Lorentzian geometry

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## Idea

__Lorentzian geometry__ is the [[geometry]] of _[[Minkowski spacetime]]_, hence essentially of a [[Euclidean space]], but equipped not with the standard Euclidean [[Riemannian metric]] of [[spacetime signature]] $(+,+,+,\ldots,+)$ (which yields [[Euclidean geometry]]) but with the [[pseudo-Riemannian metric]] of [[signature]] $(-,+,+,\ldots,+)$. This is in particular the context of the [[theory (physics)|theory]] of [[physics]] called "[[theory of special relativity]]", and it is _locally_ the context of the "[[theory of general relativity]]".

Note that while Lorentzian geometry is analogous to [[Euclidean geometry]] (as [[Minkowski space]] is analogous to a [[Euclidean space]]), a [[Lorentzian manifold]] is analogous to a [[Riemannian manifold]].  Thus, one might use 'Lorentzian geometry' analogously to [[Riemannian geometry]] (and insist on __Minkowski geometry__ for our topic here), but usually one skips all the way to [[pseudo-Riemannian geometry]] (which studies [[pseudo-Riemannian manifolds]], including both Riemannian and Lorentzian manifolds).

## Lorentzian Cartan geometry and first order gravity

Since the [[isometry group]] of [[Minkowski spacetime]] is the [[Poincaré group]], and since Minkowski spacetime is the [[quotient]] of the [[Poincaré group]] by the [[Lorentz group]]-[[subgroup]], Lorentzian geometry and the study of [[Lorentzian manifolds]] is to a large extent the [[Cartan geometry]] of the [[Poincaré group]].

Promoting this perspective from global to [[local symmetry]] yields the [[first order formulation of gravity]]. Promoting this perspective form the [[Poincaré group]] to the [[super Poincaré group]] yields _[[supergravity]]_. Promoting it further to the [[L-infinity algebra cohomology|Lie n-algebra]] extensions of the [[super Poincaré group]] (from the [[brane scan]]/[[schreiber:The brane bouquet|brane bouquet]]) yields [[type II supergravity]], [[heterotic supergravity]] and [[11-dimensional supergravity]] in [[higher Cartan geometry]]-formulation ([[D'Auria-Fré formulation of supergravity]]).

[[!include local and global geometry - table]]

## Related concepts

* [[timelike curve]], [[spacelike curve]], [[lightlike curve]]

* [[causet]]

* [[Lorentzian manifold]]

* [[hyperbolic geometry]]

## References
 {#References}

A [[synthetic geometry|synthetic]] set of axioms for Lorentzian geometry was developed in these articles

* Edwin B. Wilson & Gilbert N. Lewis (1912) _The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics_ Proceedings of the American Academy of Arts and Sciences 48:387–507

* James Cockle, _Synthetic Spacetime_, ([web](https://web.archive.org/web/20090926011519/http://ca.geocities.com/cocklebio/synsptm.html))

Introductions and surveys:

* {#Bär04} [[Christian Bär]], *Lorentzian Geometry*, lecture notes (2004) [[pdf](https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Lehre/Veranstaltungen/WS0405-SS08/LorentzianGeometryEnglish13Jan2020.pdf), [[Baer-LorentzianGeometry.pdf:file]]]

* Javayoles, Sánchez, _An introduction to Lorentzian Geometry and its application_, 2010 ([pdf](http://webs.um.es/majava/charlas/CursodeLorentz.pdf))

* Graciela Birman, Katsumi Nomizu, _Trigonometry in Lorentzian geometry_, The American Mathematical Monthly Vol. 91, No. 9 (Nov., 1984), pp. 543-549 ([JSTOR](http://www.jstor.org/stable/2323737))

See also

* Geometry Group Berlin, _[Lorentzian geometry](http://www3.math.tu-berlin.de/geometrie/lorentz/)_

[[!redirects Lorentzian geometry]]
[[!redirects Lorentzian geometries]]
[[!redirects Lorentz geometry]]
[[!redirects Lorentz geometries]]
[[!redirects Minkowski geometry]]
[[!redirects Minkowski geometries]]
[[!redirects Minkowskian geometry]]
[[!redirects Minkowskian geometries]]

[[!redirects Lorentzian signature]]
[[!redirects Lorentzian signatures]]


Lorentzian orbifold

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## Idea

A _Lorentzian orbifold_  is an [[orbifold]] with [[Lorentzian structure]], hence like a [[Lorentzian manifold]] but possibly with [[singularities]] modeled on [[fixed points]] of local [[actions]] of [[finite groups]] by [[Lorentz transformations]].

## Related concepts

* [[Riemannian orbifold]]

* [[symplectic orbifold]]

* [[complex orbifold]]

## References

Discussion of [[Lorentzian orbifolds]] as [[target spacetimes]] in [[string theory]]:

* [[Gary Horowitz]], Alan R. Steif, _Singular string solutions with nonsingular initial data_, Physics Letters B Volume 258, Issues 1–2, 4 April 1991, Pages 91-96 (doi:10.1016/0370-2693(91)91214-G)

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

* Hong Liu, [[Gregory Moore]], [[Nathan Seiberg]], _Strings in a Time-Dependent Orbifold_, JHEP 0206:045, 2002 ([arXiv:hep-th/0204168](https://arxiv.org/abs/hep-th/0204168))

* Hong Liu, [[Gregory Moore]], [[Nathan Seiberg]], _Strings in Time-Dependent Orbifolds_, JHEP 0210:031, 2002 ([arXiv:hep-th/0206182](https://arxiv.org/abs/hep-th/0206182))

* [[Boris Pioline]], _Aspects de la théorie des cordes enchamp de fond dépendant du temps_, LPTHE, Paris 2003 ([pdf](https://www.lpthe.jussieu.fr/~pioline/Seminars/semhabilit.pdf))

* [[Micha Berkooz]], Dori Reichmann, _A Short Review of Time Dependent Solutions and Space-like Singularities in String Theory_, Nucl. Phys. Proc. Suppl. 171:69-87, 2007 ([arXiv:0705.2146](https://arxiv.org/abs/0705.2146))

Classification of [[compact topological space|compact]] 2d Lorentzian orbifolds:

* N. I. Zhukova, E. A. Rogozhina, _Classification of compact lorentzian 2-orbifolds with noncompact full isometry groups_, Sib Math J 53, 1037–1050 (2012) ([doi:10.1134/S0037446612060080](https://doi.org/10.1134/S0037446612060080))

[[conformal maps]] between Lorentzian orbifolds:

* N. I. Zhukova, _The influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds_, Ufa Mathematical Journal
Volume 10, Number 2 (2018) pp. 44-57 ([doi:10.13108/2018-10-2-44](http://dx.doi.org/10.13108/2018-10-2-44))

[[!redirects Lorentzian orbifolds]]


Lorentzian polynomial

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## Idea

_Lorentzian polynomials_ are a class of [[homogeneous polynomials]] with [[real number|real]] [[coefficients]] that are used to relate continuous [[convex analysis]] and discrete convex analysis via [[tropical geometry]]. The theory of Lorentzian polynomials provides techniques to prove that various naturally-occurring sequences of non-negative real numbers, $(a_i)$, are log-concave, i.e., $a_k^2 \geq a_{k-1}a_{k+1}$ for all $k$.

## Definition

\begin{definition}
\label{LorentzianPolynomial}
Let $f \in \mathbb{R}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d$ such that all coefficients of $f$ are strictly [[positive number|positive]]. If either $d = 0, 1$, or $d\geq 2$  and the [[Hessian]] of any $(d-2)$th partial derivative $\partial_{i_1} \cdots \partial_{i_{d-2}}f$ has Hessian matrix with signature  $(+,-,-,\ldots,-)$, then $f$ is said to be **strictly Lorentzian**. If $f$ is in the closure of the set of strictly Lorentzian polynomials, then it is said to be **Lorentzian**.
\end{definition}

## Related concepts

* [[polynomial]]
* [[matroid]]

## References

* [[Petter Brändén]], [[June Huh]], *Lorentzian polynomials*, Annals of Mathematics, **192** 3 (2020) [[arXiv:1902.03719](https://arxiv.org/abs/1902.03719), [doi:10.4007/annals.2020.192.3.4](https://doi.org/10.4007/annals.2020.192.3.4)]

* {#Huh19} [[June Huh]], _Lorentzian polynomials_, Simons Lectures 2019 ([videos](https://math.mit.edu/news/seminars/simons19.php))

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]

[[!redirects Lorentzian polynomials]]


Lorenz Eberhardt

* [webpage](https://itp.phys.ethz.ch/people/person-detail.html?persid=191164)

## Selected writings

On proof of [[AdS/CFT duality]] for the special case of [[AdS3-CFT2|$AdS_3/CFT_2$]]:

* Shouvik Datta, [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Stringy $\mathcal{N} = (2,2)$ holography for $AdS_3$_ JHEP 1801 (2018) 146 ([arXiv:1709.06393](https://arxiv.org/abs/1709.06393))

* {#EberhardtGaberdiel19a} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _String theory on $AdS_3$ and the symmetric orbifold of Liouville theory_ ([arXiv:1903.00421](https://arxiv.org/abs/1903.00421))

* {#EberhardtGaberdiel19b} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], _Strings on $AdS_3 \times S^3 \times S^3 \times S^1$_ ([arXiv:1904.01585](https://arxiv.org/abs/1904.01585))

* {#EberhardtGaberdielGopakumar19} [[Lorenz Eberhardt]], [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _Deriving the $AdS_3/CFT_2$ Correspondence_ ([arXiv:1911.00378](https://arxiv.org/abs/1911.00378))

* Andrea Dei, [[Lorenz Eberhardt]], _Correlators of the symmetric product orbifold_ ([arXiv:1911.08485](https://arxiv.org/abs/1911.08485))

* [[Lorenz Eberhardt]], _$AdS_3/CFT_2$ at higher genus_ ([arXiv:2002.11729](https://arxiv.org/abs/2002.11729))

* [[Lorenz Eberhardt]], ,_Summing over Geometries in String Theory_ ([arXiv:2102.12355](https://arxiv.org/abs/2102.12355))

* [[Lorenz Eberhardt]], *A perturbative CFT dual for pure NS-NS $AdS_3$ strings*, J. Phys. A: Math. Theor. **55** 064001 [[arXiv:2110.07535](https://arxiv.org/abs/2110.07535), [arXiv:10.1088/1751-8121/ac47b2](https://doi.org/10.1088/1751-8121/ac47b2)]

On the [[Wess-Zumino-Witten model]]:

* [[Lorenz Eberhardt]], *Wess-Zumino-Witten models*, lecture notes at *[YRISW 2019: A modern primer for 2D CFT](https://conf.itp.phys.ethz.ch/esi-school/)*, Vienna (2019) [[pdf](https://conf.itp.phys.ethz.ch/esi-school/Lecture_notes/WZW%20models.pdf), [[Eberhardt-WZWModels.pdf:file]]]

On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:

* [[Scott Collier]], [[Lorenz Eberhardt]], [[Mengyang Zhang]], *3d gravity and Chern-Simons theory -- references* [[arXiv:2304.13650](https://arxiv.org/abs/2304.13650)]

category: people


Lorenz gauge

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## Idea

For [[electromagnetism]] the _Lorenz gauge condition_ is one possible [[gauge fixing]] condition: It requests that the [[divergence]] of the [[gauge potential]] $A$ [[differential 1-form]] vanishes: $d \star A = 0$.

On [[Minkowski spacetime]] in standard [[coordinates]] this is the condition $\partial_\mu A^\mu = 0$ (using [[Einstein summation convention]]).

In [[BV-BRST formalism]] this gauge (and its variants) is implement by introducing the auxiliary "[[Nakanishi-Lautrup field]]" together with its "[[antighost field]]", see at _[[quantization of Yang-Mills theory]]_.

## References

Named after [[Ludvik Lorenz]] (not to be confused with [[Hendrik Lorentz]] whose name is attached to the [[Lorentz group]]).

