Handbook of Quantum Gravity

This page compiles pointers to material contained in the (upcoming) book collection:

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* Cosimo Bambi, Leonardo Modesto and [[Ilya L. Shapiro]]:

  \linebreak

  **Handbook of Quantum Gravity**

  \linebreak

  Springer (2023)

  [doi:10.1007/978-981-19-3079-9](https://doi.org/10.1007/978-981-19-3079-9)

on aspects of [[quantum gravity]].

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**Contents:**

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On [[perturbative quantum gravity]]:

* [[Ilya L. Shapiro]], *The background information about perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.12319](https://arxiv.org/abs/2210.12319)]

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Viewed as an [[effective quantum field theory]]:

* [[John F. Donoghue]], *Quantum General Relativity and Effective Field Theory*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.09902](https://arxiv.org/abs/2211.09902)]

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On [[gauge invariance|gauge invariant]] [[renormalization]] of [[perturbative quantum gravity]]:

* [[Petr M. Lavrov]], [[Ilya Shapiro]], *Gauge invariant renormalizability of quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09271](https://arxiv.org/abs/2210.09271)]

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On  [[renormalization]] of [[perturbative quantum gravity]] with [[higher curvature corrections]]:

* Nobuyoshi Ohta, *One-loop divergences in higher-derivative gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.02583](https://arxiv.org/abs/2210.02583)]

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On [[gravitational waves]] in [[quantum gravity]]:

* Amaury Micheli, Patrick Peter, *Quantum cosmological gravitational waves?*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.00182](https://arxiv.org/abs/2211.00182)]

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On [[black hole thermodynamics]] in [[perturbative quantum gravity]]:

* Dmitri V. Fursaev, *Black Hole Thermodynamics and Perturbative Quantum Gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.06081](https://arxiv.org/abs/2210.06081)]

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On the [[soft graviton theorem]]:

* [[Martin Beneke]], Patrick Hager, Robert Szafron, *Soft-Collinear Gravity and Soft Theorems*, in section *Effective Quantum Gravity* of *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.09336](https://arxiv.org/abs/2210.09336)]

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On the [[pure spinor]] [[Berkovits superstring]]:

* [[Nathan Berkovits]], [[Carlos R. Mafra]], *Pure spinor formulation of the superstring and its applications*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.10510](https://arxiv.org/abs/2210.10510)]

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On [[moduli stabilization]] in [[string theory]]:

* [[Liam McAllister]], [[Fernando Quevedo]], *Moduli Stabilization in String Theory*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2310.20559](https://arxiv.org/abs/2310.20559)]

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On [[supergravity]]:

* [[Ergin Sezgin]], *Survey of supergravities*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2312.06754](https://arxiv.org/abs/2312.06754)]

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On the usage of [[pure spinor]] [[superfields]] in maximal [[supergravity]]:

* [[Martin Cederwall]], *Pure spinors in classical and quantum supergravity*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.06141](https://arxiv.org/abs/2210.06141)]

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On [[D=4 N=1 supergravity]] formulated via [[superspace]]:

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Superspace approaches to $\mathcal{N} = 1$ supergravity*, *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.17088](https://arxiv.org/abs/2210.17088)]

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On [[D=4 N=2 supergravity]] formulated via [[superspace]]:

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Covariant superspace approaches to $\mathcal{N}=2$ supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.11162](https://arxiv.org/abs/2211.11162)]

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

* [[Leonardo Castellani]], *Group manifold approach to supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.04318](https://arxiv.org/abs/2211.04318)]

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On [[scattering amplitudes]] in [[supergravity]] via "[[double copy]]" [[KLT relations]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Marco Chiodaroli]], [[Henrik Johansson]], [[Radu Roiban]], *Supergravity amplitudes, the double copy and ultraviolet behavior*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.07392](https://arxiv.org/abs/2304.07392)]

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Review of [[U-duality]] and [[exceptional generalized geometry]]  in [[KK-compactification]] of [[D=11 supergravity]]:

* [[Henning Samtleben]], *11D Supergravity and Hidden Symmetries*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2303.12682](https://arxiv.org/abs/2303.12682)]

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On [[quantum anomalies]] and the [[Green-Schwarz mechanism]]:

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

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On the [[super-embedding formalism]] for [[Green-Schwarz superstrings]] and [[super p-branes|super $p$-branes]]:

* [[Igor A. Bandos]], [[Dmitri P. Sorokin]], *Superembedding approach to superstrings and super-$p$-branes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10668](https://arxiv.org/abs/2301.10668)]

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On [[asymptotic safety]]:

* Riccardo Martini, Gian Paolo Vacca, Omar Zanusso, *Perturbative approaches to non-perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.13910](https://arxiv.org/abs/2210.13910)]

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* Astrid Eichhorn, Marc Schiffer, *Asymptotic safety of gravity with matter*, in: [[Handbook of Quantum Gravity]], Springer (2023) [[arXiv:2212.07456](https://arxiv.org/abs/2212.07456)]

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* Jan M. Pawlowski, Manuel Reichert, *Quantum Gravity from dynamical metric fluctuations* in *[[Handbook of Quantum Gravity]]* [[arXiv:2309.10785](https://arxiv.org/abs/2309.10785)]

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On [[black holes]] in the context of [[asymptotic safety]]:

* Alessia Platania, *Black Holes in Asymptotically Safe Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.04272](https://arxiv.org/abs/2302.04272)]

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On non-locality:

* Arnau Bas i Beneito, Gianluca Calcagni, Lesław Rachwał, *Classical and quantum nonlocal gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.05606](https://arxiv.org/abs/2211.05606)]

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On [[causal dynamical triangulation]]:

* Dario Benedetti, *Landau Theory of Causal Dynamical Triangulations*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.11043](https://arxiv.org/abs/2212.11043)]

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On [[Hořava–Lifshitz gravity]]:

* Andrei O. Barvinsky, *Horava models as palladium of unitarity and renormalizability in quantum gravity* [[arXiv:2301.13580](https://arxiv.org/abs/2301.13580)]

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On the [[perturbative algebraic quantum field theory|locally covariant pAQFT approach]] to effective [[quantum gravity]] and applications to experiment:

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], *Locally covariant approach to effective quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/2212.07800)]

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On the conceptual foundations of [[loop quantum gravity]]:

* [[Carlo Rovelli]], Francesca Vidotto, *Philosophical Foundations of Loop Quantum Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.06718](https://arxiv.org/abs/2211.06718)]

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On interactions between [[quantum gravity]] and [[quantum information]] and applications to [[loop quantum gravity]]:

* Eugenio Bianchi, [[Etera Livine|Etera R. Livine]], *Loop Quantum Gravity and Quantum Information*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.05922](https://arxiv.org/abs/2302.05922)]

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On adaptations of Open Effective Theories in [[gravity|gravitational]] settings:

* C.P. Burgess, Greg Kaplanek, *Gravity, Horizons and Open EFTs*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.09157](https://arxiv.org/abs/2212.09157)]

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Reviews the relationship between 2D [[piecewise flat spacetime|casual dynamical triangulations]] and 2D projectable HL [[quantum gravity]]:

* Yuki Sato, *CDT and Hořava-Lifshitz QG in Two Dimensions*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.03446](https://arxiv.org/abs/2212.03446)]

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Discusses the impact of [[asymptotic safety]] on [[effective action functional|quantum effective action]]:

* Benjamin Knorr, Chris Ripken, Frank Saueressig *Form Factors in Asymptotically Safe Quantum Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.16072](https://arxiv.org/abs/2210.16072)]

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Reviews the functional $f(R)$ approximations in [[asymptotic safety]]:

* Tim R. Morris, Dalius Stulga, *The functional f(R) approximation*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.11356](https://arxiv.org/abs/2210.11356)]

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On boundary charges and non-perturbative [[quantum gravity]]:

* [[Laurent Freidel]], Marc Geiller, Wolfgang Wieland, *Corner symmetry and quantum geometry*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.12799](https://arxiv.org/abs/2302.12799)]

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In terms of [[causal sets]]:

* Nomaan X, *Quantum Field Theory On Causal Sets*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2306.04800](https://arxiv.org/abs/2306.04800)]

On Covariant Growth Dynamics and causal sets theory:

* Stav Zalel, *Covariant Growth Dynamics*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.10582](https://arxiv.org/abs/2302.10582)]

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A pedagogical introduction to $N=1$ [[supergravity]]/[[supersymmetry]] in relation to [[particle physics]]:

* Florian Domingo, Michel Rausch de Traubenberg, *Supergravity: Application in Particle Physics*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2209.12541](https://arxiv.org/abs/2209.12541)]

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On the structure and couplings of [[supergravity]] theories at the component level:

* [[Gianguido Dall'Agata]], Marco Zagermann, *Simple Supergravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.10044](https://arxiv.org/abs/2212.10044)]

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A status report on [[black hole]]-like solutions in non-local theories of [[gravity]]:

* Luca Buoninfante, Breno L. Giacchini, Tibério de Paula Netto, *Black holes in non-local gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.03497](https://arxiv.org/abs/2211.03497)]

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On random geometry and applications to higher dimensional [[Euclidean gravity|Euclidean]] [[quantum gravity]]:

* Timothy Budd, *Lessons from the Mathematics of Two-Dimensional Euclidean Quantum Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.03031](https://arxiv.org/abs/2212.03031)]

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On [[infrared divergence|infrared]] and [[ultraviolet divergence|ultraviolet]] limits of four dimensional Casual Dynamical Triangulations:

* Jakub Gizbert-Studnicki, *Semiclassical and Continuum Limits of Four-Dimensional CDT*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.06068](https://arxiv.org/abs/2301.06068)]

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On how to understand long-[[wavelength]] [[quantum fluctuation|fluctuations]] produced by accelerating [[cosmology|cosmological]] backgrounds:

* Daniel Green, *EFT for de Sitter Space*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.05820](https://arxiv.org/abs/2210.05820)]

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Reviews recent constructions of different non-[[Lorentzian geometry|Lorentzian]] [[supergravity]] theories in various [[dimension]]s:

* [[Eric Bergshoeff|Eric A. Bergshoeff]], Jan Rosseel, *Non-Lorentzian Supergravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.02604](https://arxiv.org/abs/2211.02604)]

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Reviews [[numerical analysis|numerical methods]] and their applications for calculations in [[spin foam]] theory:

* Pietro Dona, Muxin Han, Hongguang Liu, *Spinfoams and high performance computing*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.14396](https://arxiv.org/abs/2212.14396)]

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On a formulation of [[entanglement entropy]] of a quantum [[scalar field]] in terms of [[n-point function|two-point]] [[correlator|correlation]] functions and its applications to casual set theory:

* Yasaman K. Yazdi, *Entanglement Entropy and Causal Set Theory*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.13586](https://arxiv.org/abs/2212.13586)]

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On [[string phenomenology]] constructions of four-dimensional [[vacuum|vacua]] of [[type II string theory]] and [[F-theory]] which yield something similar to the [[standard model of particle physics|Standard Model]] or extensions thereof ([[MSSM]]):

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

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On [[field theory|field-theoretical]] treatments of [[General Relativity]] based on an [[effective quantum field theory|EFT]] extension of the [[Einstein-Hilbert action]]:

* N. Emil J. Bjerrum-Bohr, Ludovic Planté, [[Pierre Vanhove]], *Effective Field Theory and Applications: Weak Field Observables from Scattering Amplitudes in Quantum Field Theory*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.08957](https://arxiv.org/abs/2212.08957)]

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Reviews frameworks developed for [[perturbation theory]] in loop quantum cosmology and applications:

* Ivan Agullo, Anzhong Wang, Edward Wilson-Ewing, *Loop quantum cosmology: relation between theory and observations*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10215](https://arxiv.org/abs/2301.10215)]

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On the [[effective quantum field theory|EFT]] description of gravitating compact objects:

