Ugo Dal Lago

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

## Selected writings

On [[linear logic]] and [[quantum circuit diagrams]]:

* {#LagoFaffian12} [[Ugo Dal Lago]], [[Claudia Faggian]], _On Multiplicative Linear Logic, Modality and Quantum Circuits_, EPTCS **95** (2012) 55-66 [[arXiv:1210.0613](http://arxiv.org/abs/1210.0613), [doi:10.4204/EPTCS.95.6](https://doi.org/10.4204/EPTCS.95.6)]

On ([[categorical semantics]] for) [[dynamic lifting]]  in the [[quantum programming language]] "[[Quipper]]":

* [[Andrea Colledan]], [[Ugo Dal Lago]], *On Dynamic Lifting and Effect Typing in Circuit Description Languages* [[arXiv:2202.07636](https://arxiv.org/abs/2202.07636)]

* [[Andrea Colledan]], [[Ugo Dal Lago]], *On Dynamic Lifting and Effect Typing in Circuit Description Languages*, talk at *TYPES Workshop*, Nantes (2022)  [[pdf](https://types22.inria.fr/files/2022/06/TYPES_2022_slides_13.pdf), [[ColledanLago-DynamicsLifting.pdf:file]]]

## Related entries

* [[linear logic]], [[linear type theory]]

+ [[dependent linear type theory]]

* [[quantum computation]]

category: people


Ulf Danielsson

* [webpage](https://ulfdanielsson.com/)

## Selected writings

On the apparent lack of any [[de Sitter spacetime]]-[[perturbative string theory vacua]] ([Danielsson-VanRiet 18](landscape+of+string+theory+vacua#DanielssonVanRiet18))

* _[Is string theory in crisis?](https://ulfdanielsson.com/is-string-theory-in-crisis/)_ (July 8, 2018)

On [[de Sitter spacetimes]] in the [[landscape of string theory vacua]] (or not):

* {#DanielssonVanRiet18} [[Ulf Danielsson]], [[Thomas Van Riet]],  _What if string theory has no de Sitter vacua?_, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018)  ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070))

Specifically on [[de Sitter spacetime]] [[cosmology]] [[brane world models]] in ambient $\sim$[[anti de Sitter spacetime|AdS]]-[[bulk]] [[spacetime]]:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

On [[D6-branes]]/[[KK-monopoles]] in [[M-theory]] discussed via [[G-structure]]:

* [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Adolfo Guarino, _KK-monopoles and $G$-structures in M-theory/type IIA reductions_, JHEP 1502 (2015) 096 ([arXiv:1411.0575](https://arxiv.org/abs/1411.0575))

On the [[swampland]]:

* {#Danielsson18} [[Ulf Danielsson]], _The quantum swampland_ ([arXiv:1809.04512](https://arxiv.org/abs/1809.04512))

  (argues that the issue with [[string theory|stringy]] de Sitter [[moduli stabilization]] raised in [Sethi 17](#Sethi17) is related to the de Sitter instability seen in QFT, according to the references [above](de+Sitter+spacetime#InPerturbativeQuantumGravity))

## related $n$Lab entries

* [[landscape of string theory vacua]]

* [[de Sitter spacetime]]

* [[swampland]]

category: people


Ulf Gran

* [webpage](http://www.gran.name)

## Selected writings

On [[Killing spinors]] in [[heterotic supergravity]]:

* [[Ulf Gran]], [[George Papadopoulos]], Diederik Roest, _Supersymmetric heterotic string backgrounds_, Phys.Lett.B656:119-126, 2007 ([arXiv:0706.4407](https://arxiv.org/abs/0706.4407))

## Related $n$Lab entries

* [[supergravity]]

* [[supergravity torsion constraints]]

category: people


Ulf Lindström

* [webpage](https://katalog.uu.se/profile/?id=N2-533)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ulf_Lindstr%C3%B6m)

## selected writings

* [[Paul Howe]], [[Ulf Lindström]], P. White, _Anomalies And Renormalization In The BRST-BV Framework_ , Phys. Lett. B246 (1990) 430.

* Björn Brinne, Svend E. Hjelmeland, [[Ulf Lindström]], _World-Volume Locally Supersymmetric Born-Infeld Actions_, Phys.Lett. B459 (1999) 507-514 ([arXiv:hep-th/9904175](http://arxiv.org/abs/hep-th/9904175))

* [[Ulf Lindström]], _Supersymmetric Sigma Model geometry_ ([arXiv:1207.1241](http://arxiv.org/abs/1207.1241))

* G. Bossard, [[Paul Howe]], Ulf Lindström, [[Kellogg Stelle]], L. Wulff, _Integral invariants in maximally supersymmetric Yang-Mills theories_ ([arXiv:1012.3142](http://arxiv.org/abs/1012.3142))

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

## related $n$Lab entries

* [[3-dimensional supergravity]]

* [[quantum anomaly]]

* [[supersymmetric sigma-model]]

* [[Green-Schwarz action functional]]

* [[T-duality]]

category: people

[[!redirects Ulf Lindstrom]]


Ulf Lindstöm

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

## related $n$Lab entries

* [[string theory]], [[T-duality]]

* [[supergravity]]

category: people


Ulf Norell

* [webpage](http://www.cse.chalmers.se/~ulfn/)

## Selected writings

Introducing the [[dependent type theory|dependently typed]] [[proof assistant]]/[[programming language]] *[[Agda]]*:

* [[Ulf Norell]], _Towards a practical programming language based on dependent type theory_, PhD thesis (2007) [[pdf](https://www.cse.chalmers.se/~ulfn/papers/thesis.pdf), [[Norell-PracticalDTT.pdf:file]]]

* [[Ulf Norell]], *Dependently Typed Programming in Agda*, p. 230-266 in: *Advanced Functional Programming* AFP 2008. Lecture Notes in Computer Science **5832** (2009) [[doi:10.1007/978-3-642-04652-0_5](https://doi.org/10.1007/978-3-642-04652-0_5), [pdf](https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf)]

Introduction:

* [[Ulf Norell]], *Programming in Agda*, presentation at *Oregon Programming Languages Summer School* (2014) [lecture 1: [video](https://www.youtube.com/watch?v=NrSW7YsneVg), 2: [video](https://www.youtube.com/watch?v=X0JWsoWTWnI), 3: [video](https://www.youtube.com/watch?v=HjpLZr_sirY), 4: [video](https://www.youtube.com/watch?v=Yw2VGwxn_g8)]

## Related entries

* [[Agda]]

category: people


Ulf-G. Meissner

* [webpage](https://www.itkp.uni-bonn.de/~meissner)

## Selected writings

On [[chiral perturbation theory]]:

* [[Véronique Bernard]], [[Ulf-G. Meissner]], _Chiral perturbation theory_, Ann. Rev. Nucl. Part. Sci.57:33-60, 2007 ([arXiv:hep-ph/0611231](https://arxiv.org/abs/hep-ph/0611231))

On [[nucleons]] as [[Skyrmions]] in [[quantum hadrodynamics]] with [[pions]], [[omega-mesons]] and [[rho-mesons]] ($\pi$-$\rho$-$\omega$-model):

* {#MeissnerZahed86} [[Ulf-G. Meissner]], [[Ismail Zahed]], _Skyrmions in the Presence of Vector Mesons_, Phys. Rev. Lett. 56, 1035 (1986) ([doi:10.1103/PhysRevLett.56.1035](https://doi.org/10.1103/PhysRevLett.56.1035))

  (includes also the [[A1-meson]])

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], [[Wolfram Weise]], _Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties_, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

* [[Ulf-G. Meissner]], [[Norbert Kaiser]], Andreas Wirzba, [[Wolfram Weise]], _Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons_, Phys. Rev. Lett. 57, 1676 (1986) ([doi:10.1103/PhysRevLett.57.1676](https://doi.org/10.1103/PhysRevLett.57.1676))

* [[Ulf-G. Meissner]], _Low-energy hadron physics from effective chiral Lagrangians with vector mesons_, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

On [[baryon chiral perturbation theory]]:

* [[Ulf-G. Meissner]], _Chiral QCD: Baryon dynamics_,in:  At The Frontier of Particle Physics, pp. 417-505 (2001) ([arxiv:hep-ph/0007092](https://arxiv.org/abs/hep-ph/0007092))

* Matthias Frink, [[Ulf-G. Meissner]], _On the chiral effective meson-baryon Lagrangian at third order_, Eur. Phys. J. A29:255-260, 2006 ([arXiv:hep-ph/0609256](https://arxiv.org/abs/hep-ph/0609256))

category: people

[[!redirects U.-G. Meissner]]


Uli Wagner

+ [institute page](https://ist.ac.at/en/research/wagner-group/)

## Selected writings

Discussion of [[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))

* [[Marek Filakovsky]], [[Peter Franek]], [[Uli Wagner]], Stephan Zhechev, _Computing simplicial representatives of homotopy group elements_, J Appl. and Comput. Topology (2018) 2: 177. ([arXiv:1706.00380](https://arxiv.org/abs/1706.00380))

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))

category: people


Ulrich Bauer

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

## Selected writings

On the [[stability of persistence diagrams]]:

* [[Ulrich Bauer]], [[Michael Lesnick]], *Induced Matchings and the Algebraic Stability of Persistence Barcodes*, Journal of Computational Geometry **6** 2 (2015) 162-191 $[$[arXiv:1311.3681](https://arxiv.org/abs/1311.3681), [doi:10.20382/jocg.v6i2a9](https://doi.org/10.20382/jocg.v6i2a9)$]$

On [[topological machine learning]] via [[kernel methods]] applied to [[persistence diagrams]]:

* {#KHNL15} Roland Kwitt, [[Stefan Huber]], Marc Niethammer, Weili Lin, [[Ulrich Bauer]], *Statistical Topological Data Analysis -- A Kernel Perspective*, in: *Advances in Neural Information Processing Systems*  (NIPS 2015) $[$[ISBN:9781510825024](https://proceedings.neurips.cc/paper/2015), [doi:10.5555/2969442.2969582](https://dl.acm.org/doi/10.5555/2969442.2969582)$]$

category: people


Ulrich Bunke

* [website](http://www.mathematik.uni-regensburg.de/Bunke/)

## Selected writings

On [[topological T-duality]]:

* {#BunkeSchick04} [[Ulrich Bunke]], [[Thomas Schick]], _On the topology of T-duality_, Rev. Math. Phys. **17** (2005) 77-112 [[arXiv:math/0405132](https://arxiv.org/abs/math/0405132), [doi:10.1142/S0129055X05002315](https://doi.org/10.1142/S0129055X05002315)]

* {#BunkeRumpfSchick08} {#BunkeRumpfSchick08} [[Ulrich Bunke]], P. Rumpf, [[Thomas Schick]], _The topology of $T$-duality for $T^n$-bundles_,  Rev. Math. Phys. **18** 1103 (2006) [[arXiv:math.GT/0501487](http://arxiv.org/abs/math.GT/0501487), [doi:10.1142/S0129055X06002875](https://doi.org/10.1142/S0129055X06002875)]

On [[twisted ad-equivariant K-theory]] via [[differentiable stacks]]:

* [[Ulrich Bunke]], Ingo Schroeder, _Twisted K-Theory and TQFT_ ([arXiv:math/0504458](https://arxiv.org/abs/math/0504458), [spire:681555](https://inspirehep.net/literature/681555))

On [[differential cohomology]]:

* [[Ulrich Bunke]], _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

On [[differential K-theory]]:

* {#BunkeSchickReferences} [[Ulrich Bunke]], [[Thomas Schick]], _Smooth K-Theory_, Astérisque 328 (2009), 45-135 ([arXiv:0707.0046](http://arxiv.org/abs/0707.0046), [numdam:AST_2009__328__45_0](http://www.numdam.org/item/?id=AST_2009__328__45_0))

On [[differential cohomology]] and the [[e-invariant]]:

* [[Ulrich Bunke]], Section 2 of: _Differential cohomology in geometry and analysis_, talk 2008 ([pdf](https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Bunke/Vortrag-Erlangen.pdf), [[BunkeEInvariantErlangen2008.pdf:file]])

On [[differential cobordism cohomology]], i.e. enhancement of [[MU]] ([[complex cobordism cohomology]]) to [[differential cohomology]]:

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

On the [[differential cohomology hexagon]]:

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_, Journal of Homotopy and Related Structures October 2014 ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

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

* [[Ulrich Bunke]], [[Niko Naumann]], _Secondary Invariants for String Bordism and tmf_, Bulletin des Sciences Mathématiques __138__:8 (2014) 912--970 ([doi:10.1016/j.bulsci.2014.05.002](https://doi.org/10.1016/j.bulsci.2014.05.002))

Introducing [[differential algebraic K-theory]] with application to [[higher regulator|higher]] [[Beilinson regulators]]:

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_, Advances in Mathematics Volume 285, 5 November 2015, Pages 1853-1969
([arXiv:1209.6451](http://arxiv.org/abs/1209.6451), [doi:10.1016/j.aim.2015.08.004](https://doi.org/10.1016/j.aim.2015.08.004))

* [[Ulrich Bunke]], [[David Gepner]], _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_, Memoirs of the American Mathematical Society 2021 ([arXiv:1306.0247](httpS://arxiv.org/abs/1306.0247), [ISBN:978-1-4704-4685-7](https://bookstore.ams.org/cdn-1608276327798/memo-269-1316/))

* {#BunkeTamme13} [[Ulrich Bunke]], [[Georg Tamme]], _Multiplicative differential algebraic K-theory and applications_,  Ann. K-Theory 1(3): 227--258 (2016)   ([arXiv:1311.1421](http://arxiv.org/abs/1311.1421), [doi:10.2140/akt.2016.1.227](https://doi.org/10.2140/akt.2016.1.227))

On ([[equivariant K-theory|equivariant]]) [[KK-theory]] as the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]:

* {#BunkeEngelLand21} [[Ulrich Bunke]], [[Alexander Engel]], [[Markus Land]], *A stable $\infty$-category for equivariant KK-theory* $[$[arXiv:2102.13372](https://arxiv.org/abs/2102.13372)$]$

On (topological) [[inertia stack]]s

* [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], _Inertia and delocalized twisted cohomology_, Homotopy, Homology and Applications 10(1) 129--180 (2008) [math.KT/0609576](https://arxiv.org/abs/math/0609576)
[doi](https://doi.org/10.4310/HHA.2008.v10.n1.a6)

## Related entries

* [[differential cohomology]]

  * [[differential K-theory]]

* [[topological T-duality]]

* [[quantum anomaly]]

category: people

[[!redirects Uli Bunke]]


Ulrich Güntzer

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=35709)

## related $n$Lab entries

* [[rigid analytic geometry]]

category: people


Ulrich Kraehmer

Ulrich Krähmer is a German mathematician now with position in Glasgow, Scotland. His interests include [[quantum group]]s, [[quantum homogeneous space]]s, [[Dirac operators in noncommutative geometry]] and [[cyclic homology]].

* [homepage](http://www.maths.gla.ac.uk/~ukraehmer)

category: people

[[!redirects Ulrich Krähmer]]
[[!redirects Uli Krähmer]]
[[!redirects U. Krähmer]]
[[!redirects Ulrich Kraehmer]]
[[!redirects Uli Kraehmer]]
[[!redirects U. Kraehmer]]
[[!redirects Ulrich Krahmer]]
[[!redirects Uli Krahmer]]
[[!redirects U. Krahmer]]


Ulrich Majer

* [webpage](http://www.uni-goettingen.de/en/71635.html)

## Selected writings

On [[David Hilbert]]'s lectures on [[general relativity]], [[quantum physics]] and [[epistemology]]:

* {#SauerMajer} [[Tilman Sauer]], [[Ulrich Majer]] with  Arne Schirrmacher, Heinz-Jürgen Schmidt (eds.): *David Hilbert's Lectures on the foundations of physics 1915-1927 --- Relativity, Quantum Theory and Epistemology*, Springer (2009) [[doi:10.1007/b12915](https://doi.org/10.1007/b12915)]

## Related entries

* [[general relativity]]

* [[Einstein-Hilbert action]]

category: people


Ulrich Mosel

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

* [Institute page](https://www.uni-giessen.de/fbz/fb07/fachgebiete/physik/institute/theorie/emeriti/mosel)

## Selected writings

On [[quantum field theory]], [[particle physics]] and [[hadrodynamics]]:

* [[Ulrich Mosel]], *Fields, Symmetries, and Quarks*, Springer 1999 ([doi:10.1007/978-3-662-03841-3](https://link.springer.com/book/10.1007/978-3-662-03841-3))

category: people


Ulrich Pennig

* [webpage](http://www.cardiff.ac.uk/people/view/250949-pennig-ulrich)

## Selected writings

On [[higher twisted K-theory]] via [[C-star algebras|$C^\ast$-algebras]]:

* {#Pennig16} [[Ulrich Pennig]], *A noncommutative model for higher twisted K-Theory*, J Topology (2016) 9 (1): 27-50 ([arXiv:1502.02807](https://arxiv.org/abs/1502.02807))

* [[Marius Dadarlat]], [[Ulrich Pennig]], *Unit spectra of K-theory from strongly self-absorbing $C^\ast$-algebras*, Algebr. Geom. Topol. 15 (2015) 137-168 ([arXiv:1306.2583](https://arxiv.org/abs/1306.2583))

* [[David E. Evans]], [[Ulrich Pennig]], *Spectral Sequence Computation of Higher Twisted K-Groups of $SU(n)$* [[arXiv:2307.00423](https://arxiv.org/abs/2307.00423)]

* [[Ulrich Pennig]], *Equivariant Higher Twisted K-Theory of $SU(n)$ via Exponential Functors*, [talk at](CQTS#PennigSep2023) [[CQTS]] (20 Sep 2023) [video:[YT](https://youtu.be/JXO-owcJgTE)]

category: people


Ulrich Rößler

* [Wikipedia entry](https://de.wikipedia.org/wiki/Ulrich_R%C3%B6%C3%9Fler) (in Deutsch)

## Selected writings

On [[solid state physics]]:

* [[Ulrich Rößler]], _Solid State Theory: An Introduction_, Springer (2004, 2009) $[$[doi:10.1007/978-3-540-92762-4](https://link.springer.com/book/10.1007/978-3-540-92762-4)$]$

category: people

[[!redirects Ulrich Rössler]]
[[!redirects Ulrich Roessler]]
[[!redirects Ulrich Rossler]]


Ulrich Schöpp

[webpage](https://www.tcs.ifi.lmu.de/people/ulrich-schoepp)

## Selected writings

On a form of [[bunched logic|bunched]] [[dependent linear type theory]]:

* {#SchöppStark04} [[Ulrich Schöpp]], [[Ian Stark]], *A Dependent Type Theory with Names and Binding*, in *Computer Science Logic. CSL 2004*, Lecture Notes in Computer Science **3210** (2004) 235–249 [[doi:10.1007/978-3-540-30124-0_20](https://doi.org/10.1007/978-3-540-30124-0_20), [web](https://homepages.inf.ed.ac.uk/stark/names+binding.html)]

## Related entries

* [[continuation-passing style]]

* [[Geometry of Interaction]]

* [[traced monoidal category]]

category: people

category: computer science

[[!redirects Ulrich Schoepp]]
[[!redirects Ulrich Schopp]]


Ulrik Buchholtz

* [webpage](https://ulrikbuchholtz.dk/)

## Selected writings

On [[cellular cohomology]] formulated in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[Favonia|Kuen-Bang Hou (Favonia)]], _Cellular Cohomology in Homotopy Type Theory_, Logical Methods in Computer Science, Volume 16, Issue 2 (June 1, 2020) ([arXiv:1802.02191](https://arxiv.org/abs/1802.02191), [lmcs:6518](https://lmcs.episciences.org/6518))

Discussion of [[∞-groups]] from the point of view of [[homotopy type theory]]:

* {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], *[[Higher Groups in Homotopy Type Theory]]*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 [[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)]

* [[Ulrik Buchholtz]], *Notes on higher groups and projective spaces*, 2016 ([pdf](https://www2.mathematik.tu-darmstadt.de/~buchholtz/higher-groups.pdf))

On [[symmetry]] and introducing the language of [[homotopy type theory]] for [[univalent foundations of mathematics]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

On [[synthetic (infinity,1)-category theory|$(\infty,1)$-category theory]]:

* {#BuchholtzWeinberger21} [[Ulrik Buchholtz]], [[Jonathan Weinberger]], *Synthetic fibered $(\infty,1)$-category theory* $[$[arXiv:2105.01724](https://arxiv.org/abs/2105.01724), [talk slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weinberger-2022-01-20-HoTTEST.pdf)$]$

On [[delooping]], [[H-spaces]] and [[Eilenberg-MacLane spaces]] in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

On higher structures in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], *Higher Structures in Homotopy Type Theory*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_7](https://doi.org/10.1007/978-3-030-15655-8_7), [arXiv:1807.02177](https://arxiv.org/abs/1807.02177)]

On [[epimorphisms]] and [[acyclic types]] in [[univalent type theory]]:

* [[Ulrik Buchholtz]], [[Tom de Jong]], [[Egbert Rijke]], *Epimorphisms and Acyclic Types in Univalent Mathematics*, ([arXiv:2401.14106](https://arxiv.org/abs/2401.14106))

category: people

[[!redirects UlrikBuchholtz]]


UlrikBuchholtz > history


Ulrike Tillmann

Ulrike Tillmann is professor for pure [[mathematics]] at Oxford.

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

## Selected writings

On [[configuration spaces of points]] ans the [[Cohomotopy charge map]]:

* {#ManthorpeTillmann} Richard Manthorpe, [[Ulrike Tillmann]], _Tubular configurations: equivariant scanning and splitting_, Journal of the London Mathematical Society, Volume 90, Issue 3 ([arxiv:1307.5669](https://arxiv.org/abs/1307.5669), [doi:10.1112/jlms/jdu050](https://doi.org/10.1112/jlms/jdu050))

## Related $n$Lab entries

* [[Galatius-Madsen-Tillmann-Weiss theorem]]

category: people


ultra-high energy cosmic ray

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

#Contents#
* table of contents
{:toc}

## References

* Peter L. Biermann et. al _The Nature and Origin of Ultra-High Energy Cosmic Ray Particles_ ([arXiv:1610.00944](https://arxiv.org/abs/1610.00944))

* Karl-Heinz Kampert , _Ultra-High Energy Cosmic Rays: Recent Results and Future Plans of Auger_ ([arXiv:1612.08188](https://arxiv.org/abs/1612.08188))

* Pierre Auger Collaboration, _Observation of a Large-scale Anisotropy in the Arrival Directions of Cosmic Rays above $8 \times 10^{18} ev$_ ([arXiv:1709.07321](https://arxiv.org/abs/1709.07321))

See also

* Wikipedia, _[Ultra-high-energy cosmic ray](https://en.wikipedia.org/wiki/Ultra-high-energy_cosmic_ray)_

## Related entries

* [[radiation]]

[[!redirects ultra-high energy cosmic rays]]


ultracategory

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### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

### Idea

**Ultracategories** are [[categories]] with [[extra structure]], called an _ultrastructure_ (see [Lurie, Sec 1.3](#Lurie)). For an ultracategory, $\mathcal{A}$, its ultrastructure assigns to a set of objects of $\mathcal{A}$ indexed by a set, $S$, equipped with an ultrafilter, $\mu$, the _categorical ultraproduct_, $\int_S A_s d \mu$, an object of $\mathcal{A}$.

Ultracategories were introduced in [Makkai 87](#Makkai87) in order to prove [[conceptual completeness]], but note that Lurie's definition slightly differs from Makkai's ([Lurie, Warning 1.0.4](#Lurie)).

