Yuji Tachikawa

* [webpage](http://member.ipmu.jp/yuji.tachikawa/)

## Selected writings

On the [[AGT correspondence]]:

* {#AGT09} [[Luis Alday]], [[Davide Gaiotto]], [[Yuji Tachikawa]], _Liouville Correlation Functions from Four-dimensional Gauge Theories_, Lett.Math.Phys.91:167-197, 2010 ([arXiv:0906.3219](http://arxiv.org/abs/0906.3219))

On [[5-brane webs]] and [[D=4 N=2 super Yang-Mills theory]]:

* [[Francesco Benini]], [[Sergio Benvenuti]], [[Yuji Tachikawa]], _Webs of five-branes and $\mathcal{N}=2$ superconformal field theories_, JHEP 0909:052 (2009) [[arXiv:0906.0359](https://arxiv.org/abs/0906.0359)]

On [[D=4 N=2 super Yang-Mills theory]]:

* [[Yuji Tachikawa]], *$\mathcal{N}=2$ supersymmetric dynamics for pedestrians*, Lecture Notes in Physics, vol. 890, 2014 ([arXiv:1312.2684](https://arxiv.org/abs/1312.2684), [doi:10.1007/978-3-319-08822-8](https://link.springer.com/book/10.1007%2F978-3-319-08822-8), [web version](http://www-hep.phys.s.u-tokyo.ac.jp/~yujitach/tmp/dummies/dummies.html#dummiesse7.html))

On [[anomaly cancellation]] in [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]]:

and their [[anomaly cancellation]] via the [[Green-Schwarz mechanism]]:

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], _Anomaly polynomial of E-string theories_, J. High Energ. Phys. 2014, 2 (2014) ([arXiv:1404.3887](https://arxiv.org/abs/1404.3887))

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], [[Yuji Tachikawa]], [[Kazuya Yonekura]], _Anomaly polynomial of general 6d SCFTs_, Progress of Theoretical and Experimental Physics, Volume 2014, Issue 10, October 2014, 103B07 ([arXiv:1408.5572](https://arxiv.org/abs/1408.5572))

On [[generalized global symmetries]] and their [[gauge symmetry|gauging]]:

* {#BT17} [[Lakshya Bhardwaj]], [[Yuji Tachikawa]]. *On finite symmetries and their gauging in two dimensions*, J. High Energ. Phys. **2018** 189 (2018) [[arXiv:1704.02330](https://arxiv.org/abs/1704.02330), doi:10.1007/JHEP03(2018)189]

Relation of the [[GSO projection]] to the [[K-theory classification of topological phases of matter]]:

* Justin Kaidi, Julio Parra-Martinez, [[Yuji Tachikawa]], _Topological Superconductors on Superstring Worldsheets_ ([arXiv:1911.11780](https://arxiv.org/abs/1911.11780))

On the [[WZW term]] in [[chiral perturbation theory]]:

* Yasunori Lee, [[Kantaro Ohmori]], [[Yuji Tachikawa]], _Revisiting Wess-Zumino-Witten terms_ ([arXiv:2009.00033](https://arxiv.org/abs/2009.00033))

On lifting the [[Witten genus]] of the [[heterotic string]] to [[topological modular forms]]:

* [[Yuji Tachikawa]], *Topological modular forms and the absence of a heterotic global anomaly*, Progress of Theoretical and Experimental Physics, **2022** 4 (2022) 04A107 $[$[arXiv:2103.12211](https://arxiv.org/abs/2103.12211), [doi:10.1093/ptep/ptab060](https://doi.org/10.1093/ptep/ptab060)$]$

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Topological modular forms and the absence of all heterotic global anomalies*, Comm. Math. Phys. **402** (2023) 1585-1620 [[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), [doi:10.1007/s00220-023-04761-2](https://doi.org/10.1007/s00220-023-04761-2)]

and in relation to [[Anderson duality]] of [[tmf]]:

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras* [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)]

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from the point of view of [[higher gauge theory]]:

* [[Yasunori Lee]], [[Kantaro Ohmori]], [[Yuji Tachikawa]], *Matching higher symmetries across Intriligator-Seiberg duality* ([arXiv:2108.05369](https://arxiv.org/abs/2108.05369))

On the mod-2 [[elliptic genus]], both as a [[partition function]] of a [[superstring]] as well as via the [[string-orientation of tmf]]:

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* $[$[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)$]$

## Related entries

* [[AGT correspondence]]

* [[FQFT]]

category: people


Yuji Terashima

* [Institute page](https://tohoku.pure.elsevier.com/en/persons/yuji-terashima)

## Selected writings

On [[fiber integration in differential cohomology]] (fiberwise [[Stokes theorem]] lifted to [[Deligne cohomology]]):

* {#GomiTerashima00} [[Kiyonori Gomi]], [[Yuji Terashima]], _A Fiber Integration Formula for the Smooth Deligne Cohomology_, International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf), [doi:10.1155/S1073792800000386](https://doi.org/10.1155/S1073792800000386))

On [[higher parallel transport]] for [[circle n-bundles with connection]]:

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Higher dimensional parallel transport_ Mathematical Research Letters 8, 25–33 (2001) ([pdf](http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf))

On [[twisted K-theory]] via [[vectorial bundles]]:

* {#Gomi} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))

* {#GomiTerashima} [[Kiyonori Gomi]], [[Yuji Terashima]], _Chern-Weil Construction for Twisted K-Theory_, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 ([doi:10.1007/s00220-010-1080-1](https://doi.org/10.1007/s00220-010-1080-1))

On [[discrete torsion]]:

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Discrete Torsion Phases as Topological Actions_, Commun. Math. Phys. (2009) 287: 889 ([doi:10.1007/s00220-009-0736-1](https://doi.org/10.1007/s00220-009-0736-1))

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

* [[Yuji Terashima]], [[Masahito Yamazaki]], _$SL(2,\mathbb{R})$ Chern-Simons, Liouville, and Gauge Theory on Duality Walls_, JHEP 1108:135, 2011 ([arXiv:1103.5748](https://arxiv.org/abs/1103.5748))

* [[Yuji Terashima]], [[Masahito Yamazaki]], _Semiclassical Analysis of the 3d/3d Relation_, Phys.Rev.D88:026011, 2013 ([arXiv:1106.3066](https://arxiv.org/abs/1106.3066))

category: people


Yukawa coupling

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]], the _Yukawa couplings_ encode the [[interaction]] between the fundamental [[fermion fields]] and the [[Higgs field]], and thus, via the [[Higgs mechanism]], the [[masses]] of the [[fermion field]] after [[electroweak symmetry breaking]].

## In string phenomenology

In [[intersecting D-brane models]] Yukawa couplings are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by  [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).

\begin{center}
<img src="https://ncatlab.org/nlab/files/YukawaFukaya.jpg" width="680">
\end{center}

> table grabbed from [Marchesano 03](#Marchesano03)

## Related concepts

* [[coupling constant]]

* [[minimal coupling]]

* [[standard model of particle physics]]

* [[spinors in Yang-Mills theory]]

## References

### General

See also

* Wikipedia, _[Yukawa interactions](http://en.wikipedia.org/wiki/Yukawa_interaction)_

### In string theory

On computation of Yukawa couplings in [[heterotic string theory]]:

* Stefan Blesneag, [[Evgeny Buchbinder]], [[Andrei Constantin]], [[Andre Lukas]], [[Eran Palti]], _Matter Field Kähler Metric in Heterotic String Theory from Localisation_ ([arXiv:1801.09645](https://arxiv.org/abs/1801.09645))

* Giorgi Butbaia, Damián Mayorga Peña, Justin Tan, Per Berglund, [[Tristan Hübsch]], Vishnu Jejjala, Challenger Mishra, *Physical Yukawa Couplings in Heterotic String Compactifications* [[arXiv:2401.15078](https://arxiv.org/abs/2401.15078)]

* Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, [[Andre Lukas]], [[Burt Ovrut]], *Computation of Quark Masses from String Theory* [[arXiv:2402.01615](https://arxiv.org/abs/2402.01615)]

and in [[intersecting D-brane models]] of [[type II string theory]] via [[Fukaya categories]]:

* {#CremadesIbanezMarchesano03} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Yukawa couplings in intersecting D-brane models_, JHEP 0307 (2003) 038 ([arXiv:hep-th/0302105](https://arxiv.org/abs/hep-th/0302105))

* {#Marchesano03} [[Fernando Marchesano]], section 7 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

Realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys. Rev. D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))

* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, Van E. Mayes, _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))

* Tianjun Li, Adeel Mansha, Rui Sun, _Revisiting the Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1910.04530](https://arxiv.org/abs/1910.04530))