* Walter D. Goldberger, *Effective Field Theory for Compact Binary Dynamics*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.06677](https://arxiv.org/abs/2212.06677)]

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A non-expert introduction to the [[effective quantum field theory|EFT]] of large scale [[structure formation|structure]]:

* Mikhail M. Ivanov, *Effective Field Theory for Large Scale Structure*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.08488](https://arxiv.org/abs/2212.08488)]

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On the basics of [[D=4 supergravity|$N=2$, 4D conformal and Einstein supergravities]] in the harmonic [[superspace]] approach:

* Evgeny Ivanov, *N=2 Supergravities in Harmonic Superspace*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07925](https://arxiv.org/abs/2212.07925)]

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On the state of the art of [[black hole]] entropy in [[loop quantum gravity]]:

* J. Fernando Barbero G., Daniele Pranzetti, *Black Hole Entropy in Loop Quantum Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.13469](https://arxiv.org/abs/2212.13469)]

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Describes how one can go from [[piecewise flat spacetime|dynamical triangulations]] to [[quantum gravity]]:

* Yoshiyuki Watabiki, *The causality road from dynamical triangulations to quantum gravity that describes our Universe*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.13109](https://arxiv.org/abs/2212.13109)]

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A pedagogical introduction to the gravitational [[asymptotic safety]] program with a focus on conceptual issues:

* Frank Saueressig, *The Functional Renormalization Group in Quantum Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.14152](https://arxiv.org/abs/2302.14152)]

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On [[loop quantum gravity]] and quantum [[Riemannian geometry]]:

* Hal M. Haggard, Jerzy Lewandowski, Hanno Sahlmann, *Emergence of Riemannian Quantum Geometry*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.02840](https://arxiv.org/abs/2302.02840)]

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An introduction to Unimodular [[gravity]] at the classical and quantum level and applications to the [[cosmological constant]] problem:

* Enrique Álvarez, Eduardo Velasco-Aja, *A Primer on Unimodular Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.07641](https://arxiv.org/abs/2301.07641)]

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On generic random [[tree]]s and two-dimensional causal triangulations:

* Bergfinnur Durhuus, Thordur Jonsson, [[John Wheater]], *From Trees to Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.15247](https://arxiv.org/abs/2211.15247)]

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On [[renormalization]] and [[spin foam]]s:

* Seth K. Asante, Bianca Dittrich, Sebastian Steinhaus, *Spin foams, Refinement limit and Renormalization*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.09578](https://arxiv.org/abs/2211.09578)]

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On spherically symmetric midi-[[superspace]]s in [[loop quantum gravity]]:

* Rodolfo Gambini, Javier Olmedo, Jorge Pullin, *Quantum geometry and black holes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.05621](https://arxiv.org/abs/2211.05621)]

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An overview on the use of Euclidean Dynamical Triangulations and Causal Dynamical Triangulations as lattice regularizations of [[quantum gravity]]:

* Jan Ambjorn, *Lattice Quantum Gravity: EDT and CDT*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2209.06555](https://arxiv.org/abs/2209.06555)]

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On [[string diagram]] techniques for [[spin network]] states in [[loop quantum gravity]]:

* Emanuele Alesci, Ilkka Mäkinen, Jinsong Yang, *Graphical calculus of spin networks*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2304.00268](https://arxiv.org/abs/2304.00268)]

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On [[heterotic string theory|heterotic]] [[string phenomenology]]:

* [[Saul Ramos-Sanchez]], [[Michael Ratz]], *Heterotic Orbifold Models*, in *[[Handbook of Quantum Gravity]]*, Springer (2024) [[arXiv:2401.03125](https://arxiv.org/abs/2401.03125), [doi:10.1007/978-981-19-3079-9](https://doi.org/10.1007/978-981-19-3079-9)]

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category: reference


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The page compiles pointers related to the book collection:

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* Kurt Engesser, Dov M. Gabbay and Daniel Lehmann (eds):

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  **Handbook of Quantum Logic and Quantum Structures**

  **Quantum logic**

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  Elsevier (2008)

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  [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/)

  [doi:10.1016/C2009-0-16935-0](https://doi.org/10.1016/C2009-0-16935-0)

  [pdf](http://perso.ens-lyon.fr/jacques.jayez/Cours/Logique_Classique/Handbook_of_Quantum_Logic_2009.pdf)

on [[quantum logic]] (mostly) and [[quantum systems]].

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**Contents:**

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On [[quantum protocols]] in [[quantum information theory]] (such as [[quantum teleportation]]) formalized as [[string diagrams]] in the [[dagger-compact category]] of [[finite dimensional vector spaces|finite-dimensional]] [[Hilbert spaces]] ([[finite quantum mechanics in terms of dagger-compact categories]]):

* {#AbramskyCoecke08} [[Samson Abramsky]], [[Bob Coecke]], *Categorical quantum mechanics*, in *[[Handbook of Quantum Logic and Quantum Structures]]*, Elsevier (2008) [[arXiv:0808.1023](http://arxiv.org/abs/0808.1023), [ISBN:9780080931661](https://www.sciencedirect.com/book/9780444528698/), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)]

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category: reference


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#Contents#
* table of contents
{:toc}

## Idea

(...)

(...)

## Definition

An **$(m,k)$-handlebody of genus $g$** is a [[manifold]] consisting of $g$ $k$-[[handles]] attached to the [[disk]] $D^m$.

## Related concepts

* [[cobordism hypothesis]]

## References

Textbook account:

* {#Kosinski93} [[Antoni Kosinski]], chapter VI of: _Differential manifolds_, Academic Press (1993) [[pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C)]

See also:

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

* Wikipedia, *[Handle decomposition](https://en.wikipedia.org/wiki/Handle_decomposition)*

* [[Anna Beliakova]], Marco De Renzi, *Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies* [[arXiv:2205.11385](https://arxiv.org/abs/2205.11385)]

[[!redirects handlebodies]]

[[!redirects handle body]]
[[!redirects handle bodies]]


Hank Chen

* [Institute page](https://uwaterloo.ca/scholar/c398chen/home)

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

## Selected writings

In a broader context of [[higher gauge theory]] and relating to the [[Dirac monopole]]:

* [[Hank Chen]], [[Florian Girelli]], *Gauging the Gauge and Anomaly Resolution* [[arXiv:2211.08549](https://arxiv.org/abs/2211.08549)]

category: people


Hanna Neumann

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

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=53256)

## Selected writings

Early discussion of [[amalgamated free products of groups]] (and apparently introducing that terminology):

* [[Hanna Neumann]], *Generalized Free Products with Amalgamated Subgroups*, American Journal of Mathematics, **71** 3 (1949) 491-540 [[jstor:2372346](https://www.jstor.org/stable/2372346)]

* [[B. Hanna Neumann]], *An Essay on Free Products of Groups with Amalgamations*, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences **246** 919 (1954) 503-554 [[jstor:91573](https://www.jstor.org/stable/91573)]

category: people

[[!redirects B. Hanna Neumann]]
[[!redirects B. H. Neumann]]


Hannes Diener

* [personal page](https://www.canterbury.ac.nz/engineering/contact-us/people/hannes-diener.html)

## Selected writings

On [[constructive mathematics|constructive]] [[reverse mathematics]]:

* [[Hannes Diener]], *Constructive Reverse Mathematics*, 2018 ([arXiv:1804.05495](https://arxiv.org/abs/1804.05495), [dspace:ubsi/1306](https://dspace.ub.uni-siegen.de/handle/ubsi/1306))

## Related $n$Lab entries

* [[taboo]]

category: people


Hanno Becker

Hi, I'm Hanno, a PhD student under supervision of Catharina Stroppel from the University of Bonn. I am interested in homological and homotopical algebra algebra, categorical logic, Lie theory and knot theory. My (hopefully) soon to be defended thesis is about applications of homotopy theoretic methods in Khovanov-Rozansky knot homology.

You can find my university homepage [here](http://www.math.uni-bonn.de/people/habecker/).

I hope to get some insights into higher category theory here and on the other hand to contribute to some topics which I am already familiar with.

+-- {: .query}
Welcome, Hanno!  ---[[Toby Bartels]]
=--

category: people


Hannu Honkasalo

* [webpage](https://www.helsinki.fi/fi/ihmiset/henkilohaku/hannu-honkasalo-9030149)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=57155)

## Selected writings

Relating [[Bredon cohomology]] to [[abelian sheaf cohomology]] with [[coefficients]] in "[[locally constant sheaves]] except for dependence on [[isotropy groups]]":

For [[finite groups]]:

* {#Honkasalo88} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology_,  Mathematica Scandinavia,  63, 179-195, 1988 ([doi:10.7146/math.scand.a-12232](https://doi.org/10.7146/math.scand.a-12232))

For [[compact Lie groups]]:

* {#Honkasalu90} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology for actions of compact Lie groups_, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 ([jstor:24492569](https://www.jstor.org/stable/24492569))

Discussion of the [[Serre spectral sequence]] in [[Bredon cohomology]]:

* {#Honkasalo97} [[Hannu Honkasalo]], _A sheaf-theoretic approach to the equivariant Serre spectral sequence_, J. Math. Sci. Univ. Tokyo 4 (1997), 53--65 ([pdf](http://journal.ms.u-tokyo.ac.jp/pdf/jms040103.pdf))

category: people


Hans Bethe

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

## Selected writings

Introducing [[spin chains]]:

* [[Hans Bethe]], _Zur Theorie der Metalle_, Z. Phys.71, 205 (1931) ([doi:10.1007/BF01341708](https://doi.org/10.1007/BF01341708))

## Related $n$Lab entries

* [[Bethe ansatz]]

category: people


Hans Ditmarsch

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

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

## Selected writings

On [[epistemic logic|epistemic]] [[modal logic]] and application to [[game theory]]:

* [[Hans Ditmarsch]], [[Wiebe Hoek]], Barteld Kooi, *Dynamic Epistemic Logic*, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) **337**, Springer (2008) [[doi:10.1007/978-1-4020-5839-4_2](https://doi.org/10.1007/978-1-4020-5839-4_2), [pdf](http://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Epistemic-logic.pdf)]

category: people

[[!redirects Hans van Ditmarsch]]


Hans Duistermaat

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

## Selected writings

On [[Lie groups]], [[Lie algebras]] and [[Lie integration]]:

* [[Hans Duistermaat]], [[Johan A. C. Kolk]], *Lie groups*, Springer (2000) [[doi:10.1007/978-3-642-56936-4](https://doi.org/10.1007/978-3-642-56936-4)]

## Related entries

* [[Lie group]]

* [[propagation of singularities theorem]]

category: people

[[!redirects Johann Duistermaat]]
[[!redirects Johann Jisse Duistermaat]]


Hans Euler

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

## Selected writings

On the [[Schwinger effect]] of [[vacuum polarization]]:

* {#HeisenbergEuler36} [[Werner Heisenberg]], [[Hans Euler]],  _Folgerungen aus der Diracschen Theorie des Positrons_, Z. Physik 98, 714–732 (1936) ([doi:10.1007/BF01343663](https://doi.org/10.1007/BF01343663))

category: people


Hans Fitting

* [Wikipedia entry](http://de.wikipedia.org/wiki/Hans_Fitting_(Mathematiker)

## related $n$Lab entries

* [[generalized Fitting subgroup]]

category: people


Hans Freudenthal

**Hans Freudenthal** (17 September 1905 – 13 October 1990) was a Dutch mathematician, who worked in [[algebraic topology]] and also created an influential direction in [[mathematics education]].