(For a conjecture that ultracategories are a kind of generalized multicategory, see [Shulman](#Shulman).)

In ([Clementino-Tholen 03](#ClemTholen)), a different concept of ultracategory is introduced as an instance of a [[generalized multicategory]].

### References

* {#Makkai87} [[Mihaly Makkai]], _Stone duality for first-order logic_, Adv. Math. __65__ (1987) no. 2, 97--170, [doi](http://dx.doi.org/10.1016/0001-8708%2887%2990020-X),  [MR89h:03067](http://www.ams.org/mathscinet-getitem?mr=900266)

* Marek W. Zawadowski, _Descent and duality_, Annals of Pure and Applied Logic __71__, n.2 (1995), 131–188

* {#Lurie} [[Jacob Lurie]], _Ultracategories_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Conceptual.pdf))

For a conjecture that ultracategories are a kind of generalized multicategory see

* {#Shulman} [[Mike Shulman]], [blog comment](https://golem.ph.utexas.edu/category/2010/01/generalized_multicategories.html#c058061)

For a different notion of ultracategory see

* {#ClemTholen} [[Maria Manuel Clementino]], [[Walter Tholen]], _Metric, topology and multicategory--a common approach_, J. Pure Appl. Algebra 179 (2003) 13-47, ([doi:10.1016/S0022-4049(02)00246-3](https://doi.org/10.1016/S0022-4049%2802%2900246-3)).

On a [[2-monad|2-monadic]] treatment of ultracategories:

* [[Francisco Marmolejo]], _Ultraproducts and continuous families of models_, PhD thesis (1995) [[hdl:10222/55092](https://dalspace.library.dal.ca/handle/10222/55092), [pdf](https://dalspace.library.dal.ca/bitstream/handle/10222/55092/NN08788.PDF?sequence=1&isAllowed=y)]

[[!redirects ultracategories]]


ultracommutative ring spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[commutative ring spectrum]]

## References

* [[Stefan Schwede]], Section 5 of: _[[Global homotopy theory]]_, New Mathematical Monographs 34, Cambridge University Press, 2018 ([doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161), [arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

[[!redirects ultracommutative ring spectra]]


ultradistribution

__Ultradistribution__ is an element of the space $\mathcal{Z}'$ dual to the space $\mathcal{Z}$ of test functions whose Fourier transform is in the space $\mathcal{D}$ of infinitely differentiable functions of compact support.

Related $n$Lab pages include [[distribution]]

* R. F. Hoskins, J. Sousa Pinto, _Theories of generalized functions: Dis-
tributions, ultradistributions and other generalized functions_, 2nd. edition
Woodhead Publ., Philadelphia 2011.
* Rui Vilela Mendes, _Stochastic solutions of nonlinear PDE's and an extension of superprocesses_, [arxiv/1209.3263](https://arxiv.org/abs/1209.3263)
* Morisuke Hasumi, _Note on the $n$-tempered ultra-distributions_, Tohoku Math. J. (2) 13 (1961), no. 1, 94--104 [doi](http://10.2748/tmj/1178244354) [euclid](http://projecteuclid.org/euclid.tmj/1178244354)
* J. Sousa Pinto, R. F. Hoskins, _A nonstandard definition of finite order ultradistributions_, Proc. Indian Acad. Sci. - Mathematical Sciences (1999) 109:4, pp 389–395 [doi](http://dx.doi.org/10.1007/BF02837998)

category: analysis

[[!redirects ultradistributions]]


ultrafilter

# Ultrafilters
* table of contents
{: toc}

## Definitions

An __ultrafilter__ on a set $S$ is a collection $F$ of [[subset]]s of $S$ satisfying the axiom
$$ A \in F \;\Leftrightarrow\; \forall (B_1, \ldots, B_n \in F),\; \exists (x \in A \cap B_1 \cap \cdots \cap B_n) .$$
This is the only axiom necessary; from this, you can prove that $F$ is a [[filter]].  Alternatively, if we start by requiring $F$ to be a filter, then we need add only the axiom
$$ A \in F \;\Leftrightarrow\; \forall B \in F,\; \exists x \in A \cap B .$$
Or if we start by requiring $F$ to be a [[proper filter]], then we need only the $\Leftarrow$ half of this latter axiom.

We may also define an ultrafilter to be [[maximal element|maximal]] among the proper filters.  This definition generalises from the [[power set]] of $S$ to any [[poset]] $L$; notice that we speak of an ultrafilter _on_ $S$ but an ultrafilter _in_ $L$.  In a distributive [[lattice]], every ultrafilter is [[prime filter|prime]]; the converse holds in a [[Boolean algebra]].

Using [[excluded middle]], it is equivalent to say that a filter $F$ on $S$ is an ultrafilter if, given any subset $A$ of $S$, either $A$ or its complement belongs to $F$; this version generalises to any [[Boolean algebra]].  Another way to define an ultrafilter in a Boolean algebra $L$ is as a Boolean-algebra homomorphism from $L$ to the set $\{\bot,\top\}$ of Boolean [[truth value]]s.

Given an element $x$ of a set $S$, the __principal ultrafilter__ (on $S$) at $x$ consists of every [[subset]] of $S$ to which $x$ belongs.  An ultrafilter $F$ is __fixed__ if the [[intersection]] of its elements is [[inhabited subset|inhabited]], in which case that intersection must be a [[singleton]] $\{x\}$ and $F$ is the principal ultrafilter at $x$.

In contrast, if this intersection [[empty set|empty]], then we call $F$ a __free ultrafilter__.  It is possible, if one denies the [[axiom of choice]], that every ultrafilter of subsets is fixed.  In contrast, the [[ultrafilter principle]] (a weak form of the [[axiom of choice]]) states that any proper filter (in any Boolean lattice) may be extended to an ultrafilter.  Then any [[infinite set]] has a free proper filter (such as the filter of [[cofinite subset]]s) and so a free ultrafilter.

Free ultrafilters are important in [[nonstandard analysis]] and [[model theory]], where the ultrafilter principle seems to be a necessity.

## Category-theoretic interpretations

There are in fact many interrelated ways of defining ultrafilters. We present a few here.

### Ultrafilters form a monad

For any set $X$, let $U X$ be the set of ultrafilters on $X$.  Principal ultrafilters provide an inclusion $\eta\colon X\to U X$, which turns out to be the unit of a [[monad]] on $Set$, as briefly touched upon above.  The multiplication can be described fairly explicitly.  First of all, if $A\subseteq X$, define $[A]\subseteq U X$ to be the set of all ultrafilters containing $A$.  Then given $\mathcal{F} \in U U X$, i.e. an ultrafilter of ultrafilters, we let $\mu(\mathcal{F}) = \{ A | [A] \in \mathcal{F} \}$; one can verify that this is an ultrafilter and makes $U$ into a monad. This monad is traditionally denoted $\beta$.

The ultrafilter monad can also be described as follows. The 2-element set carries a unique structure of [[Boolean algebra]] internal to the category of sets. When thus cast in the role of [[dualizing object]], it induces an adjoint pair of functors

$$(Set \stackrel{P = Set(-, \mathbf{2})^{op}}{\to} Bool^{op}) \; \; \dashv \; \; (Bool^{op} \stackrel{Bool(-, \mathbf{2})}{\to} Set)$$

whose corresponding monad $Bool(P -, \mathbf{2})$ is canonically identified with the ultrafilter monad $\beta$. Another description (due to Kennison and Gildenhuys) is that it is the [[codensity monad]] induced from the full embedding $Fin \hookrightarrow Set$ of finite sets into $Set$. See [Leinster](http://www.tac.mta.ca/tac/volumes/28/13/28-13.pdf) for a full account, and some extensions.

### Ultrafilters and coproducts

Observe that the canonical map

$$\beta(X) + \beta(Y) \to \beta(X + Y)$$

is an isomorphism (for, each ultrafilter $U$ on $X + Y$ must contain exactly one of $X, Y$; if it contains $X$, then it is generated by the ultrafilter on $X$ given by $\{X \cap A: A \in U\}$).

+-- {: .un_theorem}
###### Theorem (R. Börger)
The endofunctor $\beta$ is terminal among endofunctors $Set \to Set$ that preserve finite coproducts.
=--

A topological proof of this fact has been given by Richter (see references).

This gives one universal characterization of the ultrafilter endofunctor. This theorem implies that there is precisely one [[monad]] structure on $\beta$, and that if $F\colon Set \to Set$ is a monad which preserves finite coproducts, then the unique transformation $F \to \beta$ is a morphism of monads.

### Ultrafilters and presheaves

\begin{theorem}
([Garner, 2018](#Garner).)
The category of [[endofunctors]] $Set \to Set$ that preserve [[finite coproducts]]
is equivalent to the category of [[presheaves]] of sets on the Blass category $U F$ of ultrafilters.
The ultrapower functor with respect to an ultrafilter $U$
corresponds to the representable functor of $U$.
\end{theorem}

The Blass category of ultrafilters has as objects, pairs of a set $X$ and an ultrafilter $\mathcal{U}$ of $\beta X$. Morphisms from $(X, \mathcal{U})$ to $(Y, \mathcal{V})$ are $=_{\mathcal{U}}$-equivalence classes of [[partial function|partial]] continuous maps defined on a set in $\mathcal{U}$, continuity here meaning that for any $V \in \mathcal{V}$, $f^{-1}(V) \in \mathcal{U}$.

### Ultrafilters and codensity

Another known universal characterization of the ultrafilter monad is via the concept of [[codensity monads]]. (This fact was recently (June 2011) mentioned by [[Tom Leinster]] at the categories list, but it also appears sparsely in the literature; see for instance _Algebraic Theories_, exercise 3.2.12(e).)

Let $i: Fin \to Set$ be the usual full inclusion of [[finite sets]] into sets. The restricted [[Yoneda embedding]]

$$Set \stackrel{y^{op}}{\to} (Set^{Set})^{op} \stackrel{(Set^i)^{op}}{\to} (Set^{Fin})^{op},$$

taking a set $X$ contravariantly to the functor $Set(X, i-)$, has a [[right adjoint]]

$$Set^{Fin}(-, i): (Set^{Fin})^{op} \to Set$$

(Note: this construction is dual to the familiar [[Kan construction]], which takes as input a functor $F: A \to C$ from a small category to a cocomplete category, and produces as output an adjoint pair whose right adjoint is a functor $C \to Set^{A^{op}}$; see for instance [[nerve and realization]].)

The composite of these functors gives a monad $Set \to Set$ which in fact is just the ultrafilter monad:

$$\beta \cong Set^{Fin}(Set(X, i-), Set(1, i-))$$

One way of considering the codensity monad formulation is that the ultrafilter functor is terminal among endofunctors whose restriction to the category of finite sets is the identity functor. This likewise gives the uniqueness of the monad structure, as well as the fact that $\beta$ is the terminal monad that restricts to $1_{Fin}$.

### Ultrafilters and unbiased Boolean algebras

The codensity monad description is related to the traditional description in terms of Boolean algebras; to see this more clearly, it helps to pass to [[Boolean algebra|unbiased Boolean algebras]], which are equivalent to Boolean algebras.

Recall that an unbiased Boolean algebra is a product-preserving functor

$$Fin_+ \to Set$$

where $Fin_+$ is the category of [[inhabited set|inhabited]] [[finite set|finite]] [[sets]]. If $X$ is any set, there is an unbiased Boolean algebra

$$Set(X, i_+ -): Fin_+ \to Set$$

which may be called the _unbiased power set_. (Here $i_+: Fin_+ \to Set$ denotes the standard inclusion.) We may denote this by $(-)^X$ for short. The inclusion functor $i_+$ is identified with $(-)^1$, and corresponds to the Boolean algebra $2$.

