Ribenboim power series

# Ribenboim power series

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## Idea

A **Ribenboim power series** group (or ring, or field) is a generalization of a [[formal power series]] ring whose exponents can belong to almost any partially ordered *monoid*.  It includes [[formal power series]], [[formal Laurent series]], [[polynomials]], [[Laurent polynomials]], and [[Hahn series]] as special cases.

## Definition

Let $P$ be a [[poset]]; say that a subset of $P$ is *artinian* if it contains no strictly decreasing infinite [[chain]], and *narrow* if it contains no infinite [[antichain]].  Let $A$ be an [[abelian group]].  The space of **Ribenboim power series** from $P$ with coefficients in $A$, denoted $G(P,A)$, is the set of functions $f:P\to A$ whose support $supp(f) = \{ p\in P \mid f(p)\neq 0 \}$ is artinian and narrow.  This is an abelian group under pointwise addition.

Now let $M$ be a strict [[partially ordered|poset]] [[monoid]], i.e. a [[monoid object]] in the [[monoidal category]] $Pos_{str}$ of [[posets]] and strict functions (i.e. morphisms that preserve the strict ordering as well as the non-strict one) whose monoidal structure is the cartesian product of $Pos$ (which is not the cartesian product of $Pos_{str}$).  More explicitly, this means $M$ is a monoid and a poset and the multiplication preserves both orders $\le$ and $\lt$ on both sides.

If $M$ is such a "strict pomonoid" and $R$ a [[ring]], then $G(M,R)$ is again a ring, with multiplication

$$ (f\cdot g)(m) = \sum_{(m_1,m_2) \in X_m(f,g)} f(m_1)\cdot g(m_2) $$

where $X_m(f,g)$ is the set of pairs $(m_1,m_2)\in supp(f) \times supp(g)$ such that $m_1\cdot m_2 = m$.  (The nontrivial fact to verify is that $X_m(f,g)$ is always finite.)

## Examples

* If $M=\mathbb{N}$ with the usual order, we obtain the usual ring of [[formal power series]] with coefficients in $R$.

* If $M=\mathbb{Z}$ with the usual order, we obtain the usual ring of [[formal Laurent series]].

* If $M=\mathbb{N}$ with the discrete order, we obtain the usual ring of [[polynomials]].

* Similarly, for $\mathbb{Z}$ with the discrete order, we obtain the ring of [[Laurent polynomials]].

* If $M$ is a linearly ordered abelian group and $R$ a field, then $G(M,R)$ is the field of [[Hahn series]] with value group $M$.

## Related pages

Other rings of generalized power series include:

* [[Puiseux series]]
* [[Novikov field]]
* [[Hahn series]]

Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not.  However, they are all instances of the linearization of a [[finiteness space]].  For Puiseux and Ribenboim series, this is shown in [BJCS](#BCJS18).

## References

* P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Semin. Univ. Hambg. 61, pp. 15--3 (1991)

* P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Algebra 79, pp. 293--312 (1992)

* P. Ribenboim, Rings of generalized power series II: Units and zero-divisors, Journal of Algebra 168, pp. 71--89 (1994)

* Richard Blute, Robin Cockett, Pierre-Alain Jacqmin, and Philip Scott, *Finiteness spaces and generalized power series*, [doi](https://doi.org/10.1016/j.entcs.2018.11.002), [arXiv:1805.09836](https://arxiv.org/abs/1805.09836)
 {#BCJS18}


Ribet's theorem

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## Idea

In [[number theory]], _[[Ken Ribet]]'s theorem_ is a statement about [[Galois representations]] associated with [[modular forms]].

[[Jean-Pierre Serre]] ad [[Ken Ribet]] observed that together with assuming the [[Taniyama-Shimura modularity conjecture]] (as yet unproven at that time), Ribet's theorem implies [[Fermat's last theorem]]. The [[modularity theorem]] was then proven by [[Andrew Wiles]] and [[Richard Taylor]], thus often known now as _[[Wiles' proof of Fermat's last theorem]]_.

## Statement

Let $f$ be a weight $2$ newform of level $\Gamma_{0}(q N)$, with $q$ not dividing $N$. Let $\overline{\rho}_{f,p}$ be the associated absolutely irreducible 2-dimensional mod $p$ [[Galois representation]] which is unramified if $q\neq p$ and finite flat if $q=p$.

Then there exists another weight $2$ newform $g$, of level $\Gamma_{0}(N)$, with associated mod $p$ Galois representation $\overline{\rho}_{g,p}$, such that

$$\overline{\rho}_{f,p}\simeq\overline{\rho}_{g,p}.$$

## References

* Wikipedia, Ribet's theorem


Ricardo Campos

* [webpage](http://imag.umontpellier.fr/~campos/)

## Selected writings

On [[graph complex]]-models for [[configuration spaces of points]]:

* {#Campos17} [[Ricardo Campos]], _Batalin-Vilkovisky formality and configuration spaces of points_, 2017 ([doi:10.3929/ethz-a-010886114](https://doi.org/10.3929/ethz-a-010886114))

* {#CamposWillwacher16} [[Ricardo Campos]], [[Thomas Willwacher]], _A model for configuration spaces of points_ ([arXiv:1604.02043](https://arxiv.org/abs/1604.02043))

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, [[Najib Idrissi]], [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

## Related $n$Lab entries

* [[graph complex]]

* [[configuration space (mathematics)]]

category: people


Ricardo Lima

* [personal page](https://dreamandsciencefactory.jimdofree.com/about-me/).

## Selected Writings

On [[C-star category|$C^\ast$-categories]] and introducing the special case of [[W-star category|$W^\ast$-categories]]:

* {#GhezLinaRoberts85} [[P. Ghez]], [[Ricardo Lima]], [[John E. Roberts]], *$W^\ast$-categories*,  Pacific J. Math. **120** 1 (1985) 79-109 [[euclid:pjm/1102703884](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-120/issue-1/Wast-categories/pjm/1102703884.full)]

category: people


Ricardo Schiappa

* [webpage](https://math.tecnico.ulisboa.pt/professor.php.en?who=schiappa)

## Selected writings

On [[resurgence]] in the [[string perturbation series]]:

* [[Ricardo Schiappa]], _Resurgence Asymptotics in String Theory_, talk at IHES 2017 ([video](https://www.youtube.com/watch?v=8tw4MaBwTEA))

On [[resurgence]] and [[non-perturbative effects]] in the [[SYK model]] and [[AdS2-CFT1 duality]]:

* [[Paolo Gregori]], [[Ricardo Schiappa]], *From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes* ([arXiv:2108.11409](https://arxiv.org/abs/2108.11409))

## Related $n$Lab entries

* [[resurgence]]

category: people


Riccardo Barbieri

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

## Selected writings

On [[flavour physics]] (and [[flavour anomalies]]):

* [[Riccardo Barbieri]], _A view of flavour physics in 2021_ ([arXiv:2103.15635](https://arxiv.org/abs/2103.15635))

category: people


Riccardo Benedetti

* [Google Scholar page](https://scholar.google.com/citations?user=39nWA2oAAAAJ)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=60822)

## Selected writings

On [[differential topology]]:

* {#Benedetti19} [[Riccardo Benedetti]], _Lectures on Differential Topology_, Graduate Studies in Mathematics **218**, AMS 2021  ([arXiv:1907.10297](https://arxiv.org/abs/1907.10297), [ISBN: 978-1-4704-6674-9](https://bookstore.ams.org/cdn-1631100821985/gsm-218/))

category: people


Riccardo D'Auria

* [website](http://areeweb.polito.it/ricerca/sugragroup/members.html)

## Selected writings

The original articles on the [[D'Auria-Fre formulation of supergravity]]:

* {#DAuriaFre80} [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. 3, fasc. 12 (1980) ([spire](http://inspirehep.net/record/156191))

