Netta Engelhardt

* [Institute page](https://physics.mit.edu/faculty/netta-engelhardt/)

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

## Selected writings

On [[quantum error correcting codes]] and [[black hole entropy]]:

* [[Chris Akers]], [[Netta Engelhardt]], [[Daniel Harlow]], [[Geoff Penington]], [[Shreya Vardhan]], *The black hole interior from non-isometric codes and complexity* [[arXiv:2207.06536](https://arxiv.org/abs/2207.06536)]

category: people


networks of category theorists

This page collects the various networks and communities, in the largest possible sense, of mathematicians and non-mathematicians whose research is [[category theory]]-oriented.

It is by no means a comprehensive list, in strict alphabetic order.

* [Applied Category Theory (ACT)](https://www.appliedcategorytheory.org/) -- On [[applied category theory]].

* [CoACT](https://www.mq.edu.au/research/research-centres-groups-and-facilities/innovative-technologies/centres/centre-of-australian-category-theory-coact) -- the Centre of Australian Category Theory at Macquarie University.

* [ItaCa](https://progetto-itaca.github.io) -- a network of Italian category theorists. The network organises both a permanent on-line seminar, and a annual in-person meeting. Besides these scientific activities, they are also committed to the popularization of CT on italian soil. In 2021 they hosted on YouTube a full introductory course in italian.

* [[Math-Phys-Cat]] ([home page](https://math-phys.group)) -- _Mathematical-Physics and Categorification Group_, a Brazilian research group on [[mathematical physics]] with emphasis on [[higher category theory and physics]].

* [TallCats](https://compose.ioc.ee) -- the Compositional Systems and Methods group at TalTech University, Tallinn, Estonia.

* [Topos institute](https://topos.institute) -- an "independent scientific institute devoted to fundamental inquiry that shapes the technologies of tomorrow".

* [YaMCATS](https://www2.le.ac.uk/departments/mathematics/extranet/staff-material/staff-profiles/simona-paoli/yorkshire-and-midlands-category-theory-seminar-yamcats) -- the Yorkshire and Midlands Category Theory Seminar,  is a category theory research network between the universities of Birmingham, Leeds, Leicester and Sheffield. The network is partially funded by the London Mathematical Society.

Feel free to continue the list!

[[!redirects Networks of category theorists]]


neural network

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###Context###
#### Probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A **neural network** is a class of [[functions]] used in both [[supervised learning | supervised]] and [[unsupervised learning | unsupervised]] [[machine learning]] to approximate a correspondence between samples in a dataset and their associated labels.

## Definition
 {#Definition}

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

Where $K\subset \mathbb{R}^d$ is compact, $\{T_L\}_{L\leq N \in \mathbb{N}}$ a finite set of affine maps such that $T_L(x) = \langle W_L,x\rangle + b_L$ where $W_L$ is the $L^{th}$ layer weight matrix and $b_L$ the $L^{th}$ layer bias, $g:\mathbb{R}\to\mathbb{R}$ a non-linear activation function, a neural network  is a function $f:K\subset \mathbb{R}^d \to \mathbb{R}^m$, such that on input $x$, computes the composition:

$$f(x) = (T_L\circ g \circ T_{L-1}\circ g \circ \dots \circ T_1)(x)$$

where $g$ is applied component-wise.

=--

Typically, $T_1$ is called the input layer, $T_L$ the output layer, and layers $T_2$ to $T_{L-1}$ are hidden layers. In particular, a real-valued 1-hidden layer neural network with computes:

$$f(x) = b' + \sum_{i=1}^n a_i g(\langle W_i, x\rangle + b)$$
where $a = (a_1, \dots, a_n)$ is the output weight, $b'$ the output bias, $W_i$ the $i^{th}$ row of the hidden weight matrix, and $b$ the hidden bias. Here, the hidden layer is $n$-dimensional.

## Large network theory
For simplicity, consider a fully connected network with one layer. In terms of the notation above, consider $K\subset \mathbb{R}$ and $m=1$, also for simplicity. Denote the collection of weights ($W$'s and $b$'s) by $\theta$. Any fixed $\theta$ fixes some function $f_\theta\colon K\to \mathbb{R}$. Now instead of fixing one $\theta$, place some probability distribution over each of its components - for example, we may consider narrow probability Gaussians sitting over each weight. This in turn introduces a probability distribution over the function space ${\mathbb{R}}^{\mathbb{R}}$. For large layer width, by the **universal approximation theorem**, one can represent any $f\colon{\mathbb{R}}\to{\mathbb{R}}$ well (i.e. up to smaller and smaller numerical error), meaning such the resulting distribution may assign non-zero value to any such function. Indeed, as established in the mid 90's, considering a sequence of networks with a wider and wider hidden layer, and placing more and more sharp Gaussians over their weights, by the [[central limit theorem]] CLT, results in a probability distribution over ${\mathbb{R}}^{\mathbb{R}}$ that typically is a Gaussian process GP. These GP's are called **neural network Gaussian processes**, NNGP. (Roughly $P[f]=e^{-S[f]}$ with $S[f] = \int \rho_f$ where $\rho_f \propto f(x)f(y) dx dy$.)

### Neural Network Field Theory
Non-Gaussian non-zero cumulant effects emerge from either not taking the infinite width limit, or alternatively also from violating CLT's assumptions. The latter can practically be done by making the random sampling of certain parameters (certain $\theta$-components) depending on already sampled parameters.

Correlation functions (w.r.t. input $x$ and outputs $y$) are empirically accessible. Lately, researches have defined networks that in this way in the limit represent quantum fields, **Neural Network Field Theories**.
We may speak of neural network quantum field, e.g., when the correlation functions fulfills the [[Osterwalder-Schrader theorem | Osterwalder-Schrader axioms]].
Here, non-Gaussian effects can be expanded as non-quadratic contributions, corresponding to field interactions. (But note that with the finite-width approach, the universal approximation property will generally also break.)
From such non-trivial correlations, one may also attempt to reconstruct the corresponding field density (and thus the actions $S$.)
J. Halverson et al. specify the necessary architecture & distributions over $\theta$ to sample from the quartic actions ($\phi^4$-theory.)
Conversely, Feynman diagram tools have been used to study the expected properties of a random-initialized network.

## Learning theory
A neural network architecture specifies parameters and one then learns a prescription $f_\theta$ on how to get close to many given target positions $y_x$, given their index $x$. In the gradient descent approach, one incrementally alters the weights ($\theta_i \mapsto \theta_{i+1}$) so as to minimize a loss function $C$, which expresses how far the prescription is from $y_x$, for all the indices at once. This process thus corresponds to a trajectory of functions from network initialization till sufficient convergence ($f_{\theta_i} \mapsto f_{\theta_{i+1}}$).
In practical neural network training, improved gradient methods are employed (e.g. stochastic and batching considerations).

Improving the amount of computational shortcuts taken in implementing $\nabla_\theta$ is what *pytorch* and *tensorflow* is largely about. There is a neural tangent library on the google github.
Notable, also from google, there are some recent improvements related to baking the statistics of the collection of data ($z$) into the learning process (self-attention, transformers).

### Neural tangent kernel theory
As is common in stochastic control, one may look at the step sizes in a very fine limit, i.e. pass to calculus proper. The algorithm is then expressed as the vector equation $\theta'(t) = -\nabla_\theta\Phi$ with $\Phi = \sum_z C(f_{\theta(t)}(z), y_z)$, where $z$ ranges over the available learning data. The mathematical derivations end up having a similar flavor as the study of the motion of a particle. Here the evolution of all components of $\theta$ are governed by a gradient of a potential that depends on $\theta$ through the costs of some property, $f_\theta$. This differential equation in turn implies a formally simple formula for $\partial_t f_\theta$, which really is the network output's evolution. The main ingredient determining $\partial_t f_\theta$ is $\Theta_\theta(x,z) = \nabla_\theta f(x)\cdot \nabla_\theta f(z)$. This quantity $\Theta_\theta$ quantifies a similarity reminiscent of [[kernel method]]s. It is small when the changes of $f_\theta$ (w.r.t. a change of $\theta$) are different on $x$ resp. $y$.
In the limit of infinite width (and small step size limit and everything in distribution), the resulting mathematical network has so many degrees of freedom that (roughly speaking) learning becomes very effective.
The learning theory in the limit goes under **neural tangent kernel theory** (NTK theory). Formal relations between different learning approaches (neural networks and kernel machines in particular) become simpler here. Depending on the loss function, the structure may remain Gaussian throughout the learning.
For a shallow network of finite size, $\Theta_\theta$ may still give a quantitative idea on how that (finite) network will change upon tuning of its parameters. Some work was done to try to establish where NKT results start to significantly fail for practical networks. Note: Infinite networks (where weights barely need to be tuned to learn) also don't encode abstracted features.

## Relation to differential equations and dynamical systems

A relation between deep neural networks, [[differential equations]], and [[dynamical systems]] was proposed in ([CMHRBH17](#CMHRBH17), [LZLD17](#LZLD17), [Weinan17](#Weinan17))

Victor Lopez-Pastor and Florian Marquardt proposed that certain [[time-reversible]] [[Hamiltonian systems]] exhibit self-learning behaviour and a physical version of the [[backpropagation]] [[algorithm]].

## Relation to renormalization group flow
 {#RelationToRenormalizationGroupFlow}

A relation between deep neural networks (DNNs) based on Restricted Boltzmann
Machines (RBMs) and [[renormalization group flow]] in [[physics]] was proposed in ([MS14](#MS14)).

## Related concepts

* [[function machine]]

* [[tensor network]]

* [[string diagram]]

* [[physical neural network]]

* [[spiking neural network]]

## References

### General

Textbook account:

* Daniel A. Roberts, Sho Yaida, Boris Hanin, *The Principles of Deep Learning Theory*, Cambridge University Press 2022 ([arXiv:2106.10165](https://arxiv.org/abs/2106.10165))

Lecture notes:

* Yasaman Bahri, Boris Hanin, Antonin Brossollet, Vittorio Erba, Christian Keup, Rosalba Pacelli, James B. Simon, *Les Houches Lectures on Deep Learning at Large & Infinite Width* [[arXiv:2309.01592](https://arxiv.org/abs/2309.01592)]

On the learning algorithm as related to [[differential equations]] and [[dynamical systems]]:

* {#CMHRBH17} Bo Chang, Lili Meng, Eldad Haber, Lars Ruthotto, David Begert, Elliot Holtham, *Reversible Architectures for Arbitrarily Deep Residual Neural Networks*, ([arXiv:1709.03698](https://arxiv.org/abs/1709.03698))

* {#LZLD17} Yiping Lu, Aoxiao Zhong, Quanzheng Li, Bin Dong, *Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations*, ([arXiv:1710.10121](https://arxiv.org/abs/1710.10121)).