[[!redirects Yukawa couplings]]

[[!redirects Yukawa coupling constant]]
[[!redirects Yukawa coupling constants]]

[[!redirects Yukawa interaction]]
[[!redirects Yukawa interactions]]


Yuki Maehara

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

## Publications

* [[Yuki Maehara]], *Inner horns for 2-quasi-categories*, Advances in Mathematics, Volume 363, 25 March 2020, 107003 ([doi:10.1016/j.aim.2020.107003](https://doi.org/10.1016/j.aim.2020.107003)), [arXiv:1902.08720](https://arxiv.org/abs/1902.08720))

## Talks

* [[Yuki Maehara]], *A cubical model for weak omega-categories*, [[Homotopy Type Theory Electronic Seminar Talks]], 8 October 2020 ([video](https://www.youtube.com/watch?v=AIBTh7MIbew), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Maehara-2020-10-08-HoTTEST.pdf))

category: people


Yuli Rudyak

* [webpage](http://www.math.ufl.edu/~rudyak/)

category: people


Yunlong Jiao

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

## Selected writings

Proof that the [[Mallows kernel]] and [[Kendall kernel]] are [[positive definite bilinear form|positive definite]]:

* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Kendall and Mallows Kernels for Permutations*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 7, pp. 1755-1769, 1 July 2018 ([doi:10.1109/TPAMI.2017.2719680](https://doi.org/10.1109/TPAMI.2017.2719680), [hal:01279273](https://hal.archives-ouvertes.fr/hal-01279273/document))

Discussion of weighted variants:

* [[Yunlong Jiao]], [[Jean-Philippe Vert]], *The Weighted Kendall and High-order Kernels for Permutations*, Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2314-2322, 2018 ([arXiv:1802.08526](https://arxiv.org/abs/1802.08526), [mlr:v80/jiao18a](http://proceedings.mlr.press/v80/jiao18a.html), [hal:hal-01717385](https://hal.inria.fr/hal-01717385))

category: people


Yunze Lu

* [Institute page](https://lsa.umich.edu/math/people/phd-students/yunze.html)

## Selected writings

Discussion of [[equivariant ordinary cohomology]] ([[Bredon cohomology]]) over the point but in arbitrary [[RO(G)-degree]]:

for [[equivariance group]] a [[dihedral group]] of [[order of a group|order]] $2p$:

* [[Igor Kriz]], [[Yunze Lu]], *On the $RO(G)$-graded coefficients of dihedral equivariant cohomology*, Mathematical Research Letters **27** 4 (2020) ([arXiv:2005.01225](https://arxiv.org/abs/2005.01225), [doi:10.4310/MRL.2020.v27.n4.a7](https://dx.doi.org/10.4310/MRL.2020.v27.n4.a7))

for [[equivariance group]] the [[quaternion group]]:

* [[Yunze Lu]], *On the $RO(G)$-graded coefficients of $Q_8$ equivariant cohomology*, Topology and its Applications, 2021 ([doi:10.1016/j.topol.2021.107921](https://doi.org/10.1016/j.topol.2021.107921))

category: people


Yuri Berest

Yuri Berest is a professor at Cornell, mathematics department: [web](http://www.math.cornell.edu/People/Faculty/berest.html)

> My research interests include mathematical physics, algebraic geometry and representation theory. I am particularly interested in various interactions between these fields. Some of my recent work is related to noncommutative geometry, integrable systems, differential operators on algebraic varieties, representation theory of Cherednik algebras, and the theory of invariants of finite reflection groups.

* Yuri Berest, Oleg Chalykh, _$A_\infty$-modules and Calogero-Moser spaces_,
J. Reine Angew. Math. __607__ (2007), 69–112, [MR2009f:16019](http://www.ams.org/mathscinet-getitem?mr=2338121), [doi](http://dx.doi.org/10.1515/CRELLE.2007.046)
* Yu. Berest, P. Etingof, V. Ginzburg, _Finite-dimensional representations of rational Cherednik algebras_, Int. Math. Res. Not. 2003, no. 19, 1053-1088.
* Yu. Berest, G. Wilson, _Automorphisms and ideals of the Weyl algebra_, Math Ann 318, 127–147 (2000) [doi](https://doi.org/10.1007/s002080000115)
* Yu. Berest, G. Wilson, _Ideal classes of the Weyl algebra and noncommutative projective geometry_ (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices __26__ (2002) 1347–1396
* Yu. Berest, G. Wilson, _Mad subalgebras of rings of differential operators on curves_, Advances in Math. __212__ no. 1 (2007), 163–190.
* Yuri Berest, _Calogero-Moser spaces over algebraic curves_, Sel. math., New ser. 14, 373–396 (2009) [doi](https://doi.org/10.1007/s00029-009-0518-9) [arXiv:0809.4521](https://arxiv.org/abs/0809.4521)
* Yuri Berest, _The problem of [[lacuna]]s and analysis on root systems_, Trans. Amer. Math. Soc. __352__ (2000), 3743–3776.
* Yuri Berest, _The theory of lacunas and quantum integrable systems_,  Calogero-Moser-Sutherland models (Montréal, QC, 1997), 53–64, CRM Ser. Math. Phys., Springer 2000.
* [[Yuri Berest]], Oleg Chalykh, Farkhod Eshmatov, _Recollement of deformed preprojective algebras and the Calogero-Moser correspondence_, Mosc. Math. J. __8__:1 (2008) 21--37
[arXiv:0706.3006](https://arxiv.org/abs/0706.3006) [doi](https://doi.org/10.17323/1609-4514-2008-8-1-21-37) [MR2422265](https://www.ams.org/mathscinet-getitem?mr=2422265)
mathnet.ru/[mmj2](http://mi.mathnet.ru/mmj2)
* Yu. Berest, X. Chen, F. Eshmatov, A. Ramadoss, _Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras_, Contemp. Math. __583__ (2012) 219--246 [arXiv:1202.2717](https://arxiv.org/pdf/1202.2717)
* Yu. Berest, [[G. Felder]], A. Ramadoss, _Derived representation schemes and noncommutative geometry_, [arXiv:1304.5314](https://arxiv.org/abs/1304.5314)
* George Khachatryan, _Derived representation schemes and non-commutative geometry_, Cornell PhD thesis under guidance of Yuri Berest [online](https://hdl.handle.net/1813/29138]
* Yuri Berest, George Khachatryan, Ajay Ramadoss, _Derived representation schemes and cyclic homology_, Adv. Math.
__245,__ (2013) 625--689 [arXiv:1112.1449](https://arxiv.org/abs/1112.1449)
* [[Yuri Berest]], [[Giovanni Felder]], Sasha Patotski, Ajay C. Ramadoss, [[Thomas Willwacher]], _Representation homology, Lie algebra cohomology and derived Harish-Chandra homomorphism_, J. Eur. Math. Soc. 19:9 (2017) 2811--2893 [arXiv:1410.0043](https://arxiv.org/abs/1410.0043)

category: people

[[!redirects Yu. Berest]]


Yuri Golfand

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

See the introduction of

* [[Mikhail Shifman]] (ed.)

  _[[The Many Faces of the Superworld]]_

  [[Yuri Golfand]] memorial volume

  World Scientific, 2000

  [doi:10.1142/4332](https://doi.org/10.1142/4332)

## Selected writings

Introducing the [[super Poincaré Lie algebra]] ("[[supersymmetry]]"):

* {#GolfandLikhtman72} [[Yuri Golfand]], [[Evgeny Likhtman]],_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: [[Victor Ginzburg]] et al. (eds.) _I. E. Tamm Memorial Volume Problems of Theoretical Physics_, (Nauka, Moscow 1972), page 37,

  translated and reprinted in: [[Mikhail Shifman]] (ed.) _[[The Many Faces of the Superworld]]_ pp. 44-53,  World Scientific (2000) ([doi:10.1142/9789812793850_0006](https://doi.org/10.1142/9789812793850_0006))

category: people


Yuri Gurevich

* [personal page](https://web.eecs.umich.edu/~gurevich/)

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

## Selected writings

On [[set theory]] with commentary on [[ETCC]]:

* [[Andreas Blass]], [[Yuri Gurevich]], *Why Sets?*, Bull. Europ. Assoc. Theoret. Comp. Sci. **84** (2004) 139-156.  [[doi:10.1007/978-3-540-78127-1_11](https://doi.org/10.1007/978-3-540-78127-1_11), [pdf](https://web.eecs.umich.edu/~gurevich/Opera/172.pdf), [pdf](http://www.math.lsa.umich.edu/~ablass/set.pdf)]