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 (doi:10.1016/0550-3213(82)90376-5,  [[GeometricSupergravityErrata.pdf:file]])

* [[Riccardo D'Auria]], [[Pietro Fré]], [[Paul Townsend]],  *Invariance of actions, rheonomy and the new minimal $N=1$ supergravity in the group manifold approach*, Ann. Phys. 155 (1984) 423-446 [[cds:143990](https://cds.cern.ch/record/143990), doi:10.1016/0003-4916(84)90007-1, [pdf](https://cds.cern.ch/record/143990/files/198305027.pdf)]

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Riccardo D'Auria]], _Geometric supergravity_ ([arXiv:2005.13593](https://arxiv.org/abs/2005.13593)) in:  [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

On [[Freund-Rubin compactifications]] of [[11d supergravity]]:

* [[Riccardo D'Auria]], [[Pietro Fre]], [[Peter van Nieuwenhuizen]], _Symmetry Breaking in $d=11$ Supergravity on the Parallelized Seven Sphere_, Phys. Lett. B 122 (1983) 225 ([spire:181634](https://inspirehep.net/literature/181634), doi:10.1016/0370-2693(83)90689-5)

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

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

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

Usage of full [[supergravity]] (retaining the [[gravitino]]) for [[AdS-CMT]], with application to [[graphene]]-like systems:

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], [[Mario Trigiante]], *Unconventional Supersymmetry at the Boundary of $AdS_4$ Supergravity*,  J. High Energ. Phys. **2018** 7 (2018) [[arXiv:1801.08081](https://arxiv.org/abs/1801.08081), doi:10.1007/JHEP04(2018)007]

* [[Laura Andrianopoli]], [[Bianca L. Cerchiai]], [[Riccardo D'Auria]], A. Gallerati, R. Noris, [[Mario Trigiante]], [[Jorge Zanelli]], *$\mathcal{N}$-Extended $D=4$ Supergravity, Unconventional SUSY and Graphene*, JHEP 01 (2020) 084 [[arXiv:1910.03508](https://arxiv.org/abs/1910.03508), doi:10.1007/JHEP01(2020)084]

On the work of [[Tullio Regge]] ([[Regge calculus]], [[D'Auria-Fre formulation of supergravity]], ...):

* [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific 2019 ([doi:10.1142/11643](https://doi.org/10.1142/11643))

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

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

## Related entries

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

category: people


Riccardo Nicoletti

* [Website](http://personalpages.to.infn.it/~nicolett/)

category: people


Riccardo Pedrotti

* [Institute page](https://web.ma.utexas.edu/users/pedrotti.riccardo/)

## Writings

* {#ComplexCobordismNotes} _Complex oriented cohomology, generalized orientation and Thom isomorphism_, 2016, 2018 ([[PedrotticECohomology2018.pdf:file]])

on [[complex oriented cohomology]], [[Conner-Floyd-Chern classes]] and the [[Thom isomorphism]] in generalized cohomology.

category: people


Riccardo Piergallini

* [webpage](http://mat.unicam.it/piergallini/home/index.html)

## selected writings

On [[branched covers]] of [[4-manifolds]]:

* {#Piergallini95} [[Riccardo Piergallini]], _Four-manifolds as 4-fold branched covers of $S^4$_, Topology Volume 34, Issue 3, July 1995 (doi:10.1016/0040-9383(94)00034-I, [pdf](https://core.ac.uk/download/pdf/82379618.pdf))

* {#IoriPiergallini02} Massimiliano Iori, [[Riccardo Piergallini]], _4-manifolds as covers of the 4-sphere branched over non-singular surfaces_, Geom. Topol. 6 (2002) 393-401 ([arXiv:math/0203087](https://arxiv.org/abs/math/0203087))

## related $n$Lab entries

* [[branched cover]]

* [[4-manifold]]

category: people


Riccati equation

# Riccati equations
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## Idea

A (general) __Riccati differential equation__ is any first-order [[ordinary differential equation]] for a function $y = y(t)$ which is of the form

$$
\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2
$$

with $a_0(t)\neq 0$ and $a_2(t)\neq 0$.

## Connection to fractional linear transformations

Riccati equations are to fractional linear transformations as *linear* first-order ordinary differential equations are to affine transformations, in the following sense.  A linear first-order ordinary differential equation

$$
\frac{d y}{d t} = a_0(t) + a_1(t) y
$$

has (under suitable regularity conditions on $a_0$ and $a_1$) solutions $y(t)$ depending in an affine manner on the initial data:

$$ y(t) = f(t) y(0) + g(t) $$

for functions $f,g$ independent of $y(0)$.   More generally, any Riccati equation

$$
\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2
$$

has (under suitable regularity conditions, and at least for short times) solutions $y(t)$ depending on the initial data as follows:

$$ y(t) = \frac{f(t) y(0) + g(t)}{h(t) y(0) + i(t)} $$

for functions $f,g,h,i$ independent of $y(0)$.

Unlike solutions  of first-order linear equations, solutions of Riccati equations can blow up in finite time; however, these solutions become defined for all time if we let $y$ take values in the [[projective line]] rather than the [[affine line]]: that is, $\mathbb{R}\mathrm{P}^1$ or $\mathbb{C}\mathrm{P}^1$ rather than $\mathbb{R}$ or $\mathbb{C}$.

Indeed, the group of fractional linear transformations, the [[projective general linear group]] $\mathrm{PGL}(2,\mathbb{R})$ or $\mathrm{PGL}(2,\mathbb{C})$, acts naturally on the projective line, with the group of affine transformations being the subgroup that preserves the affine line.

The group of smooth $\mathrm{PGL}(2)$-valued functions on the real line acts on the space of Riccati equations.  First, note that this group acts on $\mathbb{P}^1$-valued functions on the real line as follows:

$$ \left( \begin{array}{cc} f(t) & g(t) \\ h(t) & i(t) \end{array} \right) \colon y(t) \mapsto \tilde{y}(t) = \frac{f(t) y(t) + g(t)}{h(t) y(t) + i(t)} $$

Then, given any Riccati equation

$$
\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2   \qquad \star
$$

there is another Riccati equation such that $\tilde{y}$ is a solution of this new Riccati equation iff $y$ is a solution of $\star$.

## Correspondence with 2nd order linear ODEs

There is a correspondence between Riccati equations and a wide class of
2nd order linear differential equations, namely Riccati equation for $y$ gives a 2nd order linear ODE for $u$ which satisfies $a_2 u' = u y$, namely

$$
u'' + a_0 a_2 u' + \left( a_1 + \frac{a_2'}{a_2} \right) u = 0.
$$
Of course, the coefficients are non-constant.

## Reduction to a Bernoulli ODE in the presence of a particular solution

Unlike the [[Bernoulli differential equation]] which has $a_0(z) = 0$ and $a_2(z) \neq 0$, the Riccati equation can not be solved in quadratures in general, unless a a particular solution $y_1$ is known. If so, then we can write $y = y_1 + u$ and
write down the equation for $u$ which appears to be an example of a Bernoulli differential equation, hence solvable in quadratures. Regarding that $w = 1/u$ is
the substitution for reducing Bernoulli to first order linear ODE, we can reduce
the original Riccati to such ODE by substituting $y = y_1 + 1/w$.

## Matrix Riccati equation

For a generalization, the matrix Riccati equation, see [the eom article](#eom).