* {#Henneaux90} [[Marc Henneaux]], section 9.1 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

[[!redirects Lorentz gauge]]

[[!redirects Gaussian-averaged Lorenz gauge]]


Lorenzo Bartolini

* [InSpire page](https://inspirehep.net/authors/1859883)

## Selected writings

On [[skyrmions]]/[[solitons]] in the [[Sakai-Sugimoto model]] ([[AdS/QCD correspondence]] via [[D=5 Maxwell theory]]):

* {#Bartolini17} [[Lorenzo Bartolini]], [[Stefano Bolognesi]], Andrea Proto, _From the Sakai-Sugimoto Model to the Generalized Skyrme Model_, Phys. Rev. D **97** 014024 (2018) \[doi:10.1103/PhysRevD.97.014024, arXiv:1711.03873\]

* [[Lorenzo Bartolini]], [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], Tommaso Rainaldi, *Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD* \[arXiv:2312.15404\]

On [[baryons]] in the [[Sakai-Sugimoto model]] of [[holographic QCD]]:

* Salvatore Baldino, [[Lorenzo Bartolini]], [[Stefano Bolognesi]], [[Sven Bjarke Gudnason]], _Holographic Nuclear Physics with Massive Quarks_,  Phys. Rev. D **103** 126015 (2021) \[doi:10.1103/PhysRevD.103.126015, arXiv:2102.00680\]

category: people


Lorenzo Casarin

* [InSpire page](https://inspirehep.net/authors/1695586)

## Selected writings

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

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

category: people


Lorenzo Maccone

* [institute page](https://fisica.unipv.it/personale/Persona.php?ID=207)

## 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


Lorenzo Ramero

* [personal page](http://math.univ-lille1.fr/~ramero/)

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

## Selected writings

On [[almost ring theory]]

* [[Ofer Gabber]], [[Lorenzo Ramero]], _Almost ring theory_, Springer 2003 ([arXiv:math/0002064](https://arxiv.org/abs/math/0002064), [doi:10.1007/b10047](https://link.springer.com/book/10.1007/b10047))

and its foundations (including [[2-category]] theory):

* [[Ofer Gabber]], [[Lorenzo Ramero]], *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))

category: people


Lorenzo Tortora de Falco

Lorenzo Tortora de Falco is a [[proof theory|proof theorist]] working at Roma Tre University.

* [webpage](http://logica.uniroma3.it/~tortora/)

## Selected writings
 {#SelectedWritings}

On [[strong normalization]]:

* {#PaganiTortora09} [[Michele Pagani]], [[Lorenzo Tortora de Falco]], *Strong Normalization Property for Second Order Linear Logic* (2009) [[pdf](https://www.irif.fr/~michele/snllTCS-1.pdf)]

category: people


Loriano Bonora

* [webpage](http://www.sissa.it/tpp/members.php?ID=37)

## Selected writings

On [[bundle gerbes]] [[connections on bundle gerbes|with connection]] and [[Čech cohomology|Čech]]-[[Deligne cohomology]] for the description of [[B-fields]] in [[string theory]]:

* [[Loriano Bonora]], [[Fabio Ferrari Ruffino]], [[Raffaele Savelli]], *Classifying A-field and B-field configurations in the presence of D-branes*, JHEP 0812:078 (2008) [[arXiv:0810.4291](https://arxiv.org/abs/0810.4291), [doi:10.1088/1126-6708/2008/12/078](https://doi.org/10.1088/1126-6708/2008/12/078)]

## Related entries

* [[heterotic supergravity]]

* [[supergravity torsion constraints]]

category: people


Loring Tu

* [website](https://ltu.pages.tufts.edu/)

## Selected writings

On [[Borel equivariant cohomology]]:

* {#Tu11} [[Loring Tu]], _What is... Equivariant Cohomology?_, Notices of the AMS, Volume 85, Number 3, March 2011 ([pdf](https://www.ams.org/notices/201103/rtx110300423p.pdf), [[TuEquivariantCohomology.pdf:file]])

* [[Loring Tu]], *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

On [[differential forms]] in [[algebraic topology]]:

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp. ([doi:10.1007/978-1-4757-3951-0](https://link.springer.com/book/10.1007/978-1-4757-3951-0), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf))

On [[Raoul Bott]]:

* [[Loring Tu]]: *[The life and works of Raoul Bott](https://legacy-www.math.harvard.edu/history/bott/bottbio/index.html)* (2002)

On [[differential geometry]] with focus on [[Chern-Weil theory]]:

* [[Loring Tu]], _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer 2017 ([ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824))

category: people


loset > history

< [[loset]]
[[!redirects loset -- history]]


Lothar Göttsche

* [personal page](https://users.ictp.it/~gottsche/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Lothar_G%C3%B6ttsche)

## Selected writings

On basics of [[algebraic geometry]] following [[Grothendieck]]'s [[FGA]]:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]]:

  **[[FGA explained]]** (ICTP, Trieste 2003--2005)

  Mathematical Surveys and Monographs __123__

  Amer. Math. Soc. 2005. x+339 pp.

  * [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s)

    [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)

    [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)

category: people

[[!redirects Lothar Goettsche]]


Lothar W. Nordheim

Lothar Wolfgang Nordheim

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

## Selected writings

On the mathematical foundations of [[quantum mechanics]]:

* {#HilbertvonNeumannNordheim} [[David Hilbert]], [[John von Neumann]], [[Lothar W. Nordheim]], *Über die Grundlagen der Quantenmechanik*,  Math. Ann. **98** (1928) 1–30 [[doi:10.1007/BF01451579](https://doi.org/10.1007/BF01451579)]

category: people

[[!redirects Lothar Nordheim]]
[[!redirects Lothar Wolfgang Nordheim]]


Lou van den Dries

**Lou van den Dries** is a model-theorist best known for his work on [[o-minimal structures]] and [[tame topology]].

* [Homepage](http://www.math.uiuc.edu/People/vddries.html)
* [Page with lecture notes on model theory, stability theory, motivic integration etc.](http://www.math.uiuc.edu/~vddries/)

## Related entries

* [[tame topology]]
* [[o-minimal structure]]

## References

* L. van den Dries, _Exponential rings, exponential polynomials and exponential functions_ , Pacific Journal of Mathematics **113** no.1 (1984) pp.51–66. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.pjm/1102709376))

* L. van den Dries, _A Generalization of the Tarski-Seidenberg
Theorem and Some Nondefinability Results_ , Bull. Am. Math. Soc.
**15** (1986) pp.189-193. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.bams/1183553469))

* L. van den Dries, _Tame Topology and O-minimal Structures_ , Cambridge UP 1998.

* [[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in Springer LNM vol. 1488 (1991).

* [[F. W. Lawvere]], _Left and right adjoint operations on spaces and data types_ , Theoretical Computer Science **316** (2004) pp.105-111.

category:people


Louis Crane

* [website](http://www.math.ksu.edu/people/personnel_detail?person_id=1330)

## Selected writings

On the [[Crane-Yetter model]] [[4d TQFT]]:

* {#CraneYetterKauffman94} [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997)  ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))

category: people


Louis Kauffman

Lou Kauffman is an American [[topology|topologist]] working in [[knot theory]] and its relationships with [[algebra]], [[combinatorics]] and [[physics]], specifically [[statistical mechanics]], [[quantum theory]].

He is based at the University of Illinois at Chicago.

* [homepage](http://homepages.math.uic.edu/~kauffman/)

## Selected writings

On [[knot theory]] related to [[physics]]/[[quantum field theory]]:

* [[Louis Kauffman]], _Knots and physics_, Series on _Knots and Everything_, Volume 1,  World Scientific, 1991 ([doi:10.1142/1116](https://doi.org/10.1142/1116))

* [[Louis Kauffman]] (ed.) _The Interface of Knots and Physics_, Proceedings of Symposia in Applied Mathematics
Volume 51 (1996) ([pdf](http://www.csee.umbc.edu/~lomonaco/kelvin/kelvin23.pdf), [doi:10.1090/psapm/051](https://doi.org/10.1090/psapm/051))

On the [[Crane-Yetter model]] [[4d TQFT]]:

* {#CraneYetterKauffman94} [[Louis Crane]], [[Louis Kauffman]], [[David Yetter]], _State-Sum invariants of 4-manifolds I_, Journal of Knot Theory and Its Ramifications Vol. 06, No. 02, pp. 177-234 (1997)  ([arXiv:hep-th/9409167](https://arxiv.org/abs/hep-th/9409167), [pdf](http://arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf), [doi:10.1142/S0218216597000145](https://doi.org/10.1142/S0218216597000145))

On [[topological quantum computation]] with [[anyons]]/[[braid group statistics]] ([[braid group representations]]):

* [[Louis Kauffman]], *Quantum Topology and Quantum Computing*, in: [[Samuel J. Lomonaco]] (ed.), *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[pdf](https://redirect.cs.umbc.edu/~lomonaco/ams/lecturenotes/Kauffman.pdf), [doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

  > (in relation to [[quantum topology]])

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Braiding Operators are Universal Quantum Gates*,  New Journal of Physics, Volume 6, January 2004 ([arXiv:quant-ph/0401090](https://arxiv.org/abs/quant-ph/0401090), [doi:10.1088/1367-2630/6/1/134](https://iopscience.iop.org/article/10.1088/1367-2630/6/1/134))

* [[Samuel J. Lomonaco]], [[Louis Kauffman]], _Topological Quantum Computing and the Jones Polynomial_, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) ([arXiv:quant-ph/0605004](https://arxiv.org/abs/quant-ph/0605004))

  > ([[topological quantum computation]] of the [[Jones polynomial]])

* [[Louis H. Kauffman]], [[Samuel J. Lomonaco]], *Topological quantum computing and $SU(2)$ braid group representations*, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) ([doi:10.1117/12.778068](https://doi.org/10.1117/12.778068), [rg:228451452](https://www.researchgate.net/publication/228451452))

* [[Louis H. Kauffman]], *Majorana Fermions and Representations of the Braid Group*, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) ([arXiv:1710.04650](https://arxiv.org/abs/1710.04650), [doi:10.1142/S0217751X18300235](https://doi.org/10.1142/S0217751X18300235))

On [[quantum computation]] and more generally on [[quantum systems]]:

* [[Goong Chen]], [[Louis Kauffman]], [[Samuel J. Lomonaco]] (eds.), *Mathematics of Quantum Computation and Quantum Technology*, Chapman and Hall/CRC (2007) (ISBN:9780367388614, [doi:10.1201/9781584889007](https://doi.org/10.1201/9781584889007))

category: people

[[!redirects L. H. Kauffman]]
[[!redirects Louis H. Kauffman]]


Louis Lyons

* [institute page](https://www2.physics.ox.ac.uk/contacts/people/lyons)

## selected writings

* {#Lyons13a} [[Louis Lyons]], _Bayes and Frequentism: a Particle Physicist's perspective_,  Journal of Contemporary Physics Volume 54, 2013 - Issue 1 ([arXiv:1301.1273](https://arxiv.org/abs/1301.1273), [doi:10.1080/00107514.2012.756312](https://doi.org/10.1080/00107514.2012.756312))

* {#Lyons13b} [[Louis Lyons]], _Discovering the Significance of 5 sigma_ ([arXiv:1310.1284](https://arxiv.org/abs/1310.1284))

## related $n$Lab entries

* [[statistical significance]]

* [[flavour anomaly]]

* [[anomalous magnetic moment]]

category: people


Louis Martini

* [institute page](https://www.ntnu.edu/employees/louis.o.martini)

## Selected writings

Development of [[(infinity,1)-category theory|$\infty$-category theory]] [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

internal [[(∞,1)-Yoneda lemma]]:

* [[Louis Martini]], *Yoneda's lemma for internal higher categories*, [[arXiv:2103.17141](https://arxiv.org/abs/2103.17141)]

internal [[(infinity,1)-limits]] and [[(infinity,1)-colimits]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Limits and colimits in internal higher category theory* [[arXiv:2111.14495](https://arxiv.org/abs/2111.14495)]

internal [[cocartesian fibrations]] and [[straightening functor]]:

* [[Louis Martini]], *Cocartesian fibrations and straightening internal to an ∞-topos* [[arXiv:2204.00295](https://arxiv.org/abs/2204.00295)]

internal [[presentable (∞,1)-categories]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Presentable categories internal to an ∞-topos* [[arXiv:2209.05103](https://arxiv.org/abs/2209.05103)]

and [[(infinity,1)-toposes|$\infty$-toposes]] internal to [[(infinity,1)-toposes|$\infty$-toposes]]:

* [[Louis Martini]], [[Sebastian Wolf]], *Internal higher topos theory* [[arXiv:2303.06437](https://arxiv.org/abs/2303.06437)]

On [[proper geometric morphisms]] between [[(infinity,1)-toposes|$(\infty,1)$-toposes]]:

* [[Louis Martini]], [[Sebastian Wolf]], _Proper morphisms of ∞-topoi_ [[arXiv:2311.08051](https://arxiv.org/abs/2311.08051)].

category: people


Louis Michel

* [IHES page](https://www.ihes.fr/~vergne/LouisMichel/)

## Selected writings

On [[lattice (discrete subgroup)|lattice theory]]:

* [[Peter Engel]], [[Louis Michel]], [[Marjorie Senechal]], *Lattice Geometry*, IHES/P/04/45 (2004) $[$[pdf](https://cds.cern.ch/record/859509/files/cer-002542451.pdf), [cds:859509](https://cds.cern.ch/record/859509)$]$

category: people


Louis Mordell

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

## Related entries

* [[Mordell conjecture]]

category: people


Louis Solomon

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=9225)

## Selected writings

On the [[representation theory]] ([[linear representations]]) of [[finite groups]] over [[algebraic number fields]]:

* [[Louis Solomon]], _The representation of finite groups in algebraic number fields_, J. Math. Soc. Japan Volume 13, Number 2 (1961), 144-164 ([euclid:euclid.jmsj/1261147137](https://projecteuclid.org/euclid.jmsj/1261147137))

On the [[Burnside ring]] (and introducing that terminology):

* {#Solomon67} [[Louis Solomon]], _The Burnside algebra of a finite group_, Journal of Combinatorial Theory Volume 2, Issue 4, June 1967, Pages 603-615 (doi:10.1016/S0021-9800(67)80064-4)

On the ([[Dolbeault cohomology|holomorphic]]) [[de Rham cohomology]] of [[configuration spaces of points]] (the result whose "[[twisted de Rham cohomology|twisted]]" generalization leads to the [[hypergeometric construction of KZ solutions]]):

* [[Peter Orlik]], [[Louis Solomon]], *Combinatorics and topology of complements of hyperplanes*, Invent Math **56** (1980) 167–189  $[$[doi:10.1007/BF01392549](https://doi.org/10.1007/BF01392549)$]$

category: people


Louis-Hadrien Robert

* [personal page](http://lrobert.perso.math.cnrs.fr/)

## Selected writings

On [[link homology]]:

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Foam evaluation and Kronheimer--Mrowka theories*, Advances in Mathematics **376**  (2021) 107433 [[arXiv:1808.09662](https://arxiv.org/abs/1808.09662), [doi:10.1016/j.aim.2020.107433](https://doi.org/10.1016/j.aim.2020.107433)]

* [[Mikhail Khovanov]], [[Louis-Hadrien Robert]], *Conical $SL(3)$ foams*, Journal of Combinatorial Algebra **6** 1/2 (2022) 79-108 [[arXiv:2011.11077](https://arxiv.org/abs/2011.11077), [doi:10.4171/jca/61](https://doi.org/10.4171/jca/61)]

category


Lov K. Grover

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

## Selected writings

Introducing [[Grover's algorithm]] for [[quantum computers]]:

* [[Lov K. Grover]], *A fast quantum mechanical algorithm for database search*, STOC '96: Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing (1996) 212-219 [[arXiv:quant-ph/9605043](https://arxiv.org/abs/quant-ph/9605043), [doi:10.1145/237814.237866](https://doi.org/10.1145/237814.237866)]

category: people

[[!redirects Lov Grover]]


low dimensional manifolds -- table

**[[manifolds]] in [[low-dimensional topology|low]] [[dimension]]**:

* [[2-manifolds]]

* [[3-manifolds]]

* [[4-manifolds]]

* [[5-manifolds]]

* [[8-manifolds]]


low dimensional rotation groups -- table

**[[rotation groups]] in [[low-dimensional topology|low]] [[dimensions]]**:

| [[classification of simple Lie groups|Dynkin label]] | [[special orthogonal group|sp. orth. group]] | [[spin group]] | [[pin group]] | [[semi-spin group]] |
|-|------------------------------|----------------|---------------|--|
| | [[SO(2)]] | [[Spin(2)]] |  [[Pin(2)]] |   |
| B1 | [[SO(3)]] | [[Spin(3)]] |  [[Pin(3)]] |   |
| D2 | [[SO(4)]] | [[Spin(4)]] |  [[Pin(4)]] |   |
| B2 | [[SO(5)]] | [[Spin(5)]] |  [[Pin(5)]] |   |
| D3 | [[SO(6)]] | [[Spin(6)]] |             |   |
| B3 | [[SO(7)]] | [[Spin(7)]] |             |   |
| [[D4]] | [[SO(8)]] | [[Spin(8)]] |             | [SO(8)](semi-spin+group#SemiSpin8)  | |
| B4 | [[SO(9)]] | [[Spin(9)]] |             |   |
| [[D5]] | [[SO(10)]] | [[Spin(10)]] |             |   |
| B5 | [[SO(11)]] | [[Spin(11)]] |             |   |
| [[D6]] | [[SO(12)]] | [[Spin(12)]] |             |   |
|  | $\vdots$  | $\vdots$    |             |   |
| D8 | [[SO(16)]] | [[Spin(16)]] |           | [[SemiSpin(16)]] |
| | $\vdots$  | $\vdots$    |             |   |
| D16 | [[SO(32)]] | [[Spin(32)]] |           | [[SemiSpin(32)]] |

see also

* [[Spin(5).Spin(3)]]

* [[finite rotation groups]]

* [[ADE-classification]]


low-dimensional topology

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## Idea

The study of the [[topology]] of [[manifolds]] of low [[dimension]], notably of [[2-manifolds]] ([[surfaces]]), [[3-manifolds]] and [[4-manifolds]], but also [[8-manifolds]]

## Related entries

* [[geometrization conjecture]]

* [[arithmetic topology]]

* [[van Kampen theorem]]

* [[quantum topology]]

* [[SnapPy]]

[[!include low dimensional manifolds -- table]]

[[!include low dimensional rotation groups -- table]]

## References

* Wikipedia, _[Low-dimensional topology](http://en.wikipedia.org/wiki/Low-dimensional_topology)_

* John Stillwell, _Classical Topology and Combinatorial Group Theory_, Springer (1980, 1993) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/stillwell1x.pdf))

[[!redirects low dimensional topology]]


Lowell Abrams

* [webpage](http://home.gwu.edu/~labrams/)

## Selected writings

On [[2d TQFT]] and its classification by [[Frobenius algebras]]:

* {#Abrams96} [[Lowell Abrams]], *Two-dimensional topological quantum field theories and Frobenius algebra*, Jour. Knot. Theory and its Ramifications **5**, 569-587 (1996) [[doi:10.1142/S0218216596000333](https://doi.org/10.1142/S0218216596000333), [ps](http://home.gwu.edu/~labrams/docs/tqft.ps)]

## Related $n$Lab entries

* [[2d TQFT]], [[Frobenius algebra]]

category: people


lower central series

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## Idea

Given a [[group]] $G$, its __lower central series__ is the [[induction|inductively]] defined descending [[sequence]] of [[subgroup]]-inclusions

$$
  G \,=\, G_0 \supset G_1\supset G_2\supset \ldots
$$

in which $G_k = [G, G_{k-1}]$ is the [[subgroup]] generated by all [[group commutators]] $g h g^{-1}h^{-1}$ where $g\in G$ and $h\in G_{k-1}$.

For a [[nilpotent group]], this series terminates in finitely many steps at the trivial subgroup and is the same length as the *[[upper central series]]*. It is the fastest descending [[central series]].

Similarly, given a [[Lie algebra]] $L$, its lower central series is the inductively defined descending sequence of Lie subalgebras $L = L_0\supset L_1\supset L_2\supset\ldots$ in which $L_k = [L, L_{k-1}]$ is the Lie subalgebra generated by all commutators $[l,h]$ where $l\in L$ and $h\in L_{k-1}$.

## Related concepts

* [[upper central series]]

* [[completion of a group]]

## References

See also

* Wikipedia, *[Central series](https://en.wikipedia.org/wiki/Central_series)*


lower set

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{: toc}

## Definition

In a [[poset]] or even [[proset]], a __lower set__ $L$ is a [[subset]] that is 'downward closed'; that is,

*  whenever $y \leq x$ and $x \in L$, then $y \in L$.

Lower sets form a [[Moore collection]] and so one can speak of the lower set _generated_ by an arbitrary subset $A$:
$$ A{\downarrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; y \leq x \} .$$

Sometimes a lower set is called an 'ideal', but that term can also mean [[ideal|something more specific]] (and always does in a [[lattice]]).

A lower set is also sometimes called a 'down set', but that term can also mean something more specific: the [[down set]] of $x$ is the lower set generated by $x$.

The [[characteristic function]] of a lower set is precisely a [[(0,1)-presheaf]].

## Examples

An [[inhabited subset|inhabited]], [[open subset|open]] lower set of [[rational numbers]] (or equivalently of [[real numbers]]) determines precisely a [[lower real number]].

## See also ##

* [[posite]]
* [[upper set]]
* [[ideal]]
* [[filter]]
* [[copresheaf]]
* [[presheaf]]

[[!redirects lower set]]
[[!redirects lower sets]]
[[!redirects lower subset]]
[[!redirects lower subsets]]

[[!redirects downward closed subset]]
[[!redirects downward closed subsets]]


Loïc Pujet

* [personal page](https://pujet.fr/)

## Selected writings

On [[synthetic homotopy theory]] via [[cubical type theory|cubical]] [[homotopy type theory]]:

* [[Anders Mörtberg]], [[Loïc Pujet]], *Cubical synthetic homotopy theory*,  CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (Jan 2020) 158–171 [[doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), [pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf)]

category: people

[[!redirects Loic Pujet]]


LSP

wikipedia: [LSP](http://en.wikipedia.org/wiki/Lightest_Supersymmetric_Particle)


LSZ reduction formula

[[!redirects LSZ theory]]

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{:toc}

## Idea

In [[perturbative quantum field theory]], the LSZ reduction formula ([Lehmann-Symanzik-Zimmermann 55](#LehmannSymanzikZimmermann55)) is a method relating [[vacuum expectation values]] of [[time-ordered products]] of field [[observables]] to [[S-matrix]] elements

## References

The original article:

* {#LehmannSymanzikZimmermann55} [[Harry Lehmann]], [[Kurt Symanzik]], [[Wolfhart Zimmermann]], _Zur Formulierung quantisierter Feldtheorien_, Nuovo Cimento 1(1), 205 (1955) ([doi:10.1007/BF02731765](https://doi.org/10.1007/BF02731765))

See also:

* Wikipedia, _[LSZ reduction formula](https://en.wikipedia.org/wiki/LSZ_reduction_formula)_


Lubin-Tate formal group

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{:toc}

## Definition

Let $k$ be a [[perfect field]] and fix a [[prime number]] $p$.

Write $W(k)$ for the [[ring of Witt vectors]] and

$$
  R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]
$$

for the ring of [[formal power series]] over this ring, in $n-1$ [[variables]]; called the _Lubin-Tate ring_.

There is a canonical [[morphism]]

$$
  p \;\colon\; R \longrightarrow k
$$

whose [[kernel]] is the [[maximal ideal]]

$$
  ker(p) \simeq (p,v_1, \cdots, v_{n-1})
  \,,
$$

This induces (...) for every [[formal group]] $f$ over $k$ a [[deformation]] $\overline{f}$ over $R$. This is the _Lubin-Tate formal group_.

## Properties

### As the universal deformation

The *[[Lubin-Tate theorem]]* asserts that the Lubin-Tate formal group $\overline{f}$ is the [[universal construction|universal]] [[deformation]] of $f$.

### As inducing Morava E-theory

The Lubin-Tate [[formal group]] is [[Landweber exactness|Landweber exact]] and hence induces a [[complex oriented cohomology theory]]. This is [[Morava E-theory]], see there for more details.

## Related concepts

* [[Lubin-Tate theory]]

* [[Lubin-Tate space]]

## References

For a general review see

* Wikipedia, _[Lubin-Tate formal group law](http://en.wikipedia.org/wiki/Lubin–Tate_formal_group_law)_

For the role in [[chromatic homotopy theory]] see

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture notes 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html)), Lecture 21 _Lubin-Tate theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf))
 {#LurieLecture}

[[!redirects Lubin-Tate formal groups]]

[[!redirects Lubin-Tate formal group law]]
[[!redirects Lubin-Tate formal group laws]]


Lubin-Tate space

#Contents#
* table of contents
{:toc}

## Idea

A _Lubin-Tate space_ is a space of [[deformations]] of a one-dimensional [[formal group]] of height $h$. One may also construct spaces $\mathcal{M}_{n}$ parametrizing deformations of formal groups with [[level structures]]. The "infinite-level" Lubin-Tate space $\mathcal{M}_{\infty}$, also known as the _Lubin-Tate tower_ is some idea of "inverse limit" of these spaces (note that the inverse limit does not actually as a [[rigid analytic space]], see Lecture 2 of [#WeinsteinNotes](#WeinsteinNotes) for the definition of the Lubin-Tate tower on the level of points).

