Christian B. Mendl

* [personal page](http://christian.mendl.net/)

* [institute page](https://www.professoren.tum.de/en/mendl-christian)

## Selected writings

On [[extremal quantum channels]]:

* [[Christian B. Mendl]], [[Michael M. Wolf]], *Unital Quantum Channels -- Convex Structure and Revivals of Birkhoff's Theorem*, Commun. Math. Phys. **289**  (2009) 1057-1096 [[arXiv:0806.2820](http://arxiv.org/abs/0806.2820)]

category: people

[[!redirects Christian Mendl]]


Christian Becker

* [GoogleScholar page](https://scholar.google.com/citations?user=TyHPk10AAAAJ&hl=nl)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=101105)

## Selected writings

On [[Cheeger-Simons differential characters]]:

* [[Christian Bär]], [[Christian Becker]], *Differential Characters and Geometric Chains*, in: *Differential Characters*, Lecture Notes in Mathematics **2112**, Springer (2014)  [[arXiv:1303.6457](https://arxiv.org/abs/1303.6457), [doi:10.1007/978-3-319-07034-6_1](https://doi.org/10.1007/978-3-319-07034-6_1)]

On [[self-dual higher gauge theory]] on [[Lorentzian spacetimes]] via [[ordinary differential cohomology]]:

* [[Christian Becker]], [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], _Abelian duality on globally hyperbolic spacetimes_, Commun. Math. Phys. **349** (2017) 361-392 [[arXiv:1511.00316](https://arxiv.org/abs/1511.00316), [doi:10.1007/s00220-016-2669-9](https://doi.org/10.1007/s00220-016-2669-9)]

category: people


Christian Blanchet

* [personal page](https://webusers.imj-prg.fr/~christian.blanchet/)

## Selected writings

On the [[ordinary homology]] of [[configuration spaces of points]] in a [[surface]] as a [[group representation]] of the [[mapping class group]]:

* [[Christian Blanchet]], [[Martin Palmer]], [[Awais Shaukat]], *Heisenberg homology on surface configurations* [[arXiv:2109.00515](https://arxiv.org/abs/2109.00515)]

* [[Awais Shaukat]], [[Christian Blanchet]], *Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action*, Archiv der Mathematik **120** (2023) 99–109 [[arXiv:2206.11475](https://arxiv.org/abs/2206.11475), [doi:10.1007/s00013-022-01793-3](https://doi.org/10.1007/s00013-022-01793-3)]

* [[Christian Blanchet]], *Heisenberg homologies of surface configurations*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#BlanchetMar2023)]

category: people


Christian Blohmann

* [website](https://people.mpim-bonn.mpg.de/blohmann/)

## Selected writings

On [[Cartan calculus]] for [[diffeological spaces]]:

* [[Christian Blohmann]], _Elastic diffeological spaces_, [arXiv:2301.02583](https://arxiv.org/abs/2301.02583).

category: people


Christian Brouder

* [webpage](http://www-ext.impmc.upmc.fr/~brouder/)

## related $n$Lab entries

* [[multiplication of distributions]]

* [[wave front set]]

* [[Feynman diagram]]

* [[renormalization]]

* [[Feynman diagrams in causal perturbation theory -- summary]]

category: people


Christian Bär

* [website](http://geometrie.math.uni-potsdam.de/index.php/de/mitarbeiter/prof-dr-christian-baer)

## Selected writings

On [[Lorentzian geometry]]:

* [[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]]]

On [[Cheeger-Simons differential characters]]:

* [[Christian Bär]], [[Christian Becker]], *Differential Characters and Geometric Chains*, in: *Differential Characters*, Lecture Notes in Mathematics **2112**, Springer (2014)  [[arXiv:1303.6457](https://arxiv.org/abs/1303.6457), [doi:10.1007/978-3-319-07034-6_1](https://doi.org/10.1007/978-3-319-07034-6_1)]

## Related entries

* [[path integral]]

* [[transgression]]

category: people

[[!redirects Christian Bar]]
[[!redirects Christian Baer]]


Christian Cuvier

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=74515)

## Selected writings

On [[Leibniz algebras]]:

* {#Cuvier94} [[Christian Cuvier]], _Algèbres de Leibnitz: définitions, propriétés_,  Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 27 (1994) no. 1, p. 1-45 ([doi:10.24033/asens.1687](https://doi.org/10.24033/asens.1687))

On [[Leibniz homology]] and [[Hochschild homology]]:

* {#Cuvier91} [[Christian Cuvier]], _Homologie de Leibniz et homologie de Hochschild_, C.R. Acad. Sci. Paris, Ser. A-B313, 569-572 (1991)

category: people


Christian Ferko

* [InSpire page](https://inspirehep.net/authors/1838025)

* [GoogleScholar page](https://scholar.google.com/citations?user=yZSlM1oAAAAJ&hl=en)

## Selected writings

On an analogue of the [[DBI action]] via [[TT deformation|$T \overline{T}$-deformation]] of 2d [[Yang-Mills theory]]:

* {#BrennanFerkoSethi20} [[T. Daniel Brennan]], [[Christian Ferko]], [[Savdeep Sethi]], *A Non-Abelian Analogue of DBI from $T \bar T$*, SciPost Phys. **8** 052 (2020) [[arXiv:1912.12389](https://arxiv.org/abs/1912.12389), [doi:10.21468/SciPostPhys.8.4.052](https://doi.org/10.21468/SciPostPhys.8.4.052)]

category: people


Christian Fronsdal

* [webpage](http://www.pa.ucla.edu/directory/christian-fronsdal)

## Selected writings

Introducing [[formal deformation quantization]]:

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

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

On [[superconformal multiplets]]

* [[Christian Fronsdal]], _Dirac supermultiplet_, Phys. Rev. D **26** (1982) 1988 [[doi:10.1103/PhysRevD.26.1988](https://doi.org/10.1103/PhysRevD.26.1988)]

and the [[singleton representation]]

* [[Moshé Flato]], [[Christian Fronsdal]], _Quantum field theory of singletons. The rac_, J. Math. Phys. **22** (1981) 1100 [[doi:10.1063/1.524993](https://doi.org/10.1063/1.524993)]

## Related entries

* [[singleton representation]]

category: people


Christian Kassel

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

* [[mixed complex]], [[cyclic homology]]

## Selected writings

On [[cyclic homology]]:

* [[Christian Kassel]], _Cyclic homology, comodules and mixed complexes_, J. Alg. 107 (1987), 195–216 (doi:10.1016/0021-8693(87)90086-X)

On [[quantum groups]]:

* [[Christian Kassel]], _Quantum groups_, Graduate Texts in Mathematics __155__, Springer 1995 ([doi:10.1007/978-1-4612-0783-2](https://link.springer.com/book/10.1007/978-1-4612-0783-2), [webpage](http://www-irma.u-strasbg.fr/~kassel/QGbk.html), [errata pdf](http://www-irma.u-strasbg.fr/~kassel/QGerrata030399.pdf))

On [[braid groups]]:

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

category:people

[[!redirects C. Kassel]]


Christian Krattenthaler

* [personal page](https://www.mat.univie.ac.at/~kratt/)

## Selected writings

On the [[hook content formula]]:

* [[Christian Krattenthaler]], *An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula*, Discrete Mathematics and Theoretical Computer Science **3**, 1998, 11–32 ([hal:hal-00958904](https://hal.inria.fr/hal-00958904), [pdf](https://hal.inria.fr/hal-00958904/document))

* [[Christian Krattenthaler]], *Another involution principle-free bijective proof of Stanley's hook-content formula*, J. Combin. Theory Ser. A 88 (1999), 66-92 ([arXiv:math/9807068](https://arxiv.org/abs/math/9807068))

category: people


Christian Lair

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=98596)

## Selected writings

Proof that the [[funny tensor product]] of [[categories]] is the only other [[symmetric closed monoidal category|symmetric closed monoidal structure]] on [[Cat]] besides the [[cartesian monoidal category|cartesian monoidal structure]]:

* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), doi:10.1016/0022-4049(80)90082-1]

Proof that not only every [[sketch|sketchable]] category is [[accessible category|accessible]] but that conversely every accessible category is sketchable:

* [[Christian Lair]], *Catégories modelables et catégories esquissables*, Diagrammes **6** 5 (1981) [[numdam:DIA_1981__6__A5_0](http://www.numdam.org/item/?id=DIA_1981__6__A5_0), [pdf](http://www.numdam.org/article/DIA_1981__6__A5_0.pdf)]

On [[Michael Barr]] & [[Charles Wells]]'s *[[Toposes, Triples, and Theories]]*:

* {#Lair86} [[Christian Lair]], _À propos de «Toposes, triples and theories» de messieurs M. Barr et C. Wells_, Diagrammes **15** 4 (1986) L1-L20 [[numdam:DIA_1986__15__A4_0](http://www.numdam.org/item/DIA_1986__15__A4_0)]

category: people


Christian Saemann

* [webpage](http://christiansaemann.de/)

## Selected writings

On the [[M2-brane 3-algebra]] and the [[BLG model]]:

* {#CherkisSaemann08} [[Sergey Cherkis]], [[Christian Saemann]], _Multiple M2-branes and Generalized 3-Lie algebras_, Phys. Rev. D78:066019, 2008 ([arXiv:0807.0808](https://arxiv.org/abs/0807.0808))

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [doi:10.1007/JHEP07(2020)003, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

On [[schreiber:Higher Structures]] in [[M-theory]]:

* {#Saemann16} [[Christian Saemann]], _Lectures on Higher Structures in M-Theory_ ([arXiv:1609.09815](https://arxiv.org/abs/1609.09815))

* [[Branislav Jurco]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]], _Higher Structures in M-Theory_, Introduction to _[Higher Structures in M-Theory, LMS-EPSRC Durham Symposium, August 2018](http://www.maths.dur.ac.uk/lms/109/index.html)_, Fortsch. d. Phys. 2019 ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807))

Proposal for understanding [[D=6 N=(1,0) SCFTs]] as [[higher gauge theories]] for gauge-[[L-infinity algebras]] which are variants of the [[string Lie 2-algebra]]:

* [[Christian Saemann]], [[Lennart Schmidt]], _The Non-Abelian Self-Dual String and the $(2,0)$-Theory_ ([arXiv:1705.02353](https://arxiv.org/abs/1705.02353))

* [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 ([arxiv:1712.06623](https://arxiv.org/abs/1712.06623))

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebra]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

Discussion of [[adjusted Weil algebras]] for [[L-infinity algebras|$L_\infty$-algebras]] and the corresponding [[higher gauge theory]]:

* {#SaemannSchmidt20} [[Christian Saemann]], [[Lennart Schmidt]], *Towards an M5-Brane Model II: Metric String Structures*, Fortschr. Phys. **68** (2020) 2000051 [[arXiv:1908.08086](https://arxiv.org/abs/1908.08086), [doi:10.1002/prop.202000051](https://doi.org/10.1002/prop.202000051)]

application to [[higher parallel transport]]:

* {#KimSaemann20} [[Hyungrok Kim]], [[Christian Saemann]], *Adjusted Parallel Transport for Higher Gauge Theories*, J. Phys. A **52** (2020) 445206 [[arXiv:1911.06390](https://arxiv.org/abs/1911.06390), [doi:10.1088/1751-8121/ab8ef2](https://doi.org/10.1088/1751-8121/ab8ef2)]

relation to [[EL-infinity algebras|$E L_\infty$-algebras]] and  [[tensor hierarchies]]:

* [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]].  *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]

survey and review:

* [[Christian Saemann]], *Adjusted Higher Gauge Theory: Connections and Parallel Transport* Lisbon (2021) [[pdf](https://math.tecnico.ulisboa.pt/seminars/download.php?fid=1485), [[Saemann-AdjustedHGT.pdf:file]]]

* [[Christian Saemann]], *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies*, talk at [SFT@Cloud 2021](https://indico.cern.ch/event/1042834/) (2021) [[pdf](https://indico.cern.ch/event/1042834/contributions/4487418/attachments/2313514/3937748/saemann_generalized_geometry.pdf), [[Saemann-ELInfinityAndTensorHierarchies.pdf:file]]]

Proposal for further enhancement of the [[higher differential geometry]]-formulation of [[T-folds]]  (such as with [[connection on a 2-bundle|2-connections]] and further [[non-geometric backgrounds]]):

* [[Hyungrok Kim]], [[Christian Saemann]], *Non-Geometric T-Duality as Higher Groupoid Bundles with Connections* [[arXiv:2204.01783](https://arxiv.org/abs/2204.01783)]

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

## Related $n$Lab entries

* [[M2-brane 3-algebra]]

* [[Nambu mechanics]]

* [[membrane matrix model]]

* [[twistor space]]

category: people


Christian Sattler

* [web page](https://www.cse.chalmers.se/~sattler/) (under construction)

* [institute page](https://www.chalmers.se/en/staff/Pages/sattler.aspx)

* [GoogleScholar page](https://scholar.google.dk/citations?user=ZJmvkfUAAAAJ&hl=da)

## Selected writings

On the [[categorical semantics]] of [[W-types]] by [[initial algebras over an endofunctor]] in [[homotopy type theory]]:

* [[Christian Sattler]], _On relating indexed W-types with ordinary ones_, in *[Types for Proofs and Programs -- TYPES 2015](https://cs.ioc.ee/types15/abstracts-book/)* (2015) 71-72 [[pdf](http://cs.ioc.ee/types15/abstracts-book/contrib31.pdf), [[Sattler-WTypes.pdf:file]]]

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

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

On the [[fibration category of semi-simplicial sets]]:

* {#Sattler18} [[Christian Sattler]], *Constructive homotopy theory of marked semisimplicial sets* ([arXiv:1809.11168](https://arxiv.org/abs/1809.11168))

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

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

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

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

## Related entries

* [[nlabmeta:technical board]]

category: people


Christian Schlichtkrull

* [webpage](http://folk.uib.no/csc021/index-eng.html)

## related $n$Lab entries

* [[Thom spectrum]]
* [[cyclotomic space]]

category: people


Christian Schubert

## Selected writings

On the appearance of [[Bernoulli numbers]] in [[perturbative quantum field theory]] and [[string theory]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Bernoulli Number Identities from Quantum Field Theory and Topological String Theory_, Communications in Number Theory and Physics, Volume 7 (2013) Number 2, 225 - 249 ([arXiv:math/0406610](https://arxiv.org/abs/math/0406610))

On the [[worldline formalism]] for [[perturbative quantum field theory]]:

* M. G. Schmidt, [[Christian Schubert]], _Worldline Green Functions for Multiloop Diagrams_, Phys.Lett. B331 (1994) 69-76 ([arXiv:hep-th/9403158](https://arxiv.org/abs/hep-th/9403158))

* {#SchmidtSchubert94} M. G. Schmidt, [[Christian Schubert]], _The Worldline Path Integral Approach to Feynman Graphs_ ([arXiv:hep-ph/9412358](http://arxiv.org/abs/hep-ph/9412358))

* {#Schubert96} [[Christian Schubert]], _An Introduction to the Worldline Technique for Quantum Field Theory Calculations_, Acta Phys. Polon.B27:3965-4001, 1996 ([arXiv:hep-th/9610108](https://arxiv.org/abs/hep-th/9610108))

* [[Christian Schubert]], _Perturbative Quantum Field Theory in the String-Inspired Formalism_, Phys.Rept. 355 (2001) 73-234 ([arXiv:hep-th/0101036](https://arxiv.org/abs/hep-th/0101036))

* Olindo Corradini, [[Christian Schubert]], James P. Edwards, Naser Ahmadiniaz, *Spinning Particles in Quantum Mechanics and Quantum Field Theory* ([arXiv:1512.08694](https://arxiv.org/abs/1512.08694))

  > (emphasis on [[spinning particles]])

* James P. Edwards, C. Moctezuma Mata, U. Müller, [[Christian Schubert]], *New techniques for worldline integration* ([arXiv:2106.12071](https://arxiv.org/abs/2106.12071))

* [[Christian Schubert]] et al., *Summing Feynman diagrams in the worldline formalism* [[arXiv:2208.06585](https://arxiv.org/abs/2208.06585) ]

Discussion of the [[Schwinger effect]] via [[worldline formalism]]:

* [[Gerald Dunne]], [[Christian Schubert]], _Worldline Instantons and Pair Production in Inhomogeneous Fields_, Phys. Rev. D72 (2005) 105004 ([arXiv:hep-th/0507174](https://arxiv.org/abs/hep-th/0507174))

* [[Christian Schubert]], *The worldline formalism in strong-field QED* [[arXiv:2304.07404](https://arxiv.org/abs/2304.07404)]

category: people


Christian Sämann

see _[[Christian Saemann]]_

category: people


Christian Terp

* [Mathematics Geneaology page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=30619)

## Selected writings

On [[compact subgroups]] of [[Lie groups]] and [[locally compact groups]]:

* [[Karl Heinrich Hofmann]], [[Christian Terp]], *Compact Subgroups of Lie Groups and Locally Compact Groups*, Proceedings of the American Mathematical Society Vol. 120, No. 2 (Feb., 1994), pp. 623-634 ([jstor:2159906](https://www.jstor.org/stable/2159906), [doi:10.2307/2159906](https://doi.org/10.2307/2159906))

category: people


Christian Voigt

* [webpage](http://www.gla.ac.uk/schools/mathematicsstatistics/staff/christianvoigt/)

category: people


Christian Zickert

* [webpage](http://www2.math.umd.edu/~zickert/)

## related $n$Lab entries

* [[hyperbolic manifold]]

category: people


Christin Bibby

* [personal page](https://www.math.lsu.edu/~bibby/)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=206178)

## Selected writings

On [[representation stability]]:

* [[Christin Bibby]], [[Nir Gadish]], *A generating function approach to new representation stability phenomena in orbit configuration spaces*, Trans. Amer. Math. Soc. Ser. B **10** (2023) 241-287 [[ams:S2330-0000-2023-00130-3](https://www.ams.org/journals/btran/2023-10-09/S2330-0000-2023-00130-3), [arXiv:1911.02125](https://arxiv.org/abs/1911.02125)]

category: people


Christina Vasilakopoulou

Finished PhD in 2014 at [Cambridge](https://www.maths.cam.ac.uk/) with [[Martin Hyland]]. She works on [[enriched categories]], [[categorical algebra]], and [[applied category theory]].
Christina is currently an Assistant Professor at the School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Greece.

Personal Website [1](https://thalis.math.upatras.gr/~cvasilak) [2](http://www.math.ntua.gr/~cvasilak)

## Selected writings

On [[categorical algebra]], [[enriched categories]], and [[Grothendieck fibrations]]:

* [[Martin Hyland]], [[Ignacio López Franco]] and [[Christina Vasilakopoulou]], _Hopf measuring C-comonoids and enrichment_. Proceedings of the London Mathematical Society __115__:5 (2017) 1118-1148. (LMS [doi](https://doi.org/10.1112/plms.12064)) ([arXiv:1509.07632](https://arxiv.org/abs/1509.07632))

* [[Christina Vasilakopoulou]], _On enriched fibrations_. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. LIX, Issue 4, 2018. ([pdf](http://cahierstgdc.com/wp-content/uploads/2018/10/Vasilakopoulou-LIX-4.pdf))

* [[Christina Vasilakopoulou]], _Enriched duality in double categories: V-categories and V-cocategories_. J. Pure Appl. Alg. __223__:7 (2019) 2889-2947 ([JPAA](https://www.sciencedirect.com/science/article/pii/S0022404918302433)) ([arXiv:1704.00329](https://arxiv.org/abs/1704.00329))

* Christina Vasilakopoulou, _Dual algebraic structures and enrichment_, 55 min presentation at Topos institute, March 2023 [yt](https://www.youtube.com/watch?v=ymKvCz6W65o)

* [[Mitchell Buckley]], [[Timmy Fieremans]], [[Christina Vasilakopoulou]], [[Joost Vercruysse]], _A Larson-Sweedler Theorem for Hopf V-Categories_ ([arXiv:1908.02049](https://arxiv.org/abs/1908.02049))

On the [[monoidal Grothendieck construction]]:

* {#Moeller_Vasilakopoulou19} [[Joe Moeller]], [[Christina Vasilakopoulou]], _Monoidal Grothendieck construction_, Theory and Applications of Categories, **35** 31 (2020) 1159-1207 $[$[arXiv:1809.00727](http://arxiv.org/abs/1809.00727), [tac:35-31](www.tac.mta.ca/tac/volumes/35/31/35-31abs.html)$]$

On [[database theory]]:

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], and [[Ryan Wisnesky]], _Algebraic databases_. Theory and Applications of Categories, 2017, Vol. 32, No. 16, pp. 547-619. ([TAC](http://www.tac.mta.ca/tac/volumes/32/16/32-16abs.html))

* [[Vasilis Spiropoulos]] and [[Christina Vasilakopoulou]], _Digree: A Middleware for a Graph Databases Polystore_. 2016 IEEE International Conference on Big Data (Big Data)

On [[dynamical systems]] and their [[categorical systems theory]]:

* [[Patrick Schultz]], [[David I. Spivak]], [[Christina Vasilakopoulou]], *Dynamical Systems and Sheaves*, Appl Categor Struct **28** (2020) 1--57 [[arXiv:1609.08086](https://arxiv.org/abs/1609.08086), [doi:10.1007/s10485-019-09565-x](https://doi.org/10.1007/s10485-019-09565-x)]

category: people


Christine Escher

* [personal page](http://people.oregonstate.edu/~escherc/)

* [institute page](https://math.oregonstate.edu/people/view/tine)

## Selected writings

On [[sphere-fiber bundles]] and specifically on [[3-sphere]]-[[fiber bundles]] over the [[4-sphere]] in relation to [[exotic 7-spheres]]:

* {#CrowleyEscher03} [[Diarmuid Crowley]], [[Christine Escher]], _A classification of $S^3$-bundles over $S^4$_, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

category: people


Christine Lescop

* [webpage](https://www-fourier.ujf-grenoble.fr/~lescop/)

## Selected writings

On the [[Knizhnik-Zamolodchikov connection]] and the [[Kontsevich integral]] for [[braids]]:

* {#Lescop00} [[Christine Lescop]], _Introduction to the Kontsevich Integral of Framed Tangles_, 2000 ([[LescopKontsevichIntegral.pdf:file]])

Comparison between the two main [[universal Vassiliev invariants]]: the [[Kontsevich integral]] and the [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbatively quantized]] [[Chern-Simons theory]]:

* [[Christine Lescop]], _On configuration space integrals for links_, Geom. Topol. Monogr. 4 (2002) 183-199 ([arXiv:math/0211062](https://arxiv.org/abs/math/0211062))

On [[universal Vassiliev invariants]] and the [[Kontsevich integral]]:

* {#Lescop20} [[Christine Lescop]], _Invariants of links and 3-manifolds from graph configurations_ ([arXiv:2001.09929](https://arxiv.org/abs/2001.09929))

category: people


Christine Paulin-Mohring

* [personal page](https://www.lri.fr/~paulin/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Christine_Paulin-Mohring)

## Selected writings

Introducing the notion of [[inductive types]] in [[type theory]]

* [[Thierry Coquand]], [[Christine Paulin]], *Inductively defined types*, COLOG-88 Lecture Notes in Computer Science **417**, Springer (1990) 50-66 [[doi:10.1007/3-540-52335-9_47](https://doi.org/10.1007/3-540-52335-9_47)]

and with it the [[calculus of inductive constructions]] used by [[Coq]]:

* [[Frank Pfenning]], [[Christine Paulin-Mohring]], *Inductively defined types in the Calculus of Constructions*, in: *Mathematical Foundations of Programming Semantics MFPS 1989*,  Lecture Notes in Computer Science **442**, Springer (1990) $[$[doi:10.1007/BFb0040259](https://doi.org/10.1007/BFb0040259)$]$

* [[Christine Paulin-Mohring]], *Inductive definitions in the system Coq -- Rules and Properties*, in: *Typed Lambda Calculi and Applications* TLCA 1993, Lecture Notes in Computer Science **664** Springer (1993) [[doi:10.1007/BFb0037116](https://doi.org/10.1007/BFb0037116)]

* [[Christine Paulin-Mohring]], *Introduction to the Calculus of Inductive Constructions*, contribution to: *Vienna Summer of Logic* (2014) [[hal:01094195](https://hal.inria.fr/hal-01094195), [pdf](https://hal.inria.fr/hal-01094195/document), [pdf slides](https://easychair.org/smart-program/VSL2014/APPA-invited-slides-6.pdf)]

category: people

[[!redirects Christine Paulin]]


Christof Wetterich

* [webpage](http://www.thphys.uni-heidelberg.de/~wetterich/)

## Selected writings

On [[asymptotic safety]]:

* Alessia Platania, [[Christof Wetterich]], _Non-perturbative unitarity and fictitious ghosts in quantum gravity_ ([arXiv:2009.06637](https://arxiv.org/abs/2009.06637))

Interpretation of the apparent [[Higgs field]]-instability in terms of [[asymptotic safety]]:

* {#ShaposhnikovWetterich09} [[Mikhail Shaposhnikov]], [[Christof Wetterich]], _Asymptotic safety of gravity and the Higgs boson mass_, Phys. Lett. B 683 (2010) 196 [[arXiv:0912.0208](https://arxiv.org/abs/0912.0208)]

## Related entries

* [[inhomogeneous cosmology]]

* [[asymptotic safety]]

category: people


Christoffel symbols

+-- {: .rightHandSide}
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
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#Christoffel symbols#
* table of contents
{:toc}

##Idea

What is called a _Christoffel symbol_ is part of a notation and language from the early times of [[differential geometry]] at the end of the 19th and the beginning of the 20th century designed to deal with what today is called an [[affine connection]]: a [[connection on a bundle|connection]] on a [[tangent bundle]] $T X \to X$.

The _Christoffel symbols_ are the components of a [[connection on a bundle|connection 1-form]] on a coordinate patch $\mathbb{R}^n \simeq U \subset X$ of the underlying [[manifold]] $X$, in terms of the basis of the tangent bundle $T U$ induded by these coordinates.

As every concrete component expression, Christoffel symbols may be useful in certain computations. Unfortunately, almost every textbook on [[gravity]] in theoretical physics follows the long-outdated tradition of describing (or not describing) the entire notion of [[connection on a bundle|connection]]s on [[tangent bundle]]s without introducing these conceptually but just describing the Yoga of how to handle the Christoffel symbol component.
This way their main effect to science nowadays is to make it harder for students of theoretical physics to understand what is really going on in the universe. It's all so simple. Speaking always in terms of Christoffel symbols and never in terms of the abstract notion of [[connection on a bundle|connection]] makes it all so hard.

## Details

The [[tangent bundle]] $T X$ of an (oriented, say) [[manifold]] $X$ is a [[vector bundle]] that may be thought of as being the [[associated bundle]] to a $SL(n)$-[[principal bundle]] $P \to X$. This means just a little than that each fiber $T_x X$ of the tangent bundle looks like $\mathbb{R}^n$ and that the [[group]] of linear transformations $GL(n)$ acts on this.

Every tangent bundle may even be regarded as the [[associated bundle]] to a $O(n)$-principal bundle, i.e. one with structure group the [[orthogonal group]]. This is often useful. But for the discussion of Christoffel symbols we need the more general general group $GL(n)$.

A [[connection on a bundle|connection]] on the tangent bundle is the same as a [[connection on a bundle|connection]] on the underlying $GL(n)$-[[principal bundle]]: locally on $X$ this is a [[Lie-algebra valued 1-form]] with values in the [[general linear Lie algebra]] $\mathfrak{gl}(n)$.

We may write such a $1$-form as

$$
  A = A^a t_a
  \,,
$$

where $\{t_a\}$ is a basis for the [[Lie algebra]] $\mathfrak{gl}(n)$. But this Lie algebra is naturally thought of as nothing but the Lie algebra of $n \times n$ matrices $Mat(n)$. Every choice of basis $\{v^\mu\}$ of $\mathbb{R}^n$ yields a corresponding choice of basis of $Mat(n)$: the matrix denoted $T^{\nu}{}_{\mu}$ is the matrix that in the basis given by the $\{v^\mu\}$ has zeros everywhere except in the $\mu$-$\nu$-position, where it has a $1$.

Using this, we may write the local connection 1-form as

$$
  A = A^\mu{}_\nu T^\nu{}_\mu
  \,.
$$

Moreover now, since $A$ is just defined on a patch $U \subset X$ which is diffeomorphic to $\mathbb{R}^n$, we may fix such a diffeomorphism in that we find coordinates on $U$. Then we can express each $1$-form $A^\mu{}_\nu$ in terms of the coordinate basis of $1$-forms $\{d x^\lambda\}$ as

$$
  A^{\mu}{}_{\nu} = A^{\mu}{}_{\lambda\nu} d x^\lambda
  \,.
$$

This way we obtain on each patch $U$ from the choice of a [[connection on a bundle|connection]] on the [[tangent bundle]] and a choice of basis of $\mathbb{R}^n$ and a choice of coordinates on $U$ a collection

$$
  \{
    A^{\mu}{}_{\lambda\nu} \in C^\infty(U)
  \}_{\lambda, \mu, \nu, = 1,\cdots n}
$$

of functions, which are the components of the local $\mathfrak{gl}(n)$-valued connection $1$-form with respect to all the choices made.

This collection is called the **Christoffel symbols** of the connection, and then traditionally not denoted by the
letter $A$ but by the letter $\Gamma$

$$
  \{
    \Gamma^{\mu}{}_{\lambda\nu} \in C^\infty(U)
  \}_{\lambda, \mu, \nu, = 1, \dots, n}
  \,.
$$

## Relation to spin connection

The literature that uses Christoffel symbols falls in two parts: one leaves it at that and never considers anything else. The other eventually talks about "spin connections" or "moving frames".

A "spin connection" is just a [[connection on a bundle|connection]] on the [[tangent bundle]] of an [[orientation|oriented]] [[manifold]] which regards the tangent bundle as being [[associated bundle|associated]] to an $SO(n)$-[[principal bundle]] with structure group the [[special orthogonal group]]. This is locally a connection $1$-form $A$ with values in the [[special orthogonal Lie algebra]] $\mathfrak{so}(n) \subset \mathfrak{gl}(n)$. By this embedding we may regard it still as a $\mathfrak{gl}(n)$-valued form, whose coefficients happen to take values in skew-symmetric matrices. For the standard basis $\{v^a\}$ of $\mathbb{R}^n$, there is as before a canonical basis for such matrtices, denoted $T^{b}{}_{a}$. So the $\mathfrak{so}(n)$-valued connection 1-form may be expanded in this basis as

$$
  A = A^{a}{}_{b} T^{b}{}_a
  \,.
$$

Using the same coordinates as before for the patch that this is defined on allows us to expand the component 1-forms $A^b{}_a$ further as

$$
  A^{b}{}_{a} = A_{\lambda}{}^{b}{}_{a} d x^\lambda
  \,.
$$
In the relevant literature a connection $1$-form as this is traditionally denoted by the letter $\omega$:

$$
  \{
    \omega_{\lambda}{}^{a}{}_{b} \in C^\infty(C)
  \}
  \,.
$$

This is what is often called the "spin connection".

There is a [[vector bundle]] [[isomorphism]]

$$
  e : T X \to T X
$$

that identifies the tangent vectors thought of locally with respect to the basis $\{v^\mu\}$ to those thought of locally in terms of the basis $\{v^a\}$. On the given patch $U$ this is over each point $x \in U$ a $GL(n)$-valued function $e \in C^\infty(X, GL(n))$ that gives pointwise a linear map $\mathbb{R}^n \to \mathbb{R}^n$ of tangent spaces with components

$$
  e : v^a \mapsto e^{a}{}_\mu v^\mu
  \,.
$$

This is traditionally called the _vielbein_ or $n$-bein (for German: _Bein_ = _leg_, with the same root as the English _bone_; _viel_ = _many_, based on the special case _vierbein_ with _vier_ = _four_).

Generally, for $\phi \in C^\infty(X,G)$ some bundle automorphism, a function with values in the structure group of the bundle takes a connection $1$-form $A$ to

$$
  A' = \phi A \phi^{-1} + \phi d \phi^{-1}
  \,,
$$

where the first term denotes pointwise the [[adjoint action]] of the Lie group $G$ on its Lie algebra $\mathfrak{g}$, and where the second term denotes the pullback $\phi^* \theta$ of the [[Maurer-Cartan form]] $\theta$ on $GL(n)$ along $\phi$.