## Literature

* Springer [[eom]] [Riccati equation](http://www.encyclopediaofmath.org/index.php?title=p/r081770)
  {#eom}

* Wikipedia: [Riccati equation](http://en.wikipedia.org/wiki/Riccati_equation)

category: analysis

[[!redirects Riccati equation]]
[[!redirects Riccati equations]]
[[!redirects Riccati differential equation]]
[[!redirects Riccati differential equations]]
[[!redirects Ricatti equation]]


Ricci curvature

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## Idea

Formally, Ricci curvature $Ric$ of a [[Riemannian manifold]] is a symmetric [[tensor|rank-2 tensor]] obtained by contraction from the [[Riemann curvature]]. Geometrically one may think $Ric(v, w)$ as the first order approximation of the infinitesimal behavior of the surface spanned by $v$ and $w$. This is made explicit by the following formula for the volume element around some point
$$
 d\mu _{g}=\left[1-{\tfrac {1}{6}}Ric_{jk}x^{j}x^{k}+O\left(|x|^{3}\right)\right]d\mu _{Euclidean}
$$

([[Einstein summation convention]]). A [[spacetime]] with vanishing Ricci curvature is also called _Ricci flat_.

## Properties

### Harmonic coordinate representation and regularity

By a trick of Lanczos, that was recovered by DeTurck and Kazdan, in harmonic coordinates the Ricci tensor can be expressed as
$$
Ric_{lm} = -\frac{1}{2} \sum_{j,k} g^{jk} \partial_j \partial_k g_{lm} + Q_{lm}(g, \nabla g)
$$
where $g^{jk}$ denotes the inverse of the metric tensor and $Q_{lm}$ is a quadratic form in $\nabla g$ with coefficients that are rational expressions in which numerators are polynomials $g$ and the denominator depends only on $\sqrt{\det g}$. Note that this formula describes the metric tensor as a quasilinear elliptic PDE. This representation is especially useful in two ways: First, there are theorems that give bounds on the regularity of the metric tensor in harmonic coordinates under geometric assumptions ([Anderson, Cheeger, and Naber](#AndersonCheegerNaber)). Second, as this expression is a quasilinear elliptic PDE, one can conclude on regularity bounds for the metric tensor from regularity estimates for the Ricci tensor. This argument allows for a regularity bootstrap in case of [[Einstein manifold|Einstein manifolds]]: given a rough Einstein metric with $k$ derivatives, the regularity theory for quasilinear PDEs gives $k+2$-regularity of the metric tensor. But the Einstein property $g = \lambda Ric$ implies the same regularity for the Ricci tensor. Hence one can apply the argument again and add infinitum.

### Cheeger-Gromoll theorem

See _[[Cheeger-Gromoll theorem]]_

## Related concepts

[[!include curvature in Riemannian geometry -- table]]

* [[Einstein-Hilbert action]]

* [[Einstein equations of motion]]

* [[Einstein manifold]]

## References

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

* Lanczos, _Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen_ Phys. Z. 23, 537-539 (1922)

* DeTurck and Kazdan, _Some regularity theorems in Riemannian geometry_ Ann. scient. Éc. Norm. Sup. (1981)

{#AndersonCheegerNaber}For regularity result see

* Anderson, _Convergence  and rigidity of manifolds under Ricci curvature bounds_, Invent. Math. (1990)
* Cheeger and Naber, _Lower bounds on Ricci curvature and quantitative behavior of singular sets_ Invent. Math. (2013)

For weaker but more general regularity results see also:

* Anderson and Cheeger, _$C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below_ J. Diff. Geo. (1992)

A conjecture that all [[compact topological space|compact]] Ricci flat manifolds either have [[special holonomy]] or else are "unstable":

* [[Bobby Acharya]], _Supersymmetry, Ricci Flat Manifolds and the String Landscape_ ([arXiv:1906.06886](https://arxiv.org/abs/1906.06886))

[[!redirects Ricci curvatures]]

[[!redirects Ricci tensor]]
[[!redirects Ricci tensors]]

[[!redirects Ricci flat]]
[[!redirects Ricci flatness]]

[[!redirects Ricci flat manifold]]
[[!redirects Ricci flat manifolds]]


Ricci flow

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## Idea

_Ricci flow_ is the [[gradient flow]] of the [[action functional]] of [[dilaton gravity]]: the [[Einstein-Hilbert action]] coupled to a [[dilaton field]].

Equivalently it is the [[renormalization group flow]] of the [[bosonic string]] [[sigma-model]] for [[background gauge field|background fields]] containing [[gravity]] and [[dilaton]] (reviewed e.g. in [Woolgra 07](#Woolgra07), [Carfora 10](#Carfora10), see also the introduction of [Tseytlin 06](#Tseytlin06)).

In ([Perelman 02](#Perelman02)) Ricci flow for dilaton gravity in 3d was shown to enjoy sufficient monotonicity properties such as to complete [[Richard Hamilton]]'s proof of the [[Poincaré conjecture]].
Hamilton's strategy was to equip a compact simply connected 3d manifold with a [[Riemannian metric]], argue that its Ricci flow will, after many "pinchings" (points where the metric degenerates) produce a collection of 3-[[spheres]] and conclude that therefor the original manifold must have been a 3-sphere, too. The technical problem is to control the number of pinchings, which may occur rapidly. Adding also the [[dilaton field]] turns out not to change the qualitative nature of the flow but make it "monotone enough" to control the pinching.

In [Rubinstein-Sinclair](#RubinsteinSinclair) a typical such Ricci flow with pinchings is visualized as follows:

<img src="http://i.stack.imgur.com/FmUuW.png" alt="" />

## References

### General

A quick survey is in the slides

* Annibale Magni, _Perelman's dilaton_ ([pdf](http://www.sissa.it/fa/difftop2010/dwnld/Magni.pdf)).

and a detailed survey is in

* [[Terry Tao]], _Ricci flow as a gradient flow, log-Sobolev inequalities, and Perelman entropy_ ([blog post](http://terrytao.wordpress.com/2008/04/24/285a-lecture-8-ricci-flow-as-a-gradient-flow-log-sobolev-inequalities-and-perelman-entropy/))

Visualization is in

* {#RubinsteinSinclair} J. Hyam Rubinstein and Robert Sinclair. "Visualizing Ricci Flow of Manifolds of Revolution", Experimental Mathematics v. 14 n. 3, pp. 257–384.

The monotonicity of the Ricci flow for the [[string]] [[sigma-model]] in [[dilaton gravity]] background was established in

* {#Perelman02} [[Grigori Perelman]], _The entropy formula for the Ricci flow and its geometric applications_ ([arXiv:math/0211159](http://arxiv.org/abs/math/0211159))

This was a key step in his completion of Hamilton's program for how to prove the [[Poincare conjecture]].

### As sigma-model renormalization group flow
 {#ReferencesAsSigmaModelRenormalizationGroupFlow}

The identification of Ricci flow with the [[renormalization group flow]] of the [[bosonic string]] [[sigma-model]] is reviewed for instance in

* Kasper Olsen, _[From Polyakov to Perelman](http://kasperolsen.wordpress.com/2010/03/19/from-polyakov-to-perelman-history-of-the-ricci-flow-and-a-millennium-prize/)_

* {#Woolgra07} E. Woolgar, _Some Applications of Ricci Flow in Physics_, Can.J.Phys.86:645,2008 ([arXiv:0708.2144](http://arxiv.org/abs/0708.2144))

* {#Carfora10} Mauro Carfora, _Renormalization Group and the Ricci flow_ ([arXiv:1001.3595](http://arxiv.org/abs/1001.3595))

and discussed in more detail for instance in

* T Oliynyk, V Suneeta, E Woolgar, _Irreversibility of World-sheet Renormalization Group Flow_, Phys.Lett. B610 (2005) 115-121 ([arXiv:hep-th/0410001](http://arxiv.org/abs/hep-th/0410001))

* T Oliynyk, V Suneeta, E Woolgar, _A Gradient Flow for Worldsheet Nonlinear Sigma Models_, Nucl.Phys. B739 (2006) 441-458 ([arXiv:hep-th/0510239](http://arxiv.org/abs/hep-th/0510239))

* {#Tseytlin06} [[Arkady Tseytlin]], _On sigma model RG flow, "central charge" action and Perelman's entropy_, Phys.Rev.D75:064024,2007 ([arXiv:hep-th/0612296](http://arxiv.org/abs/hep-th/0612296))

See also at _[string theory -- References -- Fields medal work related to string theory](string+theory#FieldMedalWork)_.