## Application to the local Langlands correspondence

As the infinite-level Lubin-Tate space has actions of the groups appearing in the statement of the [[local Langlands correspondence]] for $\GL_{n}$, it has applications to its proof. See also Theorem 7 in [#WeinsteinNotes](#WeinsteinNotes), which is originally due to Harris-Taylor.

## Generalizations

The infinite-level Lubin-Tate space is a special case of an infinite-level [[Rapoport-Zink space]] (see [#ScholzeWeinstein12](#ScholzeWeinstein12)). These are in turn special cases of local Shimura varieties and the even more general notion of moduli of local shtukas.

## Related concepts

* [[Lubin-Tate theory]]

* [[Lubin-Tate formal group]]

## References

* {#WeinsteinNotes} [[Jared Weinstein]], _The geometry of Lubin-Tate spaces_, [pdf](https://math.bu.edu/people/jsweinst/FRGLecture.pdf)

* {#ScholzeWeinstein12} [[Peter Scholze]], [[Jared Weinstein]], _Moduli of p-divisible groups_, [arXiv:1211.6357](https://arxiv.org/abs/1211.6357)


Lubin-Tate theory

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli stack of formal groups]] $\mathcal{M}_{FG}$ admits a natural [[stratification]] whose open [[strata]] $\mathcal{M}^n_{FG}$ are labeled by a [[natural number]] called the _[[height of a formal group|height of formal groups]]_.

The [[deformation theory]] around these [[strata]] is _Lubin-Tate theory_.

The universal Lubin-Tate deformation ring of a [[formal group]] of [[height of a formal group|height]] $n$ induces, via the [[Landweber exact functor theorem]] a [[complex oriented cohomology theory]], a localization of this is $n$th [[Morava E-theory]] $E(n)$.

## Lubin-Tate formal group

Let $k$ be a [[perfect field]] and fix a [[prime number]] $p$.

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

Write $W(k)$ for the [[ring of Witt vectors]] and

$$
  R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]
$$

for the ring of [[formal power series]] over this ring, in $n-1$ [[variables]]; called the _Lubin-Tate ring_.

=--

There is a canonical [[morphism]]

$$
  p \;\colon\; R \longrightarrow k
$$

whose [[kernel]] is the [[maximal ideal]]

$$
  ker(p) \simeq (p,v_1, \cdots, v_{n-1})
  \,,
$$

This induces (...) for every [[formal group]] $f$ over $k$ a [[deformation]] $\overline{f}$ over $R$. This is the _[[Lubin-Tate formal group]]_.

## Lubin-Tate theorem

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

The [[Lubin-Tate formal group]] $\overline{f}$ is the [[universal property|universal]] deformation of $f$ in that for every [[infinitesimal object|infinitesimal thickening]] $A$ of $k$, $\overline{f}$ induces a [[bijection]]

$$
  Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)
$$

between the $k$-[[associative algebra|algebra]]-[[homomorphisms]] from $R$ into $A$ and the [[deformations]] of $A$.

=--

(e.g. [Lurie 10, lect 21, theorem 5](#LurieLecture))

## Related concepts

* [[Lubin-Tate space]]

* [[Lubin-Tate formal group]]

* [[chromatic homotopy theory]]

* [[Morava E-theory]]

## References

* {#LurieLecture} [[Jacob Lurie]], Lecture 21 of: _[[Chromatic Homotopy Theory]]_, Lecture notes 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html)), Lecture 21 _Lubin-Tate theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf))

On the [[topological Hochschild homology]] of the Lubin-Tate ring spectrum via [[factorization homology]]:

* [[Geoffroy Horel]], *Higher Hochschild homology of the Lubin-Tate ring spectrum*, Algebr. Geom. Topol. **15** (2015) 3215-3252 [[arXiv:1311.2805](https://arxiv.org/abs/1311.2805), [doi:10.2140/agt.2015.15.3215](https://doi.org/10.2140/agt.2015.15.3215)]

[[!redirects Lubin-Tate spectrum]]

[[!redirects Lubin-Tate ring]]
[[!redirects Lubin-Tate rings]]

[[!redirects Lubin-Tate theorem]]


Luc Illusie

__Luc Illusie__ is a French algebraic geometer, former student of Grothendieck. He established some of the key notions and techniques including "Illusie's [[cotangent complex]]", Illusie-Fontaine positive characteristic method for proving Hodge to de Rham spectral sequence degeneration, various results on formal schemes, logarithmic geometry, crystalline cohomology, arithmetic geometry and so on.

* [webpage](http://www.math.u-psud.fr/~illusie/)
[[!redirects L. Illusie]]


Luc Menichi

* [webpage](http://perso.math.univ-angers.fr/spip.php?article15)

## Selected writings

On [[BV-algebra]] [[structure]] on [[Hochschild cohomology]]:

* {#Menichi09} [[Luc Menichi]], *Batalin-Vilkovisky algebra structures on Hochschild cohomology*, Bulletin de la Société Mathématique de France, **137** 2 (2009) 277-295
[[arXiv:0711.1946](https://arxiv.org/abs/0711.1946), [numdam:BSMF_2009__137_2_277_0/](http://www.numdam.org/item/BSMF_2009__137_2_277_0/), [pdf](http://math.univ-angers.fr/perso/lmenichi/BV_Hochschild.pdf)]

On [[Sullivan models]] in [[rational homotopy theory]]:

* {#Menichi13} [[Luc Menichi]], *Rational homotopy -- Sullivan models*, in: *Free Loop Spaces in Geometry and Topology*, IRMA Lect. Math. Theor. Phys., EMS (2015) [[arXiv:1308.6685](https://arxiv.org/abs/1308.6685), [doi:10.4171/153](https://doi.org/10.4171/153)]

and specifically regarding the [[Sullivan models of loop spaces]]:

* [[Luc Menichi]], _Sullivan models and free loop space_, A short introduction to Sullivan models, with the Sullivan model of a free loop space and the detailed proof of Vigué-Sullivan theorem on the Betti numbers of free loop space. Workshop on free loop space à Strasbourg, November 2008 (scanned notes [pdf](http://math.univ-angers.fr/perso/lmenichi/Sullivan_models_free_loop_space.pdf))

## Related $n$Lab entries

* [[Hochschild cohomology]]

* [[string topology]]

* [[J-homomorphism]]

* [[BV-algebra]]

* [[Sullivan model of free loop space]]

category: people


Luca Barbieri-Viale

* [webpage](http://users.unimi.it/barbieri/)

## related $n$Lab entries

* [[Nori motive]]

category: people


Luca Giorgetti

* [personal page](https://sites.google.com/view/luca-giorgetti)

## Selected writings

On [[2d CFT]] in terms of [[AQFT on curved spacetimes]]:

* [[Marco Benini]], [[Luca Giorgetti]], [[Alexander Schenkel]], *A skeletal model for 2d conformal AQFTs* ([arXiv:2111.01837](https://arxiv.org/abs/2111.01837))

category: people


Luca Iliesiu

* [Spire page](https://inspirehep.net/authors/1399314?ui-citation-summary=true)

* [GoogleScholar page](https://scholar.google.ca/citations?user=wHsmq4gAAAAJ&hl=en)

## Selected writings

* [[Luca Iliesiu]], *From statistical mechanics to microstate counting for extremal and near-extremal black holes*, talk at *[[String 2022]]* [[indico:4940850](https://indico.cern.ch/event/1085701/contributions/4940850), [slides](https://indico.cern.ch/event/1085701/contributions/4940850/attachments/2482365/4261646/Luca%20Iliesiu%20Tuesday-Reserach-Iliesiu-v2.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=96645a86-6d44-40ab-9abf-dd5a1920d881)]

category: people


Luca Mesiti

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

* [institute page](https://eps.leeds.ac.uk/maths/pgr/8472/luca-mesiti)

## Selected writings

On aspects of the [[2-Grothendieck construction]]:

* [[Luca Mesiti]], *The 2-Set-enriched Grothendieck construction and the lax normal conical 2-limits* [[arXiv:2302.04566](https://arxiv.org/abs/2302.04566)]

A [[2-comonad]] characterizing [[Grothendieck fibrations]]:

* [[Jacopo Emmenegger]], [[Luca Mesiti]], [[Giuseppe Rosolini]], [[Thomas Streicher]], *A comonad for Grothendieck fibrations* [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]

On [[2-colimits]] in slide 2-categories:

* [[Luca Mesiti]], *Colimits in 2-dimensional slices* [[arXiv:2305.01494](https://arxiv.org/abs/2305.01494)]

On the [[indexed Grothendieck construction]]:

* [[Elena Caviglia]], [[Luca Mesiti]], *Indexed Grothendieck construction* [[arXiv:2307.16076](https://arxiv.org/abs/2307.16076)]

Discussion of 2-classifiers for [[2-toposes]] is in

* [[Luca Mesiti]], _2-classifiers via dense generators and Hofmann-Streicher universe in stacks_ [[arXiv:2401.16900](https://arxiv.org/abs/2401.16900)]

## Related entries

* [[marked 2-limit]]

category: people


Luca Mezincescu

* [webpage](http://web2.physics.miami.edu/~mezincescu/)

## related $n$Lab entries

* [[super 1-brane in 3d]]

category: people


Luca Pol

* [personal page](https://sites.google.com/view/lucapol/)

## Selected writings

On ([[monoidal localization|monoidal]]) [[Bousfield localization of model categories|Bousfield localization]] of [[stable model categories]]:

* [[Luca Pol]], [[Jordan Williamson]], *The Left Localization Principle, completions, and cofree $G$-spectra*, J. Pure Appl. Algebra **224** 11 (2020) 106408 [[arXiv:1910.01410](https://arxiv.org/abs/1910.01410), [doi:10.1016/j.jpaa.2020.106408](https://doi.org/10.1016/j.jpaa.2020.106408)]

category: people


Luca Tagliacozzo

* [webpage](https://www.strath.ac.uk/staff/tagliacozzolucadr)

## Selected writings

Textbook on [[tensor networks]]:

* Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, [[Luca Tagliacozzo]], Gang Su, Maciej Lewenstein, _Tensor Network Contractions_, Lecture Notes in Physics, Springer (2020) ([arXiv:1708.09213](https://arxiv.org/abs/1708.09213), [doi:10.1007/978-3-030-34489-4](https://link.springer.com/book/10.1007/978-3-030-34489-4))

On [[tree tensor networks]] related to  [[entanglement entropy]]:

* [[Luca Tagliacozzo]], G. Evenbly, [[Guifre Vidal]],
_Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law_, Phys. Rev. B 80, 235127 (2009) ([arXiv:0903.5017](https://arxiv.org/abs/0903.5017))

On [[tensor networks]] applied to [[lattice gauge theory]]:

* [[Luca Tagliacozzo]], Alessio Celi, Maciej Lewenstein, _Tensor Networks for Lattice Gauge Theories with continuous groups_, Phys. Rev. X 4, 041024 (2014) ([arXiv:1405.4811](https://arxiv.org/abs/1405.4811))

* {#BBCCCDFJLMMRRTVV19} M.C. Bañuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, [[Luca Tagliacozzo]], K. Van Acoleyen, [[Frank Verstraete]], U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller:

  _Simulating Lattice Gauge Theories within Quantum Technologies_

  ([arXiv:1911.00003](https://arxiv.org/abs/1911.00003))

category: people


Lucas Mann

* [institute page](https://www.uni-muenster.de/FB10/Service/show_perspage.shtml?id=1680)

## Selected writings

On the [[yoga of six operations]] in [[rigid analytic geometry]]:

* {#Mann22} [[Lucas Mann]], *A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry* [[arXiv:2206.02022](https://arxiv.org/abs/2206.02022)]

## Related entries

* [[solid module]]

category: people


Lucas Williams

* [Institute page](https://www2.math.binghamton.edu/p/people/grads/williaml/start)

## Selected writings

On [[configuration spaces of points]] and [[braid groups]]:

* {#Williams20} [[Lucas Williams]], *Configuration Spaces for the Working Undergraduate*,  Rose-Hulman Undergraduate Mathematics Journal, **21** 1 (2020) Article 8. ([arXiv:1911.11186](https://arxiv.org/abs/1911.11186), [rhumj:vol21/iss1/8](https://scholar.rose-hulman.edu/rhumj/vol21/iss1/8))

category: people


Lucian Ionescu

* [webpage](https://my.ilstu.edu/~lmiones/)

## Selected writings

On [[coproducts]] on [[graph complexes]] and the relation of their [[cobar construction]] to [[renormalization]]:

 * [[Lucian Ionescu]], _Perturbative Quantum Field Theory and Configuration Space Integrals_, In: John Byden, _Advances in Topological Quantum Field Theory_ ([arXiv:hep-th/0307062](https://arxiv.org/abs/hep-th/0307062), [doi:10.1007/978-1-4020-2772-7](https://link.springer.com/book/10.1007/978-1-4020-2772-7))

* [[Lucian Ionescu]], _Cohomology of Feynman graphs and perturbative quantum field theory_, In: O. Kovras, _Focus on Quantum Field Theory_, Nova Publishers Inc. 2004,  ([arXiv:math/0506142](https://arxiv.org/abs/math/0506142))

category: people


Lucien Hardy

* [webpage](https://www.perimeterinstitute.ca/people/lucien-hardy)

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

## related $n$Lab entries

* [[quantum physics]]

category: people


Lucile Vandembroucq

Lucile Vandembroucq is an Associate Professor in the
University of Minho.