## As a forerunner of category theory
 {#AsAForerunnerOfCategoryTheory}

Freudenthal's influence proved decisive for the recognition of [[general abstract]] structures in [[algebraic topology]]: His 1937 paper on "[Entwicklungen von Räumen und Gruppen](#Freudenthal37)" stresses the importance of a general abstract theory for the connection between topological and algebraic systems in the passage to [[direct limits]] in order to avoid giving the same [[proofs]] over and over again:

> [[p. 147](http://www.numdam.org/item/CM_1937__4__145_0.pdf#page=4)]: "Der Zusammenhang zwischen beiden ist kaum bemerkt und niemals systematisch untersucht worden. Kap. I-II dieser Arbeit bringen die allgemeine Theorie dieser Entwicklungen. Kap. VI soll zeigen, welche Bedeutung die Harmonie der Entwicklungen von Gruppen mit denen von Räumen für die Topologie hat; es soll die logische Struktur des Gedankengeflechtes freilegen, das zahlreichen topologischen Betrachtungen gemeinsam ist. In der Tat sind es immer wieder dieselben Wege, auf denen sich bei der Definition Bettischer Gruppen usw. der Übergang vom Kombinatorischen zum Mengentheoretischen vollzieht; es sind immer wieder dieselben Beweismethoden, mit denen man zu den einfachsten Sätzen über diese Gruppen gelangt.(...) Welchen Nutzen unter solchen Umständen eine einmal entwickelte allgemeine Theorie tragen kann, leuchtet wohl ein. Konkrete neue Einsichten vermitteln die Kap. I, II, VI allerdings kaum. Der Sinn von Kap. I-II ist mehr der einer logischen Analyse, der von Kap. VI ist der des Durchexerzierens der Begriffe und Sätze
aus Kap. I-II an einer großen Anzahl (immerhin ganz allgemeiner) Beispiele."

The term "natural homomorphism" is introduced on [p. 196](http://www.numdam.org/item/CM_1937__4__145_0.pdf#page=53). This provides the blueprint for the definition of *[[natural transformations]]* in the seminal paper "General theory of natural equivalences" by [Eilenberg & MacLane (1945)](category+theory#EilenbergMacLane45) that introduced [[category theory]] (cf. their discussion of "naturality" on p .236) and resonated Freudenthal's "allgemeine Theorie" in the title.

## Selected writings

Introducing the notion of [[topological ends]] end of [[end compactification]]:

* {#Freudenthal31} [[Hans Freudenthal]], _Über die Enden topologischer Räume und Gruppen_, Mathematische Zeitschrift **33** (1931) 692-713 [[doi:10.1007/BF01174375](https://doi.org/10.1007/BF01174375), [pdf](http://dspace.library.uu.nl/bitstream/handle/1874/7437/1930-freudenthal-dissertatie.pdf?sequence=1)]

* [[Hans Freudenthal]], *Neuaufbau der Endentheorie*, Annals of Mathematics Second Series **43** 2 (1942) 261-279 [[doi:10.2307/1968869](https://doi.org/10.2307/1968869), [jstor:1968869](https://www.jstor.org/stable/1968869)]

* [[Hans Freudenthal]], *Über die Enden diskreter Räume und Gruppen*, Commentarii Mathematici Helvetici **17** (1944) 1–38 [[doi:10.1007/BF02566233](https://doi.org/10.1007/BF02566233)]

On [[intuitionistic logic]]:

* [[Hans Freudenthal]], *Zur intuitionistischen Deutung logischer Formeln*, Comp. Math. **4** (1937) 112-116 [[numdam:CM_1937__4__112_0](http://www.numdam.org/item/?id=CM_1937__4__112_0)]

Introducing the [[Freudenthal suspension theorem]] (and the terminology "[[stem of homotopy groups of spheres]]"):

* {#Freudenthal38} [[Hans Freudenthal]], first paragraph of: _Über die Klassen der Sphärenabbildungen I. Große Dimensionen_, Compositio Mathematica, Tome 5 (1938) , pp. 299-314  ([numdam:CM_1938__5__299_0](http://www.numdam.org/item/?id=CM_1938__5__299_0))

On [[Lie groups]]:

* {#Freudenthal65} [[Hans Freudenthal]], _Lie groups in the foundations of geometry_, Advances in Mathematics __1__ (1965) 145 - 190 [[dspace](http://dspace.library.uu.nl/handle/1874/17442)]

On teaching mathematics:

* [[Hans Freudenthal]], _Why to teach mathematics so as to be useful_ (1968) [[pdf](https://promathmedia.files.wordpress.com/2013/09/freudenthal-1968.pdf)]

On [[category theory]] _avant la lettre_:

* {#Freudenthal37} [[Hans Freudenthal]], _Entwicklungen von Räumen und ihren Gruppen_, Comp. Math. **4** (1937) 145-234. [[numdam](http://www.numdam.org/item/CM_1937__4__145_0/)]

On _lingua cosmica_, a tongue for intercourse with martians:

* [[Hans Freudenthal]], _Lincos: Design of a Language for Cosmic Intercourse Part I_, Elsevier Amsterdam (1960).

## Related entries

* [[Freudenthal suspension theorem]]

* [[Freudenthal magic square]]

category: people

[[!redirects Freudenthal]]


Hans Grauert

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

## Selected writings

On the [[Oka-Grauert principle]]:

* {#Grauert58} [[Hans Grauert]], _Analytische Faserungen über holomorph-vollständigen Räumen_, Math. Ann. __135__, 263–-273 (1958) ([doi:10.1007/BF01351803](http://dx.doi.org/10.1007/BF01351803))

## Related $n$Lab entries

* [[Stein manifold]]

category: people


Hans Günter Dosch

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

## Selected writings

On [[holographic light front QCD]]:

* [[Stanley Brodsky]], [[Guy de Teramond]],  [[Hans Günter Dosch]], [[Joshua Erlich]],
_Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))

* {#ZhouDosch18} Liping Zou, [[Hans Günter Dosch]], _A very Practical Guide to Light Front Holographic QCD_, ([arXiv:1801.00607](https://arxiv.org/abs/1801.00607))

* [[Stanley Brodsky]], [[Guy de Teramond]], [[Hans Günter Dosch]], _Light-Front Holography and Supersymmetric Conformal Algebra: A Novel Approach to Hadron Spectroscopy, Structure, and Dynamics_ ([arXiv:2004.07756](https://arxiv.org/abs/2004.07756))

On [[hadron supersymmetry]] via [[light cone gauge|light cone]] [[supersymmetric quantum mechanics]] in [[holographic light front QCD]];

* {#dTDB14}  [[Guy de Teramond]], [[Hans Günter Dosch]], [[Stanley Brodsky]], _Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding_, Phys. Rev. D 91, 045040 (2015) ([arXiv:1411.5243](https://arxiv.org/abs/1411.5243))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Superconformal Baryon-Meson Symmetry and Light-Front Holographic QCD_, Phys. Rev. D 91, 085016 (2015) ([arXiv:1501.00959](https://arxiv.org/abs/1501.00959))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum_, Phys. Rev. D 92, 074010 (2015) ([arXiv:1504.05112](https://arxiv.org/abs/1504.05112))

* {#BTDL16}  [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Cédric Lorcé, _Meson/Baryon/Tetraquark Supersymmetry from Superconformal Algebra and Light-Front Holography_, International Journal of Modern Physics AVol. 31, No. 19, 1630029 (2016) ([arXiv:1606.04638](https://arxiv.org/abs/1606.04638))

* [[Hans Günter Dosch]], [[Guy de Teramond]], [[Stanley Brodsky]], _Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II_, Phys. Rev. D 95, 034016 (2017) ([arXiv:1612.02370](https://arxiv.org/abs/1612.02370))

* [[Marina Nielsen]], [[Stanley Brodsky]], [[Guy de Téramond]], [[Hans Günter Dosch]], Fernando S. Navarra, [[Liping Zou]], _Supersymmetry in the Double-Heavy Hadronic Spectrum_, Phys. Rev. D 98, 034002 (2018) ([arXiv:1805.11567](https://arxiv.org/abs/1805.11567))

In view of [[tetraquarks]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_ ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

category: people


Hans Halvorson

* [website](http://www.princeton.edu/~hhalvors/)

## related $n$Lab entries

* [[AQFT]]

category: people


Hans J. Briegel

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

* [Insitute page](https://www.uibk.ac.at/th-physik/staff/briegel/)

## Selected writings

Introducing [[measurement-based quantum computation]]:

* {#RaussendorfBriegel01} [[Robert Raussendorf]], [[Hans J. Briegel]], *A One-Way Quantum Computer*, Phys. Rev. Lett. **86** (2001) 5188 [[doi:10.1103/PhysRevLett.86.5188](https://doi.org/10.1103/PhysRevLett.86.5188)]

Review:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

category: people

[[!redirects Hans Briegel]]


Hans Jockers

* [institute page](http://www.th.physik.uni-bonn.de/Groups/Klemm/members.php)

## Selected writings

On [[quantum K-theory rings]]:

* [[Hans Jockers]], [[Peter Mayr]], *Quantum K-Theory of Calabi-Yau Manifolds*, J. High Energ. Phys. **2019** 11 (2019) [[arXiv:1905.03548](https://arxiv.org/abs/1905.03548), doi:10.1007/JHEP11(2019)011]

Relation to [[D=3 N=2 super Yang-Mills theory]]:

* [[Hans Jockers]], [[Peter Mayr]], *A 3d Gauge Theory/Quantum K-Theory Correspondence*, Advances in Theoretical and Mathematical Physics **24** 2 (2020) [[arXiv:1808.02040](https://arxiv.org/abs/1808.02040), [doi:10.4310/ATMP.2020.v24.n2.a4](https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a4)]

## Related entries

* [[M-theory on G2-manifolds]]

category: people


Hans Maassen

* [personal page](https://www.math.ru.nl/~maassen)

## Selected writings

On [[quantum probability theory]]:

* [[Hans Maassen]], _Quantum Probability Theory_, Lecture notes 1998 ([pdf](https://www.math.ru.nl/~maassen/lectures/qp.pdf), [[MaassenQuantumProbability.pdf:file]])

category: people


Hans Porst

* [website](http://www.math.uni-bremen.de/~porst/)

## Selected writings

On [[locally presentable category|local presentability]] of [[CocommCoalg|categories of]] [[coalgebras]] and their [[comodules]]:

* [[Hans-Eberhard Porst]], *On corings and comodules*, Archivum Mathematicum **42** 4 (2006) 419-425 [[dmlcz:108017](https://dml.cz/handle/10338.dmlcz/108017), [pdf](http://dml.cz/bitstream/handle/10338.dmlcz/108017/ArchMathRetro_042-2006-4_7.pdf)]

On the history of the [[adjoint functor theorem]]:

* [[Hans-E. Porst]], _The history of the General Adjoint Functor Theorem_ [[arxiv:2310.19528](https://arxiv.org/abs/2310.19528)]

## Related entries

* [[duality]]

* [[Gelfand duality]]

category: people

[[!redirects Hans-E. Porst]]
[[!redirects Hans-Eberhard Porst]]


Hans Primas

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

## Selected writings

On [[quantum mechanics]] as the foundation of [[quantum chemistry]] with a focus on [[interpretation of quantum mechanics]]:

* [[Hans Primas]], *Chemistry, Quantum Mechanics and Reductionism*, Springer (1983) [[doi:10.1007/978-3-642-69365-6](https://doi.org/10.1007/978-3-642-69365-6)]

category: people


Hans Reichenbach

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

## Related entries

* [[philosophy of physics]]

category: people


Hans Reissner

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Hans_Jacob_Reissner)

* [English Wikipedia entry](https://en.wikipedia.org/wiki/Hans_Reissner) (this has no mentioning of the subject of black holes...)