For the case at hand $e$ relate the Christoffel symbols to the "spin connection". In full component beauty this is traditionally written as

$$
  \omega_{\lambda}{}^{a}{}_{b} =
  e^{a}{}_{\mu} \Gamma^{\mu}{}_{\lambda\nu} e^{\nu}{}_{b}
  +
  e^{a}{}_{\mu} \partial_\lambda  e^{\mu}{}_{b}
  \,,
$$

where $\{e^{\mu}{}_{a}\}$ denote the components of the inverse $e^{-1}$ bundle automorphism.

This is traditionally the way that the Christoffel symbols are related to the notion of connection. But really both the Christoffel symbols as well as the spin connection components are nothing but a local component expression of the general notion of a connection $1$-form on a $GL(n)$-[[principal bundle]].

## Related concepts

* [[affine connection]]

## References

Originally due to

* [[Elwin Bruno Christoffel]], _Über die Transformation der homogenen Differentialausdrücke zweiten Grades_, Journal für die reine und angewandte Mathematik 70 (1869), 46–70. [doi:10.1515/crll.1869.70.46](https://doi.org/10.1515/crll.1869.70.46), [doi:10.1515/9783112389409-003](https://doi.org/10.1515/9783112389409-003).

[[!redirects Christoffel symbol]]


Christoph Bergbauer

* [webpage](http://page.mi.fu-berlin.de/bergbau/)

## selected writings

* {#BergbauerBrunettiKreimer09} [[Christoph Bergbauer]], [[Romeo Brunetti]], [[Dirk Kreimer]], _Renormalization and resolution of singularities_ ([arXiv:0908.0633](https://arxiv.org/abs/0908.0633))

## related $n$Lab entries

* [[Feynman amplitude on compactified configuration space of points]]

category: people


Christoph Dorn

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

## Selected writings

Introducing the notion of [[associative n-categories]]:

* [[Christoph Dorn]], _Associative $n$-categories_, talk at _[103rd Peripatetic Seminar on Sheaves and Logic](https://www.math.muni.cz/~loregianf/PSSL103/PSSL103.php)_ ([pdf](https://www.math.muni.cz/~loregianf/PSSL103/slides/Dorn.pdf))

* [[Christoph Dorn]], _Associative n-categories_ PhD thesis, 2019 ([pdf](https://www.dornchristoph.com/writing/thesis_final.pdf))

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700), [latest](https://cxdorn.github.io/assets/pdfs/fct.pdf))

On [[manifold diagrams]] and [[manifold-diagrammatic n-categories]]:

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758), [latest](https://cxdorn.github.io/assets/pdfs/mdiag.pdf))

* {#DornDouglas22B} [[Christoph Dorn]], [[Christopher Douglas]], _A brief introduction to framed combinatorial topology_, 2022 ([pdf](https://cxdorn.github.io/assets/pdfs/fctintro_latest.pdf))

* {#Dorn23} [[Christoph Dorn]], _Nine short stories about geometric higher categories_, 2023 ([pdf](https://cxdorn.github.io/assets/pdfs/nine-stories.pdf))

Exposition:

* [[Christoph Dorn]], *Manifold Diagrams -- A Brief Report*, [talk at](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#DornNov2023) [[CQTS]] (15 Nov 2023) [slides:[[Dorn-ManifoldDiagramsReport.pdf:file]], video:[YT](https://youtu.be/7KK89MuxmX8)]

## Related entries

* [[associative n-category]]

* [[manifold diagram]]

* [[homotopy.io]]

category: people


Christoph Pegel

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=230341)

## Selected writings

On [[cofibrant objects]] in the [[Thomason model structure]] (Thomason-cofibrant [[posets]]):

* [[Roman Bruckner]], [[Christoph Pegel]], _Cofibrant objects in the Thomason Model Structure_, [[arXiv:0808.4108](http://arxiv.org/abs/1603.05448)]

category: people


Christoph Sachse

* [website](http://www.math.uni-hamburg.de/home/sachse/)

## Related $n$Lab entries

* [[supergeometry]]

* [[superalgebra]]

* [[supermanifold]]

category: people


Christoph Schweigert

* [website](http://www.math.uni-hamburg.de/home/schweigert/)

## Selected writings

On [[D-branes]] at an [[ADE-singularity]] and their [[open string]] [[worldsheet]] [[2d CFT]]s (see also [here](ADE+singularity#ReferencesTypeIIStringsOnADE)):

* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278  ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))

The [[FRS-theorem on rational 2d CFT]]:

* [[Ingo Runkel]], [[Jens Fjelstad]], [[Jürgen Fuchs]], [[Christoph Schweigert]], *Topological and conformal field theory as Frobenius algebras*, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemp. Math. **431** (2007) 225-248 $[$[doi:10.1090/conm/431](http://dx.doi.org/10.1090/conm/431), [arXiv:math/0512076](https://arxiv.org/abs/math/0512076)$]$

On [[QFT with defects|defects]] and [[duality in physics|duality]] ("[[generalized global symmetry]]") in [[2d CFT]] (via the [[FRS theorem on 2d rational CFT]]):

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Kramers-Wannier duality from conformal defects*, Phys. Rev. Lett. **93** (2004) 070601 [[doi:10.1103/PhysRevLett.93.070601](https://doi.org/10.1103/PhysRevLett.93.070601), [arXiv:cond-mat/0404051](http://arxiv.org/abs/cond-mat/0404051)]

* [[Jürg Fröhlich]], [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], *Duality and defects in rational conformal field theory*, Nucl. Phys. B **763** (2007) 354-430 [[arXiv:hep-th/0607247](http://arxiv.org/abs/hep-th/0607247), [doi:10.1016/j.nuclphysb.2006.11.017](https://doi.org/10.1016/j.nuclphysb.2006.11.017)]

On [[QFTs with defects]] via [[higher algebra|higher]] [[categorical algebra]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Alessandro Valentino]], *Bicategories for boundary conditions and for surface defects in 3-d TFT*, Commun. Math. Phys. **321**  (2013) 543–575 [[arXiv:1203.4568](https://arxiv.org/abs/1203.4568), [doi:10.1007/s00220-013-1723-0](https://doi.org/10.1007/s00220-013-1723-0)]

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 potential generalization to [[logarithmic CFT]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], *Full Logarithmic Conformal Field theory - an Attempt at a Status Report*, Proc. of *[[Higher Structures in M-Theory 2018]]*, Fortschr. Phys.  **67** 8-9 (2019)  $[$[arXiv:1903.02838](https://arxiv.org/abs/1903.02838), [doi:10.1002/prop.201910018](https://doi.org/10.1002/prop.201910018)$]$

Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]

Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]

Exposition:

* [[Christoph Schweigert]], *String-net methods for CFT correlators*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#SchweigertMar2023)]

Survey of [[algebra|algebraic]] [[structures]] in [[2d conformal field theory]] ([[vertex operator algebras]], [[conformal blocks]], [[modular tensor category|modular]] [[fusion categories]], [[modular functors]], [[FRS theorem]], [[string-net models]]):

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], *Algebraic structures in two-dimensional conformal field theory*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

## Related entries

* [[conformal field theory]]

* [[FRS-theorem on rational 2d CFT]]

* [[bundle gerbe]], [[Jandl gerbe]]

category: people


Christoph Sorger

* [webpage](http://www.math.sciences.univ-nantes.fr/~sorger/)

category: people


Christoph Wockel

* [website](http://www.math.uni-hamburg.de/home/wockel/)

## Selected writings

On [[Lie group cohomology]]:

* [[Friedrich Wagemann]], [[Christoph Wockel]], *A Cocycle Model for Topological and Lie Group Cohomology*, Transactions of the American Mathematical Society **367** 3 (2015) 1871-1909 ([arXiv:1110.3304](https://arxiv.org/abs/1110.3304), [jstor:24513021](https://www.jstor.org/stable/24513021))

category: people


Christophe Breuil

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

## Selected writings

On the [[modularity theorem]]:

* {#BreuilConradDiamondTaylor2001} [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], [[Richard Taylor]], *On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises*, Journal of the American Mathematical Society **14** 4 (2001) 843–939 $[$[doi:10.1090/S0894-0347-01-00370-8](https://doi.org/10.1090/S0894-0347-01-00370-8)$]$

On the [[local Langlands correspondence]]:

* {#Breuil2010} [[Christophe Breuil]], *The emerging p-adic Langlands program*, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010 ([pdf](https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/ICM2010.pdf))

category: people


Christophe Reutenauer

__Christophe Reutenauer__ is a Canadian mathematician specialized in algebraic combinatorics and applications.

Related pages include [[cyclic derivation]], [[quasi-symmetric function]], [[free Lie algebra]]

* C. Reutenauer, _Free Lie algebras_, Handbook of Algebra, vol. 3, 2003, 887-903, doi

* C. Reutenauer, _Applications of a noncommutative jacobian matrix_, Journal of Pure and Applied Algebra __77__, n. 2, 1992, p. 169-181, doi

* C. Reutenauer, _Cyclic derivation of noncommutative algebraic power series_, J. Alg. __85__, 32-39 (1983) [pdf](https://core.ac.uk/download/pdf/82506869.pdf)

* Claudia Malvenuto, C. Reutenauer, _Plethysm and conjugation of quasi-symmetric functions_, Discrete Mathematics 193 (1998) 225-233 [pdf](https://sites.math.washington.edu/~billey/colombia/references/malvenuto.reutenauer.1998.pdf)

category: people

[[!redirects C. Reutenauer]]


Christophe Soulé

__Christophe Soulé__ is a French mathematician with main interests in [[arithmetic geometry]] and [[algebraic K-theory]]. He is a pioneer of the subject of [[field with one element]].