* {#Weinan17} Weinan E, *A Proposal on Machine Learning via Dynamical Systems*, Communications in Mathematics and Statistics, 5, 1–11 (2017). ([doi:10.1007/s40304-017-0103-z](https://doi.org/10.1007/s40304-017-0103-z))

* Victor Lopez-Pastor, Florian Marquardt, *Self-learning Machines based on Hamiltonian Echo Backpropagation*, ([arXiv:2103.04992](https://arxiv.org/abs/2103.04992))

On the learning algorithm as [[gradient descent]] of the loss functional:

* {#ThierryMieg18} Jean Thierry-Mieg, _Connections between physics, mathematics and deep learning_, Letters in High Energy Physics, vol 2 no 3 (2019) ([arXiv:1811.00576](https://arxiv.org/abs/1811.00576), [doi:10.31526/lhep.3.2019.110](https://doi.org/10.31526/lhep.3.2019.110))

On the learning algorithm as analogous to the [[AdS/CFT correspondence]]:

* Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, _Machine Learning Spatial Geometry from Entanglement Features_, Phys. Rev. B 97, 045153 (2018) ([arxiv:1709.01223](https://arxiv.org/abs/1709.01223))

* W. C. Gan and F. W. Shu, _Holography as deep learning_,  Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) ([arXiv:1705.05750](https://arxiv.org/abs/1705.05750))

* J. W. Lee, _Quantum fields as deep learning_ ([arXiv:1708.07408](https://arxiv.org/abs/1708.07408))

* {#HSTT18} [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, _Deep Learning and AdS/CFT_,  Phys. Rev. D 98, 046019 (2018) ([arxiv:1802.08313](https://arxiv.org/abs/1802.08313))

[[category theory|Category theoretic]] treatments of deep learning in neural networks:

* [[Brendan Fong]], [[David Spivak]], Rémy Tuyéras, _Backprop as Functor: A compositional perspective on supervised learning_, ([arXiv:1711.10455](https://arxiv.org/abs/1711.10455))

* [[David Spivak]], _Learners' languages_, ([arXiv:2103.01189](https://arxiv.org/abs/2103.01189))

* G.S.H. Cruttwell, [[Bruno Gavranović]], [[Neil Ghani]], Paul Wilson, Fabio Zanasi, _Categorical Foundations of Gradient-Based Learning_, ([arXiv:2103.01931](https://arxiv.org/abs/2103.01931))

Neural networks field theory:

* Mehmet Demirtas, James Halverson, Anindita Maiti, Matthew D. Schwartz, Keegan Stoner, *Neural Network Field Theories: Non-Gaussianity, Actions, and Locality*, (2023)  ([arXiv:2307.03223](https://arxiv.org/abs/2307.03223))

Quantum neural networks (in [[quantum computation]] for [[quantum machine learning]]):

* Iris Cong, Soonwon Choi & Mikhail D. Lukin, *Quantum convolutional neural networks*, Nature Physics volume 15, pages 1273–1278 (2019)  ([doi:10.1038/s41567-019-0648-8](https://doi.org/10.1038/s41567-019-0648-8))

* {#MariBromleyIzaacSchuldKilloran20} Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, Nathan Killoran,  *Transfer learning in hybrid classical-quantum neural networks*, Quantum 4, 340 (2020) ([arXiv:1912.08278](https://arxiv.org/abs/1912.08278))

* Stefano Mangini, Francesco Tacchino, Dario Gerace, Daniele Bajoni, Chiara Macchiavello, *Quantum computing models for artificial neural networks*, EPL (Europhysics Letters) 134(1), 10002 (2021) ([arXiv:2102.03879](https://arxiv.org/abs/2102.03879))

Topological deep learning for data supported on topological domains such as [[simplicial complexes]], [[cell complexes]], and [[hypergraphs]]:

* Ephy R. Love, Benjamin Filippenko, Vasileios Maroulas, Gunnar Carlsson, _Topological Deep Learning_ ([arXiv:2101.05778](https://arxiv.org/abs/2101.05778))

* Mathilde Papillon, Sophia Sanborn, Mustafa Hajij, Nina Miolane, _Architectures of Topological Deep Learning: A Survey on Topological Neural Networks_ ([arXiv:2304.10031](https://arxiv.org/abs/2304.10031))

* Mustafa Hajij et al., _Topological Deep Learning: Going Beyond Graph Data_ ([pdf](https://www.researchgate.net/publication/370134352_Topological_Deep_Learning_Going_Beyond_Graph_Data))

### Relation to tensor networks

Application of [[tensor networks]] and specifically [[tree tensor networks]]:

* Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Blázquez García, Gang Su, Maciej Lewenstein, _Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure_, New Journal of Physics, 21, 073059 (2019) ([arXiv:1710.04833](https://arxiv.org/abs/1710.04833))

* Song Cheng, Lei Wang, Tao Xiang, Pan Zhang, _Tree Tensor Networks for Generative Modeling_, Phys. Rev. B 99, 155131 (2019) ([arXiv:1901.02217](https://arxiv.org/abs/1901.02217))

### Relation to renormalization group flow
 {#ReferencesRelationToRenormalizationGroupFlow}

Relation of deep learning to [[renormalization group flow]]:

* {#MS14} Pankaj Mehta, David J. Schwab - _An exact mapping between the Variational Renormalization Group and Deep Learning_, 2014 ([arXiv:1410.3831](https://arxiv.org/abs/1410.3831))

Further discussion under the relation of [[renormalization group flow]] to [[bulk]]-flow in the context of the [[AdS/CFT correspondence]]:

* Yi-Zhuang You, Zhao Yang, Xiao-Liang Qi, _Machine Learning Spatial Geometry from Entanglement Features_, Phys. Rev. B 97, 045153 (2018) ([arxiv:1709.01223](https://arxiv.org/abs/1709.01223))

* W. C. Gan and F. W. Shu, _Holography as deep learning_,  Int. J. Mod. Phys. D 26, no. 12, 1743020 (2017) ([arXiv:1705.05750](https://arxiv.org/abs/1705.05750))

* J. W. Lee, _Quantum fields as deep learning_ ([arXiv:1708.07408](https://arxiv.org/abs/1708.07408))

* [[Koji Hashimoto]], Sotaro Sugishita, Akinori Tanaka, Akio Tomiya, _Deep Learning and AdS/CFT_,  Phys. Rev. D 98, 046019 (2018) ([arxiv:1802.08313](https://arxiv.org/abs/1802.08313))

[[!redirects neural networks]]

[[!redirects deep learning]]

[[!redirects quantum neural network]]
[[!redirects quantum neural networks]]

[[!redirects deep neural network]]
[[!redirects deep neural networks]]


neutral constructive mathematics

\tableofcontents

## Idea

Neutral constructive mathematics is a branch of [[constructive mathematics]] which simply removes the [[axiom of choice]] and [[excluded middle]] from the [[axioms]] of the [[foundations]], and does not replace it either with weaker versions of the axiom of choice or with axioms which contradict the axiom of choice. Thus, neutral constructive mathematics is agnostic about the status of the [[axiom of choice]] and [[excluded middle]].

As a result, [[impredicative mathematics|impredicative]] neutral constructive mathematics is the [[internal logic]] of an [[elementary topos]] with [[natural numbers object]].

## Examples

Examples of mathematical foundations which one could do impredicative neutral constructive mathematics include:

* [[Martin-Löf type theory]], [[cubical type theory]], or ([[higher observational type theory|higher]]) [[observational type theory]] with a [[type of all propositions]].

* intuitionistic [[higher-order logic]]

* intuitionistic [[ETCS]]

* intuitionistic bounded [[SEAR]]

* [[intuitionistic Mostowski set theory]]

* the [[internal logic]] of any [[elementary topos]]

Examples of mathematical foundations which one could do predicative neutral constructive mathematics include:

* [[Martin-Löf type theory]], [[cubical type theory]], or ([[higher observational type theory|higher]]) [[observational type theory]].

* [[constructive Mostowski set theory]]

* the [[internal logic]] of any $\Pi$-$W$-pretopos.

## Consequences

* In general, for any statement $P$ which is true in [[classical mathematics]], the statement that "the [[law of excluded middle]] implies $P$" is true in neutral constructive mathematics.

* Furthermore, for any statement $P$ which is true in [[classical mathematics]] with the [[axiom of choice]], the statement that "the axiom of choice implies $P$" is true in neutral constructive mathematics.

### Real analysis

* In [[topology]], [[locales]] are usually better behaved than [[topological spaces]]. Thus, in [[real analysis]] one usually talks about the [[localic real numbers]] $\mathbb{R}_L$ rather than the [[Dedekind real numbers]] $\mathbb{R}_D$. Many statements, such as the [[Heine-Borel theorem]], the [[fundamental theorem of algebra]], et cetera, are true in the [[localic real numbers]], while they are not provable in the [[Dedekind real numbers]]. Nevertheless, the [[Dedekind real numbers]] are still of interest, being the spatial points of the [[localic real numbers]], and the two notions coincide with [[excluded middle]].

* Similarly, one usually talks about the localic complex numbers $\mathbb{C}_L$ or the localic real [[Clifford algebra]] $\mathrm{Cl}_{3, 0}(\mathbb{R}_L)$, et cetera, instead of the Dedekind [[complex numbers]] $\mathbb{C}_D$ or the Dedekind real Clifford algebra $\mathrm{Cl}_{3, 0}(\mathbb{R}_D)$.

* The [[modulated Cantor real numbers]] or [[Eudoxus real numbers]] $\mathbb{R}_C$ are a subset of the [[Dedekind real numbers]], but cannot be proven to be equivalent to the [[Dedekind real numbers]], nor can they be proven to be [[sequentially Cauchy complete]]. Nevertheless, the modulated Cantor real numbers are of interest since they are defined using [[sequences]] of [[rational numbers]], [[locators]], or almost linear homomorphisms on the [[integers]], and thus are the ones which could be computed with in an algorithm, (see [[exact real arithmetic]]).

* The [[HoTT book real numbers]] $\mathbb{R}_H$ are the minimum subset of the [[Dedekind real numbers]] for which [[analytic functions]] like the [[exponential function]], [[sine]], and [[cosine]] are well-defined, as well as the [[Jordan content]] and [[Riemann integration]]. They are defined as the sequential Cauchy completion of the [[rational numbers]].

* Unsigned decimal digits are not sufficient to represent the modulated Cantor real numbers, one has to use signed decimal digits instead.

* Continuous functions on the localic real numbers are continuous maps between the locales.

* If using the [[localic real numbers]], [[semidefinite integrals]] are fundamental and [[antiderivatives]] are a derived concept; additionally, the existence of a semidefinite integral must be proven directly in the [[fundamental theorem of calculus]], rather than indirectly through the existence of [[definite integrals]]

### Algebra

* Not every subset has an injection into the superset. Thus, subsets with an injection are usually better behaved than general subsets.

* Not every vector space has a basis. Thus, free vector spaces are usually better behaved than vector spaces in general.

## See also

* [[Jason Rute]], *What is neutral constructive mathematics*, ([MathOverflow](https://mathoverflow.net/questions/404270/what-is-neutral-constructive-mathematics))

* [[Fred Richman]], *Constructive Mathematics without Choice*,
in: *Reuniting the Antipodes -- Constructive and Nonstandard Views of the Continuum*, Synthese Library **306**, Springer (2001) 199-206 [[doi:10.1007/978-94-015-9757-9_17](https://doi.org/10.1007/978-94-015-9757-9_17)]

There was a seminar on neutral constructive mathematics:

* Mathematical Logic and Constructivity: *The Scope and Limits of Neutral Constructivism*, Stockholm, Sweden, August 20-23, 2019 ([website](http://logic.math.su.se/mloc-2019/))


neutral element > history

< [[identity element]]


neutral non-well-founded set theory

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## Idea

Neutral non-well-founded set theory is [[set theory]] without any of the [[axiom of foundation|foundation-like axioms]], including

* [[membership induction]]

* the [[axiom of regularity]]

* [[transitive closure]]

* [[Mostowski's principle]]

*  the [[axiom of well-founded materialization]]

but also without any [[axioms]] which contradict the axioms above, such as the [[axiom of anti-foundation]], or any weaker versions of the above axioms.

The [[axiom of choice]] is not in general accepted here because it implies the [[axiom of well-founded materialization]] in general, as well as [[Mostowski's principle]] for [[material set theories]].

