately finite type $II_1$ factors:

* [[Francis J. Murray]], [[John von Neumann]], *On rings of operators. IV*, Annals of Mathematics, Second Series, **44** 4 (1943) 716-808 [[doi:10.2307/1969107](https://doi.org/10.2307/1969107), [jstor:1969107](https://www.jstor.org/stable/1969107)]

On decomposing [[von Neumann algebras]] as a [[direct integral]] of factors:

* {#vNeumann49} [[John von Neumann]], _On rings of operators, reduction theory_, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) [[jstor:1969463]( http://www.jstor.org/stable/1969463)]

Recollection of the history which made von Neumann turn to discussion of these "factors", motivated from considerations in the foundations of [[quantum mechanics]] and [[quantum logic]]:

* {#Rédei96} [[Miklos Rédei]], *Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead)*, Studies in History and Philosophy of Modern Physics **27** 4 (1996) 493-510 [doi:10.1016/S1355-2198(96)00017-2]

Lecture notes:

* V.S. Sunder, _von Neumann algebras, $II_1$-factors, and their subfactors_ [[pdf](https://www.imsc.res.in/~sunder/iitb3.pdf)]

* Hideki Kosaki, _Type III factors and index theory_ (1993) [[pdf](http://pages.uoregon.edu/njp/lec-f.pdf)]

### Subfactors

The mathematics of inclusions of _subfactors_ is giving deep structural insights. See also at _[[planar algebra]]_.

* [[Vaughan Jones]],

  _Index for subfactors_, Invent. Math. __72__, I (I983);

  _A polynomial invariant for links via von Neumann algebras_, Bull. AMS __12__, 103 (1985);

  _Hecke algebra representations of braid groups and link polynomials_, Ann. Math. __126__, 335 (1987)

* [[Vaughan Jones]], [[Scott Morrison]], [[Noah Snyder]], _The classification of subfactors of index at most 5_ ([arXiv:1304.6141](http://arxiv.org/abs/1304.6141))
* Vaughan F. R. Jones, David Penneys, _Infinite index subfactors and the GICAR categories_, [arxiv/1410.0856](http://arxiv.org/abs/1410.0856)

Symmetries of depth two inclusions of subfactors may be described via associative [[bialgebroid]]s,

* Lars Kadison, Kornél Szlachányi, _Bialgebroid actions on depth two extensions and duality_, Adv. Math. __179__:1 (2003) 75-121 doi

### Relation to quantum field theory
 {#ReferencesInQuantumFieldTheory}

On von Neumann algebra factors as algebras of [[quantum observables]] in [[quantum physics]] and [[quantum field theory]] (which was their original motivation, cf. [Rédei 1996](#Rédei96)):

* [[Jakob Yngvason]], _The role of type III factors in quantum field theory_ [[arXiv:math-ph/0411058](https://arxiv.org/abs/math-ph/0411058)]

* [[Jonathan Sorce]], *Notes on the type classification of von Neumann algebras* [[arXiv:2302.01958](https://arxiv.org/abs/2302.01958)]

* [[Roberto Longo]], [[Edward Witten]], *A note on continuous entropy* [[arXiv:2202.03357](https://arxiv.org/abs/2202.03357), [spire:2029393](https://inspirehep.net/literature/2029393)]

and particularly in [[quantum field theory on curved spacetime]]:

* [[Edward Witten]], *Why Does Quantum Field Theory In Curved Spacetime Make Sense? And What Happens To The Algebra of Observables In The Thermodynamic Limit?*, in *Dialogues Between Physics and Mathematics*, Springer (2022) [[arXiv:2112.11614](https://arxiv.org/abs/2112.11614), [doi:10.1007/978-3-031-17523-7_11](https://doi.org/10.1007/978-3-031-17523-7_11)]

* Edward Witten, *Algebras, Regions, and Observers* [[arXiv:2303.02837](https://arxiv.org/abs/2303.02837)]

such as on [[de Sitter spacetime]]:

* [[Venkatesa Chandrasekaran]], [[Roberto Longo]], [[Geoff Penington]], [[Edward Witten]], *An Algebra of Observables for de Sitter Space*, Journal of High Energy Physics **2023**  82 (2023) [[arXiv:2206.10780](https://arxiv.org/abs/2206.10780), doi:10.1007/JHEP02(2023)082]

and potential application of von Neumann algebra factors to [[quantum gravity]]:

* [[Edward Witten]], *Gravity and the Crossed Product*, Journal of High Energy Physics **2022** 8 (2022) [[arXiv:2112.12828](https://arxiv.org/abs/2112.12828), doi:10.1007/JHEP10(2022)008]

* [[Edward Witten]], *A Note On The Canonical Formalism for Gravity* [[arXiv:2212.08270](https://arxiv.org/abs/2212.08270), [inspire:2615434](https://inspirehep.net/literature/2615434)]

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