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

An _ultrafilter on_ $X$ is a [[natural transformation]] $(-)^X \to (-)^1$ in the category $Prod(Fin_+, Set)$.
=--

Of course this is the same as a natural transformation $(-)^X \to (-)^1$ in the usual [[functor category]] $Set^{Fin_+}$. Such transformations are in bijection with transformations

$$Set(X, i-) \to Set(1, i-)$$

in $Set^{Fin}$ used in the codensity monad description of ultrafilters.

### Ultrafilters via other algebraic theories

There are other descriptions of ultrafilters, based on [$k$-valued Post algebras](Boolean+algebra#PostAlg). Recall the proposition there:

* Let $M_k$ be the monoid of [[endofunctions]] on a $k$-element set, for a finite set of cardinality $k \geq 3$. The forgetful functor $U_k: BoolAlg \to M_k\text{-}Set$ is full and faithful.

Let us take for example $k = 3$. Given a set $X$, the functor $U_3$ takes the unbiased Boolean algebra $Set(X, i_+ -)$ to the set $Set(X, 3) = 3^X$, regarded as a set equipped with the canonical (pointwise) action of $Set(3, 3) = M_3$. It takes an ultrafilter on $X$, i.e., an unbiased Boolean algebra map

$$Set(X, i_+ -) \to Set(1, i_+ -),$$

to a morphism of $M_3$-sets

$$Set(X, 3) \to 3.$$

The proposition implies that the mapping

$$\beta(X) \to M_3\text{-}Set(3^X, 3)$$

is a natural bijection, giving us another way of viewing ultrafilters on $X$, this time involving only _unary_ operations.

This is of course another codensity monad description in disguise; this time the codensity monad is induced by the full inclusion

$$B M_3 \to Set$$

of the category of endofunctions on the 3-element set into the category of sets. The results of this section say that the ultrafilter monad is the codensity monad for this particular inclusion.

In a post to the categories list (see references), [[Bill Lawvere]] remarks that large cardinal hypotheses can be formulated as obstructions to similar codensity monads being isomorphic to the identity. For example, the existence of [[infinite sets]] is equivalent (assuming the [[ultrafilter principle]]) to the fact that $\beta$ is not isomorphic to the identity. Or, for the full inclusion

$$M_{\mathbb{N}} \to Set$$

the existence of an uncountable [[measurable cardinal]] is equivalent (in ZFC) to the fact that the induced codensity monad is not isomorphic to the identity (i.e., the first uncountable measurable cardinal is where the first "jump" takes place). Lawvere gives a couple more examples of a more geometric nature.

## Algebras of the ultrafilter monad {#comphaus}

A theorem due to [[Ernest Manes]] is that the [[Eilenberg-Moore category]] of ultrafilter monad is the category of [[compact Hausdorff spaces]] with its obvious forgetful functor to $Set$.

* In one direction, if $X$ is a compact Hausdorff space, then the corresponding algebra structure
$$\xi: \beta X \to X$$
sends an ultrafilter $F$ on $X$ to the unique point in $X$ to which $F$ converges. ($F$ **converges to** $x$ if the filter of neighborhoods of $x$ is contained in $F$; see [[convergence space]].)

* In the other direction, given an algebra structure $\xi\colon \beta X \to X$, we define a topology by defining a set $U \subseteq X$ to be [[open subset|open]] if
$$\forall_{x \in X} \forall_{F \in \beta X} (x \in U \: and \: \xi(F) = x) \Rightarrow U \in F$$
The sense of this is that a set is open if it is a neighborhood of each of its points, and the neighborhood filter of $x$ is the intersection of all ultrafilters that converge to $x$. (The unit condition is that
$$\xi (prin(x)) = x$$
so that the neighborhood filter of $x$ is contained in $prin(x)$, i.e., each neighborhood of $x$ contains $x$.) It is not hard to verify that this condition indeed defines a topology. In fact, the topology is compact Hausdorff, essentially because [[compact space|compactness]] is equivalent to having every ultrafilter converge to some point, and [[Hausdorff space|Hausdorffness]] to having that point be unique.

In particular, $\beta X$ can be equipped with a compact Hausdorff topology which is the [[free object|free]] compact Hausdorff space generated by $X$, or equivalently the [[Stone-Čech compactification]] of the [[discrete topology]] on $X$.

The monad $\beta$ also extends to the [[bicategory]] $Rel$ of sets and binary [[relations]].  It was observed by Barr that the [[generalized multicategories]] defined relative to this extension can be identified with arbitrary [[topological spaces]]; see [[relational beta-module]].  Thus, compact Hausdorff spaces are to topological spaces as [[monoidal categories]] are to [[multicategories]].

By exploiting the connection between monads and [[algebraic theory|algebraic theories]], it is possible to define a **compact Hausdorff object** in any category $C$ with small products, as a product-preserving functor

$$Kl(\beta)^{op} \to C$$

where $Kl(\beta)$ is the [[Kleisli category]] (or the full category of compact Hausdorff spaces whose objects are of the form $\beta S$).

## Number of ultrafilters

If $X$ is a [[finite set]], then all ultrafilters on $X$ are principal and the number of them is the cardinality of $X$.

If $X$ is an [[infinite set]] of cardinality $\kappa$, then the number of ultrafilters on $X$ is $2^{2^\kappa}$. A proof may be found at [[independent family of sets]].

## References

* R. Börger, _Coproducts and Ultrafilters_, JPAA **46** (1987) pp.35-47.

* E. Manes, _Algebraic Theories_, Graduate Texts in Mathematics **26**, Springer-Verlag, 1976.

* G. Richter, _Axiomatizing the Category of Compact Hausdorff Spaces_, Categories at Work (ed. Herrlich and Porst), Heldermann-Verlag (1991), 199-215. [(link)](http://www.heldermann.de/R&E/RAE18/ctw12.pdf)

* [[Tom Leinster]], _Post to the categories list_. [(link)](http://permalink.gmane.org/gmane.science.mathematics.categories/6703)

* [[Todd Trimble]], _Post to A Dialogue on Infinity_. [(link)](http://dialinf.wordpress.com/2009/01/21/ultraproducts-the-category-theoretic-way/#comment-219)

* [[Bill Lawvere]], _Post to the categories list_. [(link)](http://facultypages.ecc.edu/alsani/ct99-00%288-12%29/msg00128.html) [archive, 21 Mar 2000](https://www.mta.ca/~cat-dist/archive/2000/00-3)

* [[Tom Leinster]], _Definition of ultrafilter_. [(blog post)](http://golem.ph.utexas.edu/category/2011/07/definitions_of_ultrafilter.html)

* [[Tom Leinster]], _Where do ultrafilters come from?_ [(blog post)](https://golem.ph.utexas.edu/category/2012/09/where_do_ultrafilters_come_fro.html)

* [[Tom Leinster]], _Codensity and the ultrafilter monad_, TAC **28** (2013) pp.332-370. [(TAC)](http://www.tac.mta.ca/tac/volumes/28/13/28-13.pdf) [arXiv:1209.3606](https://arxiv.org/abs/1209.3606).

* H. Volger, _Ultrafilters, ultrapowers and finiteness in a topos_, JPAA **6** (1975) pp.345-356.

* {#Garner} [[Richard Garner]], _Ultrafilters, finite coproducts and locally connected classifying toposes_. [arXiv:1808.08687](https://arxiv.org/abs/1808.08687).

* Jirí Adámek, Lurdes Sousa, _D-Ultrafilters and their Monads_. [arxiv:1909.04950](https://arxiv.org/abs/1909.04950).

[[!redirects ultrafilter]]
[[!redirects ultrafilters]]

[[!redirects principal ultrafilter]]
[[!redirects principal ultrafilters]]

[[!redirects fixed ultrafilter]]
[[!redirects fixed ultrafilters]]

[[!redirects free ultrafilter]]
[[!redirects free ultrafilters]]

[[!redirects ultrafilter monad]]
[[!redirects ultrafilter monads]]

[[!redirects compact Hausdorff object]]
[[!redirects compact Hausdorff objects]]
[[!redirects compact hausdorff object]]
[[!redirects compact hausdorff objects]]
[[!redirects compact Hausdorff space object]]
[[!redirects compact Hausdorff space objects]]
[[!redirects compact hausdorff space object]]
[[!redirects compact hausdorff space objects]]
[[!redirects compactum object]]
[[!redirects compactum objects]]


ultrafilter theorem

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### Context
#### $(0,1)$-Category theory
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#### Foundations
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# The ultrafilter theorem
* tic
{: toc}

## Idea

In [[classical mathematics]], the ultrafilter theorem is a theorem about ultrafilters, proved as a standard application of [[Zorn's lemma]].  In the [[foundations]] of mathematics, however, it is interesting to consider which results are implied by it or equivalent to it (very few imply it without being equivalent, other than those that imply the full axiom of choice itself).

## Statement and proof

+-- {: .un_theorem}
###### Ultrafilter theorem

Given a [[set]] $X$, every proper [[filter]] on $X$ may be extended to an [[ultrafilter]] (that is, a [[maximal element|maximal]] [[proper filter]]) on $X$.
=--

+-- {: .proof}
###### Standard proof

Given a proper filter $F$ on $X$, consider the [[partial order|poset]] of proper filters that refine (contain) $F$, ordered by inclusion (reverse refinement).  Of course, $F$ is in this poset.  Given a chain $\mathcal{C}$ of proper filters that refine $F$, the [[union]] $F \cup \bigcup \mathcal{C}$ is a proper filter that refines $F$ and is an upper bound of $\mathcal{C}$.  By [[Zorn's lemma]], there is a proper filter maximal among those that refines $F$, which is therefore maximal among all proper filters.
=--

Although this proof uses Zorn's lemma, the statement itself is weaker.  In particular, it is weaker than the [[axiom of choice]] (even assuming the principle of [[excluded middle]]).

## Strength and weakness as a "choice principle"

Although the ultrafilter theorem is weaker than the axiom of choice, it can be used to prove quite a few results that are traditionally proved using [[Zorn's lemma]] (or some other thing equivalent to the axiom of choice). For example, it can be used to prove

* the [[Hahn–Banach theorem]] (which in turn implies existence of Lebesgue non-measurable sets),

* the [[Banach-Tarski paradox|Banach–Tarski paradox]],

* the [[prime ideal theorem]] for [[rigs]], [[rings]], and [[Boolean algebras]],

* that every [[formally real field]] can be [[total order|totally ordered]],

* existence and uniqueness of [[algebraic closures]] (see [[splitting field]]),

* existence of non-trivial [[ultraproducts]],

* the [[compactness theorem]] and completeness theorem for [[first-order logic]] (see below)

as a partial list. An enlightening discussion of the things one might expect to be able to prove using the ultrafilter theorem is given [here at MO](http://mathoverflow.net/questions/202458/how-do-i-apply-the-boolean-prime-ideal-theorem).

However, in other respects, the ultrafilter theorem is very weak as a "choice principle". For example, it is too weak to prove the axiom of [[countable choice]] (so for example, it cannot prove that a countable union of countable sets is countable), a fact exhibited in Cohen's first model for the failure of the axiom of choice (where even countable choice fails but the ultrafilter theorem holds). It therefore cannot prove the [[axiom of dependent choice]] either.

It can prove *some* traditional applications of [[dependent choice]]. For example, it can be used to [[linear order|linearly order]] any set. From there it can be used to prove that a countable union of nonempty finite sets is countable, which in turn can be used to establish the weak König lemma; see [here](/nlab/show/compactness+theorem#totalorder).

In [[constructive mathematics]], the ultrafilter principle also implies [[De Morgan's law]], which means that the poset of [[truth values]] $\Omega$ is a [[De Morgan Heyting algebra]]. Any [[topos]] with an internal ultrafilter principle is thus a [[De Morgan topos]].

## Other formulations

Here are several statements equivalent (internal to an arbitrary [[topos]]) to the ultrafilter theorem.

The first three are, if you work through the definitions, almost direct rephrasings of the theorem above:

*  Every [[net]] has a universal subnet.  (We must define 'subnet' and 'universal net' properly to make this work, at which point the equivalence is immediate.)