{#ImportantExamples} Important examples of neutral non-well-founded set theory include

* $ZF^{\circlearrowleft}$ and its structural counterpart [[SEAR]],

* $IBZ^{\circlearrowleft}$ and its structural counterparts intuitionistic bounded SEAR and intuitionistic [[ETCS]].

> Here the superscript "$\scriptsize{\circlearrowleft}$" probably means "omit the [[axiom of foundation]]", and "$ZF$" refers to [[Zermelo-Fraenkel set theory]].

## See also

* [[neutral constructive mathematics]] (for the equivalent with [[excluded middle]])

## References

The foundation-like axioms are given in section 2 of

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

The set theory in this textbook does not have the axiom of regularity, and is thus equivalent to $ZFC^{\circlearrowleft}$. However, it isn't fully neutral, since Mostowski's principle still holds.

* Paul Halmos, *Naive Set Theory*. Princeton, NJ: D. Van Nostrand Company, 1960.


neutrino

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# Contents
* table of contents
{: toc}

## Idea

The _neutrino_ is one of the fundamental [[particles]]/[[matter]] [[field (physics)|fields]] in the [[standard model of particle physics]].

## Neutrino masses
 {#NeutrinoMasses}

The origin and nature of the observed tiny but non-vanishing [[neutrino]] [[masses]] remains one of the open problems of the [[standard model of particle physics]]. Review includes [Fisher-Kayser-McFarland 99](#FisherKayserMcFarland99).

On neutrino [[masses]] and the [[standard model of particle physics]] as an [[effective field theory]]:

[Weinberg 09, p. 15](#Weinberg09):

> I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$.

In fact this scale is about the [[GUT scale]] for [[grand unified theories]], see for instance ([Mohapatra 06](#Mohapatra06)) for more on this relation, and see at _[[seesaw mechanism]]_.

Detailed matching of parameters of non-supersymmetric $Spin(10)$-GUT to [[neutrino]] [[masses]] is discussed in [Ohlsson-Pernow 19](#OhlssonPernow19)

## History and prediction

The history of the prediction of the neutrino is interesting and may (or may not) contain some general lessons for theoretical physics/[[mathematical physics]].

Before 1930  [[experiments]] detected an apparent violation of the [[conservation law]] for [[energy]] in processes of [[beta decay]].

Back then [[Niels Bohr]] proposed that, therefore, apparently the fundamental [[conservation laws]] in [[physics]] may be violated and possibly only hold [[probability theory|statistically]] over many quantum mechanical processes, but possibly not microscopically.

Opposed to this was the suggestion by [[Wolfgang Pauli]], who insisted that the [[conservation laws]] ought to hold true, and that therefore there must be an undetected new [[fundamental particle]] created in [[beta decay]], which carries away the apparently missing [[energy]].

Pauli with his argument and suggestion turned out to be right. The missing particle -- called the _neutrino_ by [[Enrico Fermi]] in 1933 -- was finally directly detected in 1956, hence 26 years after its proposal. (Compare maybe to the [[Higgs boson]] for which detection came 50 years after prediction.)

Notice that back then, predicting unobserved and possibly practically unobservable [[fundamental particles]] was not taken as lightly as in some circles it is these days (e.g. in [[supersymmetry]] and/or [[string theory]]). In ([AP 06, p. 7](#AP06)) Pauli is quoted on this as having said:

> I've done a terrible thing today, something which no theoretical physicist should ever do. I have suggested something that can never be verified experimentally.

But concerning the motivation for this predictions, notice that back around 1915 [[Emmy Noether]] had proven the theorem now named after her -- [[Noether's theorem|Noether's first theorem]] -- which asserts that [[symmetries]] of [[equations of motion]] in [[physics]] correspond to [[conservation laws]]. In particular if a system (notably the [[forces]] and [[interactions]] of [[fundamental particles]]) does not change over [[time]], then [[Noether's theorem|Noether's first theorem]] asserts that the [[energy]] of the system must be conserved.
The theorem applies as soon as the [[physics]] is described by a [[local Lagrangian]]/[[local action functional]] and the [[principle of extremal action]]. This was in principle well established in 1930, even though maybe not as widely appreciated as it could have been. The generalization of this to [[quantum mechanics]] and [[quantum field theory]] was maybe fully understood only later.

But in view of this theorem, Bohr's suggestion that energy conservation fails would have implied that fundamental physics is not governed by an [[action principle]], something arguably more dramatic than some violation of some energy conservation might seem.

Today Pauli is widely acknowledged for holding up the conservation laws against Bohr's proposal, see for instance ([AP 06, p. 6](#AP06)), where it says:

> Pauli's belief in the absolute credibility of symmetry principles led him to defend conservation laws even when at that time the empirical evidence was doubtful. His prediction of the neutrino is a great example.

But on the other hand, it seems that Pauli's respect for the conservation laws was not informed by Noether's theorem, but rested rather on an intuitive feeling, for in ([AP 06, p.5](#AP06)) Pauli is quoted as late as 1953 thus:

> I am very much in favour of the general principle to bring empirical conservation laws and invariance properties in connection with mathematical groups of transformations of the laws of nature.

While this does support the correct answer, it seems to be a rather weak way of stating it, given that Noether's theorem establishes this "connection" as a theorem, already back in 1915.

Similarly, in ([AP 06, p. 6](#AP06)) Pauli is quoted as reacting to Bohr's proposal by saying:

> I am myself fairly convinced $[...]$ that Bohr with his corresponding deliberations concerning a violation of energy conservation is entirely on the wrong track! $[...]$ The idea of a violation of the conservation of energy in β-decay is and remains, in my opinion, cheap and very clumsy philosophy.

If Pauli had really been relying on symmetry and the Noether theorem, he could have said "provably wrong" instead of just "cheap and clumsy". Especially since "clumsy" suggests "possible, even if not enjoyable", where in fact it is impossible unless the whole foundations of physics are changed.

## Related concepts

* [[mass term]]

* [[sterile neutrino]]

* [[ANITA experiment]]

\linebreak

[[!include flavors of fundamental particles -- table]]

## References

### Early history

A discussion of Pauli's thoughts leading him to the prediction of the neutrino:

* {#AP06} Atmanspacher, Primas,  _Pauli's ideas..._, 2006 ([pdf](http://www.igpp.de/english/tda/pdf/paulijcs8.pdf))

* [[Francesco Vissani]], *First steps towards understanding neutrinos* [[arXiv:2310.07834](https://arxiv.org/abs/2310.07834)]

### General

* [[Goran Senjanovic]], _LHC and the neutrino paradigm_, talk at CERN EP seminar 2011 ([cern:148714](https://indico.cern.ch/event/148714), [recording](https://youtu.be/VVoMUHokn2U))

* Athar et al. *Status and Perspectives of Neutrino Physics*, [IUPAP Neutrino Panel](https://www.iupapneutrinopanel.org/) Report ([arXiv:2111.07586](https://arxiv.org/abs/2111.07586))

* M. Sajjad Athar, A. Fatima, S. K. Singh, *Neutrinos and their interactions with matter* [[arXiv:2206.13792](https://arxiv.org/abs/2206.13792)]

* *Theory of Neutrino Physics -- Snowmass TF11 (aka NF08) Topical Group Report* [[arXiv:2209.07983](https://arxiv.org/abs/2209.07983)]

See also

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

On the neutrino mass problem:

* {#FisherKayserMcFarland99} Peter Fisher, Boris Kayser, Kevin S. McFarland, _Neutrino Mass and Oscillation_ Ann. Rev. Nucl. Part. Sci.49:481-528, 1999 ([arXiv:hep-ph/9906244](https://arxiv.org/abs/hep-ph/9906244))

* {#Mohapatra06} R. N. Mohapatra, _Models of Neutrino Masses: A Brief Overview_, 2006 ([pdf](http://lss.fnal.gov/conf/C0606131/2.2mohapat.pdf), [[MohapatraNeutrinoMasses.pdf:file]])

* Boris Kayser, _Are Neutrinos Their Own Antiparticles?_,
J. Phys. Conf. Ser. 173:012013, 2009 ([arXiv:0903.0899](https://arxiv.org/abs/0903.0899))

* Guido Fantini, Andrea Gallo Rosso, [[Francesco Vissani]], Vanessa Zema, _The formalism of neutrino oscillations: an introduction_, in: *The State of the Art of Neutrino Physics* World Scientific (2018)
[[arXiv:1802.05781](https://arxiv.org/abs/1802.05781), [doi:10.1142/10600](https://doi.org/10.1142/10600)]

What it would mean if neutrinos were [[Dirac spinors]]:

* Salvador Centelles Chuliá, *Theory and phenomenology of Dirac neutrinos* ([arXiv:2110.15755](https://arxiv.org/abs/2110.15755))

With further emphasis on the [[flavor problem]]:

* {#Xing19} Zhi-zhong Xing, _Flavor structures of charged fermions and massive neutrinos_, Physics Reports Volume 854, 20 April 2020, Pages 1-147 ([arXiv:1909.09610](https://arxiv.org/abs/1909.09610), [doi:10.1016/j.physrep.2020.02.001](https://doi.org/10.1016/j.physrep.2020.02.001))

Comments on neutrinos masses as a hint for the [[standard model of particle physics]] being an [[effective field theory]] are in

* {#Weinberg09} [[Steven Weinberg]], _Effective Field Theory, Past and Future_ ([arXiv:0908.1964](http://arxiv.org/abs/0908.1964))

See also

* [[Pierre Ramond]], _Neutrinos and Particle Physics Models_ ([arXiv:1902.01741](https://arxiv.org/abs/1902.01741))

* M.C. Gonzalez-Garcia, _Neutrino Masses and Mixing_ ([arXiv:1902.04583](https://arxiv.org/abs/1902.04583))

* Shao-Feng Ge, Jing-Yu Zhu, _The Normal Neutrino Mass Hierarchy is Exactly What We Need!_ ([arXiv:1910.02666](https://arxiv.org/abs/1910.02666))

* S. Bilenky, _Neutrinos: Majorana or Dirac?_ ([arXiv:2008.02110](https://arxiv.org/abs/2008.02110))

* [[Francesco Vissani]], _What is matter according to particle physics and why try to observe its creation in lab?_, Universe **7** 3 (2021) 61 [[arXiv:2103.02642](https://arxiv.org/abs/2103.02642), [doi:10.3390/universe7030061](https://doi.org/10.3390/universe7030061)]

More on this in relation to [[GUT]] models:

* [Mohapatra 06](#Mohapatra06)

* {#Witten17} [[Edward Witten]], p. 2 of _Symmetry and Emergence_ ([arXiv:1710.01791](https://arxiv.org/abs/1710.01791))

* {#OhlssonPernow19} Tommy Ohlsson, Marcus Pernow, _Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution_ ([arXiv:1903.08241](https://arxiv.org/abs/1903.08241))

Discussion of neutrinos as [[dark matter]]-candidates:

* {#NeutrinoWhite16} _A White Paper on keV Sterile Neutrino Dark Matter_,  Journal of Cosmology and Astroparticle Physics, Volume 2017, January 2017 ([arXiv:1602.04816](https://arxiv.org/abs/1602.04816))

* D. J. Buettner, P. D. Morley, *Why not Neutrinos as the Dark Matter? A Critical Review, KATRIN and New Research Directions*, [7th Symposium on Neutrinos and Dark Matter in Nuclear Physics (NDM22)](https://indico.phy.ornl.gov/event/142/#:~:text=The%207th%20Symposium%20on%20Neutrinos,NC%2C%2028801%2C%20US)
[[arXiv:2208.06460](https://arxiv.org/abs/2208.06460)]

Discussion of [[neutrino]] [[masses]] in [[leptoquark]]-[[model (in theoretical physics)|models]] for [[flavour anomalies]]:

* {#BGV19} Innes Bigaran, John Gargalionis, Raymond R. Volkas, _A near-minimal leptoquark model for reconciling flavour anomalies and generating radiative neutrino masses_ ([arXiv:1906.01870](https://arxiv.org/abs/1906.01870))

Attempts to explain the [[flavour anomalies]] with right-handed neutrinos:

* Carlo Marzo, Luca Marzola, Martti Raidal, _Common explanation to the $R_{K^{(\ast)}}$, $R_{K^{(\ast)}}$ and $\epsilon'/\epsilon$ anomalies in a 3HDM+$\nu_R$ and connections to neutrino physics_ ([arXiv:1901.08290](https://arxiv.org/abs/1901.08290))

* Luigi Delle Rose, Shaaban Khalil, Simon J.D. King, Stefano Moretti, _$R_K$ and $R_{K^\ast}$ in an Aligned 2HDM with Right-Handed Neutrinos_ ([arXiv:1903.11146](https://arxiv.org/abs/1903.11146))

* Rusa Mandal, Clara Murgui, Ana Peñuelas, [[Antonio Pich]], _The role of right-handed neutrinos in $b \to c \tau \bar \nu$ anomalies_ ([arXiv:2004.06726](https://arxiv.org/abs/2004.06726))

Discussion of neutrino masses in [[M-theory on G2-manifolds]]:

* Eric Gonzalez, [[Gordon Kane]], Khoa Nguyen, *Neutrino mass matrices from localization in M-theory on G2 orbifold* ([arXiv:2107.12893](https://arxiv.org/abs/2107.12893))

[[!redirects neutrino]]
[[!redirects neutrinos]]
[[!redirects neutriono]]

[[!redirects electron neutrino]]
[[!redirects electron neutrinos]]

[[!redirects muon neutrino]]
[[!redirects muon neutrinos]]

[[!redirects tau neutrino]]
[[!redirects tau neutrinos]]

[[!redirects neutrino mass]]
[[!redirects neutrino masses]]

[[!redirects neutrino mass problem]]


neutron

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#### Physics
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#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _proton_ is a [[bound state]] of one [[up quarks]] and two [[down quark]], hence a [[baryon]].

The [[bound states]] of [[protons]] and [[neutrons]] (jointly: [[nucleons]]) in turn are the [[nucleus (physics)|nuclei]], the cores of [[atoms]].

[[!include flavors of fundamental particles -- table]]

## Related entries

* [[neutron star]]

* [[proton]]

* [[Lambda baryon]]

## References

See also

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

[[!redirects neutrons]]


neutron star

#Contents#
* table of contents
{:toc}

## Related concepts

* [[neutron]]

* [[black hole]]

* [[gravitational wave]]

* [[supernova]]

## References

### General

* {#Brambilla14} Brambilla et al., Section 7.3 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

See also:

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

Including [[strange quarks]]:

* V. P. Goncalves, L. Lazzari, _An introduction to strange stars_ (in Portuguese) ([arXiv:2006.02976](https://arxiv.org/abs/2006.02976))

Including [[sigma-meson]] and [[omega-meson]]:

* David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, _Studying the parameters of the extended $\sigma$-$\omega$ model for neutron star matter_ ([arXiv:2006.03676](https://arxiv.org/abs/2006.03676))

Observation of [[gravitational waves]] coincident with [[electromagnetic radiation]] from merging neutron stars:

* {#LIGOVirgo17} B. P. Abbott et al. ([[LIGO]] Scientific Collaboration and [[Virgo]] Collaboration), _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ Phys. Rev. Lett. 119, 161101, 2017 ([doi:10.1103/PhysRevLett.119.161101](https://doi.org/10.1103/PhysRevLett.119.161101))

### Via the Skyrmion model

Discussion of [[model (in theoretical physics)|models]] of [[neutron stars]] by [[Skyrmions]]:

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _BPS Skyrmions as neutron stars_, Physics Letters B
Volume 742, 6 March 2015, Pages 136-142 ([arXiv:1407.3799](https://arxiv.org/abs/1407.3799))

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _Neutron stars in the BPS Skyrme model: mean-field limit vs. full field theory_, Phys. Rev. C 92, 025802 (2015) ([arXiv:1503.03095](https://arxiv.org/abs/1503.03095))

* Xiang-Hai Liu, Yong-Liang Ma, [[Mannque Rho]], _Topology change and nuclear symmetry energy in compact-star matter_, Phys. Rev. C 99, 055808 (2019) ([arXiv:1811.10012](https://arxiv.org/abs/1811.10012))

* [[Carlos Naya]], _Neutron stars within the Skyrme model_, 	Int. J. Mod. Phys. E 28, 1930006 (2019) ([arXiv:1910.01145](https://arxiv.org/abs/1910.01145))

* C. Adam, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _Adding crust to BPS Skyrme neutron stars_ ([arXiv:2004.03610](https://arxiv.org/abs/2004.03610))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro García, Ricardo Vázquez, Andrzej Wereszczynski, _A new consistent Neutron Star Equation of State from a Generalized Skyrme model_ ([arXiv:2006.07983](https://arxiv.org/abs/2006.07983))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, _Quasi-universal relations for generalized Skyrme stars_ ([arXiv:2011.08573](https://arxiv.org/abs/2011.08573))

Taking [[higher curvature corrections]] into account ([[Starobinsky model of cosmic inflation|Starobinsky model]]):

* C. Adam, M. Huidobro, R. Vazquez, A. Wereszczynski, _BPS Skyrme neutron stars in generalized gravity_ ([arXiv:2005.10834](https://arxiv.org/abs/2005.10834))

See also:

* [[Mannque Rho]], _Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter_ ([arXiv:2004.09082](https://arxiv.org/abs/2004.09082))

* Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, *Kaon condensation in skyrmion matter and compact stars* [[arXiv:2212.00385](https://arxiv.org/abs/2212.00385)]

### Via holographic QCD

Discussion via [[holographic QCD]] (see also at [[quark-gluon plasma]]):

* Carlos Hoyos, Niko Jokela, [[Matti Järvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, _Transport in strongly coupled quark matter_ ([arXiv:2005.14205](https://arxiv.org/abs/2005.14205))

* Carlos Hoyos, Niko Jokela, [[Matti Järvinen]], Javier G. Subils, Javier Tarrio, Aleksi Vuorinen, *Holographic approach to transport in dense QCD matter* ([arXiv:2109.12122](https://arxiv.org/abs/2109.12122))

* [[Matti Järvinen]], *Holographic modeling of nuclear matter and neutron stars*,  Eur.Phys.J.C. 56 2021 ([arXiv:2110.08281](https://arxiv.org/abs/2110.08281))

* Niko Jokela, *NICER view on holographic QCD* ([arXiv:2111.07940](https://arxiv.org/abs/2111.07940))

* Tobías Canavesi, Octavio Fierro, Nicolás Grandi, Pablo Pisani, *Scalar correlators and normal modes in holographic neutron stars* ([arXiv:2111.10408](https://arxiv.org/abs/2111.10408))

[[!redirects neutron stars]]


New Dimensions in Geometry

* [[Yuri Manin]], _New Dimensions in Geometry_, talk at 25th Arbeitstagung Bonn 1984 ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/chapter04.pdf))

on [[supergeometry]] and related topics.

category: reference


New Foundations

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#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

New Foundations (NF) is a version of axiomatic [[set theory]] in which there is a set of all sets, but the [[axiom of comprehension]] is restricted by a "stratification" condition.

The [[category]] of sets and functions in NF is known to not be [[cartesian closed category|cartesian closed]], so it is not a [[topos]], but can be equipped with [[finite limits]] and a [[subobject classifier]]. One can even construct a set of all functions between two sets, but there is no evaluation map $A\times B^A \to B$ making this an exponential in the category.

There are at least three proofs of this fact: one, due to Randall Holmes, doesn't seem to have been made public; the others are due to McLarty and Forster, and derive contradictions (different in the two proofs) from assumptions on the behaviour of a pairing function used to construct binary products.

For a long time the (relative) consistency of NF was an open problem, although NF with urelements (NFU) was known to be consistent (relative to ZF plus some large cardinals).  Recently there [may be](http://mathoverflow.net/questions/132103/the-status-of-the-consistency-of-nf-relative-to-zf) a proof of its consistency.

## Related concepts

* [[foundations of mathematics]]

* [[univalent foundations for mathematics]]

## References

* Wikipedia, _[New Foundations](http://en.wikipedia.org/wiki/New_Foundations)_

* Stanford Encyclopedia of Philosophy, _[Quine's New Foundations](https://plato.stanford.edu/entries/quine-nf/)_

* [[Colin McLarty]], _Failure of cartesian closedness in NF_, Journal of Symbolic Logic, **57** no 2 (1992) pp 555-556 ([pdf](http://www.cwru.edu/artsci/phil/FCCinNF.pdf))

* Thomas Forster, _Why the Sets of NF do not form a Cartesian-closed Category_, (1997) ([pdf](https://www.dpmms.cam.ac.uk/~tf/cartesian-closed.pdf))

[[!redirects NF]]
[[!redirects NFU]]


New Math

The __New Math__ is the reform of [[mathematics education]] in USA in 1960s which was performed in the wake of Sputnik and which emphasized set theory, modular arithmetic, algebraic inequalities and abstract algebra. The reform eventually failed and the explanations why this is so are differing.

* wikipedia [New Math](https://en.wikipedia.org/wiki/New_Math)
* Ralph A. Raimi, [Whatever Happened to the New Math?](http://www.math.rochester.edu/people/faculty/rarm/smsg.html)

category: education


New Spaces for Mathematics and Physics

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### Context
#### Mathematics
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[[!include mathematicscontents]]
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#### Physics
+-- {: .hide}
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#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

This page collects pointers related to the conference and book project:

* [[Gabriel Catren]], [[Mathieu Anel]], (eds.)

  \linebreak

  **New Spaces for Mathematics and Physics**

  \linebreak

  [conference](http://ercpqg-espace.sciencesconf.org/program) (IHP Paris, Oct-Sept 2015)

  and

  [book project](http://ercpqg-espace.sciencesconf.org/resource/page/id/1):

  \linebreak

  _New spaces in mathematics: formal and conceptual reflections_,

  Cambridge University Press (2021)

  ISBN:9781108854429

  [doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429)

  [gBooks](https://books.google.us/books?id=8XUfEAAAQBAJ)

  \linebreak

  _New spaces in physics: formal and conceptual reflections_,

  Cambridge University Press (2021)

  ISBN:9781108854399

  [doi:10.1017/9781108854439](https://doi.org/10.1017/9781108854439)

[toc pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Catren-NewSpacesTOC.pdf)

on current developments regarding foundations of [[geometry]] in [[mathematics]] and [[physics]].

#Contents#
* table of contents
{:toc}

## Book

### Volume I: New Spaces in Mathematics

#### I.1. Differential geometry

On [[differential geometry]].

##### I.1.1 Diffeologies (Patrick Iglesias-Zemmour)

[[Patrick Iglesias-Zemmour]] on [[diffeological spaces]].

##### I.1.2 New methods for old spaces: synthetic differential geometry (Anders  Kock)

* [[Anders Kock]], _New methods for old spaces: [[synthetic differential geometry]]_
([arXiv:1610.00286](https://arxiv.org/abs/1610.00286))

##### I.1.3 Microlocalanalysis and beyond (Pierre Schapira)

[[Pierre Schapira]] on [[microlocal analysis]].

* [arXiv:1701.08955](https://arxiv.org/abs/1701.08955)

#### I.2. Topology and algebraic topology

On [[topology]] and [[algebraic topology]].