## Selected writings

Introducing the [[Reissner-Nordström spacetime]] describing [[electromagnetism|electromagnetically]] [[charged particle|charged]] [[black holes]]:

* [[Hans Reissner]], _Über die Eigengravitation des elektrischen Feldes nach der Einstein'schen Theorie_, Annalen der Physik 50 pp. 106–120, 1916 ([doi:10.1002/andp.19163550905]( https://doi.org/10.1002/andp.19163550905), [pdf](http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Reissner.pdf),  [atticus](https://www.atticusrarebooks.com/pages/books/1026/hans-reissner-jacob-also-jacob-johannes-reissner/uber-die-eigengravitation-des-elektrischen-feldes-nach-der-einstein-schen-theorie-offprint-from))

category: people


Hans Samelson

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

* J. J. O'Connor, E. F. Robertson, _Hans Samelson_ ([web](https://mathshistory.st-andrews.ac.uk/Biographies/Samelson/))

## Selected writings

On [[topological groups]]:

* [[Hans Samelson]], _Beiträge zur Topologie der Gruppenmannigfaltigkeiten_, Ann. of Math. 2, 42, (1941) 1091 - 1137 [[jstor:1970463](https://www.jstor.org/stable/1970463), [doi:10.2307/1970463](https://doi.org/10.2307/1970463)]

On [[transitive actions]] on [[n-spheres|$n$-spheres]] by [[compact Lie groups]]:

* {#MontogomerySamelson43} [[Deane Montgomery]], [[Hans Samelson]], _Transformation Groups of Spheres_, Annals of Mathematics Second Series **44** 3 (1943) 454-470 [[jstor:1968975](https://www.jstor.org/stable/1968975)]

Introducing the concept and terminology of the [[Pontrjagin product]] on the [[ordinary homology|homology]] of [[based loop spaces]]:

* [[Raoul Bott]], [[Hans Samelson]], *On the Pontryagin product in spaces of paths*, Commentarii Mathematici Helvetici **27** (1953) 320–337 [[doi:10.1007/BF02564566](https://doi.org/10.1007/BF02564566)]

Introducing the [[Samelson product]]:

* [[Hans Samelson]].  _A Connection Between the Whitehead and the Pontryagin Product_.  American Journal of Mathematics 75:4 (1953) 744 [[doi:10.2307/2372549](https://doi.org/10.2307/2372549)]

## Related $n$Lab entries

* [[Samelson product]]

category: people


Hans Zassenhaus

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

## Selected writings

On [[free action|free]] [[orthogonal group|orthgonal]] [[group actions on n-spheres]]:

* {#Zassenhaus35} [[Hans Zassenhaus]], *Über endliche Fastkörper*, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg. Vol. 11. No. 1. Springer-Verlag, 1935 ([pdf](https://link.springer.com/content/pdf/10.1007/BF02940723.pdf))

category: people


Hans-Christian Pauli

* [inspire page](https://inspirehep.net/authors/994032)

## Selected writings

Introducing [[discretized light-cone quantization]]:

* [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Solving field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 1993 [[doi:10.1103/PhysRevD.32.1993](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.32.1993)]

* {#PauliBrodsky85b} [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Discretized light-cone quantization: Solution to a field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 2001 [[doi:10.1103/PhysRevD.32.2001](https://doi.org/10.1103/PhysRevD.32.2001)]

and its application to [[quantum electrodynamics]]:

* {#TangBrodskyPauli91} [[Andrew C. Tang]], [[Stanley J. Brodsky]], [[Hans-Christian Pauli]], *Discretized light-cone quantization: Formalism for quantum electrodynamics*, Phys. Rev. D **44** (1991) 1842 [[doi:10.1103/PhysRevD.44.1842](https://doi.org/10.1103/PhysRevD.44.1842)]

and to [[quantum chromodynamics]]:

* [[Stanley J. Brodsky]], [[Gary McCartor]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *The challenge of light-cone quantization of gauge field theory*, Part. World **3** 3 (1993) 109-124  [[cds:240388](http://cds.cern.ch/record/240388), [inspire:335247](https://inspirehep.net/literature/335247)]

* [[Stanley Brodsky]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *Quantum Chromodynamics and Other Field Theories on the Light Cone*, Phys. Rept. **301** (1998) 299-486 [[arXiv:hep-ph/9705477](https://arxiv.org/abs/hep-ph/9705477), doi:10.1016/S0370-1573(97)00089-6]

* {#Pauli99} [[Hans-Christian Pauli]], *Discretized light-cone quantization and the effective interaction in hadrons*, AIP Conf. Proc. **494** (1999) 80-139 [[arXiv:hep-ph/9910203](https://arxiv.org/abs/hep-ph/9910203), [doi:10.1063/1.1301662](https://doi.org/10.1063/1.1301662)]

category: people


Hans-Joachim Baues

Hans-Joachim Baues (d. May 8, 2020) was a German topologist and the author of

*  _[[Algebraic Homotopy]]_, Cambridge studies in advanced mathematics 15, Cambridge University Press, (1989);
*  _[[Combinatorial Homotopy and 4-Dimensional Complexes]]_, de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991);
*  _[[Homotopy Types]]_, in I.M.James, ed., _[[Handbook of Algebraic Topology]]_, 1--72, Elsevier, (1995);
*  _Homotopy type and homology_, Oxford University Press, 1996
* (with Antonio Quintero) _Infinite Homotopy Theory_,  K-monographs in mathematics, Volume 6, Kluwer, 2001;

among other works.

A part of his work  developed the ideas of  [[Henry Whitehead|Whitehead's]] basic programme for [[algebraic homotopy]], using a mix of [[cofibration category|cofibration categories]] and categories with a particular type of [[cylinder functor]], that he called [[I-category|I-categories]]. These are treated in separate entries.

Baues  also systematically developed aspects of secondary homological algebra (secondary cohomological operations, secondary Steenrod algebra, secondary chain complexes, secondary triangulated category) which is a form of categorification of the usual homological algebra. He  developed them together to give sound applications in the computations with Adams-Novikov spectral sequence and calculations of homotopy groups of spheres.

He is also well known for his development of the "cohomology of small categories" with Wirsching, sometimes called [[Baues-Wirsching cohomology]], although versions of this theory existed prior to his contribution.

[MPIM Obituary ](https://www.mpim-bonn.mpg.de/node/10303)

## Selected writings

On [[cohomology]] of [[monoids in monoidal categories]] (specifically in [[distributive monoidal categories]]):

* [[Hans-Joachim Baues]], [[Mamuka Jibladze]], [[Andy Tonks]], *Cohomology of monoids in monoidal categories*, in: *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202**, AMS (1997) 137-166 [[doi:10.1090/conm/202](https://doi.org/10.1090/conm/202), preprint:[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/1484/1/preprint_1995_121.pdf)]

category: people

[[!redirects H-J. Baues]]
[[!redirects H. J. Baues]]
[[!redirects Hans Joachim Baues]]
[[!redirects H.-J. Baues]]
[[!redirects Baues]]
[[!redirects Hans Baues]]
[[!redirects H.-J. Baues]]
[[!redirects Hans J. Baues]]


Hans-Jürgen Borchers

* [Wikipedia entry](https://en.wikipedia.org/wiki/Hans-J%C3%BCrgen_Borchers)

## Selected writings

Introducing the [[time tube theorem]] in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], *Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre*, Il Nuovo Cimento (1955-1965) **19** (1961) 787–793 [[doi:10.1007/BF02733373](https://doi.org/10.1007/BF02733373)]

Introducing the *[[Borchers property]]* in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], _A remark on a theorem of B. Misra_,  Comm. Math. Phys. **4** 5 (1967) 315-323 [[euclid](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-4/issue-5/A-remark-on-a-theorem-of-B-Misra/cmp/1103839939.full)]

On [[self-adjoint extensions]] for [[local nets of observables]] in [[AQFT]]:

*  [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *Local nets and self-adjoint extensions of quantum field operators*, Letters in Mathematical Physics, **21** (1991) 151–155 [[doi:10.1007/BF00401649](https://doi.org/10.1007/BF00401649)]

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

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

On [[Tomita-Takesaki modular theory]] in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], *On Revolutionizing of Quantum Field Theory with Tomita's Modular Theory*, ESI Preprint 773 (1999) [[pdf](https://www.mat.univie.ac.at/~esiprpr/esi773.pdf)]

On the [[PCT theorem]] for [[local observables]] in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *On the PCT--Theorem in the Theory of Local Observables*, in: *Mathematical physics in mathematics and physics: Quantum and operator algebraic aspects*, Fields Inst. Commun. **30** (2001) 39-64  [[arXiv:math-ph/0012020](http://arxiv.org/abs/math-ph/0012020), [spire:538524](https://inspirehep.net/literature/538524)]

category: people

[[!redirects Hans-Juergen Borchers]]
[[!redirects Hans-Jurgen Borchers]]

[[!redirects H.-J. Borchers]]

[[!redirects H. J. Borchers]]


Hans-Peter Dürr

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Hans-Peter_D%C3%BCrr)

## Selected writings

Precursor of [[quantum hadrodynamics]]:

* [[Hans-Peter Duerr]], _Relativistic Effects in Nuclear Forces_, Phys. Rev. 103, 469 (1956) ([doi:10.1103/PhysRev.103.469](https://doi.org/10.1103/PhysRev.103.469))

category: people

[[!redirects Hans-Peter Duerr]]


Hans-Peter Nilles

* [webpage](http://www.th.physik.uni-bonn.de/nilles/)

## Selected writings

On [[R-parity]] of  [[supersymmetry]]-models and the [[strong CP problem]]:

* [[Hans-Peter Nilles]], [[Stuart Raby]], _Supersymmetry and the strong CP problem_, Nuclear Physics B Volume 198, Issue 1, 26 April 1982 (doi:10.1016/0550-3213(82)90547-8)

On [[gaugino condensation]] in [[heterotic string theory]] and [[Horava-Witten theory]]:

* {#Nilles04} [[Hans-Peter Nilles]], _Gaugino Condensation and SUSY Breakdown_, Lectures at Cargese School of Physics and Cosmology, Cargese, France, August 2003 ([arXiv:hep-th/0402022](https://arxiv.org/abs/hep-th/0402022))

On [[heterotic string theory|heterotic]] [[string phenomenology]]:

* [[Hans-Peter Nilles]], [[Saul Ramos-Sanchez]], Patrick K.S. Vaudrevange, Akin Wingerter, _Exploring the $SO(32)$ Heterotic String_, JHEP 0604:050 (2006) \[doi:10.1088/1126-6708/2006/04/050[arXiv:hep-th/0603086](https://arxiv.org/abs/hep-th/0603086)\]

## Related entries

* [[string phenomenology]]

* [[R-parity]]

* [[gaugino condensation]]

category: people


Hans-Volker Niemeier

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

## Selected writings

Classification of what came to be known as *[[Niemeier lattices]]*:

* [[Hans-Volker Niemeier]], *Definite quadratische Formen der Dimension 24 und Diskriminante 1*, Journal of Number Theory **5** 2 (1973) 142-178 [doi:10.1016/0022-314X(73)90068-1]

category: people


Hans-Werner Henn

* [personal page](https://irma.math.unistra.fr/~henn/)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Hans-Werner_Henn)

## Selected writings

On [[homotopy theory|homotopy theoretic]] methods in [[group cohomology]]:

* [[William G. Dwyer]], [[Hans-Werner Henn]], *Homotopy Theoretic Methods in Group Cohomology*, Advanced Courses in Mathematics - CRM Barcelona, Springer (2001) [[doi:10.1007/978-3-0348-8356-6](https://doi.org/10.1007/978-3-0348-8356-6)]

On [[chromatic homotopy theory|chromatic]] [[stable homotopy theory]]:

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

category: people


Hansjörg Geiges

* [webpage](http://www.mi.uni-koeln.de/~geiges/)

## Selected writings

On [[contact geometry]]:

* [[Hansjörg Geiges]], *Contact geometry*, in F. J. E. Dillen and L. C. A. Verstraelen, (eds.), *Handbook of Differential Geometry* vol. 2  North-Holland, Amsterdam (2006) 315-382 [[arXiv:math/0307242](http://arxiv.org/abs/math/0307242), [ISBN:978-0-444-52052-4](https://www.sciencedirect.com/handbook/handbook-of-differential-geometry/vol/2/suppl/C)]