On formal statement and [[proof]] of the [[deferred measurement principle]] of [[quantum circuits]]:

* {#GurevichBlass21} [[Yuri Gurevich]], [[Andreas Blass]], *Quantum circuits with classical channels and the principle of deferred measurements*, Theoretical Computer Science **920** (2022) 21–32 [[arXiv:2107.08324](https://arxiv.org/abs/2107.08324), [doi:10.1016/j.tcs.2022.02.002](https://doi.org/10.1016/j.tcs.2022.02.002)]

On a formal [[quantum programming language]]-perspective on [[quantum circuits]]:

* [[Yuri Gurevich]], [[Andreas Blass]], *Software science view on quantum circuit algorithms* [[arXiv:2209.13731](https://arxiv.org/abs/2209.13731)]

category: people


Yuri Makeenko

* [InSpire page](https://inspirehep.net/authors/999145)

## Selected writings:

Early discussion of [[flux tubes]]/[[Wilson lines]] as effective [[strings]] in [[Yang-Mills theory]] ([Gauge/String duality](AdS-CFT+correspondence#PolyakovGaugeStringDualityReferences)):

* {#MakeenkoMigdal81} [[Yuri Makeenko]], [[Alexander A. Migdal]], *Quantum chromodynamics as dynamics of loops*, Nuclear Physics B **188** 2 (1981) 269-316 [doi:10.1016/0550-3213(81)90258-3]

  > "So the [[world sheet]] of [[string]] should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric [[flux tube]] in the [[monopole]] plasma."

On [[gauge theory]] ([[Yang-Mills theory]]):

* [[Yuri Makeenko]], *Methods of contemporary gauge theory*, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [[doi:10.1017/CBO9780511535147]( https://doi.org/10.1017/CBO9780511535147), [gBooks](http://books.google.com/books?id=9W-W2w75ulAC)]

category: people

[[!redirects Yuri M. Makeenko]]


Yuri Manin

Yuri Ivanovich Manin (1937-2023, Russian: Юрий Иванович Манин) was a Russian-born mathematician of polymath broadness, with main works in [[number theory]] and [[arithmetic geometry]], [[noncommutative geometry]], [[algebraic geometry]] and [[mathematical physics]].

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

* [MPI obituary](https://www.mpim-bonn.mpg.de/node/11864)

His diverse work includes a classification theorem in the theory of commutative [[formal group]], early study of monoidal transformations and exposition on [[motive]]s in 1960-s, a fundamental starting work in [[quantum information theory]], proposals on [[quantum logic]]s, an approach to [[quantum group]]s, ADHM construction in [[soliton]] theory, work with [[Maxim Kontsevich]] on [[Gromov-Witten invariants]], work on [[Frobenius manifold]]s (and introduced more general "F-manifolds" with Claus Hertling). He published a number of influential monographs including on [[noncommutative geometry]], [[quantum group]]s, [[complex geometry]] and [[gauge theory|gauge theories]], introduction to [[scheme]]s, [[Frobenius manifold]]s, mathematical [[logic]]s...

Manin's students include:

* [[A. Beilinson]],
* [[V. Drinfel'd]]
* Vera Serganova
* Ivan Penkov
* [[M. Kapranov]]
* [[Yuri Tschinkel]]
* [[Ralph Kaufmann]]
* ...

## Selected writings

* _[[New Dimensions in Geometry]]_

* _[[Gauge Field Theory and Complex Geometry]]_

Introducing what came to be called the *[[Gauss-Manin connection]]*:

* {#Manin58} [[Yuri Manin]], _Algebraic curves over fields with differentiation_, Izv. Akad. Nauk SSSR Ser. Mat. **22** 6 (1958) 737-756 $[$[mathnet:izv3998](http://mi.mathnet.ru/izv3998), [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=3998&volume=22&year=1958&issue=6&fpage=737&what=fullt&option_lang=eng)$]$ (in Russian)

Introducing the notion of [[quantum computation]]:

* {#Manin1980} [[Yuri I. Manin]], Introduction to: *Computable and Uncomputable*, Sov. Radio (1980) [Russian original: [[Manin-1980.pdf:file]]], English translation on p. 69-77 of *Mathematics as Metaphor: Selected essays of Yuri I. Manin*, Collected Works **20**, AMS (2007) [[ISBN:978-0-8218-4331-4](https://bookstore.ams.org/cworks-20/)]

  > Perhaps, for a better understanding of [molecular biology], we need a mathematical theory of quantum automata.

and review of [[Shor's algorithm]]:

* [[Yuri I. Manin]], *Classical computing, quantum computing, and Shor's factoring algorithm*, Astérisque, **266** Séminaire Bourbaki 862 (2000) 375-404 [[arXiv:quant-ph/9903008](https://arxiv.org/abs/quant-ph/9903008), [numdam:SB_1998-1999__41__375_0](http://www.numdam.org/item/?id=SB_1998-1999__41__375_0)]

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

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

On [[homological algebra]] and [[homotopical algebra]] (via [[triangulated categories]] and including the [[model structure on dgc-algebras]] for [[rational homotopy theory]]):

* [[Sergei Gelfand]], [[Yuri Manin]], _[[Methods of homological algebra]]_,   transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 ([doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5))

Introduced [[quantum linear group]]s as universal coacting bialgebras (and their quotient Hopf algebras) in

* [[Yu. I. Manin]], _Quantum groups and non-commutative geometry_, CRM, Montreal 1988.

Some of these structures have repercussion on the study of [[quadratic operad]]s, as in

* [[Yuri Ivanovich Manin]], [[Bruno Vallette]], _Monoidal structures on the categories of quadratic data_, Documenta Mathematica __25__, 1727--1786

On relations of [[AdS3/CFT2]] to [[hyperbolic geometry]] and [[Arakelov geometry]] of [[algebraic curves]]:

* [[Yuri Manin]], [[Matilde Marcolli]], _Holography principle and arithmetic of algebraic curves_, Adv. Theor. Math. Phys. 5 (2002) 617-650 ([arXiv:hep-th/0201036](https://arxiv.org/abs/hep-th/0201036)

On [[quantum cohomology]] and [[Gromov-Witten invariants]]

* [[Maxim Kontsevich]], [[Yuri Manin]], _Gromov-Witten classes, quantum cohomology, and enumerative geometry_, Commun. Math. Physics __164__ (1994) 525-562 [doi](https://doi.org/10.1007/BF02101490) arXiv:[hep-th/9402147](https://arxiv.org/pdf/hep-th/9402147)

* [[Yuri Manin]], _Frobenius manifolds, quantum cohomology, and moduli spaces_, Amer. Math. Soc. Colloqium Publications __47__, 1999

* [[Maxim Kontsevich]], [[Yuri Manin]], [[Ralph Kaufmann]], _Quantum cohomology of a product_,  Invent. Math. __124__ (1996) 313-339 [doi](https://doi.org/10.1007/s002220050055) arXiv:[q-alg/9502009](https://arxiv.org/abs/q-alg/9502009)

## Quotes

> _What binds us to space-time is our rest mass, which prevents us from flying at the speed of light, when time stops and space loses meaning. In a world of light there are neither points nor moments of time; beings woven from light would live "nowhere" and "nowhen"; only poetry and mathematics are capable of speaking meaningfully about such things_

_Mathematics as Metaphor_: Selected Essays of Yuri I. Manin (ed. 2007) ([libquotes](https://libquotes.com/yuri-manin/quote/lbj0h0e))

category: people

[[!redirects Yuri I. Manin]]
[[!redirects Yuri Ivanovich Manin]]

[[!redirects Yu. Manin]]
[[!redirects Yu. I. Manin]]


Yuri Matiyasevich

__Yuri Vladimirovič Matiyasevich__ (also spelled Jurij/Yurij/Yurii Matijasevič/Matijasevich/Matiasevich, Russian Ю́рий Влади́мирович Матиясе́вич) is a Russian
mathematician who obtained a negative solution to [[Hilbert's problems|Hilbert's tenth problem]].
By a seminal theorem of Matiyasevich, for every statement in mathematics (say [[ZFC|ZF set theory]]) there is a [[Diophantine equation]] whose solvability is equivalent to the validity of the statement. This is not of much practical importance, but of large theoretical importance.