##### I.2.1 Topo-logie (Mathieu Anel & André Joyal)

On [[Grothendieck toposes]] regarded as "[[logos|logoses]]":

* [[Mathieu Anel]], [[Andre Joyal]], _Topo-logie_ ([pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf))

##### I.2.2 Spaces as infinity-groupoids (Timothy Porter)

* [[Timothy Porter]], _Spaces as infinity-groupoids_ ([preprint pdf](https://ncatlab.org/nlab/files/Spaces+as+infinity-groupoids.pdf)): on [[homotopy types]] as [[infinity-groupoids]].

#####I.2.3 Homotopy type theory: the logic of space (Mike Shulman)

On [[homotopy type theory]]:

* [[Mike Shulman]], _Homotopy type theory: the logic of space_ ([arXiv:1703.03007](https://arxiv.org/abs/1703.03007))

#### I.3. Algebraic geometry

On [[algebraic geometry]].

##### I.3.1 Sheaves and functors of points (Michel Vaquié)

[[Michel Vaquié]] on [[sheaves]] and [[functorial geometry]].

##### I.3.2 Stacks (Nicole Mestrano & Carlos Simpson)

[[Carlos Simpson]] on  [[algebraic stacks]].

##### I.3.3 The geometry of ambiguity – An introduction to derived geometry (Mathieu Anel)

On [[derived geometry]].

##### I.3.4 Geometry in dg-categories (Maxim Kontsevich)

[[Maxim Kontsevich]] on [[derived noncommutative geometry]] in terms of [[formal duals]] of [[dg-categories]]  ([[stable infinity-categories]], [[enhanced triangulated categories]], see also at _[[spectrum of a triangulated category]]_).

### Volume II: New Spaces in Physics

#### II.1. Non-commutative and super-commutative geometries

On [[noncommutative geometry]] and [[supergeometry]].

##### II.1.1 Noncommutative Geometry, the spectral standpoint (Alain Connes)

[[Alain Connes]] on [[spectral triple|spectral geometry]].

##### II.1.2 Topos quantum theory (Klaas Landsman)

[[Klaas Landsman]] on [[Bohr toposes]].

##### II.1.3 Super-geometry (Mikhail Kapranov)

[[Mikhail Kapranov]] on [[superalgebra]] as [[sphere spectrum]]-graded algebra (see also at _[[spectral super-scheme]]_):

* [[Mikhail Kapranov]], _Supergeometry in mathematics and physics_, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) _[[New Spaces for Mathematics and Physics]]_ ([arXiv:1512.07042](http://arxiv.org/abs/1512.07042))

* {#Kapranov15b} [[Mikhail Kapranov]], _Super-geometry_, talk at _[New Spaces for Mathematics & Physics](http://ercpqg-espace.sciencesconf.org/program)_, IHP Paris, Oct-Sept 2015 ([video recording](https://www.youtube.com/watch?v=bjsNwKYT8JE))

#### II.2. Symplectic geometry

On [[symplectic geometry]]:

##### II.2.1 Derived stacks in symplectic geometry (Damien Calaque)

On [[derived geometry|derived]] [[symplectic geometry]]:

* [[Damien Calaque]], _Derived stacks in symplectic geometry_ ([arXiv:1802.09643](https://arxiv.org/abs/1802.09643))

##### II.2.2 Higher pre-quantized geometry (Urs Schreiber)

On [[higher prequantum geometry]]:

* [[Urs Schreiber]], _[[schreiber:Higher Prequantum Geometry]],([arXiv:1601.05956](http://arxiv.org/abs/1601.05956), [video recording](https://www.youtube.com/watch?v=EaYMuyr9KII))

#### II.3. Space-time

On [[spacetime]].

##### II.3.1 Struggles with the continuum (John Baez)

On the [[continuum]]:

* [[John Baez]], _Struggles with the continuum_ ([arXiv:1609.01421](https://arxiv.org/abs/1609.01421))

##### II.3.2 Twistor theory (Roger Penrose)

On [[twistors]]:

* [[Roger Penrose]], *Twistor Theory: A Geometric Perspective for Describing the Physical World* ([doi:10.1017/9781108854399.011](https://doi.org/10.1017/9781108854399.011))

##### II.3.3 Loop quantum gravity (Muxin Han)

##### II.3.4 Stringy geometry and emergent space (Marcos Mariño)

[[Marcos Mariño]] on [[AdS-CFT duality]].

\linebreak

## Conference
### Smoothness and singularities

#### Schemes (Pierre Cartier)

[[Pierre Cartier]] on [[schemes]].

([video recording](https://www.youtube.com/watch?v=etdvnswIhMw))

#### Synthetic differential geometry - new methods for old spaces (Anders Kock)
 {#Kock}

[[Anders Kock]] on [[synthetic differential geometry]]

([video recording](https://www.youtube.com/watch?v=AXz7xu3WrPE))

#### Lie (or differentiable) groupoids (Jean Pradines)

[[Jean Pradines]] on [[Lie groupoids]]

([video recording](https://www.youtube.com/watch?v=Sg_dStsGcIw))

#### Diffeologies (Patrick Iglesias-Zemmour)
 {#IglesiasZemmour}

[[Patrick Iglesias-Zemmour]] on [[diffeological spaces]].

([video recording](https://www.youtube.com/watch?v=4sZDmiVOhaA))

(lecture notes [pdf](http://math.huji.ac.il/~piz/documents/AITD.pdf))

### Spaces with categories of points

#### Geometric aspects of topos theory in relation with logical doctrines (André Joyal)

[[André Joyal]] on [[topos theory]].

([video recording](https://www.youtube.com/watch?v=kaZpOEOAUzE))

#### Sheaves and functors of points (Michel Vaquié)

[[Michel Vaquié]] on [[sheaves]] and [[gros toposes]].

#### Stacks and the Artin property (Carlos Simpson)
 {#Simpson}

[[Carlos Simpson]] on [[algebraic stacks]] and the [[Artin representability theorem]].

([video recording](https://www.youtube.com/watch?v=vI-09KIkBbY))

#### The Geometry of Ambiguity: An introduction to the ideas of Derived Geometry (Mathieu Anel)
 {#Anel}

[[Mathieu Anel]] on [[derived geometry]] ([pdf](http://mathieu.anel.free.fr/mat/doc/Anel-DerivedGeometry.pdf))

([video recording](https://www.youtube.com/watch?v=198ssqlW9BI))

### Non-commutative geometry

#### Geometry in triangulated categories (Maxim Kontsevich)
 {#Kontsevich}

[[Maxim Kontsevich]] on [[derived noncommutative geometry]] in terms of [[formal duals]] of [[stable infinity-categories]] ([[enhanced triangulated categories]], see also at _[[spectrum of a triangulated category]]_).

* [[Maxim Kontsevich]], _Geometry in triangulated categories_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris, Oct-Sept 2015 ([video recording](https://www.youtube.com/watch?v=XnmO9RAbl50))

#### Quantum differential geometry (Shahn Majid)

[[Shahn Majid]] on [[noncommutative geometry]]

### Spaces up to homotopy
#### Spaces as infinity-groupoids (Timothy Porter)

[[Timothy Porter]] on thinking of spaces as infinity groupoids and the relation with the [[homotopy hypothesis]].

([video recording](https://www.youtube.com/watch?v=BH_NwgM8QjI))  ([[Spaces as infinity-groupoids.pdf|draft of article:file]])

#### Homological decomposition and motives (Denis-Charles Cisinski)

[[Denis-Charles Cisinski]] on [[motives]].

([video recording](https://www.youtube.com/watch?v=GhcxtwUJ0TY))

#### Homotopy type theory: the logic of space (Mike Shulman)
  {#Shulman}

On [[homotopy type theory]]

* [[Mike Shulman]], _Homotopy type theory: the logic of space_ ([arXiv:1703.03007](https://arxiv.org/abs/1703.03007))

### Spaces in classical & quantum mechanics

#### Derived stacks in symplectic geometry (Damien Calaque)
 {#Calaque}

* [[Damien Calaque]] on [[derived geometry|derived]] [[symplectic geometry]]:

* [[Damien Calaque]], _Derived stacks in symplectic geometry_ ([arXiv:1802.09643](https://arxiv.org/abs/1802.09643))

#### Higher pre-quantized geometry (Urs Schreiber)
 {#Schreiber}

* [[Urs Schreiber]], _[[schreiber:Higher Prequantum Geometry]],([arXiv:1601.05956](http://arxiv.org/abs/1601.05956), [video recording](https://www.youtube.com/watch?v=EaYMuyr9KII))

### Spaces in quantum gravity

#### Spin networks and spinfoams (Hanno Sahlmann)

Hanno Sahlmann on spin networks.

([video recording](https://www.youtube.com/watch?v=QDU46lK1FDU))

#### Super-geometry (Mikhail Kapranov)
 {#Kapranov}

[[Mikhail Kapranov]] on [[superalgebra]] as [[sphere spectrum]]-graded algebra (see also at _[[spectral super-scheme]]).

* [[Mikhail Kapranov]], _Supergeometry in mathematics and physics_, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) _[[New Spaces for Mathematics and Physics]]_ ([arXiv:1512.07042](http://arxiv.org/abs/1512.07042))

* {#Kapranov15b} [[Mikhail Kapranov]], _Super-geometry_, talk at _[New Spaces for Mathematics & Physics](http://ercpqg-espace.sciencesconf.org/program)_, IHP Paris, Oct-Sept 2015 ([video recording](https://www.youtube.com/watch?v=bjsNwKYT8JE))

#### Twistor theory (Roger Penrose)
 {#Penrose}

[[Roger Penrose]] on [[twistors]].

([video recording](https://www.youtube.com/watch?v=kmYfYOW0vSg))

#### Stringy geometry and emergent space (Marcos Mariño)

[[Marcos Mariño]] on [[AdS-CFT]]

([video recording](https://www.youtube.com/watch?v=MD_jj3O6IRE))

[[!redirects New Spaces in Mathematics]]
[[!redirects New Spaces in Physics]]

category: reference


Newlander-Nirenberg theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _Newlander-Nirenberg theorem_ states that an [[almost complex structure]] comes from a [[complex structure]] precisely if its [[Nijenhuis tensor]] vanishes.

For discussion of this in terms of [[integrability of G-structures]] see there  at _[Examples -- Complex structure](integrability+of+G-structures#ExampleComplexStructure)_.

## Related concepts

* [[Koszul-Malgrange theorem]]

## References

Textbook accounts include

* {#Voisin02} [[Claire Voisin]], section 2.3.3 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

Reviews include

* Ben McMillan, _The Newlander-Nirenbeg theorem_ ([[McMillanNNTheorem.pdf:file]])


Newman's diamond lemma > history


Newman's lemma

#Contents#
* table of contents
{:toc}

## Idea

*Newman's lemma* implies good properties ([[confluence]]) of a [[rewriting system]] based on satisfaction of a "diamond condition" on generating moves:

With the lemma it is sufficient to verify the weaker property of *[[local confluence]]*, once [[strong normalization]] is established (assuming that the rewriting system does possess these two properties).