*  [webpage](https://www.cmat.uminho.pt/index.php/user/671)

## Selected writings

On [[topological complexity]] of [[connected sums]]:

* {#CohenVandembroucq18} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Motion planning in connected sums of real projective spaces_ (2018), [arXiv:1807.09947](https://arxiv.org/abs/1807.09947)

On [[topological complexity]] of the [[Klein bottle]]:

* {#CohenVandembroucq16} [[Daniel C. Cohen]], [[Lucile Vandembroucq]], _Topological Complexity of the Klein bottle_ (2018), [arXiv:1612.03133v3](https://arxiv.org/abs/1612.03133v3)


Lucrezi Ravera > history

see _[[Lucrezia Ravera]]_


Lucrezia Ravera

## Selected writings

On [[D=7 supergravity]]:

* [[Pietro Fré]], [[Pietro Antonio Grassi]], [[Lucrezia Ravera]], M. Trigiante, _Minimal $D=7$ Supergravity and the supersymmetry of Arnold-Beltrami Flux branes_, JHEP 1606 (2016) 018 ([arXiv:1511.06245](https://arxiv.org/abs/1511.06245))

On the [[supergravity Lie 3-algebra]] (see also at *[[M-theory supersymmetry algebra]]*):

* {#AndrianopoliDAuriaRavera16} [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_, JHEP 1608 (2016) 095 ([arXiv:1606.07328](https://arxiv.org/abs/1606.07328))

* [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _More on the Hidden Symmetries of 11D Supergravity_ ([arXiv:1705.06251](https://arxiv.org/abs/1705.06251))

* [[Lucrezia Ravera]], *On the hidden symmetries of $D=11$ supergravity* ([arXiv:2112.00445](https://arxiv.org/abs/2112.00445))

On the [[self-dual higher gauge field]] on the [[M5-brane]] lifted to [[superspace]] in the [[D'Auria-Fré formulation of supergravity]]:

* {#ACDGMNRT22} [[Laura Andrianopoli]], [[C. A. Cremonini]], [[Riccardo D'Auria]], [[Pietro A. Grassi]], R. Matrecano, R. Noris, [[Lucrezia Ravera]], [[Mario Trigiante]], *M5-brane in the superspace approach*, Phys. Rev. D **106** 2 (2022) 026010 [[arXiv:2206.06388](https://arxiv.org/abs/2206.06388), [doi:10.1103/PhysRevD.106.026010](https://doi.org/10.1103/PhysRevD.106.026010)]

On the [[D'Auria-Fré formulation of supergravity]]:

* [[Lucrezia Ravera]], *Geometric approach to supergravity in superspace and some applications*, lecture at *Training school on Cartan Geometry*, Masaryk University (Nov 2023) [video: [YT](https://youtu.be/tuSXuAtvdRk)]

On [[super-Cartan geometry]]:

* J. François, [[Lucrezia Ravera]], *Cartan geometry, supergravity, and group manifold approach* [[arXiv:2402.11376](https://arxiv.org/abs/2402.11376)]

## Related  entries

* [[D'Auria-Fré formulation of supergravity]]

* [[M-theory super Lie algebra]]

* [[7d supergravity]]

category: people


Ludmil Hadjiivanov

* [institute page](https://www.inrne.bas.bg/index.php/tep/ludmil-hadjiivanov)

* [GoogleScholar page](https://scholar.google.com/citations?user=45Bg7VQAAAAJ&hl=en)

## Selected writings

On [[braid group representations]] via [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] on [[conformal blocks]]:

* [[Ivan Todorov]], [[Ludmil Hadjiivanov]], *Monodromy Representations of the Braid Group*, Phys. Atom. Nucl. **64** (2001) 2059-2068; Yad.Fiz. **64** (2001) 2149-2158 [[arXiv:hep-th/0012099](https://arxiv.org/abs/hep-th/0012099), [doi:10.1134/1.1432899](https://doi.org/10.1134/1.1432899)]

category: people


Ludmil Katzarkov

__Ludmil Katzarkov__ is a Bulgarian [[algebraic geometer]], holding degree from Moscow State University and positions in Miami and Vienna.

* [website](http://www.dieuniversitaet-online.at/professuren/curricula-vitae/beitrag/news/univ-prof-phd-dr-ludmil-katzarkov-1/80.html)

#related $n$Lab entries#

* [[nonabelian Hodge theory]], [[homological mirror symmetry]]

[[!redirects Katzarkov]]
[[!redirects L. Katzarkov]]

category: people


Ludvik Lorenz

> not to be cofused with [[Hendrik Lorentz]]

***

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

## related $n$Lab entries

* [[Lorenz gauge]]

category: people


Ludwig Bieberbach

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

* [MacTutor math history page](https://mathshistory.st-andrews.ac.uk/Biographies/Bieberbach/)

## Selected writings

On the classification of [[crystallographic groups]]:

* {#Bieberbach1910} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen des $n$ dimensionalen Euklidischen Raumes mit einem endlichen Fundamentalbereich*, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1910) 75-84 $[$[dml:58754](https://eudml.org/doc/58754)$]$

* {#Bieberbach1911} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung)*, Mathematische Annalen **70** (1911) 297–336  $[$[doi:10.1007/BF01564500](https://doi.org/10.1007/BF01564500)$]$

* {#Bieberbach1912} [[Ludwig Bieberbach]], *Über die Bewegungsgruppen der Euklidischen Räume (ZweiteAbhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich*, Mathematische Annalen  **72** (1912) 400-412 $[$[doi:10.1007/BF01456724]( https://doi.org/10.1007/BF01456724)$]$

category: people


Ludwig Boltzmann

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

## Related entries

* [[statistical mechanics]], [[thermodynamics]]

* [[entropy]]

* [[Boltzmann constant]]

* [[Boltzmann distribution]]

category: people

[[!redirects Boltzmann]]


Ludwig Faddeev

[[!redirects Ludvig Faddeev]]

**Ludwig Dmitrievich Faddeev** (Russian: Russian: Лю́двиг Дми́триевич Фадде́ев, translit. Ljudvig Dmitrievič Faddeev) was a Russian theoretical physicist, academician (and many years secretary of the Russian Academy of Sciences), son of the algebraist [[Dmitri Faddeev]]. With Popov he invented a remarkable method in [[quantization]] of [[gauge theories]], i.e. in the evaluation of [[path integrals]] with [[gauge field]] degrees of freedom (Faddeev-Popov method, see at _[[BRST complex]]_). With B. S. Pavlov, Fadeev has applied the [[abstract scattering theory]] to the study of [[automorphic form|automorphic function]]s. Faddeev is one of the pioneers of the Leningrad (St. Petersburg) school of [[integrable systems]] and [[quantum inverse scattering method]] which also lead to the discovery of [[quantum groups]].

* wikipedia in [English](http://en.wikipedia.org/wiki/Ludvig_Faddeev), [Russian](http://ru.wikipedia.org/wiki/%D0%A4%D0%B0%D0%B4%D0%B4%D0%B5%D0%B5%D0%B2,_%D0%9B%D1%8E%D0%B4%D0%B2%D0%B8%D0%B3_%D0%94%D0%BC%D0%B8%D1%82%D1%80%D0%B8%D0%B5%D0%B2%D0%B8%D1%87)

## Writings

On [[mathematical physics]]:

* _Modern Mathematical Physics: what it should be?_ in Fokos, Grigoryan, Kibble, Zegarlinski (eds) _Mathematical Physics 2000_, World Scientific (2000) ([arXiv:math-ph/0002018](https://arxiv.org/abs/math-ph/0002018))

On the [[mass gap problem]]:

* [[Ludvig Faddeev]], _Mass in Quantum Yang-Mills Theory_, Bull Braz Math Soc, New Series **33** 2 (2002) 201-212 [[arXiv:0911.1013](https://arxiv.org/abs/0911.1013)]

On [[solitons]]:

* [[Ludwig D. Faddeev]], [[Leon Takhtajan]], *Hamiltonian methods in the theory of solitons*, Springer (2007) [[doi:10.1007/978-3-540-69969-9](https://doi.org/10.1007/978-3-540-69969-9)]

## Related entries

* [[Fadeev-Popov ghost]]

* [[infrared divergencies]]

category: people
[[!redirects Ludwig Fadeev]]
[[!redirects Ludwig D. Fadeev]]
[[!redirects Ludwig D. Faddeev]]
[[!redirects L. D. Fadeev]]
[[!redirects Ludwig Faddeev]]


Ludwig Maurer

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

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=52321)

## Selected writings

Introducing the [[Maurer-Cartan equation]]:

* [[Ludwig Maurer]], _Über allgemeinere Invarianten-Systeme_, Münch. Ber. __18__ (1888), 103-150.

category: people


Ludwig Wittgenstein

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

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

## Selected writings

* [[Tractatus Logico-Philosophicus]]
* Philosophical investigations, see [[language game]]
* [[Blue and Brown Books]]

category: people

[[!redirects Wittgenstein]]


Luigi Accardi

* [institute page](https://uniroma2.academia.edu/LuigiAccardi/CurriculumVitae)

* [GoogleScholar page](https://scholar.google.com/citations?user=kHBI7P-eBOIC&hl=th)

## Selected writings

On aspects of [[quantum probability]] (such as the [[EPR paradox]], and [[Bell's inequality]]):

* [[Luigi Accardi]], *The Probabilistic Roots of the Quantum Mechanical Paradoxes*, in: *The Wave-Particle Dualism*,  Fundamental Theories of Physics **3** Springer (1984) [[doi:10.1007/978-94-009-6286-6_16](https://doi.org/10.1007/978-94-009-6286-6_16)]

category: people


Luigi Alfonsi

* [personal page](https://www.alfonsiluigi.com/)

#contents#
* table of contents
{:toc}

## Position

Lecturer at the [Department of Physics, Astronomy and Mathematics](https://www.herts.ac.uk/study/schools-of-study/physics-engineering-and-computer-science/physics-astronomy-and-mathematics) at the [University of Hertfordshire](https://www.herts.ac.uk/) and researcher in Mathematical Physics.

## Research

My main research interest lies in the global geometric structures underlying the [[dualities]] of [[string theory]] and [[M-theory]], to join the dots between [[double field theory]], [[gerbes]], [[generalized geometry]], [[non-geometry]] and [[T-folds]].

## Links

* [Academic webpage](https://go.herts.ac.uk/luigi_alfonsi)

* [InSpire-Hep page](https://inspirehep.net/authors/1776881)

* [ResearchGate page](https://www.researchgate.net/profile/Luigi_Alfonsi)

* email: [l.alfonsi@herts.ac.uk](mailto:l.alfonsi@herts.ac.uk)

## Short bio

PhD in Theoretical Physics at the Centre for Research in String Theory (currently known as [Centre for Theoretical Physics](https://www.qmul.ac.uk/spcs/ctp/)) at [Queen Mary University of London](https://www.qmul.ac.uk/), under the supervision of Prof. [[David Berman]].