See also

* Alexander Frenkel, [[Petr Horava]], Stephen Randall, _Topological Quantum Gravity of the Ricci Flow_ ([arXiv:2010.15369](https://arxiv.org/abs/2010.15369))

* Alexander Frenkel, [[Petr Horava]], Stephen Randall, _Perelman's Ricci Flow in Topological Quantum Gravity_ ([arXiv:2011.11914](https://arxiv.org/abs/2011.11914))


Richard A. Toupin

* [www.fdellisola.it/about-richard-toupin](http://www.fdellisola.it/about-richard-toupin)

## Selected writings

On [[classical field theory]]:

* [[Clifford Truesdell]], [[Richard A. Toupin]], *The Classical Field Theories*, in: *Principles of Classical Mechanics and Field Theory*, Encyclopedia of Physics, volume III/1, Springer (1960) 226-858 [[doi:10.1007/978-3-642-45943-6_2](https://doi.org/10.1007/978-3-642-45943-6_2)]

category: people

[[!redirects Richard Toupin]]


Richard Battye

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

## Selected writings

On [[Skyrmions]] via [[rational maps]] from the [[complex plane]], hence [[holomorphic maps]] from the [[Riemann sphere]], to itself:

* [[Richard Battye]], [[Paul Sutcliffe]], *Skyrmions, Fullerenes and Rational Maps*, 	Rev. Math. Phys. 14 (2002) 29-86 ([arXiv:hep-th/0103026](https://arxiv.org/abs/hep-th/0103026))

On [[Skyrmions]] modelling [[atomic nuclei]]:

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001)) in:  [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

category: people

[[!redirects Richard Antony Battye]]
[[!redirects Richard A. Battye]]


Richard Blute

Rick Blute is a mathematician at the University of Ottawa.  His areas of interest include the relations of category theory with quantum computing and mathematical physics.

 His primary research areas are:

  *  Linear logic, most recently differential linear logic
* Categorical Logic and proof theory
* Category theory, especially monoidal categories
*  Applications of category theory to physics

see also his

* [webpage](http://aix1.uottawa.ca/~rblute/)

## Related entries

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

* [[differential category]]

## Selected writings

On [[categorical semantics]] of [[linear type theory]] (such as for the [[exponential modality]]):

* [[Richard Blute]], [[Philip Scott]], *Category theory for linear logicians*, in: *Linear Logic in Computer Science*, Cambridge University Press (2004) 3-64 [[doi:10.1017/CBO9780511550850.002](https://doi.org/10.1017/CBO9780511550850.002), [pdf](https://www.site.uottawa.ca/~phil/papers/catsurv.web.pdf)]

* {#BluteCockettSeely96} [[R. F. Blute]] , [[J. R. B. Cockett]], [[R. A. G. Seely]], *! and ? -- Storage as tensorial strength*, Mathematical Structures in Computer Science **6** 4 (1996) 313-351 [[doi:10.1017/S0960129500001055](https://doi.org/10.1017/S0960129500001055)]

* [[R. Blute]], [[J.R.B. Cockett]], [[T. Porter]], [[R.A.G.Seely]],  _Kähler categories_,  Cahiers Top.
Géom. Diff. cat., 52 (2011) 253 – 268 ([pdf](http://www.math.mcgill.ca/rags/difftl/kahlercahiers.pdf))

* {#BlutePanangadenSeely94} [[Richard Blute]], [[Prakash Panangaden]], [[R. A. G. Seely]], _Fock Space: A Model of Linear Exponential Types_ (1994) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6825), [[BPSLinear.pdf:file]])

Introducing [[differential categories]]:

*  {#BluteCocketSeely06} [[R. F. Blute]], [[J. R. B. Cockett]], and [[R. A. G. Seely]]: _Differential categories_. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) ([doi:10.1017/S0960129506005676](https://doi.org/10.1017/S0960129506005676))

* [[Richard Blute]], [[Robin Cockett]], [[Jean-Simon Lemay]] and [[Robert Seely]], _Differential categories revisited,_ Appl. Categ. Struct. 28, 171-235 (2020). ([arXiv:1806.04804](https://arxiv.org/abs/1806.04804), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y))

category: people

[[!redirects Rick Blute]]
[[!redirects R. Blute]]
[[!redirects R.F.Blute]]
[[!redirects R. F. Blute]]
[[!redirects R Blute]]


Richard Body

send e-mail to richardDOTbodyATshawDOTca

category: people


Richard Borcherds

_Richard Ewen Borcherds_ is a professor of mathematics at University of California at Berkeley. He is a Fields medalist of British origin. Much of his work is inspired by [[mathematical physics]]. He invented vertex algebras and has deep results on [[automorphic form]]s and Kac-Moody Lie algebras. Many examples and constructions in his work were came from the theory of finite groups (especially the monster group which was responsible for the existence of the Monster Moonshine vertex operator algebra).

* Berkeley [webpage](http://math.berkeley.edu/~reb)

* [wikipedia](http://en.wikipedia.org/wiki/Richard_Borcherds)

## Selected writings

On the [[Monster vertex algebra]]:

* {#Borcherds86} [[Richard Borcherds]], _Vertex algebras, Kac-Moody algebras, and the Monster_, PNAS May 1, 1986 83 (10) 3068-3071 ([doi:10.1073/pnas.83.10.3068](https://doi.org/10.1073/pnas.83.10.3068), [pdf](https://pdfs.semanticscholar.org/a3bb/2d2772ccc0c1582f0216f9ab87c1b8a11eb6.pdf))

* {#BorcherdsConwayQueenSloane} [[Richard Borcherds]], [[John Conway]], L. Queen, N. J. A. Sloane, _A Monster Lie Algebra?_, In: _Sphere Packings, Lattices and Groups_, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol 290. Springer 1993 ([doi:10.1007/978-1-4757-2249-9_30](https://doi.org/10.1007/978-1-4757-2249-9_30), [pdf](http://neilsloane.com/doc/Me99.pdf))

On [[Lie groups]] and [[quantum groups]]:

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

[[!redirects R. E. Borcherds]]
[[!redirects Richard Ewen Borcherds]]
[[!redirects Richard E. Borcherds]]
[[!redirects R. Borcherds]]

category: people


Richard Brauer

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

## related entries

* [[Brauer group]]

* [[Brauer induction theorem]]

[[!redirects Brauer]]

category: people


Richard Cartwright

* Wikipedia entry

* [philosophy.mit.edu/cartwright](https://philosophy.mit.edu/cartwright)

## Selected writings

On [[Gottfried Leibniz]]'s  principles of  [[identity of indiscernibles]] and [[indiscernibility of identicals]]:

* {#Cartwright71} [[Richard Cartwright]], *Identity and Substitutivity*, p. 119-133 of: Milton Munitz (ed.) *Identity and Individuation*, New York University Press (1971) [[[Cartwright-IdentityAndSubstitutivity.pdf:file]]]

category: people


Richard Cleve

* [personal page](http://cleve.iqc.uwaterloo.ca/)

* [institute page](https://uwaterloo.ca/institute-for-quantum-computing/profiles/richard-cleve)

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

## Selected writings

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

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

On [[quantum complexity theory]]:

* [[Richard Cleve]], *An Introduction to Quantum Complexity Theory*, p. 103-127 of: *Quantum Computation and Quantum Information Theory*, World Scientific (2001) [[arXiv:quant-ph/9906111](https://arxiv.org/abs/quant-ph/9906111), [doi:10.1142/9789810248185_0004](https://doi.org/10.1142/9789810248185_0004)]

category: people


Richard Crowell

[[!redirects R. H. Crowell]]
Richard (Dick) Crowell was emeritus professor of mathematics at Dartmouth College.  He is best known for his book on [[knot theory]] written with [[Ralph Fox]], but also wrote on the derived module of a homomorphism, related to the [[Fox derivatives]].