*  The Boolean prime ideal theorem: every proper [[ideal]] in a [[Boolean ring]] is contained in a [[prime ideal]].  (Stronger formulations of the Boolean prime ideal theorem also follow.)

*  The [[Stone representation theorem]]: every [[Boolean algebra]] is a subalgebra of some [[function algebra]] $2^X$ into the [[initial object|initial]] Boolean algebra.  (Stronger formulations of the Stone representation theorem also follow.)

These basic results in [[logic]] are equivalent to the ultrafilter theorem:

*  Consistency: if a set $\Sigma$ of formulas in propositional [[classical logic]] is syntactically consistent (proves no contradiction), then it is semantically consistent (has a model).  (The converse is immediate; stronger formulations of the consistency theorem, including for predicate logic, also follow.)

*  Compactness: if every finite subset of $\Sigma$ has a model, then so does $\Sigma$.  (The converse is immediate; stronger formulations of the compactness theorem, including for predicate logic, also follow.)

Various characterisations of [[compact spaces]] are equivalent to the ultrafilter theorem:

*  Given a set $S$, the space $2^S$ (in the product topology) is compact.  (This can be seen as a very weak form of the Tychonoff theorem below.)

*  More generally, if every ultrafilter on a [[convergence space]] $X$ is convergent, then $X$ is compact.  (The converse is immediate.)

*  A [[uniform space]] is compact if it is [[complete space|complete]] and [[totally bounded space|totally bounded]] (i.e. it is [[Bishop-compact]]).  (The converse is immediate.)

These classical results in analysis are equivalent to the ultrafilter theorem in a [[Boolean topos]] with [[natural numbers object]]:

*  The [[Tychonoff theorem]] for [[Hausdorff spaces]]: any product of compact Hausdorff spaces is compact; equivalently, any product of compact Hausdorff spatial [[locales]] is spatial.  (If we drop the Hausdorff condition, then the result is equivalent to the full [[axiom of choice]].)

*  [[Stone–Čech compactification]]: the [[compact Hausdorff spaces]] form a [[reflective subcategory]] of [[Top]].  (The classical result that they form a reflective subcategory of the category $T_{3\frac{1}{2}}$ of completely regular Hausdorff spaces is enough; the classical result that the reflection is a dense embedding on $T_{3\frac{1}{2}}$ also follows.)

*  [[Banach–Alaoglu theorem]]: the closed unit ball of the double dual of a [[Banach space]] is weak* compact.  (The result for *separable* spaces does not require the ultrafilter theorem.)

## Failures

Of course, since the ultrafilter theorem is independent of [[ZF]], there are models in which it fails.  More strongly, however, there are apparently models of ZF in which *every ultrafilter (on every set) is principal*.  See [this MO answer](http://mathoverflow.net/questions/76495/existence-of-non-principal-ultrafilters-on-sets/76499#76499).

## References

Various equivalences with the ultrafilter theorem are stated and proved (in <b>ZF</b>) in

* Eric Schechter; [[HAF|Handbook of Analysis and its Foundations]].

See a summary (in GIF!): [page 1](http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/equivuf1.gif) and [page 2](http://www.math.vanderbilt.edu/~schectex/ccc/excerpts/equivuf2.gif).

It\'s possible that I\'ve made some mistakes above in deciding which of these use excluded middle or the existence of a natural numbers object.  (I very much doubt that any of them use replacement.)

For the ultrafilter principle in constructive mathematics:

* John Bell, "Boolean Algebras and Distributive  Lattices Treated Constructively", _Mathematical Logic  Quarterly  45_, 1999. [pdf](publish.uwo.ca/~jbell/BOOLCON.pdf)

[[!redirects ultrafilter principle]]
[[!redirects ultrafilter theorem]]
[[!redirects ultrafilter lemma]]


ultrapower

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In model theory, an ultrapower of a structure with respect to an ultrafilter is a common method for creating new structures, with applications to nonstandard arithmetic and nonstandard analysis. It is a standard construction for constructing models which embody infinite and infinitesimal quantities, as in [[Abraham Robinson]]'s nonstandard analysis.

An ultrapower is a special case of [[ultraproduct]]; see the description under the sheaf-theoretic interpretation below.

## Definition

Let $X$ be a set, and let $\mathcal{U}$ be an [[ultrafilter]] on $X$; that is, a collection of subsets such that

1. If $A \in \mathcal{U}$ and $A \subseteq B \subseteq X$, then $B \in \mathcal{U}$,

1. If $A, B \in \mathcal{U}$, then $A \cap B \in \mathcal{U}$,

1. $\emptyset \notin \mathcal{U}$,

1. For every set $A \subseteq X$, either $A$ belongs to $\mathcal{U}$ or its complement $\neg A$ belongs to $\mathcal{U}$.

(The first three axioms are the defining axioms for a [[filter]], and the last is satisfied for maximal filters, aka ultrafilters.)

Let $Y$ be a model of a (finitary, single-sorted) first-order theory $T$ (given by a [[signature]] $\Sigma$ together with a set of axioms in the language generated by $\Sigma$). Then the **ultrapower**

$$Y^X/\mathcal{U}$$

is the set of $\sim$-equivalence classes $[f]$ on the set of functions $f \in hom(X, Y)$, where $f \sim g$ if and only if

$$\{x \in X: f(x) = g(x)\} \in \mathcal{U}.$$

The ultrapower is a structure of $\Sigma$ under the evident pointwise definitions: for each $n$-ary function symbol $\phi$, define

$$\phi(f_1, \ldots, f_n)(x) = \phi_Y(f_1(x), \ldots, f_n(x))$$

and observe that $f_1 \sim g_1, \ldots, f_n \sim g_n$ implies that $\phi(f_1, \ldots, f_n) \sim \phi(g_1, \ldots, g_n)$, so that the interpretation of $\phi$ descends to equivalence classes. Similarly, for each $n$-ary relation symbol $R$, say $\vdash R([f_1], \ldots, [f_n])$ if

$$\{x \in X: \vdash_Y R(f_1(x), \ldots, f_n(x))\} \in \mathcal{U}$$

for any chosen representatives $f_1, \ldots, f_n$. Note that the well-definedness of the interpretations of functions and relations depends only on the filter axioms.

That the ultrapower is a model of the theory is a consequence of [[Los ultraproduct theorem|Los's theorem]]. (Here one needs all of the ultrafilter conditions, in particular
one needs condition 4 in order to accommodate satisfaction of formulas involving instances of negation.)

## Sheaf-theoretic interpretation

An ultrafilter $\mathcal{U}$ on a set $X$ may be regarded as a point in the Stone-Cech compactification $\beta X$ of the discrete space $X_d$, and ultrapowers may be interpreted as a special case of taking stalks.

Namely, let $i: X_d \hookrightarrow \beta X$ be the inclusion. This is a continuous function; hence it induces a geometric morphism $i_*$ between the sheaf toposes over these spaces. Then, under the composite

$$Set \stackrel{\Delta}{\to} Set/X \cong Sh(X_d) \stackrel{i_*}{\to} Sh(\beta X) \stackrel{stalk_{\mathcal{U}}}{\to} Set,$$

the set $Y$ is taken to the ultrapower $Y^X/\mathcal{U}$.

An object of $Set/X$ is just an $X$-indexed collection $Y_x$ of sets, and the more general **ultraproduct** $\prod_x Y_x/\mathcal{U}$ is the value obtained by applying the composite

$$Set/X \cong Sh(X_d) \stackrel{i_*}{\to} Sh(\beta X) \stackrel{stalk_{\mathcal{U}}}{\to} Set$$

to the object $\langle Y_x \rangle_{x \in X}$.

## Filterquotient construction

For any filter $\mathcal{U}$ on a set $X$, we can form the 2-colimit of the pseudofunctor

$$F: \mathcal{U}^{op} \to Cat: U \mapsto Set/U (\simeq Set^U)$$

Each $F(i: U \subseteq V): Set/V \to Set/U$ is the logical morphism of toposes given by the pullback functor $i^*$. The 2-colimit is itself a topos denoted $Set/\mathcal{U}$, in fact a 2-valued topos if $\mathcal{U}$ is an ultrafilter, and the functor defined by

$$Set \stackrel{\Delta}{\to} Set/X \to Set/\mathcal{U}$$

defines a logical morphism.

## Applications

If $\mathcal{U}$ is a _principal_ ultrafilter, then the ultrapower of $Y$ is isomorphic to $Y$ again. Thus the interest in ultrapowers relies on the existence of non-principal ultrafilters, which requires some sort of choice principle.

An important application is to $Y = \mathbb{R}$ as a model of the theory of ordered fields. If $\mathcal{U}$ is a non-principal ultrafilter on the set of natural numbers $\mathbb{N}$, then the model $\mathbb{R}^{\mathbb{N}}/\mathcal{U}$ provides a model for nonstandard analysis (in the sense promulgated by [[Abraham Robinson]]). There are in particular infinitesimal elements in the ultrapower, for example the $\sim$-equivalence class of the sequence $\langle 1/n \rangle_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}}$.

At the same time, the ultrapower is elementarily equivalent to $\mathbb{R}$, so that the sentences true for $\mathbb{R}$ in the first-order theory of ordered fields coincide with the sentences that are true for the ultrapower. This principle assures us that any conclusions adduced with the help of infinitesimals in the ultrapower are still valid in the standard model $\mathbb{R}$; in some cases, however, the arguments based on infinitesimals may offer more perspicuous proofs.


ultraproduct

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## Idea

The **ultraproduct** construction is an important tool in [[model theory]] that permits us to produce a new structure from an infinite family of structures. The construction has a decidedly 'algebraic' flavor and hence occurs naturally in applications of model theory to [[algebra]].

It is closely related via the [[Łoś ultraproduct theorem]] to the [[compactness theorem]]: suppose one is given a set of formulas $S$ in some first-order language and a family of structures such that any finite subset of formulas is modeled by all but a finite number of structures. Then an ultraproduct of those structures may be used to model the entire set $S$.[^compact]

[^compact]: Due to the outstanding importance of the compactness theorem, it is possible to prove 'almost all' results in model theory by the use of ultraproducts. This approach to model theory is pursued in [Bell-Slomson (1969)](#BellSlomson). For contrast, compare with the more sober view of [Hodges (1993)](#Hodges93).

In slightly greater detail, given a family of [[structures]] of the same signature in the sense of [[model theory]] (or more specially, [[universal algebra]]), one can (assuming the [[ultrafilter principle]], a weak form of the [[axiom of choice]]) use [[ultrafilters]] to form a certain [[congruence]] on the [[direct product]] and construct a [[quotient object]], a new structure of the same signature, called an __ultraproduct__. As long as the ultrafilter is [[free ultrafilter|free]] (contains the filter of cofinite subsets), the last sentence of the preceding paragraph is validated.

An ultraproduct of some number of copies of the _same_ structure is called an __ultrapower__.

Another important facet, somewhat reminiscent of [[Morita equivalence]], is the [[Keisler-Shelah theorem]]: Two structures are elementary equivalent $\mathfrak{A}\equiv\mathfrak{B}$ iff they have isomorphic ultrapowers: $\mathfrak{A}^I / \mathcal{U} \cong \mathfrak{B}^I / \mathcal{U}$ for some set $I$ and ultrafilter $\mathcal{U}$. This deep result says that the partially syntactic concept of _elementary equivalence_ can be characterized in purely semantic form with the help of ultrapowers.

## Definition

First we present the bare-bones set-theoretic construction; then we discuss structures and models.