MSc in [Quantum Fields and Fundamental Forces](https://www.imperial.ac.uk/study/pg/physics/quantum-fields-fundamental-forces/) at [Imperial College London](https://www.imperial.ac.uk/) under the supervision of Prof. [[Chris Hull]].

## Selected writings
 {#SelectedWritings}

On 2-dimensional [[Yang-Mills theory]]

* _Review of non-abelian localization for $2$d Yang-Mills Theory_, 2017 ([pdf](https://drive.google.com/file/d/1gFPv0tg7mKXQKmhB0dK-Hj3PWdTcM05K/view))

On global [[double field theory]] formulated in [[higher differential geometry]]:

* {#Alfonsi19} [[Luigi Alfonsi]], _Global Double Field Theory is Higher Kaluza-Klein Theory_, Fortschr. Phys. 2020, 68, 2000010. ([arXiv:1912.07089](https://arxiv.org/abs/1912.07089), [doi:10.1002/prop.202000010](https://doi.org/10.1002/prop.202000010))

  (relating [[Kaluza-Klein compactification]] on [[principal ∞-bundles]] to [[double field theory]], [[T-folds]], [[non-abelian T-duality]], [[type II geometry]], [[exceptional geometry]], ...)

* [[Luigi Alfonsi]], _The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective_, Fortschr. Phys. 2021, 2000102 ([arXiv:2007.04969](https://arxiv.org/abs/2007.04969), [doi:10.1002/prop.202000102](https://doi.org/10.1002/prop.202000102)))

Review:

* [[Luigi Alfonsi]], _Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory_ ([arXiv:2102.10970](https://arxiv.org/abs/2102.10970))

On the [[geometric quantization]] of a [[sigma-model]] in [[double field theory]]:

* [[Luigi Alfonsi]] and [[David Berman]], _Double Field Theory and Geometric Quantisation_ ([arXiv:2101.12155](https://arxiv.org/abs/2101.12155))

On other global geometric aspects of theoretical physics (e.g. [[classical double copy]]):

* {#AWW20} [[Luigi Alfonsi]], Chris D. White and Sam Wikeley, _Topology and Wilson lines: global aspects of the double copy_ ([arXiv:2004.07181](https://arxiv.org/abs/2004.07181))

PhD thesis:

* [[Luigi Alfonsi]], _Extended Field Theories as higher Kaluza-Klein theories_ ([arXiv:2108.10297](https://arxiv.org/abs/2108.10297))

On [[higher Kac-Moody algebras]]:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex*, Journal of Geometry and Physics **191** (2023) 104903 [[doi:10.1016/j.geomphys.2023.104903](https://doi.org/10.1016/j.geomphys.2023.104903), [math.QA/2208.06009](https://arxiv.org/abs/2208.06009)]

On [[derived differential geometry]] and [[higher Lie theory]] cast in ([[differential cohesion|differential]]) [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] theory:

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Towards non-perturbative BV-theory via derived differential cohesive geometry* [[arXiv:2307.15106](https://arxiv.org/abs/2307.15106)]

On [[higher geometry]] ([[higher differential geometry]]) in physics:

* [[Luigi Alfonsi]], *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

## Invited talks

On [[non-perturbative quantum field theory|non-perturbative]] [[BV-BRST formalism]] with an eye towards [[M-theory]]:

* _Classical non-perturbative BV-theory via derived differential geometry and the puzzle of quantisation_, seminar talk at Quantum Maths Seminar series at the University of Nottingham, Nottingham, 8th November 2023

* [[Luigi Alfonsi]], *Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Alfonsi2023)]

* _Classical BV-BFV theory as derived $n$-plectic geometry_, seminar talk at [Monthly Global Diffeology Seminar series](https://diffeology.net/index.php/seminar), 2nd December 2022

On [[higher structures]] in [[string dualities]]:

* *Higher quantum geometry and global string duality*, talk at NYU Abu Dhabi (April 2022) $[$[web](Center+for+Quantum+and+Topological+Systems#AlfonsiFeb2022)$]$

* _Extended geometry from higher geometry_, gong show at [Higher Structures in QFT and String Theory](https://sites.google.com/view/jrqftconf2021), 13th July 2021

* _Towards an extended/higher correspondence -- generalised geometry, bundle gerbes and global T-duality_, seminar at [University of Hertfordshire](https://www.herts.ac.uk/research/groups-and-units/mathematics-and-mathematical-physics), 12th May 2021

* _Global Double Field Theory and tensor hierarchy_, seminar at [Prague Mathematical Physics Seminar Series](https://www.youtube.com/channel/UCIYkS8Ei3JBIpG1az60MceQ), 1st April 2021 ([recording](https://www.youtube.com/watch?v=c0M3VAXWdTQ&t=405s&ab_channel=PragueMathematicalPhysicsSeminar))

* _Higher geometric aspects of global Double Field Theory_, seminar at [Exceptional Geometry Seminar Series](https://sites.google.com/view/egseminars), 28th August 2020 ([recording](https://drive.google.com/file/d/1Hzn_sN0GewfYaP1ZGrIw8jUunDnhud-8/view?usp=sharing), [slides](https://drive.google.com/file/d/1xa_aq8K8aBR1mxwnKFIZjtST2QBGpnSO/view?usp=sharing))

* _Global Double Field Theory is Higher Kaluza-Klein Theory_, regular seminar at Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, 16th March 2020.

* _Global Double Field Theory as gerbe geometry_, talk at [Generalized Geometry and Applications](https://www.math.uni-hamburg.de/projekte/gg2020/), University of Hamburg, 3rd March 2020.

## Related entries

* [[BV-BRST formalism]]

* [[T-duality]], [[double field theory]]

* [[non-geometry]]
  * [[T-folds]]
  * [[exotic branes]]

* [[generalized complex geometry]], [[exceptional geometry]], [[type II geometry]]

* [[Kaluza-Klein compactification]]
  * [[Kaluza-Klein monopole]]

* [[non-perturbative quantum field theory]]

category: people


Luigi Amico

Luigi Amico is Executive Director of the [Quantum Physics group](https://www.tii.ae/quantum/our-research/quantum-physics) at the [Technology Innovation Institute](https://www.tii.ae/) (TII) in Abu Dhabi.

* [webpage](https://www.tii.ae/team/prof-dr-luigi-amico)

## Selected writings

On [[atomtronics]]:

* [[Luigi Amico]], [[Malcolm G. Boshier]], *Atomtronics* [[arXiv:1511.07215](https://arxiv.org/abs/1511.07215)]

* [[Luigi Amico]], [[Malcolm G. Boshier]], [[Juan Polo Gomez]] et al.: *Roadmap on Atomtronics: State of the art and perspective*, AVS Quantum Sci. **3** 039201 (2021) [[arXiv:2008.04439](https://arxiv.org/abs/2008.04439), [doi:10.1116/5.0026178](https://doi.org/10.1116/5.0026178)]

* [[Luigi Amico]], Dana Anderson, [[Malcolm Boshier]], Jean-Philippe Brantut, Leong-Chuan Kwek, Anna Minguzzi, Wolf von Klitzing: *Colloquium: Atomtronic circuits: From many-body physics to quantum technologies*, Rev. Mod. Phys. **94** (2022) 041001 [[doi:10.1103/RevModPhys.94.041001](https://doi.org/10.1103/RevModPhys.94.041001)]

(...)

category: people


Luigi Bianchi

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

## Selected writings

On [[quadratic forms]] and apparently introducing the terminology *[[Hermitian form]]* (in §24):

* [[Luigi Bianchi]], *Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî*, Mathematische Annalen **40** (1892) 332–412 [[doi:10.1007/BF01443558](https://doi.org/10.1007/BF01443558)]

## Related entries

* [[Bianchi identity]]

category: people


Luis Alday

* [webpage](https://www.maths.ox.ac.uk/people/luis.alday)

## Selected writings

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

* {#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 [[Liouville theory]] and 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))

Discussion of [[superconformal bootstrap]] in view of [[AdS/CFT]], hence as a precise and detailed construction of [[large-N limit|large-N]] and [[asymptotic boundary|asymptotically]] [[anti-de Sitter spacetime|AdS]] [[string theory]]/[[M-theory]]:

On [[KK-compactification|extra dimensions]]:

* [[Luis Alday]], [[Eric Perlmutter]], _Growing Extra Dimensions in AdS/CFT_ ([arXiv1906.01477](https://arxiv.org/abs/1906.01477))

On [[string scattering amplitudes]]:

* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))

On the [[D=6 N=(2,0) SCFT]]:

* [[Luis Alday]], [[Shai Chester]], Himanshu Raj, _6d $(2,0)$ and M-theory at 1-loop_ ([arXiv:2005.07175](https://arxiv.org/abs/2005.07175))

On the [[conformal bootstrap]]:

* [[Luis F. Alday]], [[Shai M. Chester]], *Pure anti-de Sitter supergravity and the conformal bootstrap* [[arXiv:2207.05085](https://arxiv.org/abs/2207.05085)]

[[!redirects Luis F. Alday]]

category: people


Luis Alvarez-Gaumé

* [wikiepdia entry](http://de.wikipedia.org/wiki/Luis_%C3%81lvarez-Gaum%C3%A9)

## Selected writings

On [[quantum anomalies]]:

* [[Luis Alvarez-Gaumé]], [[Edward Witten]], _Gravitational Anomalies_ Nucl. Phys. B234 (1984) 269 (doi:10.1016/0550-3213(84)90066-X, [spire:192309](http://inspirehep.net/record/192309))

* [[Luis Alvarez-Gaumé]], [[Paul Ginsparg]], _The structure of gauge and gravitational anomalies_, Ann. Phys. 161 (1985) 423. (doi:10.1016/0003-4916(85)90087-9, [spire:202565](http://inspirehep.net/record/202565))

and specifically on the [[Green-Schwarz mechanism]]:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Anomalies and the Green-Schwarz Mechanism*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.06467](https://arxiv.org/abs/2211.06467)]

On the [Chern-Simons level renormalization](Chern-Simons+level#LevelRenormalization) (shift by the [[dual Coxeter number]]):

* [[Luis Alvarez-Gaumé]], J. M. F. Labastida, A. V. Ramallo, *A note on perturbative Chern-Simons theory*, Nuclear Physics B **334** 1 (1990) 103-124 (doi:10.1016/0550-3213(90)90658-Z)

On [[duality in physics]] and specifically [[duality in string theory]]:

* {#AlvarezGaume98} [[Luis Alvarez-Gaumé]], Frederic Zamora, Section IV.B of: _Duality in Quantum Field Theory (and String Theory)_, AIP Conference Proceedings 423, 46 (1998) ([arXiv:hep-th/9709180](https://arxiv.org/abs/hep-th/9709180))

On ([[effective field theory]]) [[quantum field theory]] and the [[standard model of particle physics]]:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Lectures on Field Theory and the Standard Model: A Symmetry-Oriented Approach* [[arXiv:2306.08097](https://arxiv.org/abs/2306.08097)]

category: people

[[!redirects Luis Alvarez-Gaume]]


Luis Español

* [institute page](https://investigacion.unirioja.es/investigadores/60/detalle)

## Selected writings

On [[quasi-toposes]]:

* {#DubucEspanol06} [[Eduardo Dubuc]], [[Luis Español]], _Quasitopoi over a base category_  ([arXiv:math.CT/0612727](http://arxiv.org/abs/math.CT/0612727))

category: people

[[!redirects Luis Espanol]]


Luis Ibáñez

* [webpage](http://www.uam.es/ss/Satellite/FisicaTeorica/en/1234888609654/1242650847708/persona/detallePDI/Ibanez_Santiago,_Luis_Enrique.htm)

* [German Wikipedia page](https://de.wikipedia.org/wiki/Luis_E._Ibáñez)

category: people

## Selected writings

On [[intersecting D-brane models]] (on [[orbifolds]]/[[orientifolds]] with [[RR-field tadpole cancellation]]):

* {#AFIRU01a} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _D=4 Chiral String Compactifications from Intersecting Branes_, J. Math. Phys. 42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _Intersecting Brane Worlds_, JHEP 0102:047, 2001 ([arXiv:hep-ph/0011132](https://arxiv.org/abs/hep-ph/0011132))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* {#IbanezMarchesanoRabadan01} [[Luis Ibáñez]], [[Fernando Marchesano]], R. Rabadan, section 7 of _Getting just the Standard Model at Intersecting Branes_, JHEP 0111:002, 2001 ([arXiv:hep-th/0105155](https://arxiv.org/abs/hep-th/0105155))

On [[bottom-up model building]] for [[intersecting D-brane models]]:

* {#AldazabalIbanezQuevedoUranga00} G. Aldazabal, [[Luis Ibáñez]], [[Fernando Quevedo]], [[Angel Uranga]], _D-Branes at Singularities: A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067v2](https://arxiv.org/abs/hep-th/0005067))

On the [[Higgs mechanism]] in [[intersecting D-brane models]]:

* {#CremadesIbanezMarchesano02} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Intersecting brane models of particle physics and the Higgs mechanism_, JHEP, 0207, 022 2002 ([arXiv:hep-th/0203160](https://arxiv.org/abs/hep-th/0203160))

On [[Yukawa couplings]]:

* {#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))

## Related $n$Lab entries

* [[string phenomenology]]

* [[intersecting D-brane model]]

* [[Higgs mechanism]]

[[!redirects Luis Ibanez]]


Luis Javier Hernandez > history

[[!redirects Luis Javier Hernandez>History]]
[[!redirects Luis Javier Hernandez]]
[[!redirects Luis Javier Hernáandez]]
[[!redirects Luis Javier Hernáandez]]
Luis Javier Hernández is a Spanish topologist, who has worked on [[proper homotopy theory]], [[shape theory
]] and on adapting ideas from those areas to give new insights into [[dynamical systems]].