* wikipedia [Yuri Matiyasevich](http://en.wikipedia.org/wiki/Yuri_Matiyasevich), [Russian version](http://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B8%D1%8F%D1%81%D0%B5%D0%B2%D0%B8%D1%87,_%D0%AE%D1%80%D0%B8%D0%B9_%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80%D0%BE%D0%B2%D0%B8%D1%87)

An accessible presentation of Matiyasevich's theorem is in

* [[Yuri Manin]], _Introduction to mathematical logic_, Springer

[[!redirects Matiyasevich]]
[[!redirects Yu. Matiyasevich]]
[[!redirects Yuri Matiyasevich]]
[[!redirects Yuri Vladimirovič Matiyasevich]]
[[!redirects Ю́рий Влади́мирович Матиясе́вич]]
[[!redirects Юрий Владимирович Матиясевич]]


Yuri N. Obukhov

* [InSpire page](https://inspirehep.net/authors/995226)

* [GoogleScholar page](https://scholar.google.com/citations?user=8wD8DMoAAAAJ&hl=en)

## Selected writings

On [[pre-metric electromagnetism]]:

* [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[doi:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]

* {#HehlItinObukhov16} [[Friedrich W. Hehl]], [[Yakov Itin]], [[Yuri N. Obukhov]], *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016  [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]

* [[Yuri N. Obukhov]], *Premetric approach in gravity and electrodynamics*, in: *The Fifteenth Marcel Grossmann Meeting* (2022) 654-659 [[doi:10.1142/9789811258251_0087](https://doi.org/10.1142/9789811258251_0087), [arXiv:1904.00180](https://arxiv.org/abs/1904.00180)]

## Related entries

* [[electromagnetism]]

[[!redirects Yuri Obukhov ]]


Yuri Tschinkel

* [webpage](http://www.cims.nyu.edu/~tschinke/)

## Selected writings

On [[elliptic fibrations]]:

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Monodromy of elliptic surfaces* [[[MonodromyOfEllipticSurfaces.pdf:file]], [arXiv:math/0002168](https://arxiv.org/abs/math/0002168)]

On [[anabelian geometry]]:

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Introduction to birational anabelian geometry* [[pdf](http://www.math.nyu.edu/~tschinke/papers/yuri/10msri/msri7.pdf), [arXiv:1011.0883](https://arxiv.org/abs/1011.0883)]

* {#Tschinkel14} [[Yuri Tschinkel]], _Introduction to anabelian geometry_, talk at _[Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html)_, Oxford 2014 ([slides pdf](http://www.cims.nyu.edu/~tschinke/.talks/oxford14/oxford14.pdf))

## Related entries

* [[anabelian geometry]]

category: people


Yuri Ximenes Martins

I am a Brazilian researcher and software developer working on abstraction processes applied to Mathematics, Physics and Philosophy. As developer I like to follow the functional paradigm.

- [webpage](https://yxm.me)
- [arXiv](https://arxiv.org/a/martins_y_1.html), [ORCID](https://orcid.org/0000-0003-3717-6774), [HAL](https://hal.archives-ouvertes.fr/search/index/?q=yuri-ximenes-martins&submit=)
- [github](https://github.com/yxm-dev)

I am co-founder of the [[Math-Phys-Cat Group]], creator and contributor of the [Math-Phys-Cat Wiki](https://wiki.math-phys.group/).

- Contact: `yxm@gmx.ie`.

category: people

[[!redirects Yuri Martins]]


Yurii Antonovich Simonov

* [spire entry](https://inspirehep.net/authors/988597)

* [Math-Net page](http://www.mathnet.ru/eng/person24572)

## Selected writings

On [[confinement]]:

* {#Simonov18} [[Yuri A. Simonov]], _Field Correlator Method for the confinement in QCD_, Phys. Rev. D 99, 056012 (2019) ([arXiv:1804.08946](https://arxiv.org/abs/1804.08946), [doi:10.1103/PhysRevD.99.056012](https://doi.org/10.1103/PhysRevD.99.056012))

* [[Yuri A. Simonov]], *The fundamental scale of QCD* ([arXiv:2103.08223](https://arxiv.org/abs/2103.08223))

category: people

[[!redirects Yuri A. Simonov]]

[[!redirects Yuri Simonov]]


Yuta Kusakabe

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

## Selected writings

On [[Oka manifolds]]:

* {#Kusakabe20} [[Yuta Kusakabe]], _Oka properties of complements of holomorphically convex sets_ ([arXiv:2005.08247](https://arxiv.org/abs/2005.08247)).

category: people


Yutaka Hosotani

* [webpage](http://kabuto.phys.sci.osaka-u.ac.jp/~hosotani/hosotani-eng.html)

## Selected writings

On [[gauge-Higgs unification]] ([[Hosotani mechanism]]):

* {#Hosotani83} [[Yutaka Hosotani]], _Dynamical Mass Generation by Compact Extra Dimensions_, Phys. Lett. 126B (1983) 309-313 ([spire:188768](http://inspirehep.net/record/188768), doi:10.1016/0370-2693(83)90170-3)

* {#Hosotani89} [[Yutaka Hosotani]], _Dynamics of non-integrable phases and gauge symmetry breaking_, Annals of Physics Volume 190, Issue 2, March 1989, Pages 233-253 (doi:10.1016/0003-4916(89)90015-8)

* {#Hosotani12} [[Yutaka Hosotani]], _Gauge-Higgs Unification Approach_, AIP Conference Proceedings 1467, 208 (2012) ([arXiv:1206.0552](https://arxiv.org/abs/1206.0552))

Specifically in [[Spin(11)]]- ("[[SO(11)]]"-) [[GUT]]-models:

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([arXiv:1504.03817](https://arxiv.org/abs/1504.03817), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotoni16} [[Yutaka Hosotani]], _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016)  ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

category: people


Yuval Grossman

* [webpage](https://physics.cornell.edu/yuval-grossman)

## Selected writings

On [[flavor physics]]:

* [[Yuval Grossman]], _Introduction to flavor physics_, CERN Yellow Report CERN-2010-002, pp. 111-144 ([arXiv:1006.3534](https://arxiv.org/abs/1006.3534))

* [[Yuval Grossman]], Philip Tanedo, _Just a Taste: Lectures on Flavor Physics_, Chapter 4 in: _Anticipating the Next Discoveries in Particle Physics (TASI 2016)_ ([arXiv:1711.03624](https://arxiv.org/abs/1711.03624),  [doi:10.1142/9789813233348_0004](https://doi.org/10.1142/9789813233348_0004))

category: people


Yuval Ne'eman

Yuval Ne'eman (1925-2006)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Yuval_Ne%27eman)

* [Obituary](https://www.science.co.il/people/Yuval-Neeman/) at *Israel Science and Technology Directory*

## Selected writings

Precursor discussion to the [[D'Auria-Fré formulation of supergravity]]:

* [[Yuval Ne'eman]], [[Tullio Regge]], *Gravity and supergravity as gauge theories on a group manifold*, Physics Letters B **74** 1–2 (1978) 54-56 [doi:10.1016/0370-2693(78)90058-8, [spire:6328](https://inspirehep.net/literature/6328)]

  also: Rivista del Nuovo Cimento **1** 5 (1978) 1–43

## Related entries

* [[hadron]]$\;$[[flavour symmetry]]

category: people


Yuval Roichman

* [personal page](https://u.cs.biu.ac.il/~yuvalr/)

## Selected writings

On the [[combinatorics]] of [[standard Young tableaux]]:

* [[Ron M. Adin]], [[Yuval Roichman]], *Enumeration of Standard Young Tableaux*, Chapter 14 in: Miklós Bóna, *Handbook of Enumerative Combinatorics*, CRC Press 2015  ([arXiv:1408.4497](https://arxiv.org/abs/1408.4497), [ISBN:9781482220858](https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858))

category: people


Yuxi Liu

I'm a mathematician trying to make the world more harmonious, friendly, and utopic. I aim to do so by pushing forwards artificial intelligence using formal mathematics.

I am the bearer of the Element of Silence.


Yves Andre

**Yves André** is a mathematician at École Normale Supérieure of Paris.

* [editor's page](http://versita.com/science/mathematics/cejm/editors/yves_andre) at Central European Math. J., [editor's page](http://rendiconti.math.unipd.it/board/andre.php?lan=english) at Rendiconti del Seminario Matematico della Università di Padova

Related items: [[slope filtration]].

[[!redirects Yves André]]


Yves Diers

* [Prabook entry](https://prabook.com/web/yves.diers/290800)

## Selected writings

On the notion of [[spectrum (geometry)|spectrum in geometry]] and  introducing the notion of [[multi-adjoints]]:

* [[Yves Diers]], _Une construction universelle des spectres, topologies spectrales et faisceaux structuraux_, Communication in Algebra Volume 12, Issue 17-18 (1984) ([doi:10.1080/00927878408823101](https://doi.org/10.1080/00927878408823101))

See also:

* [[Yves Diers]], _Catégories localisables_, PhD thesis. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977) [[[Categories localisables.pdf:file]]]

* [[Yves Diers]], _Catégories localement multiprésentables_, Archiv der Mathematik 34.1 (1980), pp. 344–356.