## Statement

Let $(X,\rightarrow)$ be an abstract [[rewriting system]] ie. a [[set]] equipped with a [[binary relation]] $\rightarrow$. The Newman lemma asserts that:

\begin{proposition}
If $\rightarrow$ is [[strongly normalizing]] and [[locally confluent]] then it is [[confluent]].
\end{proposition}

## Related concepts

* [[rewriting]]

* [[diamond]]

* [[confluence]]

* [[Church-Rosser theorem]],

## References

The original article:

* [[Maxwell H. A. Newman]], _On theories with a combinatorial definition of "equivalence"_, Annals of Mathematics **43** 2 (1942) 223–243 [[doi:10.2307/2269299](https://doi.org/10.2307/2269299)]

See also:

* [Wikipedia article](https://en.wikipedia.org/wiki/Newman%27s_lemma)

* (Širšov-Bergman) [[zoranskoda:diamond lemma]] (zoranskoda lab)

[[!redirects Newman's diamond lemma]]

[[!redirects Newman lemma]]
[[!redirects Newman diamond lemma]]


Newton identities

#Contents#
* table of contents
{:toc}

## Idea

(...)

## References

Review:

* Milan Mosse, *Newton's identities* ([pdf](http://web.stanford.edu/~marykw/classes/CS250_W19/Netwons_Identities.pdf))

See also

* Wikipedia, *[Newton identities](https://en.wikipedia.org/wiki/Newton%27s_identities)*

[[!redirects Newton's identities]]

[[!redirects Newton identity]]
[[!redirects Newton's identity]]


Newton's laws of motion

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

[Newton (1687)](#Newton1687) formulated three law of [[mechanics]] (today called "[[classical mechanics]]") whose statement constitutes the beginning of [[physics]] as a science rooted in [[mathematics]], in contrast to the more naively descriptive "[[natural philosophy]]".

## Statements

In modern formulation, Newton's three laws express the following:

### Newton's first and second law

The first and second law together says that [[equations of motion]] in [[physics]] are given by [[differential equations]] of _second_ order.

This implies that initial value data for [[equations of motion]] is given by 0th and by 1st [[derivatives]]. This data is in physics traditionally is referred to as _[[canonical coordinates]]_ and _[[canonical momenta]]_, respectively. In modern mathematical physics this is axiomatized by the notion of [[phase space]] being a [[symplectic manifold]].

### Newton's third law

The third Law expresses an invariance under the [[equations of motion]] under the [[Galilei group]].

  With the advent of the "[[theory of special relativity]]" this law was later refined to postulate invariance under the [[Poincaré group]]. With the advent of the [[theory of general relativity]] it was further refined to postulate only [[local symmetry]] under the [[Poincaré group]], which leads to the modern formulation of [[Einstein gravity]].

## Related concepts

* [[Newtonian mechanics]]

* [[equations of motion]]

  * [[Hamilton's equations]], [[de Donder-Weyl-Hamilton equation]]

  * [[Einstein's equations]]

## References

* {#Newton1687} [[Isaac Newton]], *[[Philosophiæ naturalis principia mathematica]]* (1687)

* Wikipedia, _[Newton's laws of motion](http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion)_

[[!redirects Newton's laws]]
[[!redirects Newton's laws]]
[[!redirects Newton's law of motion]]
[[!redirects Newton's laws of motion]]
[[!redirects Newton's law of motion]]
[[!redirects Newton's laws of motion]]

[[!redirects Newton's first law]]
[[!redirects Newton's first law]]
[[!redirects Newton's first law of motion]]
[[!redirects Newton's first law of motion]]

[[!redirects Newton's second law]]
[[!redirects Newton's second law]]
[[!redirects Newton's second law of motion]]
[[!redirects Newton's second law of motion]]

[[!redirects Newton's third law]]
[[!redirects Newton's third law]]
[[!redirects Newton's third law of motion]]
[[!redirects Newton's third law of motion]]

[[!redirects Newton's laws of physics]]
[[!redirects Newton's laws of mechanics]]

[[!redirects Newtonian physics]]


Newton's method

#Contents#
* table of contents
{:toc}

## Idea

What is called _Newton's method_ after [[Isaac Newton]] is an [[recursion|recursive]] procedure for computing approximations to zeros ("[[roots]]") of [[differentiable functions]] with values in the [[real numbers]].

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[differentiable function]] and for $x_0 \in \mathbb{R}$ a [[real number]] such that the first [[derivative]] $f' \colon \mathbb{R} \to \mathbb{R}$ is non-vanishing at $x_0$.

Let $\{x_n \in \mathbb{R} | n \in \mathbb{N}\}$ be defined [[recursion|recursively]] by

$$
  x_{n+1}
  \coloneqq
  x_n
  -
  \frac{f(x_n)}{f'(x_n)}
  \,.
$$

Under mild conditions, this sequence [[convergence|converges]] to a zero/[[root]] of $f$.

## Related concepts

* [[Puiseux series]]

## References

* Wikipedia, _[Newton's method](http://en.wikipedia.org/wiki/Newton%27s_method)_

[[!redirects Newton method]]


Newton-Cartan structure

#Contents#
* table of contents
{:toc}

## Idea
A Newton-Cartan structure on a [[4-manifold]] gives the manifold the structure of a non-relativistic Newtonian [[spacetime]].

## Definition

+-- {: .num_defn}
###### Definition
Let $M$ be a 4-manifold. A **Newton-Cartan structure** consists of

 * A degenerate symmetric [[bivector]] $h : M \to S^2 (TM)$ of signature $(0, +, +, +)$.
 * A [[1-form]] $\theta: M \to T^*M$ that lies in the kernel of $h$ viewed as a map $T^*M \to TM$.
 * A [[torsion of a metric connection|torsion-free]] [[affine connection]] $\nabla$ such that $\nabla h = 0$ and $\nabla \theta = 0$.
=--

Note that unlike [[Riemannian metrics]], the connection is not uniquely specified by $h$ and $\theta$.

## Related concepts

 * [[pseudo-Riemannian metric]]


Newton-Leibniz operator

[[!redirects D operator]]
[[!redirects derivative]]

+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

Given an [[Archimedean field]] $F$, an [[open interval]] $I \subseteq F$, a [[pointwise continuous function]] $f:C^0(I, F)$ is __differentiable__ if the [[difference quotient]] has a [[limit of a binary function approaching a diagonal|limit approaching the diagonal]]

$$isDifferentiable(f) := hasLimitApproachingDiagonal\left(\frac{f(x) - f(y)}{x - y}\right)$$

Let us define the type of differentiable functions $D^1(I, F)$ on $I$ as the type of pointwise continuous functions that are differentiable.
$$D^1(I, F) := \{f \in C^0(I, F) \vert isDifferentiable(f)\}$$

The __Newton–Leibniz operator__ or __D operator__ $\tilde{D}: D^1(I, F) \to (I \to F)$ is a function from the type of [[differentiable function]]s $D^1(I, F)$ to the type of functions $I \to F$, and is pointwise defined as
$$\tilde{D}(f) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(x) - f(y)}{x - y}$$
for a [[differentiable function]] $f:D^1(I, F)$. For any differentiable function $f:D^1(I, F)$, the function $\tilde{D}(f)$ is called the __derivative__ of $f$.

## Properties ##

These all follow from the [[algebraic limit theorem]]s of the [[limit of a binary function approaching a diagonal]].

### Constant function rule ###

Let $c:I \to F$ be a constant function. With constant functions the function $c:I \to F$ and the value of the function $c:F$ are usually written with the same symbol. For all constant functions $c:I \to F$,

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} \frac{c - c}{x - y}$$

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} \frac{0}{x - y}$$

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} 0$$

$$\tilde{D}(c) = 0$$

### Identity function rule ###

Where $\frac{x}{x}$ is another notation for $x \cdot x^{-1}$

$$\tilde{D}(id_F) = \lim_{(x, y) \to (x, x)} \frac{x - y}{x - y}$$

$$\tilde{D}(id_F) = 1$$

### Linearity ###

For all elements $a \in F$ and $b \in F$ and functions $f:I \to F$ and $g:I \to F$,

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot f(x) + b \cdot g(x)) - (a \cdot f(y) + b \cdot g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot f(x) - a \cdot f(y)) + (b \cdot g(x) - b \cdot g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot (f(x) - f(y)) + (b \cdot (g(x) - g(y))}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{a \cdot (f(x) - f(y)}{x - y} + \frac{b \cdot (g(x) - g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = a \cdot \lim_{(x, y) \to (x, x)} \frac{(f(x) - f(y)}{x - y} + b \cdot \lim_{(x, y) \to (x, x)} \frac{(g(x) - g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = a \cdot \tilde{D}(f) + b \cdot \tilde{D}(g)$$

### Leibniz rule ###

For all functions $f:I \to F$ and $g:I \to F$,

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot g(x)) - (f(y) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot g(x)) - (f(x) \cdot g(y)) + (f(x) \cdot g(y)) - (f(y) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot (g(x) - g(y)) + ((f(x) - f(y)) \cdot g(y))}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{f(x) \cdot (g(x) - g(y)}{x - y} + \frac{(f(x) - f(y)) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = f \cdot \left(\lim_{(x, y) \to (x, x)} \frac{g(x) - g(y)}{x - y}\right) + \left(\lim_{(x, y) \to (x, x)} \frac{f(x) - f(y)}{x - y}\right) \cdot g$$

$$\tilde{D}(f \cdot g) = f \cdot \tilde{D}(g) + \tilde{D}(f) \cdot g$$

### Power rule ###

For all functions $f:I \to F$ and natural numbers $n \in \mathbf{N}$,

$$\tilde{D}(f^n) = \lim_{(x, y) \to (x, x)} \frac{f(x)^n - f(y)^n}{x - y}$$

We proceed by induction on the natural numbers. The base case $n = 0$: For all functions $f:I \to F$,

$$\tilde{D}(f^0) = \lim_{(x, y) \to (x, x)} \frac{f(x)^0 - f(y)^0}{x - y}$$

$$\tilde{D}(f^0) = \lim_{(x, y) \to (x, x)} \frac{1 - 1}{x - y}$$

$$\tilde{D}(f^0) = \tilde{D}(1)$$

Then the inductive case: For all functions $f:I \to F$ and natural numbers $n \in \mathbf{N}$,

$$\tilde{D}(f^{n+1}) = \lim_{(x, y) \to (x, x)} \frac{f(x)^{n+1} - f(y)^{n+1}}{x - y}$$

$$\tilde{D}(f^{n+1}) = \lim_{(x, y) \to (x, x)} \frac{f(x) \cdot f(x)^{n} - f(y) \cdot f(y)^{n}}{x - y}$$

$$\tilde{D}(f^{n+1}) = \tilde{D}(f) \cdot f^n + \tilde{D}(f^n) \cdot f$$

$$\tilde{D}(f^{n+1}) = \sum_{n=0}^{n} \tilde{D}(f) \cdot f^n$$

### Reciprocal rule ###

## See also ##

* [[differentiable function]]

* [[iterated differentiable function]]

* [[smooth function]]

* [[antiderivative]]


nForum

< [[nlabmeta:nForum FAQ]]


nGrpd

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

## Definition

The [[(n,1)-category|(n+1,1)-category]] **$n Grpd$** is the collection of all [[n-groupoid]]s. It is the full [[sub-(∞,1)-category]] on the $n$-[[truncated]] objects in [[∞Grpd]].

category: category


Niall Macpherson

* [webpage](https://www.sissa.it/tpp/members.php?ID=269)

## Selected writings

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] as [[defect field theory]] "inside" [[AdS7-CFT6]]:

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))

See also

* Andrea Legramandi, [[Niall Macpherson]],  _$AdS_3$ solutions with $\mathcal{N}=(3,0)$ from $S^3 \timesS^3$ fibrations_, ([arXiv:1912.10509](https://arxiv.org/abs/1912.10509))

On [[JT-gravity]] and [[AdS2/CF1]] in [[M-theory]]:

* Junho Hong, [[Niall Macpherson]], Leopoldo A. Pando Zayas, _Aspects of $AdS_2$ classification in M-theory: Solutions with mesonic and baryonic charges_, JHEP 11 (2019) 127 ([arXiv:1908.08518](https://arxiv.org/abs/1908.08518))


Niccolò Veltri

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

## Selected writings

On [[higher inductive types]] in [[homotopy type theory]]:

* [[Niccolò Veltri]], [[Niels van der Weide]], *Constructing Higher Inductive Types as Groupoid Quotients*, Logical Methods in Computer Science **17** 2 (2021)  [[arXiv:2002.08150](https://arxiv.org/abs/2002.08150), doi:10.23638/LMCS-17(2:8)2021]

category: people

[[!redirects Niccolo Veltri]]


nice category of spaces

> For more see at *[[convenient category of topological spaces]]*.