He is based at the University of La Rioja, Spain.

* [Home Page](http://www.unirioja.es/cu/luhernan/)

category: people


Luis Javier Hernández Paricio

[[!redirects Luis Javier Hernández]]
[[!redirects Luis Javier Hernandez]]
[[!redirects Luis Javier Hernandez]]
Luis Javier Hernández Paricio is a Spanish topologist, who has worked on [[proper homotopy theory]], [[shape theory
]] and on adapting ideas from those areas to give new insights into [[dynamical systems]].

He is based at the University of La Rioja, Spain.

* [Home Page](http://www.unirioja.es/cu/luhernan/)

##Publications and preprints

include

* L.J. Hernández and [[T. Porter]], _Proper pointed maps from $\mathbb{R}$ to a $\sigma$-compact space_, Math. Proc. Camb. Phil. Soc. 103 (1988) 457-462.

* L.J. Hernández and [[T. Porter]], _Global analogues of the Brown-Grossman proper homotopy groups of an end_, Math. Proc. Camb. Phil. Soc. 104 (1988) 483-496.

* L.J. Hernández and [[T. Porter]], _Categorical Models of N-types for pro-crossed complexes and In-prospaces_, in 1990 Barcelona Conference on Algebraic Topology, Springer Lecture Notes in Mathematics, No.1509, 1992.

* L.J. Hernández and [[T. Porter]], _An embedding theorem for proper n-types_, Top. and its Applications, 48, (1992) 215-233

*  J. Ignacio Extremiana Aldana, L. Javier Hernández, [[Teresa Rivas|M. Teresa Rivas Rodríguez]], _Postnikov factorizations at infinity_,Topology and its Applications, 153,(2005), 370-393.

*  J. García Calcines, L. J. Hernández, and [[Teresa Rivas|M.T. Rivas]],  _Limit and end functors of dynamical systems via exterior spaces, _ [Preprint (2011)](http://www.unirioja.es/cu/luhernan/puben.html).

* J. I. Extremiana Aldana, [[Luis Javier Hernández Paricio]] and Maria Teresa Rivas Rodriguez, _A closed model category for $(n-1)$-connected spaces_ , Proc. AMS, 124, Number 11,  1996 ([pdf](http://www.ams.org/proc/1996-124-11/S0002-9939-96-03606-4/S0002-9939-96-03606-4.pdf))

category: people

[[!redirects Luis Javier Hernández-Paricio]]


Luis Javier Hernández-Paricio > history

see at *[[Luis Javier Hernández Paricio]]*


Luis Scoccola

* [website](http://luisscoccola.github.io/)

## Selected writings

On [[persistent homotopy|persistent homotopy theory]]:

* {#LanariScoccola22} Edoardo Lanari, [[Luis Scoccola]], _Rectification of interleavings and a persistent Whitehead theorem_, Algebraic & Geometric Topology (to appear), $[$[arXiv:2010.05378] (https://arxiv.org/abs/2010.05378)$]$

On [[localization of a space|localization]] (in [[algebraic topology]]) formulated in [[modal homotopy type theory|modal]] [[homotopy type theory]]:

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

Proof of the [[Hurewicz theorem]] in [[homotopy type theory]], hence in general [[(∞,1)-toposes]]:

* [[Daniel Christensen]], [[Luis Scoccola]], _The Hurewicz theorem in Homotopy Type Theory_ ([arXiv:2007.05833](https://arxiv.org/abs/2007.05833))

## Relevant entries

* [[mathematics presented in homotopy type theory]]

category: people


Lukas Brantner

* [personal page](https://people.maths.ox.ac.uk/brantner/)

## Selected writings

On the [[Morava E-theory]] of [[configuration spaces of points]]:

* [[Lukas Brantner]], [[Jeremy Hahn]], [[Ben Knudsen]], *The Lubin-Tate Theory of Configuration Spaces: I* ([arXiv:1908.11321](https://arxiv.org/abs/1908.11321))

On [[higher algebra|higher algebraic]] [[Lie algebras]] (spectral Lie algebras) in [[algebraic topology]]:

* [[Lukas Brantner]], *Generalised Lie algebras in Algebra and Topology* ([pdf](https://people.maths.ox.ac.uk/brantner/oberwolfachreport_brantner2.pdf))

category: people


Lukas Buhné

* [webpage](http://grk1670.math.uni-hamburg.de/researchers/buhne/)

## publications

thesis: _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_

category: people

[[!redirects Lukas Buhne]]


Lukas Heidemann

* [institute page Cambridge](https://www.cst.cam.ac.uk/people/lh765)

* [institute page Oxford](https://www.cs.ox.ac.uk/people/lukas.heidemann/)

## Selected writings

On [[associative n-categories]] and their [[formal proof|formalization]] in [[proof assistants]] (cf.: [[Globular]], [[homotopy.io]]):

* [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

category: people


Lukas Müller

* [personal page](https://sites.google.com/view/lukas-mueller/home)

## Selected writings

On [[higher parallel transport]] of [[circle n-bundles with connection]] as an [[extended functorial field theory|extended]] [[homotopy field theory]]:

* [[Lukas Müller]], [[Lukas Woike]], *Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory*,  J. Homotopy Relat. Struct. **15** (2020) 113–142  [[arXiv:1802.10455](https://arxiv.org/abs/1802.10455), [doi:10.1007/s40062-019-00242-3](https://doi.org/10.1007/s40062-019-00242-3)]

On [[QFT]] [[quantum anomalies]] as invertibled [[extended TQFTs]]:

* [[Lukas Müller]], _Extended Functorial Field Theories and Anomalies in Quantum Field Theories_ [[arXiv:2003.08217](https://arxiv.org/abs/2003.08217)]

On [[factorization homology]] for [[surfaces]] equipped with [[flat connections]] for a [[finite group]]:

* [[Corina Keller]], [[Lukas Müller]], *Finite symmetries of quantum character stacks*, Theory and Applications of Categories **39** 3 (2023) 51-97  [[arXiv:2107.12348](https://arxiv.org/abs/2107.12348), [tac:39-03](http://www.tac.mta.ca/tac/volumes/39/3/39-03abs.html)]

Exposition:

* [[Lukas Müller]], *Deformation quantization and categorical factorization homology*, talk at [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#MuellerMar2023), video:[YT](https://www.youtube.com/watch?v=kC488wISPX4)]

category: people

[[!redirects Lukas Mueller]]
[[!redirects Lukas Muller]]


Lukas Woike

* [webpage](http://grk1670.math.uni-hamburg.de/researchers/woike/)

## Selected writings

On [[extended functorial field theory|extended]] [[homotopy field theory]]:

* [[Christoph Schweigert]], [[Lukas Woike]], *Extended Homotopy Quantum Field Theories and their Orbifoldization*, Journal of Pure and Applied Algebra **224** 4 (2020) 106213 [[arXiv:1802.08512](https://arxiv.org/abs/1802.08512), doi:10.1016/j.jpaa.2019.106213]

On [[higher parallel transport]] of [[circle n-bundles with connection]] as an [[extended functorial field theory|extended]] [[homotopy field theory]]:

* [[Lukas Müller]], [[Lukas Woike]], *Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory*,  J. Homotopy Relat. Struct. **15** (2020) 113–142  [[arXiv:1802.10455](https://arxiv.org/abs/1802.10455), [doi:10.1007/s40062-019-00242-3](https://doi.org/10.1007/s40062-019-00242-3)]

## Related entries

* [[homotopical algebraic quantum field theory]]

category: people


Lukasz Fidkowski

* [Wikipedie entry (de)](https://de.wikipedia.org/wiki/Lukasz_Fidkowski)

* [institute page](https://phys.washington.edu/people/lukasz-fidkowski)

* [GoogleScholar page](https://scholar.google.com/citations?user=f-Wpl8sAAAAJ&hl=en)

## Selected writings

On the [[Kitaev spin chain]] with [[interactions]] and implications for the classification of [[topological phases of matter]]:

* [[Lukasz Fidkowski]], [[Alexei Kitaev]], *The effects of interactions on the topological classification of free fermion systems*, Phys. Rev. B **81** (2010) 134509  [[arXiv:0904.2197](https://arxiv.org/abs/0904.2197), [doi:10.1103/PhysRevB.81.134509](https://doi.org/10.1103/PhysRevB.81.134509)]

category: people


Lukáš Vokřínek

* [webpage](https://www.muni.cz/en/people/43588-lukas-vokrinek)

## Selected writings

Discussion of ([[equivariant homotopy theory|equivariant]]) [[homotopies]] and [[homotopy groups]] in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension_, SIAM J. Comput., 43(5), 1728–1780  ([arXiv:1211.3093](https://arxiv.org/abs/1211.3093), [doi:10.1137/120899029](https://doi.org/10.1137/120899029))

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Extendability of continuous maps is undecidable_, Discrete Comput Geom (2014) 51: 24 ([arXiv:1302.2370](https://arxiv.org/abs/1302.2370), [doi:10.1007/s00454-013-9551-8](https://doi.org/10.1007/s00454-013-9551-8))

Discussion of [[Cohomotopy]]-sets in [[computational topology]]:

* [[Martin Čadek]], [[Marek Krčál]], [[Jiří Matoušek]], [[Francis Sergeraert]], [[Lukáš Vokřínek]], [[Uli Wagner]], _Computing all maps into a sphere_, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 ([arxiv:1105.6257](https://arxiv.org/abs/1105.6257))

* [[Lukáš Vokřínek]], _Decidability of the extension problem for maps into odd-dimensional spheres_,  Discrete Comput Geom (2017) 57: 1 ([arXiv:1401.3758](https://arxiv.org/abs/1401.3758))

Discussion of [[equivariant homotopy groups]] in [[computational topology]]:

* [[Lukáš Vokřínek]], _Computing the abelian heap of unpointed stable homotopy classes of maps_, Archivum Mathematicum, Volume 49 (2013), No. 5 · ([arXiv:1312.2474](https://arxiv.org/abs/1312.2474), [doi:10.5817/AM2013-5-359](http://dx.doi.org/10.5817/AM2013-5-359))

category: people

[[!redirects Lukas Vokrinek]]


Lurie spectral sequences -- table

| [[tower diagram]]/[[filtered object in an (infinity,1)-category|filtering]] | [[spectral sequence of a filtered stable homotopy type]] |
|--|--|
| [[filtered chain complex]] | [[spectral sequence of a filtered complex]] |
| [[Postnikov tower]] | [[Atiyah-Hirzebruch spectral sequence]] |
| [[chromatic tower]] | [[chromatic spectral sequence]] |
| [[simplicial skeleton|skeleta]] of [[simplicial object in an (∞,1)-category|simplicial object]] | [[spectral sequence of a simplicial stable homotopy type]] |
| [[simplicial skeleton|skeleta]] of [[Sweedler coring]] of [[E-∞ algebra]] | [[Adams spectral sequence]] |
| [[filtration by support]] | ... |
| [[slice filtration]] | [[slice spectral sequence]] |


Lutian Zhao

* [webpage](http://www.math.uiuc.edu/~lzhao35/index.html)

category: people


Luuk Stehouwer

* [Personal webpage](https://sites.google.com/view/luuk-stehouwer)

## Selected writings

On the [[K-theory classification of topological phases of matter]] for the case of [[topological crystalline insulators]] classified by [[equivariant K-theory]]:

* [[Luuk Stehouwer]], [[Jan de Boer]], [[Jorrit Kruthoff]], [[Hessel Posthuma]], *Classification of crystalline topological insulators through K-theory*, Adv. Theor. Math. Phys, **25** 3 (2021) 723–775 [[arXiv:1811.02592](https://arxiv.org/abs/1811.02592), [doi:10.4310/ATMP.2021.v25.n3.a3](https://doi.org/10.4310/ATMP.2021.v25.n3.a3)]

On [[hermitian functorial field theory]] and the [[spin-statistics theorem]]:

* [[Lukas Müller]], [[Luuk Stehouwer]], _Reflection Structures and Spin Statistics in Low Dimensions_ [[arXiv:2301.06664](https://arxiv.org/abs/2301.06664)]

On [[dagger categories]]:

* [[Luuk Stehouwer]], [[Jan Steinebrunner]], *Dagger categories via anti-involutions and positivity* [[arXiv:2304.02928](https://arxiv.org/abs/2304.02928)]

category: people


Lyle Eugene Pursell

Lyle Eugene Pursell was a mathematician at the University of Missouri at Rolla (currently known as Missouri University of Science and Technology).