* Yves Diers, _Categories of Boolean Sheaves of Simple Algebras_, Springer, Lecture Notes in Mathematics, vol. 1187, (1986).

* Yves Diers,   _Familles universelles  de   morphismes_,   Ann.   Soc.   Sci.   Bruxelles, 93  (1979),  175-195.

* Yves Diers,  _Multimonads   and   multimonadic   categories_,  J.   Pure   Appl.   Algebra   17  (1980),  153-170.

category: people


Yves Félix

[[!redirects Yves Felix]]

* [webpage](http://uclouvain.be/yves.felix)

## Selected writings

On [[rational homotopy theory]]:

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

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

* {#FelixHalperin17} [[Yves Félix]], [[Steve Halperin]], _Rational homotopy theory via Sullivan models: a survey_,  Notices of the International Congress of Chinese Mathematicians Volume 5 (2017) Number 2 ([arXiv:1708.05245](https://arxiv.org/abs/1708.05245), [doi:10.4310/ICCM.2017.v5.n2.a3](https://dx.doi.org/10.4310/ICCM.2017.v5.n2.a3))

On [[rational homotopy theory]] with general [[fundamental groups]]:

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

* {#FelixHalperinThomas15} [[Yves Félix]], [[Steve Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory II_, World Scientific 2015 ([doi:10.1142/9473](https://doi.org/10.1142/9473))

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

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

On [[rational parametrized stable homotopy theory]]:

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

On [[rational models of mapping spaces]]:

* {#BuijsFelixMurillo12} [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], _$L_\infty$-rational homotopy of mapping spaces_,  published as _$L_\infty$-models of based mapping spaces_,  J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524 ([arXiv:1209.4756](https://arxiv.org/abs/1209.4756), [doi:10.2969/jmsj/06320503](https://doi.org/10.2969/jmsj/06320503))

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

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

On the [[cobar construction]]:

* [[Yves Felix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Adams' Cobar Equivalence_, Transactions of the American Mathematical Society, Vol. 329, No. 2 (1992), pp. 531-549 ([jstor:2153950](https://www.jstor.org/stable/2153950))

## Related $n$Lab entries

* [[rational homotopy theory]]

* [[Sullivan model of free loop space]]

category: people


Yves Guiraud

Yves Guiraud is a French mathematician and theoretical computer scientist.  He is Chargé de recherche INRIA attached to [INRIA Rocquencourt](http://www.inria.fr/centre/paris-rocquencourt), but based in Irif [[PPS]], at Paris 7. His main research is in rewriting systems, especially in higher dimensions.

His webpage is [here](http://www.pps.univ-paris-diderot.fr/~guiraud/).

The list of his preprints on the Arxiv is [here](http://arxiv.org/find/math/1/au:+Guiraud_Y/0/1/0/all/0/1)

His thesis (2004) was

*  [Présentations d'opérades et systèmes de réécriture](http://tel.archives-ouvertes.fr/docs/00/04/70/96/PDF/tel-00006863.pdf)

####Thesis abstract

_This thesis studies the computational properties of operad presentations, or Penrose diagrams rewrite systems, together with their links with classical types of rewrite systems. With new criteria for termination and confluence, the convergence of the presentation $L(\mathbb{Z}_2)$ of $\mathbb{Z}/2\mathbb{Z}$-vector spaces, a commutative equational theory, is proved. Operad presentations are shown to be generalizations of both word rewrite systems and Petri nets; furthermore, they provide explicit resource management calculi for left-linear term rewrite systems. This work is concluded by the description of obstructions for proving the same result for the lambda-calculus. Two appendices present the links between operads and other structures from universal algebra, together with a calculus of explicit substitutions._

category : people


Yves Lafont

Yves Lafont is a Professor at the _Université de la Méditerranée (Aix-Marseille 2)_ and a researcher at the _Institut de Mathématiques de Luminy_. He is well known for work in Linear Logic, and more recently on polygraphic resolutions.

* [website](https://www.i2m.univ-amu.fr/perso/yves.lafont/)

## Selected writings

* Doctoral thesis: [Logiques, catégories & machines : implantation de langages de programmation guidée par la logique catégorique](https://www-apr.lip6.fr/~tasson/doc/reading/Yves_Laffont_phd.pdf)

On [[formal logic|form]] [[logic]] and [[formal proof|formal]] [[proof]] [[proof theory|theory]] ([[type theory|typed]] [[lambda-calculus|$\lambda$-calculus]], [[linear logic]],  [[coherence spaces]], ...)

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

category: people


Yves Laszlo

* [webpage](http://www.math.u-psud.fr/~laszlo/)

category: people


Yvette Kosmann-Schwarzbach

* [webage](http://www.math.polytechnique.fr/~kosmann/)

category: people


Yvonne Choquet-Bruhat

* [wikipedia entry](http://en.wikipedia.org/wiki/Yvonne_Choquet-Bruhat)

## Selected wrirings

On [[analysis]] and [[differential geometry]] on [[smooth manifolds]] with application to [[mathematical physics]]:

* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$

## Related entries

* [[Einstein's equations]]

* [[The Cauchy Problem in Classical Supergravity]]

category: people


Z

The [[integers]] $\mathbb{Z}$.

## Related entries

* [[S]]


Z'-boson

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[particle physics]] the term _Z'-boson_ refers to a hypothetical higher-[[mass]] cousin of the [[Z-boson]] which is predicted by some [[model (in theoretical physics)|models]] that extend the [[standard model of particle physics]], such as [[grand unified theories]] (e.g. [Sahoo 06](#Sahoo06)).

The existence of the $Z'$ has been proposed as one possible explanation of the apparently observed [[flavour anomalies]] ([Gauld-Goertz-Haisch 13](#GauldGoertzHaisch13),  [D'Ambrosio-Iver-Piccinini-Polosa 19](#DAmbrosioIverPiccininiPolosa19))

## References

### General

* Paul Langacker, _The Physics of Heavy $Z'$ Gauge Bosons_, Rev.Mod.Phys.81:1199-1228, 2009 ([arXiv:0801.1345](https://arxiv.org/abs/0801.1345))

* [The Review of Particle Physics 2017](http://pdg.lbl.gov/2017/), _59. $Z'$-boson searches_ ([pdf](http://pdg.lbl.gov/2017/reviews/rpp2017-rev-zprime-searches.pdf))

* [kjende.web.cern.ch/kjende/de/zpath_zprime.htm](https://kjende.web.cern.ch/kjende/de/zpath_zprime.htm)

* _[The Z' Hunter's Guide](https://sites.google.com/site/zprimeguide/)_

Experimental constraints:

* [[CMS collaboration]], _Search for an $L_\mu - L_\tau$ gauge boson using $Z \to 4\mu$ events in proton-proton collisions at $\sqrt{s} = 13TeV$_ ([arXiv:1808.03684](https://arxiv.org/abs/1808.03684))

As arising in [[GUTs]]:

* {#Sahoo06} S. Sahoo, _The prediction of mass of $Z'$-boson in an $SO(10)$-based model_, Indian J. Phys. 80 (2), 191-194 (2006) ([pdf](http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6019/1/The%20Prediction%20of%20Mass%20of%20Z%27-Boson_By%20S%20Sahoo.pdf))

### In relation to flavour anomalies

As a possible solution to the [[flavour anomalies]]:

* {#GauldGoertzHaisch13} Rhorry Gauld, Florian Goertz, Ulrich Haisch, _An explicit Z'-boson explanation of the $B \to K^\ast \mu^+ \mu^-$ anomaly_, JHEP01(2014)069 ([arXiv:1310.1082](https://arxiv.org/abs/1310.1082))

* Richard H. Benavides, Luis Muñoz, William A. Ponce, Oscar Rodríguez, Eduardo Rojas, _Minimal $Z^\prime$ models for flavor anomalies_ ([arXiv:1812.05077](https://arxiv.org/abs/1812.05077))

* {#DAmbrosioIverPiccininiPolosa19} Giancarlo D'Ambrosio, A. M. Iyer, F. Piccinini, A.D. Polosa, _Confronting $B$ anomalies with atomic physics_ ([arXiv:1902.00893](https://arxiv.org/abs/1902.00893))

* P. Ko, Takaaki Nomura, Chaehyun Yu, _$b \to s \mu^+ \mu^-$ anomalies and related phenomenology in $U(1)_{B_{3-x_\mu L_\mu - x_\tau L_\tau}}$ flavor gauge models_ ([arXiv:1902.06107](https://arxiv.org/abs/1902.06107))