By a "nice" or "[[convenient category of topological spaces|convenient]]" category of space one means a [[category]] that subsumes at least most of the given kinds of [[spaces]] of interest (typically: [[topological space]]) but possibly also less well-behaved spaces, such that the category itself becomes well behaved.

The point is that the default category [[Top]] of [[topological spaces]] lacks many good category-theoretic properties: It is [[complete category|complete]], [[cocomplete category|cocomplete]] and [[extensive category|extensive]], but:

* not [[cartesian closed category|cartesian closed]] or [[locally cartesian closed category|locally cartesian closed]],

* not [[locally presentable category|locally presentable]], and

* not a [[topos]] or even a [[quasitopos]],

* nor is it even a [[pretopos]] or an [[exact category|exact]] or even a [[regular category]].

The lack of cartesian closure and, to a lesser extent, local presentability, is especially problematic for [[homotopy theory]].  Many different solutions for repairing lack of cartesian closure have been proposed, generally involving either restricting to a subcategory of [[Top]] (usually [[reflective subcategory|reflective]] or coreflective, so that it inherits completeness and cocompleteness), enlarging it to a supercategory, or some combination thereof.  Most involve restricting the topologies to those that can be specified on "small" (and in particular, [[compact space|compact]]) subsets.

In particular, a [[convenient category of topological spaces]] is, in the technical sense of the [nLab](http://ncatlab.org/nlab/show/HomePage), a cartesian-closed category of spaces together with some other useful properties (q.v.).

## Examples

* The frequent choice among algebraic topologists today is to use the subcategory of [[compactly generated space|compactly generated spaces]], which is cartesian closed, but not locally cartesian closed.  It is a coreflective subcategory of a reflective subcategory of $Top$.

* Homotopy theorists often find the category of [[simplicial sets]] to be an especially nice environment. It is for example a [[Grothendieck topos]], thus a [[locally cartesian closed category]] and satisfying all [[exactness properties]] one expects of toposes, and is a [[locally presentable category]]. The fact that every topological space has a simplicial set as its [[singularization]] then becomes an application of the homotopy theory of simplicial sets to the study of topological spaces, rather than a way to use simplicial sets to study the homotopy theory of topological spaces.

  For more on this see _[[Top]]_, _[[homotopy theory]]_ and [[infinity-groupoid]].

* The subcategory of [[Delta-generated topological spaces]] is also both cartesian closed and locally presentable.

* An approach of mainly historical interest is to use [[quasitopological space|quasitopological spaces]], an enlargement of $Top$ which is cartesian closed.

* The category $PsTop$ of [[pseudotopological space|pseudotopological spaces]] (also called Choquet spaces) is a quasitopos containing $Top$ as a full reflective subcategory.  In particular, $PsTop$ is [[locally cartesian closed category|locally cartesian closed]] (but not locally presentable).

* In his paper _On a topological topos_, Peter Johnstone described a [[Grothendieck topos]] $E$ which contains the category of [[sequential space|sequential]] topological spaces as a full reflective subcategory which is closed under many colimits (including all those used to define [[CW complex|CW complexes]]).  Again, since $E$ is a Grothendieck topos, it is locally presentable and locally cartesian closed.  Moreover, the [[geometric realization]] and [[singular complex]] functors form a [[geometric morphism]] between $E$ and the category of [[simplicial set|simplicial sets]].  The "underlying set" functor $E\to Set$ is not [[faithful functor|faithful]], but it is faithful on the full subcategory of [[subsequential space|subsequential spaces]], which contain the sequential spaces and form a [[quasitopos]].  See [[topological topos]].

* The category of [[compact Hausdorff spaces]] is perfectly nice for some purposes. While neither cartesian closed nor locally presentable, it is however a complete and cocomplete [[pretopos]]. In this way it is both a category of [[nice topological spaces]] and a nice category of topological spaces, thus an exception proving the "rule" described at [[dichotomy between nice objects and nice categories]].

* The category of [[locales]] and the full subcategory of [[sober spaces]] can be considered nice for certain purposes. The category of locales is extensive and is opposite to the category of [[frames]] (which is monadic over $Set$ and thus [[exact category|exact]]). The category of locales is however neither cartesian closed nor locally presentable, although there is a nice description of [[exponential object|exponentiable locales]]. Johnstone's Stone Spaces gives an account of topology via locale theory.

* John Milnor proposed the category of spaces having the homotopy type of a CW complex as a nice category of spaces. If $X$ and $Y$ are objects and $X$ is compact, then there is an exponential object $Y^X$ in this category. This by the way is also a category of [[nice topological spaces]].

* The category of [[algebraic lattices]], considered as a full subcategory of $T_0$-[[separation axioms|spaces]], is a nice cartesian closed category of spaces in which to do [[domain theory]]. Related to this is the category of [[equilogical spaces]], which is locally cartesian closed (and thus also regular) and arises as the [[exact completion|reg/ex completion]] of the category of $T_0$ spaces.

* The category of [[condensed sets]] is a nice category of spaces which is a [[well-powered category|well-powered]], [[locally cartesian closed]] [[pretopos|infinitary-pretopos]] in which one can do [[condensed mathematics]]. Related to this is the category of [[pyknotic sets]], which form a [[topos]].

## References

* [[Peter May]], _[[A Concise Course in Algebraic Topology]]_ (Chapter 5, for compactly generated spaces)

* O. Wyler, _Convenient categories for topology_

* L. Fajstrup and J. Rosicky, _A convenient category for directed homotopy_ (for Delta-generated spaces)

* E. Spanier, _Quasi-topologies_ (for quasi-topological spaces)

* O. Wyler, _Lecture notes on topoi and quasitopoi_ (for pseudotopological spaces)

* [[Peter Johnstone]], _On a [[topological topos]]_

* [[Peter Johnstone]], [[Stone Spaces]]

* J. Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90, no. 2 (1959), 272-280.

[[!redirects nice categories of spaces]]


nice simplicial topological space

+-- {: .rightHandSide}
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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _nice simplicial topological space_ is a [[simplicial topological space]] that satisfies certain extra properties that make it well behaved in [[homotopy theory]], notably so that its [[geometric realization of simplicial spaces]] is its [[homotopy colimit]].

## Definition

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

Let $X : \Delta^{op} \to Top$ be a [[simplicial topological space]].

Such $X$ is called

* **good** if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are all [[closed cofibrations]];

* **proper** if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a [[closed cofibration]], where $s X_n = \bigcup_i s_i(X_{n-1})$.

In other words this says: $X_\bullet$ is proper if it is cofibrant in the [[Reedy model structure]] $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on [[simplicial object]]s with respect to the [[Strøm model structure]] on [[Top]].

=--

The notion of good simplicial topological space goes back to ([Segal 1973](#Segal73)), that of proper simplicial topological space to ([May](#May)).

## Properties

### General

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

A good simplicial topological space is proper.

=--

A proof appears as [Lewis, corollary 2.4 (b)](#Lewis). A generalization of this result is in [RobertsStevenson](#StevensonRoberts).

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

For $X_\bullet$ any simplicial topological space, then ${|Sing X_\bullet|}$ is good, hence proper, and the natural morphism

$$
  {|Sing X_\bullet|} \to X_\bullet
$$

is degreewise a [[weak homotopy equivalence]].

=--

This follows by results in ([Lewis](#Lewis)).

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

Since for $X \in Top$ the map $|Sing X| \to X$ is a cofibrant [[resolution]] in the standard [[Quillen model structure on topological spaces]], we have that

$$
  |Sing X_\bullet| \to X_\bullet
$$

is a degreewise [[weak homotopy equivalence]]. In particular each space $|Sing X_n|$ is a [[CW-complex]], hence in particular a [[locally equi-connected space]]. By ([Lewis, p. 153](#Lewis)) inclusions of [[retract]]s of locally equi-connected spaces are [[closed cofibration]]s, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.

=--

### Models for the homotopy colimit

+-- {: .num_prop}

###### Proposition

Let $X_\bullet$ be a [[simplicial topological space]]. Then there is a [[natural transformation|natural]] [[weak homotopy equivalence]]

$$
 \Vert X_\bullet\Vert \simeq hocolim_{n \in \Delta} X_n
$$

from its fat [[geometric realization of simplicial topological spaces]] to the [[homotopy colimit]] over the simplicial diagram $X : \Delta^{op} \to Top$.

If moreover $X_\bullet$ is [proper](#ProperSimplicialSpace), then the [[natural transformation|natural morphism]]  $ {\Vert X\Vert} \to {|X|}$ is a [[weak homotopy equivalence]], and hence also the ordinary geometric realization is a model for the homotopy colimit.

=--

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

That the [[geometric realization of simplicial topological spaces]] of a proper simplicial space is is homotopy colimit follows from the [above fact](#ProperIsReedyCofibrant) that proper spaces are Reedy cofibrant, and using the general statement discussed at [[homotopy colimit]] about description of homotopy colimits by [[coend]]s.

=--

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

In the case $X_\bullet$ that is a [good](#GoodSimplicialSpace) [[simplicial topological space]], a direct (i.e., not using the fact that goodness implies properness) proof that $ \Vert X\Vert  \to |X|$ is a weak homotopy equivalence has been sketched by [[Graeme Segal]] and then refined by [[Tammo tom Dieck]].

=--

## Related concepts

* [[well-pointed simplicial topological group]]

* [[simplicial topological space]]

* [[simplicial topological group]]

* [[geometric realization of simplicial spaces]].

## References

The definition of _proper_ simplicial space goes back to

* [[Peter May]], _The Geometry of Iterated Loop Spaces_ , Lecture Notes in Mathematics, 1972, Volume 271(1972), 100-112 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf))

May originally said _strictly proper_ for what now is just called _proper_ .

The definition of _good_ simplicial space goes back to

* {#Segal73} [[Graeme Segal]], *Configuration-spaces and iterated loop-spaces*, Invent. Math. __21__ (1973), 213–221.  ([doi:10.1007/BF01390197](https://doi.org/10.1007/BF01390197), [pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf), MR 0331377)

The implication $good \Rightarrow proper$ seems to be a [[folk theorem]]. Its origin is maybe in

* {#Lewis} [[L. Gaunce Lewis, Jr.]], _When is the natural map $X \to \Omega \Sigma X$ a cofibration?_ , Trans. Amer. Math. Soc. 273 (1982), 147--155.

A generalization of the statement that good implies proper to other [[topological concrete categories]] and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a [[simplicial topological group]] is in

* [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundle in parameterized spaces_ ([arXiv:1203.2460](http://arxiv.org/abs/1203.2460))
 {#RobertsStevenson}

* [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](http://arxiv.org/abs/1203.2461))
 {#Stevenson}

Comments on the relation between properness and cofibrancy in the [[Reedy model structure]] on $[\Delta^{op}, Set]$ are made in

* [[Paul Goerss]], [[Kristen Schemmerhorn]], _Model Categories and Simplicial Methods_ ([arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537)).