* {#Geiges} [[Hansjörg Geiges]], *An Introduction to Contact Topology*, Cambridge Studies in Advanced Mathematics **109**, Cambridge University Press (2008) [[doi:10.1017/CBO9780511611438](https://doi.org/10.1017/CBO9780511611438)]

category: people

[[!redirects Hansjoerg Geiges]]


Hao Geng

* [GoogleScholar page](https://scholar.google.com/citations?user=W7s5KbsAAAAJ&hl=en)

## Selected writings

Argument that "islands" can resolve the [[black hole information paradox]] only for massive gravity:

* [[Hao Geng]], [[Andreas Karch]], *Massive Islands*,  J. High Energ. Phys. **2020** 121 (2020) [[arXiv:2006.02438](https://arxiv.org/abs/2006.02438), doi:10.1007/JHEP09(2020)121]

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Inconsistency of islands in theories with long-range gravity*, J. High Energ. Phys. **2022** 182 (2022) [doi:10.1007/JHEP01(2022)182]

* [[Hao Geng]], *Recent Progress in Quantum Gravity: Karch-Randall Braneworld, Entanglement Islands and Graviton Mass*, Washington (2022) [[hdl:1773/49427](http://hdl.handle.net/1773/49427)]

* [[Hao Geng]], *Graviton Mass and Entanglement Islands in Low Spacetime Dimensions* \[arXiv:2312.13336\]

and what to do about it:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Information Transfer with a Gravitating Bath*, SciPost Phys. **10** 103 (2021)  \[arXiv:2012.04671, [doi:10.21468/SciPostPhys.10.5.103](https://scipost.org/10.21468/SciPostPhys.10.5.103)\]

category: people


HAP

HAP is a homological algebra library for use with the [[GAP]] computer algebra system, and is still under development. Its initial focus is on computations related to the cohomology of groups. Both finite and infinite groups are handled, with emphasis on integral coefficients.

It includes some functions for computing homology of crossed modules and simplicial groups, and also some functions for handling simplicial complexes, cubical complexes and regular CW-complexes in the context of topological data analysis.

It has been developed by [[Graham Ellis]] and colleagues in Galway.

###Link page

* [HAP homepage](http://hamilton.nuigalway.ie/Hap/www/)


HaPPY code

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## Idea

The *hyperbolic pentagon code* or *HaPPY code* is a [[quantum error correction code]] (a class of such codes really, indexed by a "cutoff" [[natural number]]) which is thought to exhibit characteristic properties akin to the encoding of [[bulk]]-[[quantum states]] by [[boundary field theory|boundary]]-states expected in the [[AdS/CFT correspondence]]. In particular, the HaPPY code (or rather the [[tensor network]] that defines it) exhibits a discretized form of the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]].

Concretely, the the HaPPY [[code subspace]] is the [[image]] of the [[linear map]] formed by:

\begin{imagefromfile}
    "file_name": "HaPPYCodeTesselation.jpg",
    "float": "right",
    "width": 440,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Harlow 18](#Harlow18)"
\end{imagefromfile}

1. picking a [[perfect tensor]] $T$ of [[rank]] 6;

1. picking a finite cutoff of the [[pentagon|pentagonal]] [[tesselation]] $\{5,4\}$ of the [[hyperbolic plane]];

1. regarding its [[Poincaré duality|Poincaré dual]] [[graph]] as a [[tensor network]] ([[string diagram]] in [[finite-dimensional vector spaces]]) by

   1. assigning $T$ to each [[vertex]] at the center of the [[pentagons]] (show in blue), with 5 of its indices contracted with its neighbours in the hyperbolic plane,

   1. and its 6th uncontracted index remaining as an input (shown in red);

   1. regading the uncontrated edges at the cutoff boundary as output (shown in white)

and thus as a [[linear map]] form the [[tensor product]] over the [[bulk]]-[[vertices]] to the [[tensor product]] over the edges sticking out over the boundary.

## Related concepts

* [[stabilizer code]]

* [[bit flip code]]

* [[Majorana dimer code]]

## References

Tha HaPPY code is due to

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

following a precursor observation in

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

Review in:

* {#Harlow18} [[Daniel Harlow]], _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_ ([arXiv:1802.01040](https://arxiv.org/abs/1802.01040))

See also:

* Elliott Gesteau, Monica Jinwoo Kang, *The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics* ([arXiv:2005.05971](https://arxiv.org/abs/2005.05971))

[[!redirects HaPPY codes]]


Happy Family

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## Idea

What has been called the _Happy Family_ in [Griess 82](#Griess82) is the [[finite set]] of those 20 of the 26 [[sporadic finite simple groups]], which are [[subquotients]] of the [[monster group]], in contrast to the remaining 6 [[pariah groups]].

The _Happy Family_ is taken to be formed of three generations:

* First: five subgroups of the [[Mathieu group]] $M_{24}$ = symmetries of the [[binary Golay code]], namely the [[Mathieu groups]].
* Second: seven subquotients of the [[Conway group]] $Co_0$ = symmetries of the [[Leech lattice]], including the three simple [[Conway groups]], $Co_1, Co_2, Co_3$, and the Hall-Janko, Suzuki, McLaughlin and Higman-Sims groups.
* Third: eight subquotients of the [[Monster group]] = conformal symmetries of the [[monster vertex operator algebra]], including the [[Fischer groups]] and the [[Thompson sporadic group]].

It has been suggested ([Lentner 12](#Lentner12)) that these three generations are generated by an iterated process of forming the [[centralizer]] of an [[involution]] of a simple group, starting out from the simple group of Lie-type $SL_3(4)$.

## References

* {#Griess82} [[Robert Griess]], Jr. _The Friendly Giant_, Invent. Math. 69 (1982) 1-102 ([doi:10.1007/BF01389186](https://doi.org/10.1007/BF01389186))

* {#Lentner12} Simon Lentner, _Response to "Why are the sporadic simple groups HUGE?"_, [MO answer](https://mathoverflow.net/a/93459/447).

See also

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


Harald Eichenherr

* [inspire page](https://inspirehep.net/authors/2240730)

## Selected writings

Introducing the [[CP^N sigma-model|$\mathbb{C}P^N$-sigma model]]:

* [[Harald Eichenherr]], *$SU(N)$ invariant non-linear σ models*, Nuclear Physics B **146** 1 (1978) 215-223 [doi:10.1016/0550-3213(78)90439-X, errata:doi:10.1016/0550-3213(79)90287-6]

* [[Harald Eichenherr]], Michael Forger, *Higher local conservation laws for nonlinear sigma models on symmetric spaces*, Commun. Math. Phys. **82** (1981) 227–255 [[doi:10.1007/BF02099918](https://doi.org/10.1007/BF02099918)]

category: people


Harald Fritzsch

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

## Selected writings

Introducing "SO(10)" ([[Spin(10)]]) [[grand unified theory]]:

* [[Harald Fritzsch]], [[Peter Minkowski]], _Unified interactions of leptons and hadrons_, Annals of Physics Volume 93, Issues 1–2, 5 September 1975, Pages 193-266 (doi:10.1016/0003-4916(75)90211-0)

Introducing the concept of [[glueballs]]:

* [[Harald Fritzsch]], [[Peter Minkowski]], _Ψ-resonances, gluons and the Zweig rule_, Nuov. Cim. A, 1975,30: 393 ([doi:10.1007/BF02730295](https://doi.org/10.1007/BF02730295))

category: people


Harald Grosse

* [webpage](http://physik.univie.ac.at/index.php?id=1597/)

## Selected writings

Introducing the [[fuzzy 4-sphere]]:

* [[Harald Grosse]], [[Ctirad Klimcik]], P. Presnajder, _On Finite 4D Quantum Field Theory in Non-Commutative Geometry_, Commun. Math. Phys.180:429-438, 1996 ([arXiv:hep-th/9602115](https://arxiv.org/abs/hep-th/9602115))

category: people


Harald Lindner

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=21475)

## Selected writings

Early discussion of [[dual objects]] in a [[monoidal category]] as [[adjunctions]] in its [[delooping]] [[2-category]]:

* [[Harald Lindner]], *Adjunctions in monoidal categories*, Manuscripta Mathematica **26** 1-2 (1978)  123-139 [[doi:10.1007/BF01167969](https://doi.org/10.1007/BF01167969)]


Harald Weinfurter

* [personal page](https://xqp.physik.uni-muenchen.de/people/professor/weinfurter/index.html)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Harald_Weinfurter)

## Selected writings

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter W. Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

On [[quantum teleportation]] realized in [[experiment]]:

* [[Dirk Bouwmeester]], Jian-Wei Pan, Klaus Mattle, Manfred Eibl, [[Harald Weinfurter]], [[Anton Zeilinger]], *Experimental quantum teleportation*, Nature **390** 575–579 (1997) [[doi:10.1038/37539](https://doi.org/10.1038/37539)]

category: people


hard Lefschetz theorem

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## Idea

For $(X,\omega)$ a [[Kähler manifold]] the operation of forming the wedge product with the [[symplectic form]] $\omega \in \Omega^{1,1}(X)$ induces on [[de Rham cohomology]] the _Lefschetz operator_

$$
  L \;\colon\; H^k(X) \longrightarrow H^{k+2}(X)
  \,.
$$

The **hard Lefschetz theorem** asserts that if $X$ is [[compact topological space|compact]] with complex [[dimension]] $dim_{\mathbb{C}}(X)= d$, then for all $k \geq 0$ the $k$th power of the $L$-operation induces an [[isomorphism]]

$$
  L^k \;\colon\; H^{d-k}(X) \stackrel{\simeq}{\longrightarrow} H^{d+k}(X)
  \,.
$$

Using that $\omega$ is a $(1,1)$-form this means equivalently in terms of [[Dolbeault cohomology]] that for all $(p+q) \leq d$ we have an isomorphism

$$
  L^{d-(p+q)}
  \;\colon\;
  H^{p,q}(X) \stackrel{\simeq}{\longrightarrow} H^{d-q,d-p}(X)
  \,.
$$

This exhibits the symmetry of the [[Hodge diamond]] under reflection about the horizontal diagonal.

The hard Lefschetz theorem induces the _[[Lefschetz decomposition]]_ (see there) of the de Rham cohomology of $X$.

The hard Leftschetz theorem also applies to other objects, for instance, to [[matroids]] (see [Huh 22](#Huh22)).

## References

* {#Voisin02} [[Claire Voisin]], section 6 of _[[Hodge theory and Complex algebraic geometry]] I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

* _Notes on Lefschetz decomposition_ [pdf](http://www.math.sunysb.edu/~azinger/mat545/LDnotes.pdf)

* MO discussion _[Intuition for primitive cohomology](http://mathoverflow.net/a/14727/381)_
[[!redirects Lefschetz decompositions]]

In the generality of [[matroids]]:

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

[[!redirects Lefschetz operator]]
[[!redirects Lefschetz operators]]


Harder-Narasimhan theorem

[[!redirects Harder-Narasimhan filtration]]

## Related concepts

* [[Narasimhan-Seshadri theorem]]

* [[Donaldson-Uhlenbeck-Yau theorem]]

## References

* [[Jonathan Evans]],  _Aspects of Yang-Mills theory_, Lecture notes on [[Yang-Mills theory]] ([web](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm)) ([lecture 12 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture12.pdf))

* Jonathan Pottharst, _Harder-Narasimhan theory_ ([arXiv:2003.11950](https://arxiv.org/abs/2003.11950))


Harish-Chandra module

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## Idea

A kind of [[module]]/[[representation]] over a pair consisting of a [[semisimple Lie algebra]] and an [[algebraic group]].