### Set-theoretic

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###### Definition
Let $X$ be a set, and let $U$ be an [[ultrafilter]] on $X$, which may be regarded as an element $U: 1 \to \beta X$ of the [[Stone-Cech compactification]] $\beta X = \hom_{Bool}(2^X, 2)$ with its usual topology. The **ultrapower functor** over $U$ is the functor

$$Set/X \simeq Sh(X_{disc}) \stackrel{i_\ast}{\to} Sh(\beta X) \stackrel{U^\ast}{\to} Set$$

where $i_\ast$ is the [[direct image functor]] between categories of [[sheaves]] induced from the [[inclusion map|inclusion]] $i: X \to \beta X$ of $X$ as a [[discrete space|discrete]] [[subspace]], and the [[inverse image functor]] $U^\ast$ is also known as the "taking the [[stalk]]" functor at the point $U \in \beta X$.
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Let us extract a more concrete description. Let $\{Y_x: x \in X\}$ be an $X$-indexed family of sets, which we can view as an object $Y \to X$ of $\Set/X$. We may view this object as a sheaf over $X$ as discrete space; as a presheaf, it takes an open set (an arbitrary subset $A \subseteq X$) to the set of sections over $A$ which is $\prod_{x \in A} Y_x$. The direct image functor $i_\ast$ takes this presheaf to the presheaf which sends an arbitrary open set $\mathcal{V} \subseteq \beta X$ to $\prod_{x \in X \cap \mathcal{V}} Y_x$, and then the stalk functor $U^\ast$ sends this to the [[filtered colimit]]

$$colim_{\mathcal{V} \ni U} \prod_{x \in X \cap \mathcal{V}} Y_x$$

and since the basic opens $\mathcal{V}_A = \{U' \in \beta X: A \in U'\}$ with $A \in U$ are [[cofinal functor|cofinal]] in the system of open neighborhoods of $U$ (ordered by reverse inclusion), the colimit may be written more simply as

$$colim_{A \in U} \prod_{x \in A} Y_x$$

and this is the *ultraproduct*, often written as $\prod_{x \in X} Y_x/U$.[^note] When all the fibers $Y_x$ are the same set $Z$, this is written as $Z^X/U$ and called an *ultrapower* of $Z$.

[^note]: The quotient notation is traditional but (ever so slightly) misleading. If all the $Y_x$ are [[inhabited sets]], then all the maps $\prod_{x \in A} Y_x \to \prod_{x \in B} Y_x$ ($B \subseteq A$) are quotient maps and the ultraproduct is a quotient of the full product $\prod_{x \in X} Y_x$; there the notation is apt. If one of the $Y_x$ is empty, then the full product is empty and thus the ultraproduct is not such a quotient. But it is useful to allow for empty models (when they exist)! The correct definition which works for all cases is as we have it: a filtered colimit over restricted products. See [p. 186](#Makkai1980) and [this MO Discussion](#MathOverflow2012) for a discussion of this topic.

### Ultraproducts of structures

Now suppose each $Y_x$ carries a structure specified by a [[signature (in logic)|signature]]; in other words, for each function symbol $f$ and predicate symbol $R$ of the signature there are specified operations and subsets

$$\omega_f: Y_x^{ar(f)} \to Y_x, \qquad \rho_R \rightarrowtail Y_x^{ar(R)}$$

where $ar$ denotes arity (constants being considered function symbols of arity zero). Then of course the products $\prod_{a \in A} Y_x$ carry structures canonically induced from those on the $Y_x$. As all of the function symbols $f$ and predicate symbols $R$ have finite arity, and since the taking of filtered colimits commutes with finite limits in $Set$, we get a canonically induced structure on the ultraproduct $\prod_{x \in X} Y_x/U$. More briefly: since $U^\ast i_\ast$ preserves finite limits, it takes an $X$-indexed finitary structure $Y_x$ in $Set/X$ to a finitary structure of the same type in $Set$.

Intuitively (and adopting language used by Lawvere), what is going on is that we have a variable structure varying over some domain $X$, and we consider a point $U$ of its compactification $\beta X$ as some kind of "ideal point at infinity", and then we "freeze" (or localize) the variation by passing to germs of the variation at that point $U$.

Notice that if we replace the word "ultrafilter" by the word "filter", the general structural facts mentioned here would remain true. That is to say: if $F$ is a filter consisting of subsets of $X$, then the colimit $colim_{A \in F} \prod_{x \in A}$ (with such $A$ ordered by reverse inclusion) is still directed or filtered, and we would still get a structure of given finitary type by passing to the colimit. Here, instead of ultrapowers and ultraproducts, one speaks of *reduced powers* and *reduced products*.

### Ultraproducts of models

To be written.

## Examples

The [[hyperreal number]]s ([wikipedia](http://en.wikipedia.org/wiki/Hyperreal_numbers)) and nonstandard integers in [[nonstandard analysis]] are obtained as [[denumerable set|countable]] ultrapowers with help of [[free ultrafilters]] on $\mathbb{N}$. Such ultrafilters contain all [[cofinite subset|cofinite]] subsets of integers, but not only them. See [wikipedia:ultraproduct](http://en.wikipedia.org/wiki/Ultraproduct).

From Michael Barr's [Models of Sketches](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1986__27_2/CTGDC_1986__27_2_93_0/CTGDC_1986__27_2_93_0.pdf)

>Unlike [[limits]] and [[colimits]], an ultraproduct is not defined by any [[universal mapping property]]. Of course, if the category has limits and ([[filtered colimit|filtered]]) colimits, then it has ultraproducts constructed as colimits of [[products]] ... But usually the category of models of a [[coherent theory]] (such as the theory of [[discrete field|fields]]) lacks products and  hence does not have categorical ultraproducts.

### In functional analysis

(Appropriately-defined) ultrapowers of [[Banach space|Banach spaces]] allow one to embed an infinite-dimensional space $X \hookrightarrow X^{\mathcal{U}}$ inside a larger Banach space such that for every bounded family $(x_i)_{i \in I}$ and every ultrafilter $\mathcal{U}$ on $I$, the $\mathcal{U}$-limit $\lim_{i, \mathcal{U}}$ exists (just take the [[germ]] $[(x_i)]$).

There are various ways to formalize this: Henson's [[positive bounded logic]], [[continuous logic]], or ad hoc variants (e.g. "Banach space structures", "normed space structures", etc.)

As one might expect, things which are only "approximately true" inside $X$ become "exactly true" in $X^{\mathcal{U}}$. For example, if $\lambda$ is an [[approximate eigenvalue]] (e.g. on the boundary of the [[spectrum of an operator|spectrum]]) for a bounded operator $T$, then the germ of the sequence of approximate eigenvectors of $\lambda$ becomes a _bona fide_ [[eigenvector]] with eigenvalue $\lambda$ for $T^{\mathcal{U}}$ on $X^{\mathcal{U}}$.

## Related entries

* [[model theory]]

* [[ultrafilter monad]]

* [[large cardinal]]

* [[ultracategory]]

## References

Standard references in model theory for ultraproducts are

* {#BellSlomson}[[John Bell|J. L. Bell]], A. B. Slomson, _Models and Ultraproducts: An Introduction_ , North-Holland Amsterdam 1969. (Dover reprint)

* P. C. Eklof, _Ultraproducts for Algebraists_ , pp.105-137 in Barwise (ed.), _Handbook of Mathematical Logic_ , Elsevier Amsterdam 1977.

For a more recent textbook treatment see

* {#Hodges93} [[Wilfrid Hodges]], _Model Theory_ , Cambridge University Press 1993. (sec. 9.5)

The categorical perspective on ultraproducts goes back to

* T. Okhuma, _Ultrapowers in categories_ , Yokohama Math. J. **14** (1966) pp.17-37.

* S. Fakir, L. Haddad, _Objets cohérents et ultraproduits dans les catégories_  , Journal
of Algebra **21** (1972) pp.410–421.

For the vacuum engineering point of view of 'freezing variation' [^cont] in the context of non-standard analysis see

* [[F. William Lawvere]], _Variable sets etendu and variable structure in topoi_ , Lecture notes University of Chicago 1975.

[^cont]: This perspective was apparently inspired by joined work with [[Miles Tierney]] on the use of [[forcing]] in the context of the [[continuum hypothesis]] in early topos theory.

The following papers are relevant for understanding the ultraproduct construction via  [[codensity monad|codensity monads]]:

* D. P. Ellerman, _Sheaves of structures and generalized ultraproducts_ , Annals of
Mathematical Logic **7** (1974) pp.163–195.

* J. F. Kennison, _Triples and compact sheaf representation_ ,  JPAA **20** (1981) pp.13-38.

* [[Tom Leinster]], _Codensity and the Ultrafilter Monad_ , TAC **28** no. 13 (2013) pp.332-370. ([tac](http://www.tac.mta.ca/tac/volumes/28/13/28-13abs.html))

For a fine point concerning the definition of ultraproducts:

* {#MathOverflow2012} [MathOverflow2012](https://mathoverflow.net/q/105397)

* {#Makkai1980} [[M. Makkai]], _On full embeddings_ , JPAA, 1980

See also

* [[M. Makkai]], _Ultraproducts and categorical logic_ , Lectures Notes in Math. __1130__, Springer 1985, pp. 222–309.

For ultraproducts in functional analysis see e.g.

* Jose Iovino, _Applications of model theory to functional analysis_

For a way to study ultraproducts in a homotopical setting see

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

[[!redirects ultraproduct]]
[[!redirects ultraproducts]]
[[!redirects ultrapower]]
[[!redirects ultrapowers]]
[[!redirects reduced product]]
[[!redirects reduced power]]
[[!redirects reduced products]]
[[!redirects reduced powers]]


ultrareal numbers

where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.
Addition (a + n) is the most basic operation, multiplication (an) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a, and exponentiation (an) can be thought of as a chained multiplication involving n numbers a. Analogously, tetration (na) can be thought of as a chained power involving n numbers a. The parameter a may be called the base-parameter in the following, while the parameter n in the following may be called the height-parameter (which is integral in the first approach but may be generalized to fractional, real and complex heights


ultraroot

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## Idea

An [[elementary substructure]] $N \preceq M$ of a [[model]] of a [[first-order theory]] is an _ultraroot_ of $M$ if $M$ is [[isomorphism|isomorphic]] to an [[ultrapower]] of $N$.

## Definition

Formally, recall that any structure $N$ admits a diagonal [[elementary embedding]] $N \overset{\Delta}{\to} N^{\mathcal{U}}$ to any [[ultrapower]] $N^{\mathcal{U}}$ of $N$.

If $i \colon N \to M$ is an [[elementary embedding]], we say that $N$ is an ultraroot of $M$ if there is some [[ultrafilter]] $\mathcal{U}$ on some index [[set]] such that there is an [[isomorphism]]

$$
\left(N \overset{i}{\to} M\right) \simeq \left(N \overset{\Delta}{\to} N^{\mathcal{U}}\right)
$$
in the [[co-slice category]] under $N$.

## Examples

* $N$ is an ultraroot of $N^{\mathcal{U}}$.

* Assuming the [[continuum hypothesis]], any [[countable set|countable]] model of a complete-[[first order theory]] will be an ultraroot of that theory's $\aleph_1$-saturated continuum-sized model.

* Similarly, taking a countably-indexed ultrapower of a slightly-expanded copy (like a nonstandard model generated by throwing in a single infinitesimal) of the reals still yields the [[hyperreal numbers]].

## Remarks
Closure under taking ultraroots, along with closure under ultraproducts and elementary embeddings, gives a criterion for detecting [[elementary class of structures|elementary classes]]. See also at _[[Birkhoff's HSP theorem]]_.

## Related concepts

* [[ultraproduct]]

* [[ultrapower]]

## References

* See e.g. page 7 of [these notes](http://www.math.uiuc.edu/~henson/cfo/UCLA09text.pdf) by Ward Henson on continuous model theory,

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ultraviolet catastrophe

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## Idea

What is called the _ultraviolet catastrophe_ is a [[paradox]] in [[classical field theory]]/[[statistical mechanics]] which predicts an infinite amount of energy in high [[frequency]] (whence: ultraviolet) [[black body radiation]].

The observed absence of the [[ultraviolet catastrophe]] in [[experiment]] (a kind of [[null result]]) led to the hypothesis of [[energy]]/[[frequency]]-quantization (now expressed by [[Planck's constant]]) and hence to the development of [[quantum physics]].