[[!redirects nice simplicial topological spaces]]

[[!redirects good simplicial space]]
[[!redirects proper simplicial space]]

[[!redirects good simplicial topological space]]
[[!redirects good simplicial topological spaces]]

[[!redirects proper simplicial topological space]]
[[!redirects proper simplicial topological spaces]]


nice topological space

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The [[category]] [[TopSp]] of all [[topological spaces]] serves in the foundations of many subfields of [[mathematics]], as reflected in the widely (but not universally) accepted convention that "space" in [[mathematics]] means "[[topological space]]", by default. However, the great flexibility of the notion of topological spaces also has its drawbacks, as without further qualification it tends to be a little too general for usage in any given sub-field.  At the same time, the relevant further conditions typically required on topological spaces in any given application varies, sometimes considerably, and many further such conditions have been and are being introduced. Hence what counts as a well-behaved ("nice") topological spaces strongly depends on context, and the main point to be aware of is the existence of key *classes* of possible extra conditions on topological spaces.

The foremost among these are certainly the *[[separation axioms]]*, which may be understood as prescribing how exotic (pathological) a topological space is allowed to be, locally, as compared to the archetypical example of *[[Euclidean space]]* (which, it is worthwhile to remember, was the default meaning of "space" for most of human history). In consequence, [[separation conditions]] are typically required of  topological spaces if and to the extent that these are used as models for *space* in the original sense of [[geometry]]. Probably the most prominent among the separation axioms is the [[Hausdorff space|Hausdorff]]-condition "[[T2-space|$T_2$]]", and many authors will by default mean "[[Hausdorff topological space]]" when saying "topological space".

Another class of nicety conditions on topological spaces are not primarily motivated by the look of the individual spaces, but by ensuring that the (preferably ([[coreflective subcategory|co-]])[[reflective subcategory|reflective]]) [[full subcategory]] of spaces satisfying these conditions has good [[category theory|category theoretic]] properties, such as [[cartesian closed category|cartesian closure]]
(but beware that there is a [[dichotomy between nice objects and nice categories]]).
Such *[[convenient categories of topological spaces]]* are of foundational importance in [[algebraic topology]] where it is tolerable to change the [[homeomorphism]]-type of a space as long as its [[homotopy type]] is retained. Prominent examples of classes of topological spaces which are "nice" in a way that their categories are "convenient", in this sense, are the "[subcategory-generated spaces](convenient+category+of+topological+spaces#CategoriesOfColimitsOfGeneratingSpaces)", namely those that may be obtained as [[colimits]] of building block spaces taken from a given [[full subcategory]]. For example, if these building blocks are taken to be all [[compact spaces]] then one speaks of *[[compactly generated topological spaces]]*, while if they are taken to be all [[Euclidean spaces]] one speaks of [[Delta-generated topological spaces]].

Further in this vein of [[algebraic topology]] and [[homotopy theory]] are conditions on a topological space which ensure that it is a good model for its ([[weak homotopy type|weak]]) [[homotopy type]]. For example, the condition that a topological space be a [[CW-complex]] (hence  [[cofibrant object|cofibrant]] in the [[classical model structure on topological spaces|classical model structure]]) ensures that [[continuous functions]] out of it model all [[homotopy classes]] of morphisms out of the homotopy type (as witnessed by [[Whitehead's theorem]]), while the condition that a [[topological group]] be *[[well-pointed topological group|well-pointed]]* (hence [[cofibrant object|cofibrant]] in the [[model category of pointed objects|pointed]] [[Strøm model category]]) ensures that the [[geometric realization of simplicial topological spaces|topological realization]] of its [[nerve]] has the expected [[homotopy type]]. It is this last condition which leads over to the notion of [[nice simplicial topological spaces]].

## Some example conditions

The following is a list -- currently somewhat random and highly incomplete -- of nicety conditions that are considered on topological spaces.

* There is a whole slew of _[[separation conditions]]_, of which the most common is the [[Hausdorff space|Hausdorff]] condition (any two points can be separated by disjoint opens).  Hausdorff is also called $T_2$ and fits in a hierarchy ranging from $T_1$ through $T_4$ originally and now (at least) $T_0$ to $T_6$ (with some fractional subscripts too).

* [[sober space|Sobriety]] is a separation condition living in between $T_0$ and $T_2$ (but incomparable with $T_1$), which guarantees a good relationship with [[locale]]s.

* A space is [[compact space|compact]] if any open cover of it has a finite subcover.  Variations include locally compact, countably compact, sequentially compact, etc.

* _[[locally compact Hausdorff spaces]] deserve special mention since they are [[exponential object|exponentiable]] in [[Top]].

* A [[compactly generated space]] is, essentially, one whose topology is determined by its restriction to compact subspaces.  These are notable because the category of compactly generated spaces is [[cartesian closed category|cartesian closed]].

* A _metrizable_ space is one whose topology can be defined by a [[metric space|metric]].  We also have pseudometric spaces, quasimetric spaces, uniformizable spaces, etc.

* A [[sequential space]] is one whose topology is determined by convergence of sequences.  Note that any topology is determined by convergence of [[net]]s or [[filter]]s.

* A [[CW complex]] is a space built out of nothing by progressively attaching cells of higher and higher dimension.  More generally, a [[cell complex]] is a space built by attaching cells, without regard to dimension (that is, lower-dimensional cells may be attached to higher-dimensional ones), and an [[m-cofibrant space]] is one that is [[homotopy equivalence|homotopy equivalent]] to a CW complex (or equivalently a cell complex).  These types of spaces are important for [[homotopy theory]] because they turn [[weak homotopy equivalence]]s into [[homotopy equivalence]]s.

* A [[locally Euclidean space]] such as a [[topological manifold]] is a space that is locally homeomorphic to $\mathbb{R}^n$ for some $n$.

* (add your favorite!)

+-- {: .query}
[[David Roberts]]: How about mentioning [[Alexander Grothendieck]]'s notion of _tame topology_? I saw in the video of [Scharlau's talk](http://www.dailymotion.com/video/x8juek_colloque-grothendieck-winfried-scha_tech) that G. wrote a lot on this, but the manuscript is lost. Do we have any idea past a vague description? (in Recoltes et Semailles or La longue marche I think.)

[[Todd Trimble]]: Quite a few people have thought long and hard about Grothendieck's speculations on tame topology in his _Esquisse d'un Programme_. The approach with which I am most familiar comes from model theory, and falls under the rubric of "o-minimal structures" (the "o" standing for "order"). See the book _Tame topology and o-minimal structures_ by van den Dries. A space which belongs to an o-minimal structure is a subspace of some Euclidean space $\mathbb{R}^n$ and turns out to be indeed nice (it admits nice triangulations for instance).

In a sense this is more of a "nice categories" approach than a "nice spaces" approach, because there is no known global property which would express what it means for a space to be tame. That is, there are many examples of o-minimal structures, but (it is conjectured) there is no maximal o-minimal structure, therefore no overarching meaning of what it would mean for a space to be tame.

Basically, an o-minimal structure $T$ is a collection $T_n \subseteq P(\mathbb{R}^n)$ which is closed under all first-order logic operations (e.g., complements, finite intersections, direct images under projections = existentially quantified sets, equality predicates, and the binary predicate $\lt$ on $\mathbb{R}$), and which satisfies the all-important o-minimality condition: the only sets belonging to $T_1$ are finite unions of points and intervals. The elements of $T$ may be called $T$-*definable sets*; the archetypal example is where $T$ is the collection of semi-algebraic sets (loci of polynomial inequalities) -- cf. the Tarski-Seidenberg theorem. The thrust of the o-minimality condition is to forbid sets like $\mathbb{N} \subseteq \mathbb{R}$ from being $T$-definable, which (following Gödel, Turing, Robinson, Matiyasevich, and others) would open the door to all sorts of pathological sets being $T$-definable as well. So you could think of o-minimality as a kind of logical "monster-barring" device, which happens to be quite effective. See van den Dries's book for a very illuminating discussion.

There are other approaches to tame topology (such as via Shiota's $\mathcal{X}$-sets), but I am less familiar with them.
=--

## Related concepts

* [[nice simplicial topological space]]

* [[convenient category of topological spaces]]

## References

Discussion of notorious subtleties:

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

[[!redirects nice topological spaces]]


Nichol Furey

Nichol (Cohl) Furey

* [personal page](https://www.furey.space/)

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

## Selected writings

On [[symmetry breaking]] in [[Spin(10)]]-[[grand unified theory]] to the exact [[standard model of particle physics|standard model]] [[gauge group]], via [[real normed division algebra]] ([[octonions]], [[quaternions]]):

* [[Nichol Furey]], M. J. Hughes, *Division algebraic symmetry breaking*, Phys. Lett. B **831** (2022)
 [[arXiv:2210.10126](https://arxiv.org/abs/2210.10126), [doi:10.1016/j.physletb.2022.137186](https://doi.org/10.1016/j.physletb.2022.137186)]

On [[division algebra|division-algebraic]] relations in [[GUT]]-[[broken symmetry|symmetry breaking]]:

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part I: General construction* [[arXiv:2312.12377](https://arxiv.org/abs/2312.12377)]

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints* [[arXiv:2312.12799](https://arxiv.org/abs/2312.12799)]

* [[Nichol Furey]], *An Algebraic Roadmap of Particle Theories, Part III: Intersections* [[arXiv:2312.14207](https://arxiv.org/abs/2312.14207)]

category: people

[[!redirects Cohl Furey]]


Nicholas Asher

* [webpage](https://www.irit.fr/~Nicholas.Asher/)

## related $n$Lab entries

* [[Science of Logic]]

category: people


Nicholas Katz

* [webpage](https://web.math.princeton.edu/~nmk/)

## Selected writings

On [[Gauss-Manin connections]]:

* [[Nicholas M. Katz]], *On the differential equations satisfied by period matrices*, Publications Mathématiques de l'IHÉS, Tome **35** (1968) 71-106 $[$[numdamPMIHES_1968__35__71_0](http://www.numdam.org/item/PMIHES_1968__35__71_0)$]$

* [[Nicholas M. Katz]], [[Tadao Oda]], *On the differentiation of De Rham cohomology classes with respect to parameters*, J. Math. Kyoto Univ. **8** 2 (1968) 199-213 $[$[doi:10.1215/kjm/1250524135](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-8/issue-2/On-the-differentiation-of-De-Rham-cohomology-classes-with-respect/10.1215/kjm/1250524135.full)$]$

On [[elliptic curves]] over general [[commutative rings]] (in [[arithmetic geometry]]):

* [[Nicholas Katz]], [[Barry Mazur]], _Arithmetic moduli of elliptic curves, Annals of Mathematics Studies_, vol. 108, Princeton University Press, 1985 ([jstor:j.ctt1b9s05p](https://www.jstor.org/stable/j.ctt1b9s05p))

category: people

[[!redirects Nicholas M. Katz]]


Nicholas Kuhn

* [webpage](http://www.math.virginia.edu/people/njk4x)

## Selected writings

On [[Goodwillie calculus]] and [[chromatic homotopy theory]]:

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

On [[generic representation theory]]:

* [[Nicholas Kuhn]], _Generic representations of the finite general linear groups and the Steenrod algebra: I, II, III_, Amer. J. Math. 116(1994), 327-360, [doi:10.2307/2374932](https://doi.org/10.2307/2374932), K-theory J. 8(1994), 395-428, [doi:10.1007/BF00961409](http://dx.doi.org/10.1007/BF00961668), K-theory J. 9(1995), 273-303, [doi:10.1007/BF00961668](http://dx.doi.org/10.1007/BF00961668).

category: people

[[!redirects Nicholas J. Kuhn]]