* [Obituary](https://paw.princeton.edu/memorials/3/83/index.xml)

##References

* R. H. Crowell and R. H. Fox, 1977, _Introduction to Knot Theory_, number 57 in Graduate Texts, Springer.

* R. H. Crowell, _The derived module of a homomorphism_, Advances
in Math., 5, (1971), 210–238.

category: knot theory

category:people
[[!redirects Crowell]]


Richard Dalitz

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

## Selected writings

Introducing [[Dalitz decays]]:

* {#Dalitz51} [[Richard Dalitz]], _On an alternative decay process for the neutral $\pi$-meson_, Proceedings of the Physical Society. Section A 64 (7), 667, 1951 ([doi:10.1088/0370-1298/64/7/115](https://doi.org/10.1088/0370-1298/64/7/115))

category: people


Richard Dawid

* [home page](https://homepage.univie.ac.at/richard.dawid/)

## Selected writings

On [[constructive empiricism]] in view of contemporary [[particle physics]] and [[string theory]]:

* {#Dawid05} [[Richard Dawid]], _High Energy Physics and Constructive Empiricism_ ([philsci:2243](http://philsci-archive.pitt.edu/2243/))

* {#Dawid07} [[Richard Dawid]], _Scientific realism in the age of string theory_,  Physics and Philosophy (2007) ([philpapers:DAWSRI](https://philpapers.org/rec/DAWSRI))

## Related $n$Lab entries

* [[constructive empiricism]]

* [[philosophy of science]], [[philosophy of physics]]

category: reference


Richard Dedekind

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

* SEP, *[Dedekind’s Contributions to the Foundations of Mathematics](https://plato.stanford.edu/entries/dedekind-foundations/)*

## Selected writings

In which the notion of a [[field]] was first introduced, but only for subfields of the [[complex numbers]]:

* {#DirichletDedekind71} [[Gustav Lejeune Dirichlet]] (1871), [[Richard Dedekind]] (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn ([Google Books](https://books.google.com/books?id=SRJTAAAAcAAJ&pg=PA424))

Introducing the notion of [[Dedekind cuts]] for the definition of the [[real numbers]]:

* {#Dedekind1872} [[Richard Dedekind]], _Stetigkeit und irrationale Zahlen_ (1872) [[Google Books](http://books.google.com/books?id=AO4GAAAAYAAJ), retyped: [web](http://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html)]

* [[Richard Dedekind]] (transl. by W. Beman), _Continuity and irrational numbers_, Chapter I in: *Essays on the Theory of Numbers*, Chicago (1901) [[Project Gutenberg](http://www.gutenberg.org/ebooks/21016), [pdf](https://www.gutenberg.org/files/21016/21016-pdf.pdf)]

On [[axioms]] for the [[natural numbers]] and [[infinite sets]]:

* [[Richard Dedekind]], *Was sind und was sollen die Zahlen?* (1888) [scan: [pdf](https://rcin.org.pl/Content/37485/WA35_52499_2264brosz_Was-sind.pdf), [doi:10.1007/978-3-663-02788-1](https://doi.org/10.1007/978-3-663-02788-1)]

* [[Richard Dedekind]] (transl. by W. Beman), _The nature and meaning of numbers_, Chapter II in: *Essays on the Theory of Numbers*, Chicago (1901) [[Project Gutenberg](http://www.gutenberg.org/ebooks/21016), [pdf](https://www.gutenberg.org/files/21016/21016-pdf.pdf)]

These two essays contain the definitions of [[Dedekind cuts]] and [[infinite sets]] (this definition is now taken to define _Dedekind_ infinite sets) an axiomatization of the [[natural numbers]] adapted by Peano into [[Peano arithmetic]] ([[Dedekind-Peano axioms]]).

category: people


Richard Dien Winfield

* [webpage](http://www.phil.uga.edu/directory/richard-dien-winfield)

## Related entries

* [[Science of Logic]]

category: people

[[!redirects Richard Winfield]]


Richard E. Hodel

* [institute page](https://scholars.duke.edu/person/hodel)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=574)

## Selected writings

On [[set theory|set theoretic]] [[topology]]:

* [[Richard E. Hodel]] (ed.), *Set-Theoretic Topology*, Academic Press (1977) \[doi:10.1016/C2013-0-11355-4\]

## Related entries

* [[Urysohn metrization theorem]]

category: people

[[!redirects Richard Hodel]]


Richard Elman

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

* Wikipedia entry

## Selected writings

On [[quadratic forms]]:

* [[Richard Elman]], [[Nikita Karpenko]], [[Alexander Merkurjev]], *Algebraic and Geometric Theory of Quadratic Forms*, Colloquium Publication **56**, AMS (2008) [[ams:coll-56](https://bookstore.ams.org/coll-56), [pdf](https://www.math.ucla.edu/~rse/old_book/Kniga.pdf)]

category: people


Richard Feynman

* [wikipedia](http://en.wikipedia.org/wiki/Richard_Feynman)

## Selected writings

Introducing the [[path integral]]-formulation of [[quantum mechanics]]:

* {#Feynman42} [[Richard P. Feynman]], *The Principles of Least Action in Quantum Mechanics*, PhD thesis (1942), reprinted in [Brown 2005](#Brown05) [[doi:10.1142/9789812567635_0001](https://doi.org/10.1142/9789812567635_0001)]

* [[Richard Feynman]], *Space-Time Approach to Non-Relativistic Quantum Mechanics*, Rev. Mod. Phys. **20** (1948) 367 [[doi:10.1103/RevModPhys.20.367](https://doi.org/10.1103/RevModPhys.20.367), [pdf](http://www-f1.ijs.si/~ramsak/km1/feynman.pdf), [pdf](https://faculty.washington.edu/seattle/physics541/2012-path-integrals/paper.pdf)]

reprinted with historical commentary in:

* {#Brown05} [[Laurie M. Brown]], *Feynman's Thesis — A New Approach to Quantum Theory*, World Scientific (2005) [[doi:10.1142/5852](https://doi.org/10.1142/5852)]

and the first monograph on the subject:

* [[Richard Feynman]], [[Albert R. Hibbs]], _Quantum Mechanics and Path Integrals_, New York: McGraw-Hill (1965)

* [[Daniel F. Styer]], [[Richard Feynman]], [[Albert R. Hibbs]], *Quantum Mechanics and Path Integrals: Emended Edition*, Dover (2010) [[ISBN:0486477223](https://store.doverpublications.com/0486477223.html), [pdf](http://www-f1.ijs.si/~ramsak/km1/FeynmanHibbs.pdf)]

Early discussion of [[quantum computation]]:

* {#Feynman82} [[Richard Feynman]], *Simulating physics with computers*,  Int J Theor Phys **21** (1982) 467–488 [[doi:10.1007/BF02650179](https://doi.org/10.1007/BF02650179)]

  > "because nature isn't classical, dammit, if you want to make a simulation of nature, you'd better make it quantum mechanical"

and [[quantum gates]]/[[quantum circuits]]:

* [[Richard Feynman]], *Quantum mechanical computers*, Foundations of Physics **16** (1986) 507–531 [[doi:10.1007/BF01886518](https://doi.org/10.1007/BF01886518)]

## Related entries

* [[perturbative quantum field theory]]

* [[Schwinger-Tomonaga-Feynman-Dyson]]

* [[path integral]]

* [[Feynman propagator]]

* [[Feynman diagram]]

* [[Feynman perturbation series]]