## References

* Wikipedia, _[Ultraviolet catastrophe](http://en.wikipedia.org/wiki/Ultraviolet_catastrophe)_


ultraviolet divergence

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## Idea

In the construction of [[quantum field theory]] various naive mathematical constructions give rise to ill-defined "[[divergent series|divergent]]" expressions.

Some of these occur in the [[limit of a sequence|limit]] that [[interaction]] point are taken _close_ to each other. Since the short distances involved in this limit translates, under [[Fourier transform]], to high [[frequencies]] enoded in the [[wave front set]] of the [[distributions]]; these are called _ultraviolet divergences_. The correct way to deal with them is called _[[renormalization]]_.

Other divergences occcur in the [[limit of a sequence|limit]] that the spacetime [[support]] of the [[interaction]] (the support of the "[[adiabatic switching]]") tends without bounds to all of spacetime, also called the [[adiabatic limit]]. Since this large distance involved in this limit translate, under [[Fourier transform]] to low [[frequencies]], these are called _[[infrared divergences]]_.

## Related concepts

* [[infrared divergence]]

* [[renormalization]]

* [[microlocal analysis]]

## References

See also

* Wikipedia, [Ultraviolet divergence](https://en.wikipedia.org/wiki/Ultraviolet_divergence)

[[!redirects ultraviolet divergences]]

[[!redirects UV divergence]]
[[!redirects UV divergences]]

[[!redirects UV-divergence]]
[[!redirects UV-divergences]]


ultraweak topology

why is the Von neuman algebra is closed uneder weak* topology in B(H)


Umberto Bottazzini

__Umberto Bottazzini__ is a historian of mathematics well known for his great attention to primary sources; his main attention is to the general history of infinitesimal calculus in 19th century and to the history of Italian mathematics.

*  _Il flauto di Hilbert: Storia della matematica_, UTET Università, 473pp.

* _The higher calculus: A history of real and complex analysis from Euler to Weierstrass_, Springer 1986

For Bottazzini's achievements see the Notices article

* 2015 Whiteman Prize, Notices Amer. Math. Soc. [pdf](http://www.ams.org/notices/201504/rnoti-p431.pdf)

category: people


Umberto Eco

_Umberto Eco_ (1931-2016) was an Italian [[philosophy|philosopher]] and writer, researcher of culture, society and cultural history.

He was one of the founders of general [[semiotics]].

* wikipedia [Umberto Eco](https://en.wikipedia.org/wiki/Umberto_Eco)
* Umberto Eco, _A theory of semiotics_, Advances in Semiotics, Indiana University Press 1976
* Umberto Eco, _Semiotics and the philosophy of language_, Indiana University Press 1984

category: people


umbral calculus

* wikipedia: [umbral calculus](http://en.wikipedia.org/wiki/Umbral_calculus)
* [[eom]]: [umbral calculus](http://eom.springer.de/U/u095050.htm)
* Steven M. Roman, Gian-Carlo Rota, _Umbral calculus_,  Advances in Math. __27__ (1978), no. 2, 95--188, [pdf until p. 120](http://www.romanpress.com/MathArticles/Umbral1.pdf), doi, [MR0485417](http://www.ams.org/mathscinet-getitem?mr=0485417)
* Nigel Ray, _Universal constructions in umbral calculus_, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 343--357, Progr. Math. __161__, Birkhäuser 1998, [MR99e:05015](http://www.ams.org/mathscinet-getitem?mr=99e:05015), [pdf](http://www.ma.man.ac.uk/~nige/ucuc.pdf), [citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.2434)
* G.-C. Rota, B. D. Taylor, SIAM J. Math. Anal. __25__ (1994), no. 2, 694--711; [MR95d:05014](http://www.ams.org/mathscinet-getitem?mr=95d:05014), [doi](http://dx.doi.org/10.1137/S0036141093245616)
* Thomas J. Robinson, _Formal calculus and umbral calculus_, The electronic journal of combinatorics 17 (2010), $\sharp R95$, [pdf](http://www.combinatorics.org/Volume_17/PDF/v17i1r95.pdf)
* [[Doron Zeilberger]], _The umbral transfer-matrix method. I. Foundations._, In memory of Gian-Carlo Rota. J. Combin. Theory Ser. A 91 (2000), no. 1-2, 451--463, [MR2001g:05018](http://www.ams.org/mathscinet-getitem?mr=2001g:05018), [doi](http://dx.doi.org/10.1006/jcta.2000.3110)
* A. Di Bucchianico, D. Loeb, _A selected survey of umbral calculus._ Electronic Journal of Combinatorics __2__ (1995), 28, [pdf](https://www.combinatorics.org/files/Surveys/ds3/ds3v2-2000.pdf)
* Li Guo, _Baxter algebras and the umbral calculus_,  Adv. in Appl. Math. __27__ (2001), 405-426; cf. also [[Rota-Baxter algebra]]
* G. Dattoli, E. Palma, , E. Sabia, K. Górska, A. Horzela, K. A. Penson, _[[operational calculus|Operational]] versus umbral methods and the Borel transform_, Int. J. Appl. Comput. Math (2017) 3: 3489 [doi](https://doi.org/10.1007/s40819-017-0315-7)

category: analysis, combinatorics, algebra


umbral moonshine

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## Idea

The **Umbral Moonshine Conjectures** assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. It relates the 23 [[Niemeier lattices]], even unimodular positive-definite lattices of rank 24 with non-trivial root systems, to [[mock theta functions]].

Umbral moonshine is a generalization of the [[Mathieu moonshine]] phenomenon which relates representations of the [[Mathieu group]] $M_24$ with [[K3 surfaces]], and which corresponds to the Niemeier lattice with the simplest root system $X = A_1^{24}$. As noted in 2010 by Eguchi, Ooguri, and Tachikawa, dimensions of some representations of $M_24$, the largest sporadic simple Mathieu group, are multiplicities of superconformal algebra characters in the K3 elliptic genus.

## Related entries

* [[Niemeier lattice]]

* [[Moonshine]]

* [[K3 surface]]

## References

* [[Miranda C. N. Cheng]], John F. R. Duncan, Jeffrey A. Harvey, _Umbral Moonshine_ ([arXiv:1204.2779](http://arxiv.org/abs/1204.2779))

* Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, _Umbral Moonshine and the Niemeier Lattices_ ([arXiv:1307.5793](https://arxiv.org/abs/1307.5793))

* John F. R. Duncan, Michael J. Griffin and Ken Ono, _Proof of the Umbral Moonshine Conjecture_ ([arXiv:1503.01472](http://arxiv.org/abs/1503.01472))

* John F. R. Duncan, Sander Mack-Crane, _Derived Equivalences of K3 Surfaces and Twined Elliptic Genera_, ([arXiv:1506.06198](http://arxiv.org/abs/1506.06198))

* Miranda C. N. Cheng, Sarah Harrison, _Umbral Moonshine and K3 Surfaces_, ([arXiv:1406.0619](http://arxiv.org/abs/1406.0619))

* {#KachruPaquetteVolpato16} [[Shamit Kachru]], [[Natalie Paquette]], Roberto Volpato, _3D String Theory and Umbral Moonshine_ ([arXiv:1603.07330](http://arxiv.org/abs/1603.07330))


umbrella category

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###### Definition

An __umbrella category__ is an [[inhabited set|inhabited]] [[category]] $C$ such that for every [[object]] $X$ in $C$, there is at least one [[subterminal object]] $T$ such that the [[hom-set]] $C(X,T)$ is [[inhabited set|inhabited]] (hence being a singleton).

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[[!redirects umbrella categories]]


Umesh Vazirani

* [personal page](https://people.eecs.berkeley.edu/~vazirani/)

* [institute page](https://www2.eecs.berkeley.edu/Faculty/Homepages/vazirani.html)

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

## Selected writings

Introducing the notion of [[quantum complexity theory]]:

* [[Ethan Bernstein]], [[Umesh Vazirani]], *Quantum Complexity Theory*, Siam Journal of Computation **26** 5 (1997) 1411-1473 [[doi:10.1137/S0097539796300921](https://doi.org/10.1137/S0097539796300921), [pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.655.1186&rep=rep1&type=pdf)]

Introduction and review:

* [[Umesh Vazirani]], *A survey of quantum complexity theory*, 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) [[doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

category: people


Umut Gürsoy

* [Institute page](https://www.uu.nl/medewerkers/UGursoy)

* [GoogleScholar page](https://scholar.google.nl/citations?user=-L4TiuIAAAAJ&hl=en)

## Selected writings

On applications of [[AdS/CFT duality]] to many-body [[quantum systems]], such as to high-$T_c$ [[superconductivity]] ([[AdS/CMT correspondence]]) and to [[QCD]] ([[AdS/QCD correspondence]])

* [[Umut Gürsoy]], *Recent developments in gauge-gravity duality applied to quantum many-body systems*, talk at *[[Strings 2022]]*  [[indico:4940863](https://indico.cern.ch/event/1085701/contributions/4940863), [pdf](https://indico.cern.ch/event/1085701/contributions/4940863/attachments/2483542/4263813/Slides_Gursoy.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=69bc5a22-b77a-4830-97b8-49c7d6aa1c29) ]

category: people

[[!redirects Umut Gursoy]]


unbound operator s adjoint domain equal 0

unbound operator adjoint equal 0


unbounded operator

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# Contents
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## Idea

This page is about unbounded [[linear operator]]s on [[Hilbert space]]s. For operators on Hilbert spaces, "bounded" and "continuous" are synonymous, so the first question to be answered is: Why consider unbounded, i.e., discontinuous operators in a category that is a [[subcategory]] of [[Top]]? The reason is simple: It is forced upon us both by applications, such as [[quantum mechanics]], and by the fact that simple and useful operators like differentiation are not bounded. Happily, in most applications the operators considered retain some sort of "limit property", namely the property of being "closed". Although that seems to be negligible compared to continuity, it allows the development of a rich and useful theory, and as a consequence there is a tremendous amount of literature devoted to this subject.

One way of dealing with unbounded operators is via [[affiliated operator]]s, see there.

### Example: differentiation is unbounded

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$, and let $T$ be the differentiation operator defined on the dense subspace of Schwartz functions $f$ by $Tf(x) \coloneqq f'(x)$. One might hope $T$ has a continuous extension to all of $L^2$, but consider the sequence $f_k(x) := \exp(-k|x|)$ for $k \in \mathbb{N}$. Then we have $\frac{\|Tf_k\|}{\|f_k\|} = k$, so $T$ is unbounded.

Note that the domain of definition of an unbounded operator will generally be given only on a dense subspace, as in this example. Indeed, the existence of unbounded operators defined everywhere (on a Hilbert space) is non-[[constructive mathematics|constructive]], relying on the [[Hahn–Banach theorem]] and refutable in [[dream mathematics]].

## Summary

After the definition we will look at some concepts that can be transferred from the bounded context [here](/nlab/show/unbounded+operator#closedness).

We will talk about the [[von Neumann algebra]] that contains all [[spectral projections]] [here](/nlab/show/unbounded+operator#affiliation).

Finally we give some counterexamples, i.e., phenomena that contradict the intuition built from bounded operators [here](/nlab/show/unbounded+operator#counterexamples).

## Definition

An **unbounded operator** $T$ on a Hilbert space $\mathcal{H}$ is a linear operator defined on a subspace $D$ of $\mathcal{H}$. $D$ is necessarily a linear submanifold. Usually one assumes that $D$ is dense in $\mathcal{H}$, which we will do, too, unless we indicate otherwise.

In particular every bounded operator $A: \mathcal{H} \to \mathcal{H}$ is an unbounded operator ([[red herring principle]]).