## Related concepts

* [[coinduced representations]] on Harish-Chandra modules are also referred to as _[[Zuckerman induction]]_

## References

* wikipedia: [(g,K)-module](http://en.wikipedia.org/wiki/%28g,K%29-module), [Harish-Chandra module](http://en.wikipedia.org/wiki/Harish-Chandra_module)
* David Vogan, Jr., Unitary Representations of Reductive Lie Groups, Annals of Mathematics Studies __118__, Princeton Univ. Press 1987
* A. W. Knapp, D. A. Vogan, _Cohomological induction and unitary
representations_, Princeton Univ. Press, 1995.

* Harish-Chandra, _Representations of semi-simple lie groups I_, Trans. Amer. Math. Soc. 75, 185–243, 1953; II, Trans. AMS 76, 26–65, 1954; III, Trans. AMS 76, 245–253, 1954.
* Fabian Januszewski, _Algebraic characters for Harish-Chandra modules_, [arxiv/1201.2447](http://arxiv.org/abs/1201.2447)

[[!redirects Harish-Chandra module]]
[[!redirects Harish-Chandra modules]]
[[!redirects Harish–Chandra module]]
[[!redirects Harish–Chandra modules]]
[[!redirects Harish--Chandra module]]
[[!redirects Harish--Chandra modules]]

[[!redirects Harish-Chandra theory]]
[[!redirects Harish–Chandra theory]]
[[!redirects Harish--Chandra theory]]


Harley Eades

* [personal page](https://metatheorem.org/)

## Selected writings

On [[type theory]] and [[verified programming]]:

* [[Harley Eades]], *Type Theory and Applications*, 2012 ([pdf](https://metatheorem.org/includes/pubs/comp.pdf), [[EadesTypeTheoryAndApplications.pdf:file]])

On [[dependent linear types]] and [[graded modalities]]:

* [[Benjamin Moon]], [[Harley Eades III]], [[Dominic Orchard]], *Graded Modal Dependent Type Theory*. In: N. Yoshida (ed.) *Programming Languages and Systems ESOP 2021*, Lecture Notes in Computer Science **12648**, Springer (2021) 462-490 [[doi:10.1007/978-3-030-72019-3_17](https://doi.org/10.1007/978-3-030-72019-3_17), [arxiv:2010.13163](https://arxiv.org/abs/2010.13163)]

## Related entries

* [[computational trilogy]]

category: people

[[!redirects Harley Eades III]]


harmonic

\tableofcontents

\section{Idea}

In [[acoustics]] and [[signal processing]], given a [[periodic function|periodic]] [[waveform]] $P$, a **harmonic** is a periodic waveform whose [[frequency]] is an [[integer]] [[multiple]] of the [[fundamental frequency]] of $P$.

\section{See also}

* [[fundamental frequency]]

* [[waveform]]

* [[periodic function]]

* [[harmonic oscillator]]

* [[Fourier analysis]]

\section{References}

See also:

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

[[!redirects harmonics]]


harmonic analysis

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## Idea

The basic problem of __harmonic analysis__ is the decomposition of elements in some [[topological vector space]] of [[functions]] in some [[linear basis]] which is typically distinguished by some nice [[representation theory|representation theoretical]] properties. This decomposition can be a sum, and a basis a topological basis, but more general it is a decomposition in the sense of an integral. The elements of the distinguished bases were in historical examples thought of as "basic waves" or "harmonics".
Some standard examples are [[Fourier analysis]] on locally compact abelian groups, [[wavelet analysis]], [[quantum group Fourier transform]] etc. In some cases the elements of the "basis" are not linearly independent, e.g. in the case of decomposition into [[coherent state]]s.

## Related entries

* [[Fourier transform]], [[Fourier-Laplace transform]]

* [[Fourier transform of distributions]]

* [[Fourier analysis]],

* [[Bochner's theorem]]

* [[Tannaka-Krein theorem]],

* [[Peter-Weyl theorem]],

* [[Gelfand-Raikov theorem]],

* [[sampling theory]]

* [[Restriction Theory]]

## References

* [[Edwin Hewett]], [[Kenneth Ross]], *Abstract Harmonic Analysis*

  Volume I: *Structure of Topological Groups, Integration Theory, Group Representations*,  Grundlehren der mathematischen Wissenschaften **115**, Springer 1979 ([doi:10.1007/978-1-4419-8638-2](https://link.springer.com/book/10.1007/978-1-4419-8638-2))

  Volume II: *Structure and Analysis for Compact Groups, Analysis on Locally Compact Abelian Groups*,  Grundlehren der mathematischen Wissenschaften **152**, Springer 1970 ([doi:10.1007/978-3-662-26755-4](https://link.springer.com/book/10.1007/978-3-662-26755-4))

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

* Elias Stein, Guido Weiss, _Introduction to Fourier analysis on Euclidean spaces_, Princeton Univ. Press 1971

See also:

* Wikipedia, [Harmonic analysis](http://en.wikipedia.org/wiki/Harmonic_analysis)

* MSRI, [Categorical Structures in Harmonic Analysis](http://www.msri.org/workshops/708), Nov 2014. (videos)

category: analysis


harmonic analysis - contents

**[[harmonic analysis]]**

**[[waves]]**

* [[plane wave]]

  [[wave vector]], [[wavelength]], [[wave number]]

  [[frequency]]

  [[amplitude]]

  [[dispersion relation]]

[[Fourier analysis]]

* [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]]

* [[Fourier transform]], [[Fourier-Laplace transform]]

* [[Fourier transform of a distribution]]

**Theorems**

* [[Fourier inversion theorem]]

* [[Parseval's theorem]]

* [[Paley-Wiener-Schwartz theorem]]

**[[sampling theory]]**

* [[wavelet]], [[curvelet]]

* [[coherent state]]

* [[signal processing]]

* [[time series]]

* [[information geometry]]


harmonic differential form

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## Definition

A [[differential form]] $\omega \in \Omega^n(X)$ on a [[Riemannian manifold]] $(X,g)$ is called a **harmonic form** if it is in the [[kernel]] of the [[Laplace operator]] $\Delta_g$ of $X$ in that $\Delta \omega = (d + d^\dagger)^2 \omega = 0$.

## Properties

### Relation to Dolbeault cohomology

On a compact [[Kähler manifold]] the _[[Hodge isomorphism]]_ (see there)
identifies harmonic differential forms  with [[Dolbeault cohomology]] classes.

### Relation to Hodge theory

For the moment see at _[[Hodge theory]]_

## References

* [[Georges de Rham]], Chapter V of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

* {#Voisin02} [[Claire Voisin]], section 5 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* Springer Online Dictionary, _Harmonic form_ ([web](http://eom.springer.de/h/h046460.htm))

[[!redirects harmonic differential forms]]

[[!redirects harmonic form]]
[[!redirects harmonic forms]]


harmonic map

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## Idea

A _harmonic map_ is a [[smooth function]] $f \colon X \longrightarrow Y$ between two [[Riemannian manifolds]] $(X,g_X)$, $(X,g_Y)$ which is a [[critical point]] of the Dirichlet [[kinetic energy]] [[functional]]

$$
  E(f) \coloneqq \int_{X} \Vert d f\Vert^2 dVol_X
  \,.
$$

where $d f \in \Gamma (T^\ast X \otimes \phi^\ast T Y)$ is the [[derivative]], where the [[norm]] is given jointly by the metrics of $X$ and $Y$ and where the [[volume form]] is that of $X$.

This is a standard [[kinetic action]] [[action functional]] for [[sigma models]], the _[[Polyakov action]]_.

## References

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

Discussion in the context of [[action functionals]] for [[theory (physics)|theories of physics]] includes

* [[Charles Misner]], _Harmonic maps as models for physical theories_, Phys. Rev. D 18, 4510 (1978) ([spire](http://inspirehep.net/record/137431/))

Discussion in the context of [[integrable systems]] includes

* [[Shabnam Beheshti]], A. Shadi Tahvildar-Zadeh, _Integrability and Vesture for Harmonic Maps into Symmetric Spaces_ ([arXiv:1209.1383](http://arxiv.org/abs/1209.1383))

[[!redirects harmonic maps]]

[[!redirects Dirichlet energy]]
[[!redirects Dirichlet energies]]


Harmonic oscillator

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## Idea

In [[physics]] the _Harmonic oscillator_ refers to the [[mechanical system]] consisting of a (non-relativistic) [[particle]] in a square potential.

## Properties

### Geometric quantization

(...)

* [Bohr-Sommerfeld leaf -- Harmonic oscillator](Bohr-Sommerfeld+leaf#HarmonicOscillator)

[[!redirects Harmonic oscillators]]


harmonic series

> This entry is about a notion in [[mathematics]]. For the related notion i  [[physics]] and [[music]], see at *[[overtone series]]*.

***

# Contents
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{: toc}

## Definition

Let $f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ defined as

$$f(n)(x) \coloneqq \sum_{i = 1}^n \frac{\cos(n x)}{n}$$

be a [[Fourier series]]. The **harmonic series** is defined as

$$h(n) \coloneqq f(n)(2 \pi m) = \sum_{i = 1}^n \frac{\cos(2 \pi m n)}{n} = \sum_{i = 1}^n \frac{1}{n}$$

for any [[integer]] $m \in \mathbb{Z}$ and the **alternating harmonic series** is defined as

$$h_\mathrm{alt}(n) \coloneqq - f(n)(2 \pi m + \pi) = - \sum_{i = 1}^n \frac{\cos(2 \pi m n + \pi n)}{n} = - \sum_{i = 1}^n \frac{(-1)^{n}}{n} = \sum_{i = 1}^n \frac{(-1)^{n + 1}}{n}$$

for any [[integer]] $m \in \mathbb{Z}$.

The harmonic series diverges:

$$\lim_{n \to \infty} h(n) = \infty$$

while the alternating harmonic series converges:

$$\lim_{n \to \infty} h_\mathrm{alt}(n) = \ln(2)$$

where $\ln$ is the [[natural logarithm]]. This is because the Fourier series above [[converges]] everywhere in $\mathbb{R}$ that is [[tight apartness relation|apart]] from an integer multiple of $2 \pi$.

## See also

* [[series]]

* [[convergence]]

* [[Fourier analysis]]

## References

* Wikipedia, *Harmonic series (mathematics)*


Harold Erbin

* [webpage](https://www.lpthe.jussieu.fr/~erbin/)

## Selected writings

On (2d [[quantum gravity]] and) [[Liouville theory]]:

* [[Harold Erbin]], *Notes on 2d quantum gravity and Liouville theory* (2015) [[pdf](http://www.lpthe.jussieu.fr/~erbin/files/liouville_theory.pdf), [[Erbin-LiouvilleTheory.pdf:file]]]

On [[string field theory]]:

* {#Erbin20a} [[Harold Erbin]], *String Field Theory – A Modern Introduction*, Lecture Notes in Physics **980**, Springer (2021) [[arXiv:2301.01686](https://arxiv.org/abs/2301.01686), [doi:10.1007/978-3-030-65321-7](https://doi.org/10.1007/978-3-030-65321-7)]

* [[Harold Erbin]], _String theory: a field theory perspective_, 2020 ([pdf](http://www.lpthe.jussieu.fr/~erbin/files/reviews/string_theory.pdf))

* [String Field Theory -- Website](http://string-field-theory.org/)

category: people


Harold Hastings

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=9351)

## related entries

* [[stable homotopy theory]], [[function spectra]]

* [[strong shape theory]]

category: people
[[!redirects H. M. Hastings]]


Harold Hilton

* [obituary pdf](https://ncatlab.org/nlab/files/Harold-HiltonSlashSimpson-obituary.pdf)

## Selected writings

On [[crystallography]] and [[crystallographic groups]]:

* [[Harold Hilton]], *Mathematical crystallography and the theory of groups of movements*, Oxford Clarendon Press (1903) $[$[web](https://archive.org/details/mathematicalcry03hiltgoog/page/n6/mode/2up)$]$

category: people


Harold Parks

* [Institute page](https://math.oregonstate.edu/directory/harold-r-parks)

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

## Selected writings

On [[geometric integration theory]]:

* [[Steven G. Krantz]], [[Harold Parks]], *Geometric integration theory*, Springer (2008) [[pdf](http://www.math.wustl.edu/~sk/books/root.pdf)]

## Related entries

* [[current (distribution theory)]]

category: people


Harold Simmons

__Harold Simmons__ was a mathematician at the University of Manchester. His works include topics as logic, model theory, [[lattice]]s, [[ring]]s, [[semiring]]s, [[localization]], [[pointless topology]], sheaf representations, recursion theory, $\lambda$-calculus, and proof theory.