* Joe Davighi, _Connecting neutral current $B$ anomalies with the heaviness of the third family_, Contribution to the [2019 QCD session](http://moriond.in2p3.fr/2019/QCD/) of the [54th Rencontres de Moriond](http://moriond.in2p3.fr/2019/) ([arXiv:1905.06073](https://arxiv.org/abs/1905.06073))

* [[Wolfgang Altmannshofer]], Joe Davighi, Marco Nardecchia, _Gauging the accidental symmetries of the Standard Model, and implications for the flavour anomalies_ ([arXiv:1909.02021](https://arxiv.org/abs/1909.02021))

* Rigo Bause, [[Gudrun Hiller]], Tim Höhne, Daniel F. Litim, Tom Steudtner, *B-Anomalies from flavorful $U(1)'$ extensions, safely* ([arXiv:2109.06201](https://arxiv.org/abs/2109.06201))

  > (in view of [[Higgs field]] [metastability](Higgs+field#MassAndVacuumInstability))

* Disha Bhatia, Nishita Desai, Amol Dighe, Frugal $U(1)_X$ models with non-minimal flavor violation for $b \to s \ell \ell$ anomalies and neutrino mixing* ([arXiv:2109.07093](https://arxiv.org/abs/2109.07093))

Realization in [[F-theory]] of [[GUT]]-models with [[Z'-bosons]] and/or [leptoquarks]] addressing the [[flavour anomalies]] and the [(g-2) anomalies](anomalous+magnetic+moment#Anomalies):

* Miguel Crispim Romao, Stephen F. King, George K. Leontaris, _Non-universal $Z'$ from Fluxed GUTs_, Physics Letters B Volume 782, 10 July 2018, Pages 353-361 ([arXiv:1710.02349](https://arxiv.org/abs/1710.02349))

* A. Karozas, G. K. Leontaris, I. Tavellaris, N. D. Vlachos, _On the LHC signatures of $SU(5) \times U(1)'$ F-theory motivated models_ ([arXiv:2007.05936](https://arxiv.org/abs/2007.05936))

* [[Ben Allanach]], _$U(1)_{B_3-L_2}$ Explanation of the Neutral Current $B$−Anomalies_ ([arXiv:2009.02197](https://arxiv.org/abs/2009.02197))

* [[Andreas Crivellin]], Claudio Andreas Manzari, Marcel Alguero, Joaquim Matias, _Combined Explanation of the $Z \to b \bar b$ Forward-Backward Asymmetry, the Cabibbo Angle Anomaly, $\tau \to \mu \nu \nu$ and $b \to s \ell^+ \ell^-$ Data_ ([arXiv:2010.14504](https://arxiv.org/abs/2010.14504))

* {#ACMM22} Marcel Algueró, [[Andreas Crivellin]], Claudio Andrea Manzari, Joaquim Matias, *Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$ Mass*, Physical Review D (2022) [[arXiv:2201.08170](https://arxiv.org/abs/2201.08170)]

[[!redirects Z'-bosons]]

[[!redirects Z' boson]]
[[!redirects Z' bosons]]

[[!redirects Z']]


Z+ ring

#Contents#
* table of contents
{:toc}

## Idea

$\mathbb{Z}_+$ rings (also known as $\mathbb{N}$-rings or fusion rings) are a sort of generalization of finite groups. A $\mathbb{Z}_+$ ring consists of a finite set of objects, and a set "fusion rules" between them. This generalizes groups in the sense that two objects don't necessarily fuse to create a third object from the set. Instead, they will fuse into a "direct sum" of the other elements. The compatibility conditions on these fusion rules can be concisely phrased as saying ring-theoretically. Namely, extending the fusion rules $\mathbb{Z}$-linearly one arrives at a ring whose underlying abelian group is a freely generated $\mathbb{Z}$-module.

$\mathbb{Z}_+$ rings naturally appear in the context of [[fusion categories]]. This is because in a fusion categories there is a notion of [[tensor product]], [[direct sum]], and [[simple object]]. The tensor product of simple objects will be the direct sum of other simple objects. This gives a "fusion rule" on the finite set of isomorphism classes of simple objects, which in turn induces the structure of a $\mathbb{Z}_+$.

Fusion rings appear very naturally in the algebraic theory of non-abelian [[anyon | anyons]]. Namely, the fusion of two non-abelian anyons has a non-deterministic result. The spectrum of possible results of the fusion and their relative probabilities gives a fusion rule.

## Definition

There are various definitions of $\mathbb{Z}_+$ ring. Our treatment here follows closely [EGNO15](#EGNO15).

\begin{definition}Let $A$ be a ring whose underlying abelian group is a finitely generated free $\mathbb{Z}$-module. A __$\mathbb{Z}_+$ basis__ for $A$ is a basis for $B=\{b_i\}_{i\in I}$ for $A$ as a $\mathbb{Z}$-module such that $b_ib_j$ is a non-negative linear combination of elements of $B$ for all $i,j\in I$. That is, there exists $c^{k}_{i,j}\in \mathbb{Z}_+$ such that

$$b_i b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k.$$
\end{definition}

Simple put, a __$\mathbb{Z}_+$-ring__ is a ring equipped with a $\mathbb{Z}_+$-basis, with the extra condition that the multiplicative identity element $1$ be a non-negative linear combination of basis elements.

The relation of the multiplicative identity to the $\mathbb{Z}_+$-basis is a very subtle one. In particular, the fact that the the identity is not part of the basis can cause troubles. For this reason, we define a __unital $\mathbb{Z}_+$-ring__ to be a $\mathbb{Z}_+$ ring whose identity element is part of the distinguished $\mathbb{Z}_+$-basis.

A $\mathbb{Z}_+$ ring is uniquely defined by its fusion rules. Hence, the definition of $\mathbb{Z}_+$-ring can be restated in terms of a collection of explicit properties that the fusion rules must satisfy. Seeing as the fusion rules are often what is of most interest, this restatement can be very useful.

\begin{proposition} Let $B=\{b_i\}_{i\in I}$ be a finite set, equipped with integers $(c^{k}_{i,j})_{i,j,k\in I}$. Let $A$ be the free $\mathbb{Z}$-module generated by $B$. Define a binary operation $A\times A\to A$ by linearly extending the rule

$$b_i\cdot b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k,$$

for all $i,j\in I$.

* (Associativity) Given $i,j,k\in I$, $b_i \cdot (b_j\cdot b_k)=(b_i\cdot b_j)\cdot b_k$ if and only if
$$\sum_{s\in I}c^{s}_{i,j}c^t_{s,k}=\sum_{s\in I}c^{s}_{j,k}c^{t}_{i,s}$$
for all $t\in I$.

* (Identity) Given $i \in I$, we have that $b_1\cdot b_{i}=b_{i}\cdot b_1=b_{i}$ if and only if
$$c^{j}_{1,i}=c^{j}_{i,1}=
\begin{cases}
1 & i=j\\
0 & \text{otherwise}
\end{cases}$$
for all $j\in I$.

\end{proposition}

Thus, the fusion rule $(c^{k}_{i,j})$ induces a $\mathbb{Z}_+$ ring structure if and only if the properties listed are satisfied.

## Fusion rings

Most of the subtlety and power in the theory of finite groups comes from the existence of inverses. Similarly, a good theory of $\mathbb{Z}_+$-rings should incorporate some idea of inverses. On the level of fusion categories this corresponds to [[rigid monoidal category | rigidity]], and on the level of anyons this corresponds to the existence of [[antiparticles]].

\begin{definition} Let $A$ be a unital $\mathbb{Z}_+$-ring with $\mathbb{Z}_+$-basis $B$. We call $A$ a __fusion ring__ if for all $i\in I$ there exists a unique $i^*\in I$ such that

$$c^{1}_{i,j}=
\begin{cases}
1 & j=i^*\\
0 & \text{otherwise}
\end{cases}$$

and the quantity $c^{k^*}_{i,j}$ is invariant under cyclic permutations of $i,j,k$.
\end{definition}

In this sense, $b_{i^*}$ is the "dual" or "inverse" of $b_i$. This induces an involution $*: A\to A$ by sending

$$a:=\sum_{i\in I}a_i\cdot b_i\mapsto a^*:=\sum_{i\in I}a_i\cdot b_{i^*}.$$

## References

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], section 3.1 in _Tensor categories_, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 ([pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf
))


Z-infinity-module

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

#Contents#
* table of contents
{:toc}

## Idea

The idea is that there should be some kind of "complete [[local ring]]" $\mathbb{Z}_\infty$ corresponding to the archimedean [[valuation]] on $\mathbb{Q}$, by analogy with the (genuine!) complete local rings $\mathbb{Z}_p$ corresponding to the (non-archimedean) $p$-adic valuations on $\mathbb{Q}$: the [[p-adic integers]]. However, the naïve approach taking
$$\mathbb{Z}_\infty = \lbrace x \in \mathbb{R} : -1 \le x \le 1 \rbrace$$
fails because this is not a subring of $\mathbb{R}$!