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

* [[Feynman slash notation]]

## Selected popular writings
 {#PopularWritings}

* {#Feynman55} [[Richard Feynman]], *The value of science*, Engineering and Science, Volume XIX, Dec. 1955 ([pdf](https://calteches.library.caltech.edu/40/2/Science.pdf)), reproduced as [Chapter 6](http://www.sitpor.org/wp-content/uploads/2015/03/Helix-Books-Richard-P.-Feynman-The-Pleasure-of-Finding-Things-Out_-The-Best-Short-Works-of-Richard-Feynman-Perseus-Publishing-Company-1999.pdf#page=161) in [Feynman 1999](#Feynman99)

* {#Feynman64} [[Richard Feynman]], *What Is and What Should Be the Role of Scientific Culture in Modern Society*, Galileo Symposium 1964, Italy, recorded as [Chapter 4](http://www.sitpor.org/wp-content/uploads/2015/03/Helix-Books-Richard-P.-Feynman-The-Pleasure-of-Finding-Things-Out_-The-Best-Short-Works-of-Richard-Feynman-Perseus-Publishing-Company-1999.pdf#page=117) in [Feynman 1999](#Feynman99)

* {#Feynman66} [[Richard Feynman]], *What is Science?*, speech at *15th annual meeting of the National Science Teachers Association*, New York City (1966), reprinted in *The Physics Teacher* **7** 6 (1969) 313-320 [[doi:10.1119/1.2351388](https://doi.org/10.1119/1.2351388), [webpage](http://www.feynman.com/science/what-is-science)]

* {#Feynman88} [[Richard Feynman]], _What Do You Care What Other People Think?_, W.W. Norton, 1988 ([Wikipedia entry](https://en.wikipedia.org/wiki/What_Do_You_Care_What_Other_People_Think%3F))

* {#Feynman99} [[Richard Feynman]], edited by J. Robbins, *The Pleasure of Finding Things Out -- The Best Short Works of Richard P. Feynman*, Perseus Books 1999 ([pdf](http://www.sitpor.org/wp-content/uploads/2015/03/Helix-Books-Richard-P.-Feynman-The-Pleasure-of-Finding-Things-Out_-The-Best-Short-Works-of-Richard-Feynman-Perseus-Publishing-Company-1999.pdf))

## Quotes

### On Science and Doubt
 {#OnDoubt}

From [Feynman 1955](#Feynman55):

> **Of all its** $[$science's$]$ **many values, the greatest  must be the freedom to doubt**

> We have found it of paramount importance that in order to progress we must recognize the ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degree of certainty - some most unsure, some nearly sure, none *absolutely* certain.

> Now, we scientists are used to this, and we take it for granted that it is perfectly consistent to be unsure -- that it is possible to live and *not* know. But I don't know whether everyone realizes that this is true.

> Our freedom to doubt was born of a struggle against authority in the early days of science. It was a very deep and strong struggle. Permit us to question -- to doubt, that's all -- not to be sure.

> And I think it is important that we do not forget the importance of this struggle and thus perhaps lose what we have gained. Here lies a responsibility to society.

> This is not a new idea; this is the idea of the age of reason. $[\ldots]$ that we should arrange a system by which new ideas could be developed, tried out, tossed out, more new ideas brought in; a trial and error system. $[\ldots]$ the openness of the possibilities was an opportunity, and that doubt and discussion were essential to progress into the unknown.

> if we suppress all discussion, all criticism, saying "This is it, boys, man is saved!" $[$ we would $]$ doom man for a long time to the chains of authority, confined to the limits of our present imagination. It has been done so many times before.

> It is our responsibility as scientists, knowing $[\ldots]$ the great progress that is the fruit of freedom of thought, to proclaim the value of this freedom, to teach how doubt is not to be feared but welcomed and discussed, and to demand this freedom as our duty to all incoming generations.

From [Feynman 1964](#Feynman64) (from [this page](http://www.sitpor.org/wp-content/uploads/2015/03/Helix-Books-Richard-P.-Feynman-The-Pleasure-of-Finding-Things-Out_-The-Best-Short-Works-of-Richard-Feynman-Perseus-Publishing-Company-1999.pdf#page=131) and the [next](http://www.sitpor.org/wp-content/uploads/2015/03/Helix-Books-Richard-P.-Feynman-The-Pleasure-of-Finding-Things-Out_-The-Best-Short-Works-of-Richard-Feynman-Perseus-Publishing-Company-1999.pdf#page=132)):

> Now the next subject $[\ldots]$ I really consider the most important and the most serious. And that has to do with the question of uncertainty and doubt.

> We absolutely must leave room for doubt or there is no progress and there is no learning. There is no learning without having to pose a question. And a question requires doubt. People search for certainty. But there is no certainty.

From [Feynman 1966](#Feynman66):

> Science is the belief in the ignorance of experts.

> When someone says, "Science teaches such and such," he is using the word incorrectly. Science doesn’t teach anything; experience teaches it. If they say to you, "Science has shown such and such," you might ask, "How does science show it? How did the scientists find out? How? What? Where?"

> It should not be "science has shown" but "this experiment, this effect, has shown." And you have as much right as anyone else, upon hearing about the experiments -- but be patient and listen to all the evidence -- to judge whether a sensible conclusion has been arrived at.

From [Feynman 1988](#Feynman88) ([here](https://books.google.de/books?id=vbMIlkpQXEkC&lpg=PT30&vq=Have%20no%20respect&pg=PT30#v=onepage&q=%22Have%20no%20respect%20whatsoever%20for%20authority%22&f=false)):

> Have no respect _whatsoever_ for authority; forget who said it and instead look at what he starts with, where he ends up, and ask yourself, "Is it reasonable?"

cf. [[Carl Sagan]] [on scepticism](Carl+Sagan#OnScepticism)

category: people

[[!redirects Feynman]]

[[!redirects Richard P. Feynman]]
[[!redirects R. P. Feynman]]


Richard G. Milner

* [personal page](https://physics.mit.edu/faculty/richard-milner/)

* [inspire page](https://inspirehep.net/authors/997400)

## Selected writings

On the history of the concept of intrinsic [[spin]] of [[fundamental particles]] (such as [[electrons]]):

* [[Richard G. Milner]], *A Short History of Spin*, [Proceedings of PSTP2013](https://pos.sissa.it/182/PSTP2013), POS **182** (2014) [[arXiv:1311.5016](https://arxiv.org/abs/1311.5016), [doi:10.22323/1.182.0003 ](https://doi.org/10.22323/1.182.0003 )]

category: people

[[!redirects Richard Milner]]


Richard Garner

* [website](http://comp.mq.edu.au/~rgarner/)

## Selected writings

On [[weak factorization systems]] such as for [[cofibrantly generated model categories]]:

* R. Garner, _Cofibrantly generated natural weak factorisation systems_, [arXiv:math.CT/0702290](http://arXiv.org/abs/math.CT/0702290).

On the [[small object argument]]:

* [[Richard Garner]], _Understanding the small object argument_, Applied Cat. Structures, [arXiv:math.CT/0712.0724](http://arXiv.org/abs/0712.0724).