* Webpage at [Manchester](http://staff.cs.manchester.ac.uk/~hsimmons) (currently problems with authorization); there is a backup at internet archive [2019](https://web.archive.org/web/20190714073511/http://staff.cs.manchester.ac.uk/~hsimmons)

category: people


Harold Steinacker

* [webpage](https://homepage.univie.ac.at/harold.steinacker/)

## Selected writings

On the [[IKKT matrix model]]:

* {#Steinacker10} [[Harold Steinacker]], _Emergent Geometry and Gravity from Matrix Models: an Introduction_, Class. Quant. Grav.27: 133001, 2010 ([arXiv:1003.4134](https://arxiv.org/abs/1003.4134))

On [[standard model of particle physics|standard model]] [[phenomenology]] within the [[IKKT matrix model]]:

* [[Harold Steinacker]], Jochen Zahn, _An extended standard model and its Higgs geometry from the matrix model_, PTEP 2014 (2014) 8, 083B03 ([arXiv:1401.2020](http://arxiv.org/abs/1401.2020))

and with emphasis on [[noncommutative geometry]] ([[fuzzy spheres]], [[D-brane geometry]]):

* [[Harold Steinacker]], _Non-commutative geometry and matrix models_, PoS QGQGS2011 (2011) 004 ([arXiv:1109.5521](https://arxiv.org/abs/1109.5521))

* [[Harold Steinacker]], *Gravity as a Quantum Effect on Quantum Space-Time* ([arXiv:2110.03936](https://arxiv.org/abs/2110.03936))

Emergence of [[higher spin gauge theory|higher spin]] [[gravity]] from the [[IKKT matrix model]]:

* [[Harold Steinacker]], _Higher-spin gravity and torsion on quantized space-time in matrix models_ ([arXiv:2002.02742](https://arxiv.org/abs/2002.02742))

* [[Yuhma Asano]], [[Harold Steinacker]], *Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model* ([arXiv:2112.08204](https://arxiv.org/abs/2112.08204))

* [[Harold C. Steinacker]], [[Tung Tran]], *Soft limit of higher-spin interactions in the IKKT model* [[arXiv:2311.14163](https://arxiv.org/abs/2311.14163)]

* [[Harold C. Steinacker]], [[Tung Tran]], *Spinorial description for Lorentzian $\mathfrak{hs}$-IKKT* [[arXiv:2312.16110](https://arxiv.org/abs/2312.16110)]

On [[cosmology]] within the [[IKKT matrix model]]:

* Emmanuele Battista, [[Harold C. Steinacker]], *One-loop effective action of the IKKT model for cosmological backgrounds* [[arXiv:2310.11126](https://arxiv.org/abs/2310.11126)]

## Related entries

* [[IKKT matrix model]]

category: people

[[!redirects Harold C. Steinacker]]


Harry Appelgate

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=18949)

Beware that in his PhD dissertation and one of his papers (*[Iterated cotriples](#ApplegateTierney70)*), the last name is misspelled as “Applegate”.  In all other papers it is spelled “Appelgate”.

## Selected writings

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Appelgate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

On ([[idempotent monad|idempotent]]) [[monads]]:

* [[Harry Appelgate]], [[Myles Tierney]], *Categories with models*, p. 156-244 In: [[Beno Eckmann]] (ed.) *[[Seminar on Triples and Categorical Homology Theory]]*, Lecture Notes in Mathematics, **80**, Springer (1969)  [[doi:10.1007/BFb0083086](https://doi.org/10.1007/BFb0083086), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0083086)]

On [[comonads]]:

* {#ApplegateTierney70} [[Harry Applegate]], [[Myles Tierney]], _Iterated cotriples_, Lecture Notes in Math. **137** (1970) 56-99 ([doi:10.1007/BFb0060440](https://doi.org/10.1007/BFb0060440))

category: people

[[!redirects H. Appelgate]]
[[!redirects Harry W. Appelgate]]
[[!redirects Harry Wesley Appelgate]]

[[!redirects Harry Applegate]]
[[!redirects H. Applegate]]
[[!redirects Harry W. Applegate]]
[[!redirects Harry Wesley Applegate]]


Harry Gindi

* [MO website](http://mathoverflow.net/users/1353/harry-gindi)

category: people

[[!redirects Harry]]


Harry J. Lipkin

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

## Selected writings

On [[exotic hadrons]] and [[hadron supersymmetry]]:

* [[Harry J. Lipkin]], *Exotic Hadrons in the Constituent Quark Model*, Prog. Theor. Phys. Suppl. **168** (2007) 15-22 ([arXiv:hep-ph/0703190](https://arxiv.org/abs/hep-ph/0703190), [doi:10.1143/PTPS.168.15](https://doi.org/10.1143/PTPS.168.15))

category: people

[[!redirects Harry Lipkin]]


Harry Lehmann

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

## Selected writings

Introducing the [[LSZ reduction formula]]:

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

## Related $n$Lab entries

* [[LSZ reduction formula]]

* [[retarded product]]

category: people


Harsh Pittie

* [institute page](https://www.ias.edu/scholars/harsh-v-pittie)

* [Mathematics Genealogy page](http://www.genealogy.ams.org/id.php?id=9349)

## selected writings

* {#Pittie91} [[Harsh Pittie]], _The integral homology and cohomology rings of SO(n) and Spin(n)_, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

  (on the [integral cohomology]] of the [[classifying spaces]] of [[special orthogonal groups]] and [[spin group]])

## related $n$Lab entries

* [[special orthogonal group]]

* [[spin group]]

category: people


Hartogs number

[[!redirects Hartogs's number]]

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

#Contents#
* table of contents
{:toc}

## Idea

The _Hartogs number_ of a [[cardinal number]] $\kappa$ is the number of ways to [[well-order]] a set of cardinality at most $\kappa$, up to isomorphism.[^term]

[^term]: The spelling _Hartogs' number_, or even _Hartog's number_, is also in use. The latter based presumably on a misapprehension since the concept was originally proposed by the German mathematician Friedrich Hartogs (1874-1943) in 1915.

Assuming the [[axiom of choice]], it is the smallest [[ordinal number]] whose cardinality is greater than $\kappa$ and therefore the [[successor]] of $\kappa$ as a cardinal number.  But even without the axiom of choice, it makes sense and is often an effective substitute for such a successor.

## Definition

We will define the Hartogs number as a [[functor|functorial]] operation from [[sets]] to [[well-ordered sets]].  The operation on numbers is just a round-about way of talking about the same thing.

So let $S$ be a set.  Without the axiom of choice (or more precisely, the [[well-ordering theorem]]), it may not be possible to well-order $S$ itself, but we can certainly well-order some [[subsets]] of $S$.  On the other hand, if we can well-order $S$ (or a subset), then there may be many different ways to do so, even non[[isomorphism|isomorphic]] ways.  So to begin with, let us form the collection of all well-ordered subsets of $S$, that is the subset of
$$ \coprod_{A: \mathcal{P}S} \mathcal{P}(A \times A) ,$$
where $\coprod$ indicates [[disjoint union]] and $\mathcal{P}$ indicates [[power set]], consisting of those pairs $(A,R)$ such that $R$ is a well-ordering.  Then form a [[quotient set]] by identifying all well-ordered subsets that are isomorphic as well-ordered sets.  This gives a set of well-order types, or [[ordinal numbers]], which can itself be well-ordered by the general theory of ordinal numbers.

The __Hartogs number__ of $S$ is this well-ordered set, the set of all order types of well-ordered subsets of $S$.  It is denoted $\aleph(S)$.  If $\kappa$ is the cardinality of $S$, then let $\kappa^+$ be the cardinality or ordinal rank (as desired) of the Hartogs number of $S$; this is called the __Hartogs number__ of $\kappa$.

There are other ways to encode $\aleph(S)$. A subset $A \subseteq S$ with a well-ordering can be specified through the collection of its initial segments or [[lower set|down-sets]], i.e., a suitable family of subsets of $S$ named by an [[element]] of $P P(S)$. Thus the set of well-orderings of subsets of $S$ can be defined as a suitable subset of $P P(S)$. Similarly, an equivalence class of such structures corresponding to a given order type is also definable as a subset of $P P(S)$, i.e., is named by a suitable element of $P P P(S)$, so $\aleph(S)$ which is the set of such equivalence classes is definable as a subset of $P P P(S)$.

All of the defining formulas in this description involve only bounded quantification, i.e., can be expressed in the language of [[ZFC|BZC]] or in [[ETCS]]. The proof that $\aleph(S)$ is well-ordered (by existence of $P$-[coalgebra](https://ncatlab.org/nlab/show/well-founded+relation#coalgebraic_formulation) morphisms between representatives of order types) can be enacted in these theories without using the axiom of choice, hence the basic theory of $\aleph(S)$ can be carried out in BZ (bounded Zermelo set theory, without choice).

## Properties

+-- {: .num_theorem}
###### Theorem
There is no [[injection]] $\aleph(S) \to S$.
=--

This theorem is to [[Cantor's theorem]] as [[Burali-Forti's paradox]] is to [[Russell's paradox]]. This observation alone hints how the proof goes:

+-- {: .proof}
###### Proof
As observed before, $\aleph(S)$ is well-ordered. Suppose there were an injection $i: \aleph(S) \to S$; then $i$ gives a bijection of $\aleph(S)$ onto its image $I$. By transport of structure, $I$ uniquely admits a well-ordering for which $i$ is an isomorphism of well-ordered sets. Using this well-ordering, there is an initial segment inclusion

$$j: I \hookrightarrow \aleph(S)$$

which takes $x \in I$ to (the order type of) $I_x = \{y \in I: y \lt x\}$. Since there can be at most one initial segment inclusion between well-orders (cf. [simulation](https://ncatlab.org/nlab/show/well-founded+relation#simulations)), $j$ must coincide with the isomorphism $i^{-1}$, which is onto. This means $I$, representing its isomorphism class $[I] \in \aleph(S)$, is order-isomorphic to a well-ordered set $I_x$ for some $x \in I$, say by $\phi: I_x \cong I$. Again, by uniqueness of initial segment inclusions, $\phi$ coincides with the inclusion $I_x \hookrightarrow I$, so this inclusion is onto and so $I_x = I$. This in turn implies $x \lt x$, a contradiction.
=--

According to this theorem, using the usual ordering of cardinal numbers, $\kappa^+ \nleq \kappa$.  So if this $\leq$ is a [[total order]] (a statement equivalent to the [[axiom of choice]]), we can say that $\kappa^+ \gt \kappa$.

Even without choice, however, we can say this:  If $\alpha$ is an ordinal number such that $|\alpha| \nleq \kappa$, then $\kappa^+ \leq \alpha$.  (Notice that we\'ve shifted our thinking of the Hartogs number from a cardinal to an ordinal.)  That is, $\kappa^+$ is the smallest ordinal number whose cardinal number is not at most $\kappa$.  This doesn\'t use any form of choice except for [[excluded middle]]; we only need choice to conclude that $|\kappa^+| \gt \kappa$.