He got his PhD in 1952 from Purdue University, advised by Merrill E. Shanks.

## Selected works

On the [[embedding of smooth manifolds into formal duals of R-algebras]]:

* [[Lyle Eugene Pursell]], _Algebraic structures associated with smooth manifolds_, PhD dissertation, Purdue University, 1952.  93 pp.  ISBN: 978-1392-88143-9.  [PDF](https://www.proquest.com/docview/2327629257).

category: people

[[!redirects Lyle Pursell]]
[[!redirects Lyle E. Pursell]]
[[!redirects L. Pursell]]
[[!redirects L. E. Pursell]]


Lyne Moser

* [website](https://lynemoser.com/)

### Selected writings

On [[model structures on DblCat]]:

* {#MSV2Cat} [[Lyne Moser]], Maru Sarazola, Paula Verdugo, *A $2Cat$-inspired model structure for double categories*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **LXIII**2 (2022) [[arxiv:2004.14233](https://arxiv.org/abs/2004.14233), journal:[pdf](http://cahierstgdc.com/wp-content/uploads/2022/04/MoserSarazolaVerdugo-LXIII-2.pdf)]

* {#MSVHInvar} [[Lyne Moser]], Maru Sarazola, Paula Verdugo,  *A model structure for weakly horizontally invariant double categories*, Algebraic & Geometric Topology **23** 4 (2023) 1725-1786 [[arxiv:2007.00588](https://arxiv.org/abs/2007.00588), journal:[pdf](https://msp.org/agt/2023/23-4/agt-v23-n4-p08-p.pdf)]

Review:

* [[Lyne Moser]], *Model Structures for Double Categories*, [talk at](CQTS#MoserOct2023) *[Workshop on Homotopy Theory and Applications](CQTS#WorkshopOnHomotopyTheory2023)*, [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/8j1WvFesnQM)]

Discussion of [[straightening and unstraightening]] for [[(∞,n)-categories]]:

* [[Lyne Moser]], [[Nima Rasekh]], [[Martina Rovelli]], _An (∞,n)-categorical straightening-unstraightening construction_ ([arXiv:2307.07259](https://arxiv.org/abs/2307.07259)).


Lynn Arthur Steen

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

## Selected writings

On [[topology]]:

* [[Lynn Arthur Steen]], [[J. Arthur Seebach]], _[[Counterexamples in Topology]]_, Springer 1970/1978 ([doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9))

[[!redirects Lynn Steen]]

category: people


László Babai

* [Wikipedia entry](https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Babai)

## Selected writings

On [[Cayley graph]] spectra:

* [[László Babai]], *Spectra of Cayley graphs*, Journal of Combinatorial Theory, Series B Volume 27, Issue 2, October 1979, Pages 180-189 (doi:10.1016/0095-8956(79)90079-0)

category: people

[[!redirects Laszlo Babai]]


László Fuchs

* [Wikipedia entry](https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Fuchs)

## Selected writings

On [[abelian groups]]:

* [[László Fuchs]], *Abelian Groups*, Springer (2015) [[doi:10.1007/978-3-319-19422-6](https://doi.org/10.1007/978-3-319-19422-6)]

[[!redirects Laszlo Fuchs]]


László Lovász

* [Wikipedia entry](https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz)

## Selected writings

On spectra of [[Cayley graphs]]:

* [[László Lovász]], *Spectra of graphs with transitive groups*, Period. Math. Hungar., 6(2):191–195,1975 ([doi:10.1007/BF02018821](https://doi.org/10.1007/BF02018821), [pdf](https://web.cs.elte.hu/~lovasz/scans/spectra-of-graphs.pdf))

category: people

[[!redirects Laszlo Lovasz]]


Léon Brillouin

* [Wikipedia entry](https://en.wikipedia.org/wiki/L%C3%A9on_Brillouin)

## Selected writings

On [[Bragg diffraction]] in [[crystals]] and introducing the notion of *[[Brillouin zone]]*:

* [[Léon Brillouin]], *Les électrons libres dans les métaux et le role des réflexions de Bragg*, J. Phys. Radium **1** (1930) 377-400 [[doi:10.1051/jphysrad:01930001011037700](https://doi.org/10.1051/jphysrad:01930001011037700), [hal:jpa-00233038](https://hal.archives-ouvertes.fr/jpa-00233038), [pdf](https://hal.archives-ouvertes.fr/jpa-00233038/document)]

category: people

[[!redirects Leon Brillouin]]


Léon Gondelman

* [personal page](https://cs.au.dk/~gondelman/)

## Selected writings

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

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

[[!redirects Leon Gondelman]]


Lévy hierarchy

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

# The Lévy hierarchy
* table of contents
{: toc}

## Idea

In [[logic]], [[model theory]], and [[set theory]], the **Lévy hierarchy** is a [[stratification]] of [[formulas]], [[definable sets]], and (definable) [[classes]] according to the complexity of the unbounded [[quantifiers]].

## Definition

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

We define classes of formulas $\Sigma_n$, $\Pi_n$, and $\Delta_n$ by [[induction]] on $n$.

* A formula is $\Sigma_0$ iff it is $\Pi_0$ iff it is $\Delta_0$, by definition if it is equivalent to a formula all of whose [[quantifiers]] are [[bounded quantifier|bounded]], i.e. of the form $\forall x\in A$ or $\exists x\in A$.

* A formula is $\Sigma_{n+1}$ if it is equivalent to one of the form $\exists \vec{x}. \phi$, where $\vec{x}$ is a list of variables and $\phi$ is $\Pi_n$.

* A formula is $\Pi_{n+1}$ if it is equivalent to one of the form $\forall \vec{x}. \phi$, where $\vec{x}$ is a list of variables and $\phi$ is $\Sigma_n$.

* A formula is $\Delta_n$ if it is both $\Sigma_n$ and $\Pi_n$.

A class is given one of these labels if it can be defined by a formula which has that label.

=--

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

The notation "$\Sigma$" and "$\Pi$" can be explained by the fact that the [[existential quantifier]] is related to a [[dependent sum]], while the [[universal quantifier]] is related to the [[dependent product]].

=--

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

These definitions are most useful in [[classical mathematics]], in which every formula is equivalent to one all of whose unbounded quantifiers are in the front, that is, a formula in [[prenex normal form]], so that every formula belongs to some $\Sigma_n$ or $\Pi_n$.

=--

[[!redirects Levy hierarchy]]
[[!redirects Sigma-n formula]]
[[!redirects Pi-n formula]]
[[!redirects Delta-n formula]]
[[!redirects ∞-n formula]]
[[!redirects ∞-n formula]]
[[!redirects ∞-n formula]]
[[!redirects Sigma-0 formula]]
[[!redirects Pi-0 formula]]
[[!redirects Delta-0 formula]]
[[!redirects ∞-0 formula]]
[[!redirects ∞-0 formula]]
[[!redirects ∞-0 formula]]
[[!redirects Sigma-1 formula]]
[[!redirects Pi-1 formula]]
[[!redirects Delta-1 formula]]
[[!redirects ∞-1 formula]]
[[!redirects ∞-1 formula]]
[[!redirects ∞-1 formula]]


Löb's theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
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#### Type theory
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#Contents#
* table of contents
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## Idea

_Löb's theorem_ in its original form is a generalization of the [[Gödel incompleteness theorem]] whose formulation lends itself to the tools of [[type theory]] and [[modal logic]].

Löb's theorem states that to prove that a [[proposition]] is [[provability logic|provable]], it is sufficient to prove the proposition under the assumption that it  is  provable.  Since the  [[Curry-Howard isomorphism]] identifies [[formal proofs]] with abstract [[syntax]] trees of [[programs]]; Löb's theorem implies, for total [[languages]] which validate it, that self-interpreters are impossible. ([Gross-Gallagher-Fallenstein 16](#GrossGallagherFallenstein16))

In [[provability logic]] the abstract statement is considered in itself as an [[axiom]] on a [[modal operator]] $\Box$ interpreted as the [[modality]] "is provable". In this form the statement reads formally:

$$
  \Box(\Box P \to P) \to \Box P
$$

for any [[proposition]] $P$ ("Löb's axiom").

This reduces to an [[incompleteness theorem]] when taking $P = $ [[false]] and using that

1. [[negation]] is $\not P = (P \to false)$;

1. [[consistency]] means that $\Box P \to \not \Box \not P$

$$
  \begin{aligned}
    & \Box (\Box false \to false) \to \Box false
    \\
    \Rightarrow \;\; & \Box ( \not \Box false ) \to \Box false
    \\
    \Rightarrow \;\;
    & \Box ( \not \Box false ) \to \not \Box \not false
    \\
    \Rightarrow \;\;
    & \Box ( \not \Box false ) \to false
    \\
    \Rightarrow \;\;
    & \not \Box \not \Box false
  \end{aligned}
$$

Where the last line reads in words "It is not provable that false is not provable."

## Guarded Recursion Variant

A variant of the Löb axiom is used in [[guarded recursion]] and [[synthetic guarded domain theory]], which uses a modality $\blacktriangleright$, usually pronounced "later". Then the Löb induction axiom is for any proposition $P$,

$$(\blacktriangleright P \to P) \to P$$

In a setting with the principle of [[unique choice]], this can be used to prove the existence of all guarded [[fixed points]].

Note that the assumptions about the later modality $\blacktriangleright$ are usually quite different from the provability modality $\Box$. For instance, $\Box$ is usually assumed to have some subset of the properties of a [[comonadic]] [[modality]], but $\blacktriangleright$ typically satisfies $P \to \blacktriangleright P$.

## Related concepts

* [[fixed-point combinator]]

* [[Lawvere's fixed point theorem]]

* [[guarded recursion]]

* [[synthetic guarded domain theory]]

## References

* {#GrossGallagherFallenstein16} [[Jason Gross]], Jack Gallagher, Benya Fallenstein, _Löb's theorem -- A functional pearl of dependently typed quining_, 2016 ([pdf](https://jasongross.github.io/lob-paper/nightly/lob.pdf))

* [[Neelakantan Krishnaswami]], _[Löb's theorem is (almost) the Y-combinator](http://semantic-domain.blogspot.de/2016/05/lobs-theorem-is-almost-y-combinator.html)_, 2016

* [[Jason Gross]], [MO comment](https://mathoverflow.net/a/273951/381) on incompletenss theorems in type theory, 2017

See also

* Wikipedia, _[Löb's theorem](https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem)_

Unified account of [[Löb's theorem]]/[[Gödel's second incompleteness theorem]], [[Kripke semantics]] for [[provability logic]], and [[guarded recursion]], via [[essentially algebraic theories]] which "self-internalize":

* [[Sridhar Ramesh]], *Introspective Theories and Geminal Categories*, PhD thesis, Berkeley (2023) [[escholarship:3mn0c475](https://escholarship.org/uc/item/3mn0c475), [[Ramesh-IntrospectiveTheories.pdf:file]]]

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