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

category: people
[[!redirects C. Soule]]
[[!redirects C. Soulé]]
[[!redirects Christophe Soule]]


Christopher Beem

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

## Selected writings

On [[holomorphic blocks]]:

* {#BeemDimoftePasquetti12} [[Christopher Beem]], [[Tudor Dimofte]], [[Sara Pasquetti]], _Holomorphic Blocks in Three Dimensions_ ([arXiv:1211.1986](http://arxiv.org/abs/1211.1986))

On [[conformal bootstrap]] for [[SCFT]]:

* [[Christopher Beem]], Madalena Lemos, Pedro Liendo, [[Leonardo Rastelli]], Balt C. van Rees, _The $N=2$ superconformal bootstrap_ ([arXiv:1412.7541](https://arxiv.org/abs/1412.7541))

* [[Christopher Beem]], Madalena Lemos, [[Leonardo Rastelli]], Balt C. van Rees, _The $(2,0)$ superconformal bootstrap_ ([arXiv:1507.05637](https://arxiv.org/abs/1507.05637))

* {#BRR29} [[Christopher Beem]], [[Leonardo Rastelli]], Balt C. van Rees, _More $N=4$ superconformal bootstrap_ ([arXiv:1612.02363](https://arxiv.org/abs/1612.02363))

Discussion of [[topological quantum field theory]] with an eye towards [[supersymmetry|supersymmetric]] field theory, in terms of the [[ordinary homology]] of [[configuration spaces of points]] (see also at _[[correlators as differential forms on configuration spaces of points]]_):

* [[Christopher Beem]], [[David Ben-Zvi]], [[Mathew Bullimore]], [[Tudor Dimofte]], [[Andrew Neitzke]], _Secondary products in supersymmetric field theory_ ([arXiv:1809.00009](https://arxiv.org/abs/1809.00009))

category: people


Christopher Brav > history

see [[Chris Brav]]


Christopher D. Hollings

[Website](https://sites.google.com/site/cdhollings/home)

category: people


Christopher Deninger

* [webpage](http://wwwmath.uni-muenster.de/reine/inst/deninger/about/)

## related $n$Lab entries

* [[Beilinson conjectures]]

* [[arithmetic topology]]

category: people


Christopher Douglas

[[!redirects Chris Douglas]]

Chris Douglas is a professor for mathematics at Oxford

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

## Selected writings

On [[tmf]]:

* [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]], _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

Introducing a notion of [[fusion 2-categories]]:

* [[Christopher L. Douglas]], [[David J. Reutter]], *Fusion 2-categories and a state-sum invariant for 4-manifolds* [[arXiv:1812.11933](https://arxiv.org/abs/1812.11933)]

## Related $n$Lab entries

* [[2d CFT]], [[conformal nets]]

* [[elliptic cohomology]]

category: people

[[!redirects Christopher L. Douglas]]


Christopher Drupieski

* [webpage](http://math.depaul.edu/cdrupies/)

## related $n$Lab entries

* [[group extension]], [[supergroup]]

category: people


Christopher Dwyer

* [institute page](https://www.saintmarys.edu/academics/faculty/christopher-dwyer)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=97712)

## Selected writings

On [[twisted equivariant K-theory]] via [[Grothendieck groups]] of [[equivariant bundle|equivariant]] [[twisted vector bundles]]:

* [[Christopher Dwyer]], _Twisted equivariant K-theory for proper actions of discrete groups_, K-Theory v. 38, n. 2 (January, 2008): 95-111 ([arXiv:0710.2136](https://arxiv.org/abs/0710.2136), [doi:10.1007/s10977-007-9016-z](http://dx.doi.org/10.1007/s10977-007-9016-z))

category: people


Christopher Fewster

* [webpage](http://www-users.york.ac.uk/~cjf3/)

## Selected writings

On [[AQFT on curved spacetimes]]:

* [[Christopher Fewster]], [[Rainer Verch]], _Algebraic quantum field theory in curved spacetimes_ ([arXiv:1504.00586](https://arxiv.org/abs/1504.00586))

On the [[split property]] in [[AQFT]]:

* [[Christopher J. Fewster]], *The split property for quantum field theories in flat and curved spacetimes* [[arXiv:1601.06936](https://arxiv.org/abs/1601.06936)]

Introduction to [[AQFT]]:

* {#FewsterRejzner19} [[Christopher Fewster]], [[Kasia Rejzner]], _Algebraic Quantum Field Theory - an introduction_ ([arXiv:1904.04051](https://arxiv.org/abs/1904.04051))

On [[homotopical AQFT]]:

* [[Simen Bruinsma]], [[Christopher J. Fewster]], [[Alexander Schenkel]], *Relative Cauchy evolution for linear homotopy AQFTs* ([arXiv:2108.10592](https://arxiv.org/abs/2108.10592))

On [[quantum measurement]] in the refined context of ([[local field theory|local]] [[algebraic quantum field theory|algebraic]]) [[quantum field theory]] [[AQFT on curved spacetimes|on Lorentzian manifolds]]:

* [[Christopher J. Fewster]], [[Rainer Verch]], *Measurement in Quantum Field Theory* [[arXiv:2304.13356](https://arxiv.org/abs/2304.13356)]

## Related entries

* [[AQFT]]

* [[AQFT on curved spacetimes]]

category: people

[[!redirects Christopher J. Fewster]]


Christopher Fuchs

# Christopher Fuchs
* table of contents
{: toc}

* [Faculty webpage](http://www.physics.umb.edu/Research/QBism/chris.html)

* [Quantum Dreams Page](http://www.perimeterinstitute.ca/personal/cfuchs/)

## Selected writings

On [[quantum Bayesianism]]/[[QBism]]

* {#CFS2001} [[Carlton Caves]], [[Christopher Fuchs]], [[Ruediger Schack]], *Unknown Quantum States: The Quantum de Finetti Representation* (2001) [[arXiv:quant-ph/0104088](http://arxiv.org/abs/quant-ph/0104088)]

* {#Fuchs2002} [[Christopher Fuchs]], *Quantum Mechanics as Quantum Information (and only a little more)* (2002) [[arXiv:quant-ph/0205039](https://arxiv.org/abs/quant-ph/0205039)]

* {#FMS2014} [[Christopher Fuchs]], [[David Mermin]], [[Ruediger Schack]], *An Introduction to QBism with an Application to the Locality of Quantum Mechanics*, American Journal of Physics **82**  8 (2014) 749--754 [[arXiv:1311.5253](https://arxiv.org/abs/1311.5253)]

* {#FuchsSchlosshauerStacey14} [[Christopher A. Fuchs]],  [[Maximilian Schlosshauer]] (foreword), [[Blake C. Stacey]] (ed.) *My Struggles with the Block Universe* [[arXiv:1405.2390](https://arxiv.org/abs/1405.2390)]

## Related entries

* [[quantum Bayesianism]]

* [[POVM]]

* [John Wheeler commanding equations to fly](John+Wheeler#CommandingEquationsToFly)

category: people

[[!redirects Christopher A. Fuchs]]


Christopher Herzog

* [webpage](https://www.kcl.ac.uk/people/christopher-herzog)

## Selected writings

On [[Seiberg duality]]:

* [[Christopher Herzog]], _Seiberg Duality is an Exceptional Mutation_ ([arXiv:hep-th/0405118](http://arxiv.org/abs/hep-th/0405118))

Discussion of [[superconductivity]] via [[AdS/CFT in condensed matter physics]]:

* [[Sean Hartnoll]], [[Christopher Herzog]], [[Gary
 Horowitz]], _Building an AdS/CFT superconductor_, Phys. Rev. Lett. 101:031601, 2008 ([arXiv:0803.3295](https://arxiv.org/abs/0803.3295))

category: people


Christopher Hull

[[!redirects Chris Hull]]

Christopher Hull is professor for theoretical physics at Imperial College London.

* [webpage](http://www.imperial.ac.uk/people/c.hull)

* [wikipedia entry](http://en.wikipedia.org/wiki/Chris_Hull_%28physicist%29)

## Selected writings

On [[D=4 supergravity]]:

* [[Chris Hull]], Phys. Rev. D30 (1984) 760;

* [[Chris Hull]], Phys. Lett. 142B (1984)

* [[Chris Hull]], Phys. Lett. 148B (1984) 297;

* [[Chris Hull]], Physica 15D (1985) 230; Nucl. Phys. B253 (1985) 650.

* [[Chris Hull]], Class. Quant. Grav. 2 (1985) 343.

* [[Chris Hull]], _New Gauged $N=8$, $D=4$ Supergravities_, _Class.Quant.Grav.20:5407-5424,2003_ ([arXiv:hep-th/0204156](https://arxiv.org/abs/hep-th/0204156))

On [[sigma-models]]:

* [[Christopher Hull]], *Lectures on Non-Linear Sigma-Models and Strings*, in *Super field theories*, NATO Science Series **160**, Springer (1987) 77-168 [[doi:10.1007/978-1-4613-0913-0_4](https://doi.org/10.1007/978-1-4613-0913-0_4)]

On [[BPS states]]:

* {#GauntlettHull99} [[Jerome Gauntlett]], [[Chris Hull]], _BPS States with Extra Supersymmetry_, JHEP 0001 (2000) 004 ([arXiv:hep-th/9909098](http://arxiv.org/abs/hep-th/9909098))

On [[gauge enhancement]]:

* {#HullTownsend95} [[Chris Hull]], [[Paul Townsend]], _Enhanced Gauge Symmetries in Superstring Theories_, Nucl.Phys. B451 (1995) 525-546 ([arXiv:hep-th/9505073](http://arxiv.org/abs/hep-th/9505073))

* [[Chris Hull]], _Duality, Enhanced Symmetry and Massless Black Holes_, in Proceedings of Strings'95: Future Perspectives in String Theory (World Scientific, 1996, I. Bars et al. eds.), p. 230 ([pdf](http://physics.usc.edu/Strings95/Proceedings/pdf/hull.pdf))

* [[Mirjam Cvetic]], [[Chris Hull]], _Wrapped Branes and Supersymmetry_, Nucl.Phys.B519:141-158,1998 ([arXiv:hep-th/9709033](http://arxiv.org/abs/hep-th/9709033))

On [[T-duality]] for the [[supersymmetric sigma-model]]:

* M. Abou-Zeid, [[Chris Hull]], [[Ulf Lindström]], [[Martin Roček]], _T-Duality in $(2,1)$ Superspace_ ([arXiv:1901.00662](https://arxiv.org/abs/1901.00662))

On [[branes]] at [[cone (Riemannian geometry)|conical singularities]]:

* [[Bobby Acharya]], [[Jose Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, _Branes at conical singularities and holography_, Adv. Theor.Math. Phys.2:1249-1286, 1999 ([arXiv:hep-th/9808014](https://arxiv.org/abs/hep-th/9808014))

On [[duality in string theory]] ([[U-duality]]):

* [[Chris Hull]], [[Paul Townsend]], Section 6 of _Unity of Superstring Dualities_, Nucl.Phys.B438:109-137, 1995 ([arXiv:hep-th/9410167](https://arxiv.org/abs/hep-th/9410167))

Introducing the [[generalized Scherk-Schwarz mechanism]]:

* [[Atish Dabholkar]], [[Chris Hull]], _Duality Twists, Orbifolds, and Fluxes_, JHEP 0309:054, 2003 ([arXiv:hep-th/0210209](https://arxiv.org/abs/hep-th/0210209))

On [[D=5 supergravity]]:

* {#GGHPR02} [[Jerome Gauntlett]], [[Jan Gutowski]], [[Christopher Hull]], Stathis Pakis, [[Harvey Reall]], _All supersymmetric solutions of minimal supergravity in five dimensions_, Class. Quant. Grav. 20 (2003) 4587-4634 [[arXiv:hep-th/0209114](http://arxiv.org/abs/hep-th/0209114), [doi:10.1088/0264-9381/20/21/005](https://doi.org/10.1088/0264-9381/20/21/005)]

On [[non-geometric string vacua]]:

* [[Chris Hull]], _A Geometry for Non-Geometric String Backgrounds_ ([arXiv:hep-th/0406102](https://arxiv.org/abs/hep-th/0406102))

On [[automorphism of a vertex operator algebra|automorphisms of]] [[vertex operator algebras]] regarded as [[symmetries]] of non-geometric [[perturbative string theory vacua]] (e.g. [[Gepner models]]):