+-- {: .proof}
###### Proof
Define a relation $R$ from $\kappa^+$ to $\alpha$ where $x R y$ if $\kappa_x^+$ and $\alpha_y$ are isomorphic as well-ordered sets. Then $R$ defines a monic partial function whose domain $\dom(R)$ is an initial segment of $\kappa^+$. If $\dom(R) = \kappa^+$, then $R$ witnesses $\kappa^+ \leq \alpha$. Otherwise there is a least $x$ such that the restriction of $R$ to $\kappa_x^+$ has no upper bound in $\alpha$ (implying $\dom(R) = \kappa_x^+$). By downward closure, it follows that $R$ is onto, hence $R^{op}$ witnesses $\alpha \leq \kappa_x^+$, where the codomain is isomorphic to a well-ordered subset of $\kappa$ by definition of $\kappa^+$, and contradicting $|\alpha| \nleq \kappa$.
=--

The axiom of choice also implies the well-ordering theorem, that any set can be well-ordered.  Thus with choice, $\kappa^+$ is (now as a cardinal again) the smallest cardinal number greater than $\kappa$; this explains the notation $\kappa^+$.

### GCH implies AC

In [[ZF]] (without C, the [[axiom of choice]]), a set $X$ can be well-ordered if there is an injection $i: X \to \aleph(X)$, for any subset of a well-ordered set inherits a well-order. As an amusing illustration of this principle, we will prove an old result due to Sierpiński, following [Gillman](#Gill):

+-- {: .num_theorem}
###### Theorem
If the generalized [[continuum hypothesis]] holds, then the
[[axiom of choice]] holds, i.e., ZF + GCH implies AC.
=--

(Recall the GCH as formulated in ZF: for infinite sets $Y, Z$, if $|Y| \leq |Z| \leq |P(Y)|$, then $|Y| = |Z|$ or $|Z| = |P(Y)|$, letting $|Y|$ denote the cardinality as usual and where $|Y| \leq |Z|$ means there is a [[monomorphism]] $Y \to Z$.)

+-- {: .proof}
###### Proof
In fact we will show that any set $Y$ can be well-ordered, which implies AC (see the article [[Zorn's lemma]]). We start with a technical trick: any $Y$ can be embedded in a set $X$ with the property $|X| = |X + X|$ (consider for example $X = P(\mathbb{N} + Y)$), and then it suffices to well-order such $X$, since we can then pull back the well-ordering along the embedding to well-order $Y$. Iterated power sets of $X$ also have that property.

Now, the Hartogs number $\aleph(X)$ can be constructed explicitly in ZF as a subset of the triple power set $P^3(X) = P P P(X)$. We will show for $n \in \{1, 2, 3\}$ that under GCH, the hypothesis $|\aleph(X)| \leq |P^n(X)|$, true for $n = 3$, leads to one of two conclusions:

1. $|X| \leq |\aleph(X)|$ (whence $X$ can be well-ordered), or

1. $|\aleph(X)| \leq |P^{n-1}(X)|$.

If we land in case 2, then we loop around and feed that conclusion back into the hypothesis and iterate. Iterating this, we cannot descend through case 2 all the way down to $|\aleph(X)| \leq |P^0(X)| = |X|$, so after a few iterations we wind up in case 1 anyway: $X$ can be well-ordered.

To show this, we need a spot of cardinal arithmetic in ZF. First, we already observed that any $P = P^n(X)$ for $n \geq 0$ has the property $|P| = |2P|$. Second, we **claim**

* If $P, A$ are sets such that $|P| = |2P|$ and $|A+P| = |2^P|$, then $|2^P| \leq |A|$

whose ZF-proof we give in a moment. Granting the claim, let us continue. We have

$$|P^{n-1}(X)| \leq |\aleph(X)| + |P^{n-1}(X)| \stackrel{hypoth}{\leq} |P^n(X)| + |P^{n-1}(X)| \leq 2|P^n(X)| = |P^n(X)|$$

or just

$$|P^{n-1}(X)| \leq |\aleph(X)| + |P^{n-1}(X)| \leq |P^n(X)|.$$

By GCH, one of those two inequalities is an equality. If the first inequality is an equality, then we conclude $|\aleph(X)| \leq |P^{n-1}(X)|$, which lands us in case 2. If the second inequality is an equality, then the claim applies and we conclude $|P^n(X)| \leq |\aleph(X)|$, in which case $|X| \leq |P^n(X)| \leq |\aleph(X)|$ and we land in case 1.

So we are done except for the claim. Given bijections $A + P \cong 2^P$, $P + P \cong P$, let $f$ be the evident composite

$$P \stackrel{incl}{\to} A + P \cong 2^P \cong 2^{P + P} \cong 2^P \times 2^P \stackrel{\pi_1}{\to} 2^P$$

where $\pi_1$ denotes first projection. Let $C$ be an element not in the image of $f$, e.g., $C = \{x \in P: x \notin f(x)\}$ ([[Cantor's theorem]]). It follows that under the bijection $A + P \cong 2^P \times 2^P$, those elements of $A + P$ that map to elements of the fiber $\pi_1^{-1}(\{C\})$ can only belong to the summand $A$. Thus it is a subset of $A$ that maps bijectively onto $\pi_1^{-1}(\{C\}) \cong 2^P$, and this proves $|2^P| \leq |A|$.
=--

### Equivalent forms of AC

Methods somewhat similar to those used to prove GCH implies AC can be used to establish equivalent forms of AC.

+-- {: .num_theorem}
###### Theorem
**(Tarski)**
Over ZF (or ETCS without choice), the axiom of choice holds iff every infinite set $Y$ can be put into bijection with its square: $Y^2 \cong Y$.
=--

+-- {: .proof}
###### Proof
The more interesting direction is the "if", so we assume every infinite set can be put into bijection with its square.

It suffices to prove that every $X$ can be well-ordered. WLOG we prove this for (Dedekind) infinite $X$, since every $X$ embeds in a Dedekind infinite set (such as $X + \mathbb{N}$), and we can pull back a well-order on the latter along such an embedding to one on the former.

For infinite $X$, let $Y$ be the disjoint union $X + \aleph(X)$, and let $\phi: Y \times Y \to Y$ be a bijection. We claim that for all $x \in X$, there exist $\alpha, \beta \in \aleph(X)$ such that $\phi(x, \alpha) = \beta$. If that were not true, then there would exist $x \in X$ such that for all $\alpha \in \aleph(X)$ we have $\phi(x, \alpha) \notin \aleph(X)$, or in other words for all $\alpha \in \aleph(X)$ we have $\phi(x, \alpha) \in X$, so that $\phi(x, -)$ defines an injection $\aleph(X) \rightarrowtail X$. This is impossible.

By the claim, for each $x \in X$ there is a least pair $(\alpha_x, \beta_x) \in \aleph(X)^2$, considered in [[lexicographic order]], such that $\phi(x, \alpha_x) = \beta_x$. This assignment $x \mapsto (\alpha_x, \beta_x)$ defines an embedding $X \to \aleph(X)^2$ into a well-ordered set, so that $X$ inherits a well-order by restriction along the embedding, and we are done.
=--

A very similar method establishes the following claim.

+-- {: .num_prop}
###### Proposition
Over ZF, AC is equivalent to the statement that every inhabited set admits a group structure.
=--

+-- {: .proof}
###### Proof
Every inhabited finite set admits a cyclic group structure, and under AC any infinite set can be put in bijection with the collection of its finite subsets, which forms a group under symmetric difference.

Conversely, suppose $Y = X + \aleph(X)$ admits a group structure. Then for every $x \in X$, there exist $\alpha, \beta \in \aleph(X)$ such that $x\alpha = \beta$. For if that were not true, then there exists $x \in X$ such that for all $\alpha \in \aleph(X)$, we have $x \alpha \notin \aleph(X)$, i.e., for all $\alpha \in \aleph(X)$ we have $x\alpha \in X$, which implies left multiplication by $x$ defines an injection $\aleph(X) \rightarrowtail X$, contradiction.

So for each $x \in X$, there exists a least $(\alpha_x, \beta_x) \in \aleph(X)^2$ in lexicographic order such that $x \alpha_x = \beta_x$. This again defines an embedding $x \mapsto (\alpha_x, \beta_x)$ into a well-ordered set $\aleph(X)^2$, and we are done.
=--

## Examples

For $n$ a [[natural number]] regarded as the cardinal number of a [[finite set]], $n^+$ is the usual [[successor]] $n + 1$.  This result uses excluded middle; else we get the _plump_ successor of $n$, which may be rather larger.

For $\aleph_0$ the cardinality of the set of all [[natural number]]s, the Hartogs number $\aleph_0^+ = \omega_1$ is the smallest [[uncountable set|uncountable]] ordinal.  Assuming the axiom of choice ([[countable choice]] and excluded middle are enough), we have $\aleph_0^+ = \aleph_1$ as a cardinal.

In general, we get a sequence $\omega_\alpha$ of infinite cardinalities of well-orderable sets; assuming excluded middle, every infinite well-orderable cardinality shows up in this sequence.  Assuming the axiom of choice, every infinite cardinal shows up, and we have $|\omega_\alpha| = \aleph_\alpha$.  (Actually, there\'s no real need to begin with infinite cardinals; if we started with $\omega_0 = 0$ instead of $\omega_0 = \mathbf{N}$ and $\aleph_0 = 0$ instead of $\aleph_0 = |\mathbf{N}|$, then absolutely *every* cardinality or well-orderable cardinality would appear.)

## Related page

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

## Related entries

* [[Lindenbaum number]]

* [[well-ordering theorem]]

* [[axiom of choice]]

* [[diagonal argument]]

## References

* Friedrich Hartogs, _Über das Problem der Wohlordnung_ , Math. Ann. **76** no.4 (1915) pp.438-443. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002266105))

* [[Radu Diaconescu]], _On Comparability in a Topos_ , Proc. AMS **98** no.3 (1986) pp.389-393. ([pdf](http://www.ams.org/journals/proc/1986-098-03/S0002-9939-1986-0857927-9/S0002-9939-1986-0857927-9.pdf))

* Leonard Gillman, _Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis_, Amer. Math. Monthly 109 (June-July 2002), 544-553. ([pdf](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gillman544-553.pdf))
 {#Gill}

The last section of Gillman's paper is based on Sierpiński's 1947 proof which may be found [here](http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm3411.pdf).

[[!redirects Hartog's number]]
[[!redirects Hartogs number]]
[[!redirects Hartogs's number]]
[[!redirects Hartogs' number]]
[[!redirects Hartogg's number]]
[[!redirects Hartog's number]]


Harvendra Singh

* [webpage](http://www.hri.res.in/~anlweb/academic_reports_physics/p_singh_harvendra/p_singh_harvendra.html)

## Selected writings

On [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

specifically on [[KK-compactification]] on [[K3]]:

* Michael Haack, [[Jan Louis]], [[Harvendra Singh]], _Massive Type IIA Theory on K3_, JHEP 0104:040, 2001 ([arXiv:hep-th/0102110](https://arxiv.org/abs/hep-th/0102110))

specifically on [[black brane|black]] [[D8-branes]] and [[D6-D8-brane bound states]]:?

* [[Harvendra Singh]], _Duality symmetric massive type II theories in $D=8$ and $D=6$_, JHEP 0204:017, 2002 ([arXiv:hep-th/0109147](https://arxiv.org/abs/hep-th/0109147))

* [[Harvendra Singh]], _Note on (D6,D8) Bound State, Massive Duality and Non-commutativity_, Nucl. Phys. B661 (2003) 394-408 ([arXiv:hep-th/0212103](https://arxiv.org/abs/hep-th/0212103))

specifically in relation to [[heterotic string theory]]:

* [[Harvendra Singh]], _Romans type IIA theory and the heterotic strings_, Phys.Lett.B545:403-410, 2002 ([arXiv:hep-th/0201206](https://arxiv.org/abs/hep-th/0201206))

category: people


Harvey Friedman

* [webpage](http://www.math.osu.edu/~friedman.8/)

category: people