## Domains: a first look

Unbounded operators are not defined on the whole Hilbert space, so it is essential that, when talking about a specific unbounded operator, we are actually talking about the pair $(T, D_T)$ of an operator $T$ together with its domain $D_T$. In particular two unbounded operators $T, S$ are equal iff their domains are equal, $D_T = D_S$, and for all $x \in D_S=D_T$ we have $Tx = Sx$.

If the domain is not specified, the default definition of the domain of a given operator $T$ is simply $D_T := \{x \in \mathcal{H} \vert Tx \in \mathcal{H} \}$.

**Warning:** if one composes two unbounded operators $T$ and $S$, it may happen that $D_T \cap D_S = \{0\}$. If we insist that all our unbounded operators are densely defined, we need as an additional assumption that $D_T \cap D_S$ is dense to make sense of the composite $T S$.

## The Hellinger-Toeplitz theorem

* Theorem: Let $A$ be an everywhere defined linear operator on a Hilbert space $\mathcal{H}$ that is symmetric. Then A is bounded.

For the definition of symmetric see below.

The Hellinger-Toeplitz theorem is a no-go theorem for [[quantum mechanics]]. Since it is known that operators essential for [[quantum mechanics]] are both symmetric and unbounded, we are led to conclude that they cannot be everywhere defined. This means that the problems that accompany only densely defined operators cannot be avoided.

### Reference

This is a corollary to the closed graph theorem III.2 in the book

* Reed, M.; Simon, B.: _Methods of modern mathematical physics_. Volume 1, Functional Analysis

## Closedness, selfadjointness, resolvent

### Closedness

Recall that the graph of an operator $T$ (or any function, in general) is the subset $\mathcal{G}_T := \{(x, y) \in \mathcal{H} \times \mathcal{H} \vert Tx = y \}$. The graph of a given operator need not be closed (in the product topology of $\mathcal{H} \times \mathcal{H}$). The notion that will be a surrogate for continuity is "closable", defined as follows:

* Definition: Given an operator T with domain $D_T$, any operator $T'$ with larger domain that is equal to $T$ on $D_T$ is called an **extension** of T, we write $T \subset T'$.

* Definition: An operator is **closed** if its graph is closed.

* Definition: An operator is **closable** if it has a closed extension. The smallest such extension is called the **closure** of $T$ and is denoted by $\overline{T}$.

* Proposition (closure of graph is graph of closure): If an operator $T$ is closable, then the closure of its graph $\overline{\mathcal{G}}$ is the graph of an operator, and this operator is its closure.

The last part deserves some elaboration: Given an operator $T$, we can always form the closure of its graph $\mathcal{G}$. How can the closure not be the graph of an operator? Given a sequence $(x_n)$ in $\mathcal{G}$, such that both limits $x \coloneqq \lim_{n \to \infinity} x_n$ and $y \coloneqq \lim_{n \to \infinity} T x_n$ exist, we have that $(x, y)$ is in the closure of $\mathcal{G}$. Now it may happen that there is another point $(x, y')$ in the closure with $y \neq y'$, which implies that the closure cannot be the graph of a single valued function.

We may assume without loss of generality that $\lim_{n \to \infinity} x_n = 0$, so that we get as a characterisation of closability: if $y \coloneqq \lim_{n \to \infinity} T x_n$ exists, then $y=0$. It $T$ were continuous, we would not have to assume that $(Tx_n)$ is convergent, so this additional assumption tells us in what respect closability is weaker than continuity.

* Definition: For a closed operator $T$ a subset $D$ of $D_T$ is called a **core** of $T$ if $\overline{T \vert_{D}} = T$. In other words: if we restrict $T$ to $D$ and take the closure, we obtain again $T$.

### Example of an operator that is not closable
...

### Selfadjointness

We let $T^*$ be the adjoint of an operator $T$. Note that for an only densely defined $T$, the domain of the adjoint may be strictly larger.

* Definition (selfadjoint et al.): An operator is **symmetric** (or **Hermitian**) if $T = T^* \vert_{D_T}$ (the adjoint is restricted to the domain of $T$). It is **selfadjoint** if it is symmetric and $D_T = D_{T^*}$. A symmetric operator is **essentially selfadjoint** if its closure is selfadjoint.

The difference of being symmetric and being selfadjoint is crucial, although there is a famous anecdote that seems to indicate otherwise:

* Anecdote of selfadjointness: Once upon a time John von Neumann thanked Werner Heisenberg for the invention of [[quantum mechanics]], because this had led to the development of so much beautiful mathematics, adding that mathematics paid back a part of the debt by clarifying for example the difference of a selfadjoint operator and one that is only symmetric. Heisenberg replied: "What is the difference?"

Nevertheless theorems that assume an operator to be selfadjoint will be not applicable to an operator that is only symmetric. One example is the [[spectral theorem]].

#### Example of a symmetric, but not selfadjoint, operator

...

## Resolvent

The definition of the resolvent does not pose any problems compared to the bounded case:

* Definition: let $T$ be a closed operator on a [[Hilbert space]] $\mathcal{H}$. A complex number $\lambda$ is in the **resolvent set** $\rho(T)$ if $\lambda \mathbb{1} - T$ is a bijection of $D_T$ and $\mathcal{H}$ with bounded inverse. The inverse operator is called
the **resolvent** $R_{\lambda}(T)$ of $T$ at $\lambda$.

* Theorem: The resolvent set is an open subset of $\mathbb{C}$ on which the resolvent is an analytic operator valued function. Resolvents at different points commute and we have
$$
      R_{\lambda}(T) - R_{\mu}(T) = (\mu - \lambda) R_{\mu}(T) R_{\lambda}(T)
$$

The proof can be done as in the bounded case.

### Commuting operators

The concept of commuting operators, which is of no problem in the bounded case, presents a conceptual difficulty in the unbounded one: for given operators $A$, $B$ we would like to be able to say whether they commute, although their composite may not have dense domain. For selfadjoint operators there is a solution to this problem: We know that in the bounded case, two selfadjoint operators commute iff their spectral projections commute. This suggests the

* Definition: two (possibly unbounded) selfadjoint operators commute iff all their spectral projections commute.

The [[spectral theorem]] shows that all bounded Borel functions of two commuting operators will also commute.

The following theorem states the reverse for two of the most important functions and shows that the definition of commutation above is reasonable:

* Theorem: let $A$ and $B$ be selfadjoint operators on a Hilbert space $\mathcal{H}$. Then the following three statements are equivalent:

(a) $A$ and $B$ commute

(b) if $\operatorname{Im} \lambda$ and $\operatorname{Im} \mu$ are $\neq 0$, then $R_{\lambda}(A)$ and $R_{\mu}(B)$ commute, as it is defined for bounded operators: $R_{\lambda}(A) R_{\mu}(B) = R_{\mu}(B) R_{\lambda}(A)$

(c) for all $t, s \in \mathbb{R}$ we have $\exp(itA) \exp(isB) = \exp(isB) \exp(itA)$

## Strongly continuous one-parameter semigroups

A _strongly continuous one-parameter semigroup_ is an unitary representation of $\mathbb{R}$ on $\mathcal{H}$ where $\mathbb{R}$ is seen as a topological group with respect to multiplication, see [[topological group]]. An explicit definition recalling these concepts is this:

* Definition: a function $U : \mathbb{R} \to \mathcal{B}(\mathcal{H})$ is a one-parameter [[semigroup]] if the semigroup condition $U(t+s) = U(t)U(s)$ holds. If for every $x \in \mathcal{H}$ and $t \to t_0$ we have $U(t)x \to U(t_0)x$ then it is **strongly continuous**. If every $U$ is an unitary operator, it is an unitary semigroup.

In the following a semigroup will be understood to be a one-parameter unitary strongly continuous semigroup.

In physics, one-parameter semigroups of this kind often represent the time evolution of a physical system described by an [[evolution equation]].

* Theorem: a selfadjoint operator $A$ generates a semigroup via $U(t) := \exp(itA)$

* Theorem (Stone's theorem): let $U$ be a semigroup, then there is a selfadjoint operator $A$ such that $U(t) = \exp(i t A)$. This operator is often called the **infinitesimal generator** of $U$.

These two theorems are essential for the Schrödinger picture of quantum mechanics, which describes a system by the [[Schrödinger equation]], we have now a one-to-one correspondence of selfadjoint operators which can be seen as [[Hamilton operator]]s (only special operators will be seen as describing actual physical systems, of course), and semigroups which describe the time evolution generated by the [[Hamilton operator]].

As a trivial observation we add that Stone's theorem is a (huge) generalization of the Taylor series, let $f: \mathbb{R} \to \mathbb{R}$ that is (real) analytic in a neighborhood of $0$, then we get for $x$ small enough:
$$
f(h) = \sum_{k=0}^{\infty} \frac{f^{n}(0)}{n!} h^n = \exp(i h(-i \frac{d}{dx} ))
$$

This shows that the operator $-i \frac{d}{dx}$ generates the semigroup of translations on the real line. Now we could, for example, use Stone's theorem to prove that $-i \frac{d}{dx}$ is selfadjoint by proving that the translation group is strongly continuous.

## Subtleties resulting from domain issues

(This rather generic title will have to be revised.)

### Nelson's example of noncommuting exponentials

#### Idea

Nelson's example shows that the rather involved definition of commutativity of two unbounded operators is well motivated, because a more naive one will have unwanted consequences. It is a counterexample to the following conjecture:

* Conjecture (**false!**): Let $A$ and $B$ be selfadjoint operators on a Hilbert space $\mathcal{H}$ and $D$ a dense subset such that both $A$ and $B$ restricted to $D$ are essentially selfadjoint. Suppose that for all $x \in D$ we have $ A B x - B A x = 0$. Then $A$ and $B$ commute.

We have already seen that on $\mathbb{R}^2$ we can define two essentially selfadjoint operators $-i \partial_x$ and $-i \partial_y$ that generate translations along the x-axis and the y-axis respectively. Both the generated translations and the operators commute (the latter if applied to differentiable functions, of course).

The central idea of Nelson's counterexample is to replace $\mathbb{R}^2$ by a [[Riemann surface]] with two sheets, such that walking east, then walking north takes you to one sheet, while walking north then walking east takes you to the other sheet.

#### Definition

We use the Riemann surface $M$ of $f(z) = \sqrt(z)$. We give a brief exposition of its construction: take two copies of $\mathbb{C}$, two "sheets", and call them I and II. Cut both sheets along $(0, \infty)$, label the edge of the first quadrant along the cut as $+$ and the edge of the fourth quadrant as $-$. Then *attach* the $+$ edge of I with the $-$ edge of II and the $-$ of I with the $+$ edge of II.

Let $\mathcal{H} = L^2(M)$ with respect to Lebesgue measure. As indicated above we define A = $-i \partial_x$ and B = $-i \partial_y$, this is with respect to the canonical chart on each sheet that projects it onto $\mathbb{C}$ and then identifies $\mathbb{C}$ with $\mathbb{R}^2$.

Let $D$ be the set of all smooth functions with compact support not containing $0$. Then:

(a) $A$ and $B$ are essentially selfadjoint on $D$

(b) $A$ and $B$ map $D$ onto $D$

(c) for all $x \in D$ we have $A B x = B A x$

(d) $\exp(i t A)$ and $\exp(i s B)$ do not commute.

Only part (a) needs further explanation...

## Related concepts

* [[linear operator]]
* [[bounded linear operator]] / **unbounded linear operator**
* [[self-adjoint extension]]
* [[operator algebra]]

## References

* Konrad Schmuedgen, _Unbounded self-adjoint operators on Hilbert space_, Springer GTM 265, 2012

Chapter VIII of the following classic volume is devoted to unbounded operators:

* Reed, M.; Simon, B.: _Methods of modern mathematical physics_. Volume 1, Functional Analysis

Nelson's example is taken from the above reference, the original reference is this:

* Edward Nelson: _Analytic Vectors_ Ann.Math. 70, p.572-615, 1959

category: analysis

[[!redirects unbounded operator]]
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[[!redirects unbounded linear operators]]

[[!redirects closable operator]]
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