* {#HullIsraelSarti17} [[Chris Hull]], Dan Israel, Alessandra Sarti, _Non-geometric Calabi-Yau Backgrounds and K3 automorphisms_, JHEP11(2017)084 ([arXiv:1710.00853](https://arxiv.org/abs/1710.00853))

On [[exceptional generalized geometry]]:

* {#Hull} [[Chris Hull]], _Generalised Geometry for M-Theory_, JHEP 0707:079, 2007 ([arXiv:hep-th/0701203](http://arxiv.org/abs/hep-th/0701203))

On [[T-folds]]:

* {#Hull04} [[Chris Hull]], _A Geometry for Non-Geometric String Backgrounds_, JHEP0510:065,2005 ([arXiv:hep-th/0406102](http://arxiv.org/abs/hep-th/0406102))

* {#Hull06} [[Chris Hull]], _Doubled geometry and T-folds_ JHEP0707:080,2007  ([arXiv:hep-th/0605149](http://arxiv.org/abs/hep-th/0605149))

* [[Chris Hull]], _Global Aspects of T-Duality, Gauged Sigma Models and T-Folds_ ([arXiv:hep-th/0604178](http://arxiv.org/abs/hep-th/0604178))

On [[double field theory]]:

* {#Hull06} [[Chris Hull]], _Doubled geometry and T-folds_ JHEP0707:080,2007  ([arXiv:hep-th/0605149](http://arxiv.org/abs/hep-th/0605149))

* [[Chris Hull]], [[Barton Zwiebach]], _Double Field Theory_, JHEP 0909:099,2009 ([arXiv:0904.4664](http://arxiv.org/abs/0904.4664))

On the [[dual graviton]]:

* {#Hull01} [[Chris Hull]], _Duality in Gravity and Higher Spin Gauge Fields_, JHEP 0109:027,2001 ([arXiv:hep-th/0107149](http://arxiv.org/abs/hep-th/0107149))

On the lift of [[D8-branes]] to [[M-theory]] [[M-branes]] by [[generalized Scherk-Schwarz reduction]], relating to [[D7-branes]] in [[F-theory]]:

* {#Hull98} [[Chris Hull]], _Massive String Theories From M-Theory and F-Theory_, JHEP 9811:027, 1998 ([arXiv:hep-th/9811021](https://arxiv.org/abs/hep-th/9811021))

On the [[D9-brane]]:

* {#Hull97} [[Chris Hull]], _Gravitational Duality, Branes and Charges_ ([arXiv:hep-th/9705162](https://arxiv.org/abs/hep-th/9705162))

On the [[M-wave]]:

* {#Hull84} [[Chris Hull]], _Exact pp wave solutions of eleven-dimensional supergravity_, Phys. Lett. B 139 (1984) 39 ([spire](http://inspirehep.net/record/14499?ln=en))

On [[type II string theory]] and [[M-theory]] on [[spacetime]] with different [[signature of a metric|signatures]]:

* [[Chris Hull]]. *Duality and the signature of space-time*, Journal of High Energy Physics **1998** 11 (1999) 017 [[arXiv:hep-th/9807127](https://arxiv.org/abs/hep-th/9807127), [doi:10.1088/1126-6708/1998/11/017](https://doi.org/10.1088/1126-6708/1998/11/017)]

On [[D=6 supergravity]] with [[number of supersymmetries|$\mathcal{N} = (4,0)$]]:

* [[Chris Hull]], *Strongly Coupled Gravity and Duality*, Nucl.Phys. B **583** (2000) 237-259 [[arXiv:hep-th/0004195](https://arxiv.org/abs/hep-th/0004195), doi:10.1016/S0550-3213(00)00323-0]

* [[Chris Hull]], *Symmetries and Compactifications of $(4,0)$ Conformal Gravity*, JHEP 0012: 007 (2000) [[arXiv:hep-th/0011215](https://arxiv.org/abs/hep-th/0011215), [doi:10.1088/1126-6708/2000/12/007](https://doi.org/10.1088/1126-6708/2000/12/007)]

* [[Chris Hull]], *Gravity, Duality and Conformal Symmetry* [[arXi:2209.11716](https://arxiv.org/abs/2209.11716)]

On [[Lagrangian densities]] for [[self-dual higher gauge theory|self-dual form fields]]:

* {#Hull23} [[Chris Hull]], *Covariant Action for Self-Dual $p$-Form Gauge Fields in General Spacetimes* [[arXiv:2307.04748](https://arxiv.org/abs/2307.04748)]

## Related entries

* [[string theory]]

* [[T-fold]]

* [[M5-brane]]

* [[5d supergravity]]

category: people


Christopher J. Dean

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

## Selected writings

On [[weak omega-category|weak $\omega$-categories]] via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

category: people

[[!redirects Christopher Dean]]


Christopher M. Dawson

* [arXiv page](https://arxiv.org/search/quant-ph?searchtype=author&query=Dawson%2C+C+M)

## Selected writings

On the [[Solovay-Kitaev theorem]]:

* [[Christopher M. Dawson]], [[Michael A. Nielsen]], *The Solovay-Kitaev algorithm*, Quantum Information & Computation **6** 1 (2006) 81–95 [[arXiv:quant-ph/0505030](https://arxiv.org/abs/quant-ph/0505030), [doi:10.5555/2011679.2011685](https://dl.acm.org/doi/10.5555/2011679.2011685)]

category: people

[[!redirects Christopher Dawson]]
[[!redirects Chris Dawson]]


Christopher Pope

* [webpage](http://people.physics.tamu.edu/pope/)

## Selected writings

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

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

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

On [[Kaluza-Klein compactification]] in [[supergravity]]:

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), doi:10.1016/0370-1573(86)90163-8)

On [[G2-conifolds]]:

* [[Gary Gibbons]], [[Don Page]], [[Christopher Pope]], _Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles_, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 ([euclid:cmp/1104180218](https://projecteuclid.org/euclid.cmp/1104180218))

On [[higher curvature corrections]] to [[D=11 supergravity]] and [[M-theory on G2-manifolds]]:

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory_, JHEP 0410:019, 2004 ([arXiv:hep-th/0312002](https://arxiv.org/abs/hep-th/0312002))

* H. Lu, [[Christopher Pope]], [[Kellogg Stelle]], [[Paul Townsend]], _String and M-theory Deformations of Manifolds with Special Holonomy_, JHEP 0507:075, 2005 ([arXiv:hep-th/0410176](https://arxiv.org/abs/hep-th/0410176))

On the work of [[Mike Duff]]:

* [[Leron Borsten]], [[Alessio Marrani]], [[Christopher N. Pope]], [[Kellogg Stelle]], *Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday*, Proceedings of the Royal Society A,
**478** (2022) 2259 [[doi:10.1098/rspa.2022.0166](https://doi.org/10.1098/rspa.2022.0166)]

## Related entries

* [[string scattering amplitudes]]

category: people

[[!redirects Christopher N. Pope]]


Christopher Reedy

Christopher Leonard Reedy was a PhD student of [[Donald W. Anderson]], obtaining his PhD from UCSD in 1974.

He originated [[Reedy categories]].

* [genealogy](http://genealogy.math.ndsu.nodak.edu/id.php?id=43120)

## Selected writings

* [[C. L. Reedy]], _Homotopy theory of model categories_, [PDF](http://www-math.mit.edu/~psh/reedy.pdf).

## Related entries

* [[Reedy category]]

* [[Reedy model structure]]

* [[model structure on simplicial algebras]]

category: people

[[!redirects Chris Reedy]]
[[!redirects Christopher Leonard Reedy]]
[[!redirects C. L. Reedy]]


Christopher Strachey

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

## Selected writings

On the theory of [[computation]] and introducing the notion of [[denotational semantics]] of [[programming languages]] by what came to be called *[[domain theory]]*:

* [[Dana S. Scott]], [[Christopher Strachey]], *Toward a Mathematical Semantics for Computer Languages*, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [[pdf](https://www.cs.ox.ac.uk/files/3228/PRG06.pdf), [[ScottStrachey-MathematicalSemantics.pdf:file]]]

category: people


Christos Papadimitriou

Christos Papadimitriou is a professor of computer science at Columbia University.

* [webpage](https://www.engineering.columbia.edu/faculty/christos-papadimitriou)

## Selected writings
 {#SelectedWritings}

Textbook on [[complexity theory]]:

* [[Christos Papadimitriou]] (1994), _Computational Complexity_ (1st ed.), Addison Wesley.

category: people


Christy Hazel

* [personal page](https://www.math.ucla.edu/~chazel/)

## Selected writings

On [[commutative ring]]-structure ([[E-infinity ring|$E_\infty$-ring]]-structure) in [[rational equivariant stable homotopy theory]] in general and in [[rational equivariant K-theory]] in particular:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

On [[Kronholm's freeness theorem]] in [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]

* [[Daniel Dugger]], [[Christy Hazel]], [[Clover May]], *Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$*, Journal of Pure and Applied Algebra **228** 3 (2024) 107473 [[arXiv:2203.05287](https://arxiv.org/abs/2203.05287), [doi:10.1016/j.jpaa.2023.107473](https://doi.org/10.1016/j.jpaa.2023.107473)]

category: people


chromatic convergence theorem

+-- {: .rightHandSide}
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###Context###
#### Stable Homotopy theory
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=--
#### Higher algebra
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[[!include higher algebra - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

For each [[prime]] $p \in \mathbb{N}$ and for each [[natural number]]  $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

$$
  L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)}
  \,,
$$

where $K(n)$ is the $n$th [[Morava K-theory]] (for the given [[prime]] $p$). These arrange into the _[[chromatic tower]]_ which for each spectrum $X$ is of the form

$$
  X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
$$

The _chromatic convergence theorem_ states mild conditions under which the [[homotopy limit]] over this tower is the $p$-[[Bousfield localization of spectra|localization]]

$$
  X \to X_{(p)}
$$

of $X$.

In particular if $X$ is a [[finite spectrum|finite]] [[p-local spectrum]] then the chromatic convergence theorem says that the [[homotopy limit]] over the [[chromatic tower]] of $X$ reproduces $X$.

Since moreover $L_n X$ is the [[homotopy fiber product]]

$$
  L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X
$$

(see at _[[smash product theorem]]_ and see [this remark](fracture+theorem#ExampleOfChromaticFracturing) at _[[fracture square]]_ )
it follows that in principle one may study any [[spectrum]] $X$ by understanding all its "chromatic pieces" $L_{K(n)} X$. This is the topic of [[chromatic homotopy theory]].

## Related concepts

* [[chromatic layer]]

* [[smash product theorem]]

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 32 _The chromatic convergence theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture32.pdf))


Chromatic Homotopy Theory

+-- {: .rightHandSide}
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#### Cohomology
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[[!include stable homotopy theory - contents]]
=--
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+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

This page collects links related to the lecture notes

* [[Jacob Lurie]],

  **Chromatic Homotopy Theory**

  Lecture series 2010

  [web](http://www.math.harvard.edu/~lurie/252x.html), [[LurieChromaticHomotopyTheory.pdf:file]]

on [[complex oriented cohomology]], the [[Adams spectral sequence]] and [[chromatic homotopy theory]] from the modern point of view of [[E-infinity geometry]].