[[Nikolai Durov]]'s definition of $\mathbb{Z}_\infty$ is inspired by classical [[Arakelov geometry]] and starts with the observation that any $\mathbb{Z}_p$-lattice $\Lambda$ in a finite-dimensional $\mathbb{Q}_p$-[[vector space]] $E$ defines a maximal compact subgroup submonoid $M_\Lambda$ of $End(E)$, and $M_\Lambda = M_{\Lambda'}$ if and only if $\Lambda$ and $\Lambda'$ are similar lattices; accordingly, a $\mathbb{Z}_\infty$-lattice up-to-similarity should correspond to maximal compact submonoids of $End(E)$ for a finite-dimensional $\mathbb{R}$-vector space $E$, i.e. the monoid of $\mathbb{R}$-linear endomorphisms of $E$ that are [[short map|short]] with respect to some [[norm]] on $E$. Thus, Durov defines a $\mathbb{Z}_\infty$-lattice to be a finite-dimensional _normed_ $\mathbb{R}$-vector space, and a morphism of $\mathbb{Z}_\infty$ lattices is defined to be a [[short map|short]] $\mathbb{R}$-[[linear map]]. This gives a category $\mathbb{Z}_{\infty}\mathbf{\text{-Lat}}$ with finite limits and colimits. (Note, however, that it is not an additive category!)

Now, notice that every [[flat module|flat]] $\mathbb{Z}_p$-module is the [[filtered colimit]] of its finite-dimensional $\mathbb{Z}_p$-submodules (which are necessarily free because $\mathbb{Z}_p$ is a local ring), and in fact the category of flat $\mathbb{Z}_p$-modules is equivalent to the category of [[ind-object|ind-objects]] of $\mathbb{Z}_{p}\mathbf{\text{-Lat}}$. So we may define a flat $\mathbb{Z}_\infty$-module to be an ind-object of $\mathbb{Z}_{\infty}\mathbf{\text{-Lat}}$. One may also give the following explicit description: a flat $\mathbb{Z}_\infty$-module $E$ is a (possibly infinite-dimensional) $\mathbb{R}$-vector space $E_{\mathbb{R}}$ together with a symmetric convex body $E_{\mathbb{Z}_\infty}$, and a morphism $E \to E'$ is an $\mathbb{R}$-linear map $E_{\mathbb{R}} \to E'_{\mathbb{R}}$ that restricts to a map $E_{\mathbb{Z}_\infty} \to E'_{\mathbb{Z}_\infty}$. This defines a category $\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}}$.

Finally, noting that the forgetful functor $U : \mathbb{Z}_{\infty}\mathbf{\text{-FlMod}} \to \mathbf{\text{Set}}$ taking a flat $\mathbb{Z}_\infty$-module $E$ to its underlying symmetric convex body $E_{\mathbb{Z}_\infty}$ has a [[left adjoint]] $F$, Durov defines a (not necessarily flat) $\mathbb{Z}_\infty$-module to be a [[module over a monad|module]] for the induced [[monad]] $\Sigma_\infty = U F$. The comparison functor embeds $\mathbb{Z}_{\infty}\mathbf{\text{-FlMod}}$ as a full subcategory of $\mathbb{Z}_{\infty}\mathbf{\text{-Mod}}$.

## Definition

Let $\Sigma_\infty : \mathbf{\text{Set}} \to \mathbf{\text{Set}}$ be the functor sending a set $S$ to the set
$$\left\lbrace \vec{v} \in \mathbb{R}^{(S)} : \left\| \vec{v} \right\|_1 = \sum_{s \in S} \left| v_s \right| \le 1 \right\rbrace$$
i.e. the solid regular cross-polytope with $S$-many vertices. The action of $\Sigma_\infty$ on maps of sets is the obvious one. Let $\eta : id \Rightarrow \Sigma_\infty$ be the natural transformation given by insertion of generators, and let $\mu : \Sigma_\infty \Sigma_\infty \Rightarrow \Sigma_\infty$ be the natural transformation given by "evaluation" of "octahedral" combinations:
$$\sum_i \alpha_i \eta \left( \sum_j \beta_{i,j} \eta (s_j) \right) \mapsto \sum_{i, j} \alpha_i \beta_{i, j} \eta (s_j)$$
One may verify that this defines a [[monad]] $(\Sigma_\infty, \eta, \mu)$ on $\mathbf{\text{Set}}$. A **$\mathbb{Z}_\infty$-module** is defined to be a [[module over a monad|module]] for this monad.

## Properties

The $\Sigma_\infty$ monad is a [[monad with arities]]: the category of arities may be taken to be $\mathbf{\text{FinSet}}$.

The $\mathbb{Z}_\infty$-module structure on a set $M$ is entirely determined by the map $\alpha_2 : \Sigma_\infty (2) \times M^2 \to M$ given by $((\lambda_1, \lambda_2), (x_1, x_2)) \mapsto \lambda_1 x_1 + \lambda_2 x_2$. Conversely, a set $M$ together with an element $\alpha_0$ and a map $\alpha_2 : \Sigma_\infty (2) \times M^2 \to M$ satisfying certain equations is a $\mathbb{Z}_\infty$-module. A map commuting with $\alpha_0$ and $\alpha_2$ is a homomorphism of $\mathbb{Z}_\infty$-modules, thus the theory of $\mathbb{Z}_\infty$-modules is a finitary [[algebraic theory]], with all that this implies.

The category $\mathbb{Z}_{\infty}\mathbf{\text{-Mod}}$ has the following properties:

* It is a [[complete category]] (with the forgetful functor creating all limits).

* It is a [[cocomplete category]] (with the forgetful functor creating all filtered colimits).

* It has a [[zero object]].

* It has a [[tensor product]] and an [[internal hom]], making it into a [[symmetric monoidal category|symmetric]] [[monoidal closed category]].

## Examples

Every normed $\mathbb{R}$-vector space $V$ induces a flat $\mathbb{Z}_\infty$-module $E$ where $E_{\mathbb{R}} = V$ and $E_{\mathbb{Z}_\infty} = \left\lbrace \vec{v} \in V : \left\| \vec{v} \right\| \le 1 \right\rbrace$. (In the other direction, every flat $\mathbb{Z}_\infty$-module induces a _seminorm_ on the underlying $\mathbb{R}$-vector space.)

A finitely-generated flat $\mathbb{Z}$-module is the symmetric convex hull of a finite set of vectors.

## References

* [[Nikolai Durov]], _New approach to Arakelov theory_. Ph.D. thesis (2007) [arXiv:0704.2030](http://arxiv.org/abs/0704.2030/).


Z-theory

#Contents#
* table of contents
{:toc}

## Idea

A term used for aspects of [[topological M-theory]].

## References

* [[Nikita Nekrasov]], _Z Theory_, ([arXiv:hep-th/0412021](http://arxiv.org/abs/hep-th/0412021))

[[!redirects Z theory]]


Zach Goldthorpe

* [blog](https://leftadjoint.wordpress.com/about/)

* [institute page](https://apps.ualberta.ca/directory/person/zgoldtho)

## Selected writings

An [[(∞,1)-category]] of [[(∞,∞)-categories]] in terms of [[induction|inductively]] and [[coinduction|coinductively]] defined [[equivalences]]:

* [[Zach Goldthorpe]], _Homotopy theories of $(\infty,\infty)$-categories as universal fixed points with respect to enrichment_, International Mathematics Research Notices **2023** 22 (2023) 19592–19640  [[arXiv:2307.00442](https://arxiv.org/abs/2307.00442), [doi:10.1093/imrn/rnad196](https://doi.org/10.1093/imrn/rnad196)]

category: people


Zachary Murray

* [LinkedIn page](https://ca.linkedin.com/in/zachary-murray-dal)

## Selected writings

On [[constructive analysis]] with the [[Cauchy real numbers]] according to [Bishop (1967)](#Bishop67), formalized in the [[Agda]] [[proof assistant]]:

* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

review:

* [[Zachary Murray]], *Constructive Real Numbers in the Agda Proof Assistant*, [talk at *CQTS*](Center+for+Quantum+and+Topological+Systems#MurrayFeb2023), (Feb 2023) [video:[YT](https://www.youtube.com/watch?v=7Q_sjfyPJqU)]

category: people


Zachery Lindsey

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=262620)