Based on the program of

* [[Frank Adams]], _[[Stable homotopy and generalised homology]]_, 1974

as nicely laid out in more detail in

* [[Stanley Kochmann]], _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

First indications of the big picture developed here are due to

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_, 1999

Via the chromatic stratification of the [[moduli stack of formal groups]] one recovers as, roughly, the second chromatic stage the [[moduli stack of elliptic curves]]. For the story in that case see

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_.

See also

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986/2003

**Lectures**

* Lecture 1 _Introduction_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture1.pdf))

* Lecture 2 _[[Lazard's theorem]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf))

* Lecture 3 _Lazard's theorem (continued)_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf))

* Lecture 4 _[[complex oriented cohomology theory|Complex-oriented cohomology theories]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf))

* Lecture 5 _[[complex bordism|Complex bordism]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf))

* Lecture 6 _[[MU and complex orientations]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf))

* Lecture 7 _[[homology of MU|The homology of MU]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf))

* Lecture 8 _The [[Adams spectral sequence]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture8.pdf))

* Lecture 9 _The Adams spectral sequence for MU_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture9.pdf))

* Lecture 10 _The proof of [[Quillen's theorem]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture10.pdf))

* Lecture 11 _[[formal groups|Formal groups]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf))

* Lecture 12 _[[height of a formal group|Heights and formal groups]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture12.pdf))

* Lecture 13 _The stratification of $\mathcal{M}_{FG}$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture13.pdf))

* Lecture 14 _Classification of formal groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture14.pdf))

* Lecture 15 _Flat modules over $\mathcal{M}_{FG}$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture15.pdf))

* Lecture 16 _The [[Landweber exact functor theorem]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture16.pdf))

* Lecture 17 _[[phantom maps|Phantom maps]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf))

* Lecture 18 _Even periodic cohomology theories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf))

* Lecture 19 _[[Morava stabilizer groups]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf))

* Lecture 20 _[[Bousfield localization of spectra|Bousfield localization]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture20.pdf))

* Lecture 21 _Lubin-Tate theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf))

* Lecture 22 _Morava E-theory and Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf))

* Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf))

* Lecture 24 _Uniqueness of Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf))

* Lecture 25 _The Nilpotence lemma_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf))

* Lecture 26 _Thick subcategories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf))

* Lecture 27 _The periodicity theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture27.pdf))

* Lecture 28 _Telescopic localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture28.pdf))

* Lecture 29 _Telescopic vs $E_n$-localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture29.pdf))

* Lecture 30 _Localizations and the Adams-Novikov spectral sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))

* Lecture 31 _The smash product theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture31.pdf))

* Lecture 32 _The chromatic convergence theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture32.pdf))

* Lecture 33 _Complex bordism and $E(n)$-localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture33.pdf))

* Lecture 34 _Monochromatic layers_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture34.pdf))

* Lecture 35 _The image of $J$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf))

#Contents#
* table of contents
{:toc}

## Lecture 1 _Introduction_

## Lecture 2 _Lazard's theorem_

* [[Lazard ring]]

* [[Lazard's theorem]]

## Lecture 3 _Lazard's theorem (continued)_

* [[Lazard's theorem]]

## Lecture 4 _Complex-oriented cohomology theories_

* [[complex oriented cohomology theory]]

* [[splitting principle]]

* [[generalized Chern classes]]

## Lecture 5 _Complex bordism_

* [[complex cobordism cohomology theory]], [[MU]]

##  Lecture 6 _MU and complex orientations_

* [[universal complex orientation on MU]]

## Lecture 7 _The homology of MU_

* [[Boardman homomorphism]]

* [[homology of MU]]

##  Lecture 8 _The Adams spectral sequence_

* [[Adams resolution]]

* [[Adams spectral sequence]]

## Lecture 9 _The Adams spectral sequence for MU_

* [[Adams-Novikov spectral sequence]]

##  Lecture 10 _The proof of Quillen's theorem_

* [[Quillen's theorem on MU]]

## Lecture 11 _Formal groups_

* [[formal group]]

## Lecture 12 _Heights and formal groups_

* [[height of a formal group]]

## Lecture 13 _The stratification of $\mathcal{M}_{FG}$_

* [[chromatic filtration]]

## Lecture 14 _Classification of formal groups_

* [[moduli stack of formal groups]]

## Lecture 15 _Flat modules over $\mathcal{M}_{FG}$_

* [[quasicoherent sheaf]], [[flat module]], [[exact functor]]

* [[Landweber exactness]]

## Lecture 16 _The Landweber exact functor theorem_

* [[Landweber exact functor theorem]]

## Lecture 17 _Phantom maps_

* [[Brown representability theorem]], [[Landweber exact spectrum]]

* [[phantom map]]

## Lecture 18 _Even periodic cohomology theories_

* [[periodic cohomology theory]]

* [[periodic ring spectrum]]

## Lecture 19 _Morava stabilizer groups_

* [[Morava stabilizer group]]

## Lecture 20 _Bousfield localization_

* [[Bousfield localization]]

* [[Bousfield localization of spectra]]

* [[p-localization]]

## Lecture 21 _Lubin-Tate theory_

* [[Lubin-Tate formal group]]

* [[Lubin-Tate theory]]

## Lecture 22 _Morava E-theory and Morava K-theory_

* [[Morava K-theory]]

* [[Morava E-theory]]

## Lecture 23 _The Bousfield Classes of $E(n)$ and $K(n)$_

* [[Bousfield equivalence]]

* [[smash product theorem]]

##  Lecture 24 _Uniqueness of Morava K-theory_

* [[∞-field]]

## Lecture 25 _The Nilpotence lemma_

* [[nilpotence lemma]]

## Lecture 26 _Thick subcategories_

* [[thick subcategory]]

## Lecture 27 _The periodicity theorem_

* [[periodicity theorem]]

## Lecture 28 _Telescopic localization_

* [[telescopic localization]]

## Lecture 29 _Telescopic vs $E_n$-localization_

* [[smashing localization]]

* [[chromatic tower]]

## Lecture 30 _Localizations and the Adams-Novikov spectral sequence_

* [[Adams-Novikov spectral sequence]]

## Lecture 31 _The smash product theorem_

* [[smash product theorem]]

## Lecture 32 _The chromatic convergence theorem_

* [[chromatic convergence theorem]]

## Lecture 33 _Complex bordism and $E(n)$-localization_

## Lecture 34 _Monochromatic layers_

* [[monochromatic layer]]

## Lecture 35 _The image of $J$_

* [[J-homomorphism]]

* [[Adams operation]]

category: reference


chromatic homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[Ravenel 1992, p. 24](#Ravenel92):

> We use the word 'chromatic' here for the following reason. The $n$-th subquotients in the chromatic filtration consists of $v_n$-periodic elements. As illustrated in 2.4.2, these elements fall into periodic families. The chromatic filtration is thus like a spectrum in the astronomical sense in that it resolves the stable homotopy groups of a finite complex into periodic families of various periods. Comparing these to the colors of the rainbow led us to the word 'chromatic'.

By the construction of [[complex oriented cohomology theories]] from [[formal groups]] (via the [[Landweber exact functor theorem]]), the [[height of a formal group|height filtration]] of formal groups induces a "chromatic" [[filtration]] on [[complex oriented cohomology theories]].
_Chromatic homotopy theory_ is the study of [[stable homotopy theory]] and specifically of [[complex oriented cohomology theories]] by means of and along this [[chromatic filtration]].

More abstractly, this filtering is induced by the [[prime spectrum of a symmetric monoidal stable (∞,1)-category]] of the [[(∞,1)-category of spectra]] for [[p-localization|p-local]] [[finite spectra]], which is labeled by the [[Morava K-theories]].

More in detail, for each [[prime]] $p \in \mathbb{N}$ and for each  $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

$$
  L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)}
  \,,
$$

where $K(n)$ is the $n$th [[Morava K-theory]] (for the given [[prime]] $p$). These arrange into the [[chromatic tower]] which for each spectrum $X$ is of the form

$$
  X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
$$

The [[chromatic convergence theorem]] states mild conditions under which the [[homotopy limit]] over this tower is the $p$-[[Bousfield localization of spectra|localization]]

$$
  X \to X_{(p)}
$$

of $X$.

Since moreover $L_n X$ is the [[homotopy fiber product]] (see at _[[smash product theorem]]_ and see [this remark](fracture+theorem#ExampleOfChromaticFracturing) at _[[fracture square]]_ )

$$
  L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X
$$

it follows that in principle one may study a spectrum $X$ by understanding all its "chromatic pieces" $L_{K(n)} X$.

## Examples

[[!include chromatic tower examples - table]]

## Related concepts

* motivation: [[J-homomorphism and chromatic homotopy]]

* [[stable homotopy theory]]

* [[stable homotopy groups of spheres]], [[J-homomorphism]]

* [[complex oriented cohomology theory]]

* [[complex cobordism cohomology theory]], [[MU]]

  * [[Lazard's theorem]]

  * [[Quillen's theorem on MU]]

  * [[Landweber exact functor theorem]]

  * [[Landweber-Novikov theorem]]

  * [[Brown-Peterson spectrum]]

* [[formal group law]]

  * [[moduli stack of formal groups]]

  * [[height of a formal group]]

  * [[Morava stabilizer group]]

* [[Morava K-theory]]

  * [[∞-field]]

  * [[integral Morava K-theory]]

  * [[K(n)-local stable homotopy theory]]

  * [[Bousfield-Kuhn functor]]

* [[Morava E-theory]]

  * [[Lubin-Tate theory]]

* [[chromatic filtration]]

  * [[chromatic localization]]

  * [[chromatic tower]]

  * [[monochromatic layer]]

  * [[periodicity theorem]]

  * [[chromatic convergence theorem]]

* [[telescopic complexity]], [[telescopic localization]]

* [[Adams spectral sequence]]

  * [[Steenrod algebra]], [[Adams operation]]

  * [[chromatic spectral sequence]], [[May spectral sequence]]

* [[algebraic K-theory]], [[red-shift conjecture]]

* [[transchromatic character]]

* [[motivic chromatic homotopy theory]]

## References
 {#References}

### Stable case

Original articles:

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

* {#Ravenel92} [[Doug Ravenel]], _Nilpotence and Periodicity in Stable Homotopy Theory_, Annals of Mathematics Studies **128**, Princeton University Press (1992) [[ISBN:9780691025728](https://press.princeton.edu/books/paperback/9780691025728/nilpotence-and-periodicity-in-stable-homotopy-theory-am-128-volume), [pdf](https://people.math.rochester.edu/faculty/doug/mybooks/nilpb.pdf), [webpage](https://people.math.rochester.edu/faculty/doug/nilp.html)]

A quick idea:

* [[Mark Mahowald]], [[Doug Ravenel]], Section 6 of: _Towards  a Global Understanding of the Homotopy Groups of Spheres_, pages 57-74 in Part II of: _The Lefschetz Centennial Conference -- Proceedings on Algebraic Topology_, Proceedings of the Lefschetz Centennial Conference held December 10-14, 1984, Contemporary Mathematics 58, American Mathematical Society, 1987.  ([ISBN:978-0-8218-5063-3](https://bookstore.ams.org/conm-58-2), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf), [[MahowaldRavenelHomotopyGroupsOfSpheres.pdf:file]])

Historical introduction:

* [[Jack Morava]], _Complex cobordism and algebraic topology_ ([arXiv:0707.3216](http://arxiv.org/abs/0707.3216))

Comprehensive lecture notes:

* {#LurieLectures} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html))

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_

Survey:

* [[Haynes Miller]], _"Chromatic" homotopy theory_ May 2011 ([pdf](http://www-math.mit.edu/~hrm/papers/chromatic.pdf))

* [[Tobias Barthel]], [[Agnès Beaudry]], *Chromatic structures in stable homotopy theory*, in [[Handbook of Homotopy Theory]], Chapman and Hall/CRC Press (2019) [[arXiv:1901.09004](https://arxiv.org/abs/1901.09004), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

A lightning review of results by Henn with Goerss, Mahowald, Rezk, and Karamanov:

* [[Hans-Werner Henn]], _Recent developments in chromatic stable homotopy theory_ [[pdf](https://etale.site/livetex/saft/henn.pdf)&rbrack

A collection of various lecture notes and other material is kept at

* [Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/conferences/talbot/index.php?year=2013), _[2013 Pre-Talbot Seminar Chromatic homotopy theoy](https://www.hiroleetanaka.com/pretalbot2013/index.php)_

See also:

* Glossary for stable and chromatic honotopy theory ([[StableChromaticGlossary.pdf:file]])

* {#Mehrle18} [[David Mehrle]], _Chromatic homotopy theory: Journey to the frontier_, Graduate workshop notes, Boulder 16-17 May 2018, ([pdf](http://pi.math.cornell.edu/~dmehrle/notes/conferences/cht/cht-notes.pdf), [[Mehrle_ChromaticHomotopyTheory.pdf:file]])

Discussion of generalization of [[elliptic cohomology]] to higher chromatic homotopy theory is discussed in

* [[Doug Ravenel]], _Toward higher chromatic analogs of elliptic cohomology_ [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/high1.pdf)

* [[Doug Ravenel]], _Toward higher chromatic analogs of elliptic cohomology II_, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1-36 ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/high2.pdf), [pdf slides](http://www.math.rochester.edu/people/faculty/doug/mypapers/halifax.pdf))

Relation of chromatic homotopy theory to [[Goodwillie calculus]] is discussed in

* [[Greg Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, Inventiones mathematicae, February 1999, Volume 135, Issue 3, pp 743-788 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf), [doi:10.1007/s002220050300](https://doi.org/10.1007/s002220050300))

* [[Nicholas Kuhn]], _Goodwillie towers and chromatic homotopy: an overview_,   Proceedings of the Nishida Fest (Kinosaki 2003),  245--279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007 ([arXiv:math/0410342](https://arxiv.org/abs/math/0410342), [doi:10.2140/gtm.2007.10.245](http://dx.doi.org/10.2140/gtm.2007.10.245))

Categorical [[ultraproducts]] are used to provide asymptotic approximations in

* [[Tobias Barthel]], [[Tomer Schlank]], [[Nathaniel Stapleton]], _Chromatic homotopy theory is asymptotically algebraic_, ([arXiv:1711.00844](https://arxiv.org/abs/1711.00844))

### Unstable case

There are also chromatic approximations in ordinary (not [[stabilization|stabilized]]) [[homotopy theory]]:

* [[Aldridge Bousfield]], _Localization and periodicity in unstable homotopy theory_, J. Amer. Math. Soc. 7 (1994), 831-873 ([doi:10.1090/S0894-0347-1994-1257059-7](https://doi.org/10.1090/S0894-0347-1994-1257059-7), [jstor:2152734](https://www.jstor.org/stable/2152734))

* {#Heuts18} [[Gijs Heuts]], _Lie algebras and $v_n$-periodic spaces_ ([arXiv:1803.06325](https://arxiv.org/abs/1803.06325))

* {#LurieHopkins18} [[Jacob Lurie]], [[Michael Hopkins]], _Unstable Chromatic Homotopy Theory_, lecture notes 2018 ([web](http://math.harvard.edu/~lurie/Thursday2017.html))

[[!redirects chromatic level]]
[[!redirects chromatic filtration]]
[[!redirects chromatic levels]]
[[!redirects chromatic stable homotopy theory]]


chromatic localization

#Contents#
* table of contents
{:toc}

## Idea

The [[Bousfield localization of spectra]] $L_{E(n)}$ at $n$th Morava E-theory is called _chromatic localization_. The [[tower diagram|tower]] of chromatic localizations as $n$ ranges is called the _[[chromatic tower]]_, leading to the _[[chromatic filtration]]_. This is the subject of [[chromatic homotopy theory]].

## Properties

### Relation to formal groups

Chromatic localization on [[complex oriented cohomology theories]] is like the restriction to the closed substack

$$
  \mathcal{M}_{FG}^{\leq n+1}
  \hookrightarrow
  \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}
$$

of the [[moduli stack of formal groups]] on those of [[height of a formal group|height]] $\geq n+1$.

(e.g. [Lurie, lect 22, above theorem 1](#LurieLect22))

In this way the localization tower at the Morava E-theories
exhibits the [[chromatic filtration]] in [[chromatic homotopy theory]].

## Examples

[[!include chromatic tower examples - table]]

## Related concepts

* [[chromatic homotopy theory]]

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html))
 {#LurieLectures}

* [[Mark Hovey]], Hal Sadofsky, _Invertible spectra in the $E(n)$-local stable homotopy category_  ([pdf](http://hopf.math.purdue.edu/Hovey-Sadofsky/lnpic.pdf))

[[!redirects chromatic localizations]]


chromatic spectral sequence

#Contents#
* table of contents
{:toc}

## Idea

A _chromatic spectral sequence_ is a [[spectral sequence of a filtered stable homotopy type]] for the case of a [[filtered object in an (infinity,1)-category|filtering]] given by a [[chromatic tower]]. ([Wilson 13](#Wilson13))

This is a type of [[spectral sequence]] useful for computing the $E^1$-term in an [[Adams-Novikov spectral sequence]]. The traditional formulation is due to [[Doug Ravenel]].

## Related concepts

[[!include Lurie spectral sequences -- table]]

## References

A review/introduction of the traditional account is around prop. 1.4.8 of

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_

more is in chapter 5, section 1 there.

The formulation in terms of the [[spectral sequence of a filtered stable homotopy type]] is in

* [[Dylan Wilson]] _Spectral Sequences from Sequences of Spectra: Towards the
Spectrum of the Category of Spectra_ lecture at [2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php) ([pdf](https://web.archive.org/web/20140909224838/http://www.math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-03-21-Dylan-Wilson-ANSS.pdf))
 {#Wilson13}

See also

* Wikipedia, _[Chromatic spectral sequence](http://en.wikipedia.org/wiki/Chromatic_spectral_sequence)_

[[!redirects chromatic spectral sequences]]


chromatic tower

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

#Contents#
* table of contents
{:toc}

## Idea

For each [[prime]] $p \in \mathbb{N}$ and for each [[natural number]]  $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

$$
  L_n \coloneqq L_{E(n)}
  \,,
$$

where $E(n)$ is the $n$th [[Morava E-theory]] (for the given [[prime]] $p$), the $n$th _[[chromatic localization]]_. These arrange into the _chromatic tower_ which for each spectrum $X$ is of the form

$$
  X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
$$

The [[homotopy fibers]] of each stage of the tower

$$
  M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))
$$

is called the $n$th _[[monochromatic layer]]_ of $X$.

The _[[chromatic convergence theorem]]_ states mild conditions under which the [[homotopy limit]] over this tower is the $p$-[[Bousfield localization of spectra|localization]]

$$
  X \to X_{(p)}
$$

of $X$.

Since moreover $L_n X$ is the [[homotopy fiber product]]

$$
  L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X
$$

(see at [[smash product theorem]]) it follows that in principle one can study a spectrum $X$ by understanding all its "chromatic pieces" $L_{K(n)} X$. This is the topic of [[chromatic homotopy theory]].

## Properties

### Chromatic spectral sequence

The [[spectral sequence of a filtered stable homotopy type]] associated with the chromatic tower (regarded as a [[filtered object in an (infinity,1)-category]]) is the _[[chromatic spectral sequence]]_ ([Wilson 13, section 2.1.2](#Wilson13))

[[!include Lurie spectral sequences -- table]]

## Examples

[[!include chromatic tower examples - table]]

## Related concepts

* [[Postnikov tower]], [[Taylor tower]]

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,
Lecture 29 _Telescopic vs $E_n$-localization_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture29.pdf))

* [[Dylan Wilson]] _Spectral Sequences from Sequences of Spectra: Towards the
Spectrum of the Category of Spectra_ lecture at [2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php) ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-03-21-Dylan-Wilson-ANSS.pdf))
 {#Wilson13}

[[!redirects chromatic towers]]


chromatic tower examples - table

**[[chromatic homotopy theory]]**

| [[chromatic level]] | [[complex oriented cohomology theory]] | [[E-∞ ring]]/[[A-∞ ring]] | [[real oriented cohomology theory]] |
|--|--|--|--|
| 0 | [[ordinary cohomology]] | [[Eilenberg-MacLane spectrum]] $H \mathbb{Z}$ | [[HZR-theory]] |
|  | 0th [[Morava K-theory]] | $K(0)$ |  |
| 1 | [[complex K-theory]] | [[complex K-theory spectrum]] $KU$ | [[KR-theory]] |
|   | first [[Morava K-theory]]  | $K(1)$ |  |
|   | first [[Morava E-theory]]  | $E(1)$ |  |
| 2 | [[elliptic cohomology]] | [[elliptic spectrum]] $Ell_E$  |   |
|   | second [[Morava K-theory]] | $K(2)$ |  |
|   | second [[Morava E-theory]] | $E(2)$ |  |
|   | [[algebraic K-theory]] of [[KU]] | $K(KU)$ | |
| 3 ...10 | [[K3 cohomology]] | [[K3 spectrum]] |  |
| $n$ | $n$th [[Morava K-theory]] | $K(n)$ |   |
|   | $n$th [[Morava E-theory]] | $E(n)$ | [[BPR-theory]] |
| $n+1$ | [[algebraic K-theory]] applied to chrom. level $n$ | $K(E_n)$ ([[red-shift conjecture]]) | |
| $\infty$ | [[complex cobordism cohomology]] | [[MU]]  |  [[MR-theory]] |