## Selected writings

On the [[model structure for quasi-categories|Joyal-type]] [[model structure for cubical quasi-categories]] on [[cubical sets]] [[connection on a cubical set|with connections]]:

* [[Brandon Doherty]], [[Chris Kapulkin]], [[Zachery Lindsey]], [[Christian Sattler]], _Cubical models of (∞,1)-categories_, Memoirs of the AMS ([accepted 2022](https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo)) [[arXiv:2005.04853](https://arxiv.org/abs/2005.04853)]

On a [[calculus of fractions]] generalized from categories to [[quasi-categories]] ("$(\infty,1)$-calculus of fractions", for *[[localization of an (infinity,1)-category|localization of $(\infty,1)$-categories]]*):

* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]

category: people


Zamolodchikov equation

see _[[Knizhnik-Zamolodchikov equation]]_

[[!redirects Zamolodchikov equation -- history]]


Zariski geometry

Zariski geometry is a structure defined by Boris Zilber. See his book

* [[Boris Zilber]], _Zariski Geometries_, Cambridge University Press, 2010 (publisher's [book page](http://www.cambridge.org/gb/knowledge/isbn/item2713251/?site_locale=en_GB))

> Zariski Geometries are abstract structures in which a suitable generalisation of Zariski topology makes sense. Algebraic varieties over an algebraically closed field and compact complex spaces in a natural language are examples of Zariski geometries. The main theorem by Hrushovski and the lecturer states that under certain non-degeneracy conditions a 1-dimensional Zariski geometry can be identified as an algebraic curve over an algebraically closed field. The proof of the theorem exhibits, as a matter of fact, a way to develop algebraic geometry from purely geometric abstract assumptions not involving any algebra at all. Recent works in model theory of complex manifolds, differential fields and non-commutative geometry point to exciting perspectives for the theory.

[[!redirects Zariski geometries]]
[[!redirects Zariski geometries]]


Zariski site

#Contents#
* table of contents
{:toc}

## Idea

There is a [[little site]] notion of the [[Zariski topology]], and a [[big site]] notion. As for the little site notion: the Zariski topology on the set of prime ideals of a commutative ring $R$ is the smallest topology that contains, as open sets, sets of the form $\{p\; \text{prime}: a \notin p\}$ where $a$ ranges over elements of $R$.

As for the big site notion, the _Zariski topology_ is a [[coverage]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative rings. This article is mainly about the big site notion.

## Definition

For $R$ a commutative [[ring]], write $Spec R \in CRing^{op}$ for its [[spectrum of a commutative ring]], hence equivalently for its incarnation in the [[opposite category]].

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

For $S \subset R$ a [[multiplicative subset]], write $R[S^{-1}]$ for the corresponding [[localization of a ring|localization]] and

$$
  Spec(R[S^{-1}]) \longrightarrow Spec(R)
$$

for the dual of the canonical ring homomorphism $R \to R[S^{-1}]$.

=--

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

The maps as in def. \ref{StandardOpenImmersion} are not [[open immersion of schemes|open immersions]] for arbitrary multiplicative subsets $S$ (see [a MathOverflow discussion](http://mathoverflow.net/questions/20782/ring-theoretic-characterization-of-open-affines)). They are for subsets of the form $S = \{ f^0, f^1, f^2, \ldots \}$, in which case they are called the _standard opens_ of $Spec(R)$.

=--

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

A family of [[morphisms]] $\{Spec A_i \to Spec R\}$ in $CRing^{op}$ is a Zariski-[[covering]] precisely if

* each ring $A_i$ is the [[localization]]

  $$
    A_i = R[r_i^{-1}]
  $$

  of $R$ at a single element $r_i \in R$

* $Spec A_i \to Spec R$ is the canonical inclusion, dual to the canonical ring homomorphism $R \to R[r_i^{-1}]$;

* There exists $\{f_i \in R\}$ such that

  $$
    \sum_i f_i r_i = 1
    \,.
  $$

=--

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

Geometrically, one may think of

* $r_i$ as a function on the [[space]] $Spec R$;

* $R[r_i^{-1}]$ as the [[open subset]] of points in this space on which the function is not 0;

* the covering condition as saying that the functions form a [[partition of unity]] on $Spec R$.

=--

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

Let $CRing_{fp} \hookrightarrow CRing$ be the [[full subcategory]] on [[finitely presented object]]s. This inherits the Zariski [[coverage]].

The [[sheaf topos]] over this [[site]] is the [[big topos]] version of the  **Zariski topos**.

=--

## Properties

### Points

The [[maximal ideal]] in $R$ correspond precisely to the [[closed subset|closed]] points of the [[prime spectrum]] $Spec(R)$ in the Zariski topology.

### As a site

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

The Zariski coverage is [[subcanonical coverage|subcanonical]].

=--

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

* $CRing_{fp}^{op}$ is the [[syntactic category]] of the [[cartesian theory]] of commutative rings;

* $CRing_{fp}^{op}$ equipped with the Zariski topology is the [[syntactic site]] of the [[geometric theory]] of [[local ring]]s.

Hence

* the big Zariski topos, def. \ref{ZariskiTopos}, is the [[classifying topos]] for [[local ring]]s.

* a [[locally ringed topos]] is equivalently a topos $\mathcal{E}$ equipped with a [[geometric morphism]] into the big Zariski topos.

=--

See [[classifying topos]] and [[locally ringed topos]] for more details on this.

### Sheafification

If $F$ is a presheaf on $CRing^{op}$ and $F^{++}$ denotes its [[sheafification]], then the canonical morphism $F(R) \to F^{++}(R)$ is an isomorphism for all [[local ring|local rings]] $R$. This follows from the explicit description of the [[plus construction]] and the fact that a local ring admits only the trivial covering.

## Kripke–Joyal semantics
 {#KripkeJoyal}

Writing $R \models \varphi$ for the interpretation of a formula $\varphi$ of the [[internal language]] of the big Zariski topos over $Spec(R)$ with the [[Kripke–Joyal semantics]], the forcing relation can be expressed as follows.

$$
\begin{array}{lcl}
  R \models x = y : F &\Longleftrightarrow&
    x = y \in F(R). \\
  R \models \top &\Longleftrightarrow&
    1 = 1 \in R. \\
  R \models \bot &\Longleftrightarrow&
    1 = 0 \in R. \\
  R \models \phi \wedge \psi &\Longleftrightarrow&
    R \models \phi \,\text{and}\, R \models \psi. \\
  R \models \phi \vee \psi &\Longleftrightarrow&
    \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
      R[f_i^{-1}] \models \phi \,or\, R[f_i^{-1}] \models \psi. \\
  R \models \phi \Rightarrow \psi &\Longleftrightarrow&
    \text{for any}\, R\text{-algebra}\, S \,\text{it holds that}\, (S \models \phi) \,implies\, (S \models \psi). \\
  R \models \forall x:F. \phi &\Longleftrightarrow&
    \text{for any}\, R\text{-algebra}\, S \,\text{and any element}\, x \in F(S) \,\text{it holds that}\, S \models \phi[x]. \\
  R \models \exists x.F. \phi &\Longleftrightarrow&
    \text{there is a partition}\, \sum_i f_i = 1 \in R \,\text{such that for all}\, i,
    \text{there exists an element}\, x_i \in F(R[f_i^{-1}]) \,\text{such that}\,
    R[f_i^{-1}] \models \phi[x_i].
\end{array}
$$

The only difference to the Kripke–Joyal semantics of the _little_ Zariski topos is that in the clauses for $\Rightarrow$ and $\forall$, one has to restrict to $R$-algebras $S$ of the form $S = R[f^{-1}]$.

## Related concepts

[[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ **Zariski site**

* [[spectrum of a commutative ring]]

## References

Examples A2.1.11(f) and D3.1.11 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

Section VIII.6 of

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
 {#MacLaneMoerdijk}

* [[The Stacks Project]], chapter 33 _Topologies on Schemes_

* Nick Duncan, _Gros and Petit Toposes_, talk notes, [88th Peripatetic Seminar on Sheaves and Logic](http://cheng.staff.shef.ac.uk/pssl88/), [pdf](http://cheng.staff.shef.ac.uk/pssl88/pssl88-duncan.pdf).

* Daniel Murfet, [The Zariski Site](http://therisingsea.org/notes/ZariskiTopology.pdf)

category: algebraic geometry

[[!redirects Zariski topos]]
[[!redirects big Zariski topos]]

[[!redirects Zariski sheaf]]
[[!redirects Zariski sheaves]]

[[!redirects Zariski descent]]

[[!redirects theory of local rings]]
[[!redirects geometric theory of local rings]]