On [[infinity-groupoid]]/[[omega-groupoid]]-[[structure]] on [[types]] in [[homotopy type theory]]:

* [[Benno van den Berg]], [[Richard Garner]], *Types are weak omega-groupoids*, Proceedings of the London Mathematical Society **102** 2 (2011) 370-394 [[arXiv:0812.0298](https://arxiv.org/abs/0812.0298), [doi:10.1112/plms/pdq026](https://doi.org/10.1112/plms/pdq026)]

On [[dependent product types]] and [[function extensionality]]:

* [[Richard Garner]], *On the strength of dependent products in the type theory of Martin-Löf*, Annals of Pure and Applied Logic **160** 1 (2009) 1-12 [[arXiv:0803.4466](https://arxiv.org/abs/0803.4466), [doi:10.1016/j.apal.2008.12.003](https://doi.org/10.1016/j.apal.2008.12.003)]

On [[2-type theory]]

* {#Garner09} [[Richard Garner]], *Two-dimensional models of type theory*, Mathematical structures in computer science 19.4 (2009): 687-736 ([doi:10.1017/S0960129509007646](https://doi.org/10.1017/S0960129509007646),[pdf](https://www.irif.fr/~mellies/mpri/mpri-ens/articles/garner-two-dimensional-models-of-type-theory.pdf))

On [[ionads]]:

*  [[Richard Garner]], _Ionads_,  J. Pure Appl. Algebra 216 (2012), no. 8-9, 1734–1747. ([arXiv](http://arxiv.org/abs/0912.1415)) ([doi](http://dx.doi.org/10.1016/j.jpaa.2012.02.013)) ([author-archived version of published copy](http://comp.mq.edu.au/~rgarner/Papers/Ionads.pdf))

On [[adhesive categories]]:

* {#GarnerLack12} [[Richard Garner]], [[Steve Lack]]: *On the axioms for adhesive and quasiadhesive categories*, Theory and Applications of Categories, **27**  3 (2012) 27-46 [[arXiv:1108.2934](http://arxiv.org/abs/1108.2934), [tac:27-03](http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html)]

On the [[categorical semantics]] of [[dependent type theory]] with [[function types]] in [[locally cartesian closed categories]] (see at *[[relation between category theory and type theory]]*):

* {#CurienGarnerHofmann} [[Pierre-Louis Curien]], [[Richard Garner]], [[Martin Hofmann]], *Revisiting the categorical interpretation of dependent type theory*, Theoretical Computer Science **546** 21 (2014) 99-119 [[doi:10.1016/j.tcs.2014.03.003](https://doi.org/10.1016/j.tcs.2014.03.003), [[CurienGarnerHofmann.pdf:file]]]

On [[transferred model structures]]:

* {#GKR18} [[Richard Garner]], [[Magdalena Kedziorek]], [[Emily Riehl]], *Lifting accessible model structures*, J. Topology **13** 1 (2020) 59-76 [[arXiv:1802.09889](https://arxiv.org/abs/1802.09889), [doi:10.1112/topo.12123](https://doi.org/10.1112/topo.12123)]

category: people


Richard Hain

* [webpage](http://fds.duke.edu/db/aas/math/hain)

## Selected writings

On the [[full subcategory]]-inclusion of [[Banach manifolds]] into the [[category]] of [[diffeological spaces]]

* {#Hain} [[Richard Hain]], _A characterization of smooth functions defined on a Banach space_,  Proc. Amer. Math. Soc. 77 (1979), 63-67 ([web](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html), [pdf](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf))

On [[regulators]] and [[characteristic classes]] of [[flat connections]]:

* [[Johan Dupont]], [[Richard Hain]], Steven Zucker, _Regulators and characteristic classes of flat bundles_ ([arXiv:alg-geom/9202023](http://arxiv.org/abs/alg-geom/9202023))

On [[iterated integrals]] and [[algebraic cycles]]:

* [[Richard Hain]], _Iterated Integrals and Algebraic Cycles: Examples and Prospects_ ([arXiv:math/0109204](http://arxiv.org/abs/math/0109204))

On the [[moduli space of elliptic curves]]:

* {#Hain08} [[Richard Hain]], _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

On [[Kuo Tsai Chen]]:

* [[Richard Hain]], [[Philippe Tondeur]], _The life and work of Kuo-Tsai Chen_,  Illinois J. Math. Volume 34, Issue 2 (1990), 175-190. ([euclid:ijm/1255988263](http://projecteuclid.org/euclid.ijm/1255988263))

category: people


Richard Hamilton

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

## Selected writings

On the [[implicit function theorem]] in the generality of [[Fréchet manifolds]]:

* [[Richard S. Hamilton]], *The inverse function theorem of Nash and Moser*, Bull. Amer. Math. Soc. **7** (1982) 65-222 [[doi:1982-07-01/S0273-0979-1982-15004-2](https://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2)]

category: people

[[!redirects Richard S. Hamilton]]


Richard Healey

* [webpage I](http://www.u.arizona.edu/~rhealey/)

* [webpage II](https://philosophy.arizona.edu/user/richard-healey)

[[philosophy of physics]]

## related $n$Lab entries

* [[theta vacuum]]

category: people


Richard Hepworth

* [website](http://www.math.ku.dk/~hepworth/)

category: people


Richard J. Wood

Richard Wood is a mathematician specializing in category theory and lattice theory. He is currently a professor at Dalhousie University.

* [Faculty page at Dalhousie](https://www.mscs.dal.ca/Faculty/rjwood.html)

## Selected writings

Introducing the notion of [[locally graded categories]] (under the name *large $V$-categories*):

* [[Richard Wood]], *Indicial methods for relative categories*, PhD thesis (1976) [[hdl:10222/55465](http://hdl.handle.net/10222/55465)]

* [[Richard Wood]],  *$V$-indexed categories*, in *Indexed Categories and Their Applications*, Lecture Notes in Mathematics **661** (1978) 126-140 [[doi:10.1007/BFb0061362](https://link.springer.com/chapter/10.1007/BFb0061362)]

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

On [[total categories]]:

* {#Wood82} [[Richard J. Wood]], *Some remarks on total categories*, J. Algebra **75** 2 (1982) 538-545 [doi:10.1016/0021-8693(82)90055-2]

Introducing the notion of [[2-categories equipped with proarrows]] ("[[proarrow equipments]]"):

* {#Wood82} [[Richard J. Wood]], *Abstract Proarrows I*, Cahiers de topologie et géométrie différentielle **23** 3 (1982)  279-290 [[numdam:CTGDC_1982__23_3_279_0](http://www.numdam.org/item/CTGDC_1982__23_3_279_0)]

* {#Wood85} [[Richard J. Wood]], *Proarrows II*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **26** 2 (1985) 135-168 [[numdam:CTGDC_1985__26_2_135_0](http://www.numdam.org/item/CTGDC_1985__26_2_135_0)]

On [[completely distributive lattices]]:

* {#WoodFawcett} [[Barry Fawcett]], [[Richard J. Wood]], *Constructive complete distributivity I* , Math. Proc. Camb. Phil. Soc. **107** (1990) 81-89 [[doi:10.1017/S0305004100068377](https://doi.org/10.1017/S0305004100068377)]

Characterizing the [[category of sets]] as that whose [[Yoneda embedding]] extends to the [[left adjoint|left]] to an [[adjoint quintuple]] of [[adjoint functors]]:

* [[Robert Rosebrugh]], [[Richard J. Wood]], *An adjoint characterization of the category of sets*. PAMS **122** 2 (1994) 409-413 [[jstor:2161031](https://www.jstor.org/stable/2161031)]

On [[distributive laws]] and [[factorization systems]]:

* {#RosebrughWood02} [[Robert Rosebrugh]], [[Richard J. Wood]], *Distributive Laws and Factorization*, Journal of Pure and Applied Algebra **175** 1–3 (2002) 327-353 [doi:10.1016/S0022-4049(02)00140-8[pdf](http://www.mta.ca/~rrosebru/articles/fac.pdf)]

On [[Cartesian bicategories]]:

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac:19-06](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))

On [[monads]] as [[extension systems]] (as in [[monads in computer science]]):

* {#MarmolejoWood10} [[Francisco Marmolejo]], [[Richard J. Wood]], *Monads as extension systems -- no iteration is necessary* [[TAC]] **24** 4 (2010) 84-113 [[24-04](http://www.tac.mta.ca/tac/volumes/24/4/24-04abs.html)]

and generalizing the perspective of [[extension systems]] to [[pseudomonads]]:

* [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]

* [[Francisco Marmolejo]], [[Richard J. Wood]], *No-iteration pseudomonads*, [[TAC]] **28** 14 (2013) 371-402 [[tac:28-14](http://www.tac.mta.ca/tac/volumes/28/14/28-14abs.html)]

On [[lenses (in computer science)]]:

* {#JohnsonRosebrughWood10} [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard Wood]], _Algebras and Update Strategies_, Journal of Universal Computer Science **16** (2010) [[doi:10.3217/jucs-016-05-0729](http://dx.doi.org/10.3217/jucs-016-05-0729)]

* [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard J. Wood]], *Lenses, fibrations and universal translations* Math. Structures Comput. Sci. **22** 1 (2012) 25–42 [[doi:10.1017/S0960129511000442](https://doi.org/10.1017/S0960129511000442)]

[[!redirects Richard Wood]]

[[!redirects R. J. Wood]]


Richard Jozsa

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

* [institute page](https://www.damtp.cam.ac.uk/person/rj310)

## Selected writings

On [[algorithms]] for which [[quantum computation]] is expected to exhibit [[quantum supremacy]] (specifically what is now called the *[[Deutsch-Jozsa algorithm]]*):

* [[David Deutsch]], [[Richard Jozsa]], *Rapid solution of problems by quantum computation*. Proc. R. Soc. Lond. A **439** 1907 (1992) 553-558 [[pdf](https://www.isical.ac.in/~rcbose/internship/lectures2016/rt08deutschjozsa.pdf), [doi:10.1098/rspa.1992.0167](https://doi.org/10.1098/rspa.1992.0167)]

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

category: people


Richard Kenyon

* [Institute page](https://math.yale.edu/people/richard-kenyon)

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

## Selected writings

On [[dimers]] in [[quantum many-body physics]]:

* [[Richard Kenyon]], [[Andrei Okounkov]], *What is... a dimer?*, Notices of the AMS, Vol. 52 No 3 (2005) ([pdf](https://www.ams.org/notices/200503/what-is.pdf))

* [[Richard Kenyon]], *An introduction to the dimer model* ([arXiv:math/0310326](https://arxiv.org/abs/math/0310326))

category: people


Richard Lashof

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

## Selected writings

On [[Bott periodicity]]:

* [[Eldon Dyer]], [[Richard Lashof]], *A topological Proof of the Bott Periodicity Theorems*, Annali di Matematica Pura ed Applicata **54** (1961) 231–254 [[doi:10.1007/BF02415354](https://doi.org/10.1007/BF02415354)]

On [[Poincare duality]], [[cobordism theory]] and formulating the modern form of the [[Pontryagin-Thom isomorphism]]:

* {#Lashof63} [[Richard Lashof]], _Poincaré duality and cobordism_, Trans. AMS 109 (1963), 257-277 ([doi:10.1090/S0002-9947-1963-0156357-4](https://doi.org/10.1090/S0002-9947-1963-0156357-4))

On the [[principal bundles]], [[spherical fibrations]], [[Hopf construction]], ...:

* {#DoldLashof59} [[Albrecht Dold]], [[Richard Lashof]], _Principal quasifibrations and fibre homotopy equivalence of bundles_, Illinois J. Math. Volume 3, Issue 2 (1959), 285-305 ([euclid:1255455128](https://projecteuclid.org/euclid.ijm/1255455128))

On [[homology of iterated loop spaces]]:

* [[Eldon Dyer]], [[Richard Lashof]], _Homology of Iterated Loop Spaces_,  American Journal of Mathematics Vol. 84, No. 1 (Jan., 1962), pp. 35-88 ([jstor:2372804 ](https://www.jstor.org/stable/2372804))

On [[equivariant Tietze extension theorems]]:

* [[Richard Lashof]], _The Equivariant Extension Theorem_, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 138-140 ([jstor:2043909](https://www.jstor.org/stable/2043909))

On [[microbundles]]:

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles I. Elementary Theory_, Invent. Math., 1, (1966), 1 – 17.

* [[N. H. Kuiper]] and [[Richard Lashof|R. K. Lashof]], _Microbundles and Bundles II. Semisimplicial Theory_, Invent. Math., 1, (1966), 243 – 259.

On [[equivariant bundles]] (equivariant [[fiber bundles]]):

* {#LashofRothenberg78} [[Richard Lashof]], [[Melvin Rothenberg]], Section 1 of: _$G$-Smoothing theory_, p. 211-266 in: _Algebraic and Geometric Topology_, Proceedings of Symposia in Pure Mathematics, Vol. XXXII, Part 1, American Mathematical Society 1978 ([doi:10.1090/pspum/032.1](https://doi.org/10.1090/pspum/032.1), [[LashofRothenbergGSmoothingTheory.pdf:file]])

* [[Richard Lashof]], _Equivariant bundles_, Illinois J. Math. 26(2): 257-271 ([doi:10.1215/ijm/1256046796.full](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-26/issue-2/Equivariant-bundles/10.1215/ijm/1256046796.full))

* [[Richard Lashof]], [[Peter May]], _Generalized equivariant bundles_, Bull. Soc. Math. Belg. Sér. A 38, 265-271, 1986 ([pdf](http://math.uchicago.edu/~may/PAPERS/55.pdf), [[LashofMayGeneralizedEquivariantBundles.pdf:file]])

On [[equivariant bundles]] with [[abelian group|abelian]] [[structure group]]:

* {#LashofMaySegal83} [[Richard Lashof]], [[Peter May]], [[Graeme Segal]], _Equivariant bundles with abelian structural group_, Contemporary Mathematics,  Volume 19, 1983 ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019), [pdf](http://www.math.uchicago.edu/~may/PAPERS/45.pdf), [[LashofMaySegalEquivariantBundles.pdf:file]])

category: people


Richard Matthews

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

## Selected writings

On [[constructible universes]]:

* [[Richard Matthews]], [[Michael Rathjen]], _Constructing the Constructible Universe Constructively_ [[arxiv:2206.08283](https://arxiv.org/abs/2206.08283)]

category: people


Richard Melrose

* [webpage](http://www-math.mit.edu/~rbm/)

## related $n$Lab entries

* [[Dirac operator on smooth loop space]]

* [[microlocal analysis]]

category: people


Richard Montague

**Richard Montague** (1930-1971) was an American logician. A student of [[Alfred Tarski]], Montague made important contributions to [[set theory]] and [[model theory]].

#### Contributions to natural language semantics

He is best known for his groundbreaking work in [[linguistics]]: his introduction of [[type theory]] and [[generalized quantifier|generalized quantifiers]] achieved for _natural language semantics_ the same what [[Gottlob Frege]] had achieved almost a century earlier for the quantification theory of formal languages.

In fact, whereas Frege saw the surface position of quantifiers as a logical defect of natural languages which had to be amended for his _Begriffsschrift_, the powerful techniques of [[lambda calculus]] employed by Montague permitted to interpret quantifiers in situ as well, leading to a theory of linguistic interpretation, called [[Montague grammar]], of a mathematical precision and elegance.

By restricting the analysis to language fragments, Montague overcome obstacles like the semantic paradoxes that prevented his teacher Tarski from applying model-theoretic methods to natural languages.

## Related pages

* Wikipedia, [Richard Montague](https://en.wikipedia.org/wiki/Richard_Montague)

* B. Partee, _Richard Montague (1930-1971)_ , pp.255-257 in Brown (ed.) _Encyclopedia of Language and Linguistics_ , Elsevier Oxford 2006². ([preprint](http://people.umass.edu/partee/docs/Richard_Montague_by_Partee_05.pdf))

## Selected writings

On [[Montague semantics]] and [[compositionality]]:

* {#Montague70a} [[Richard Montague]], _English as a formal language_, Linguaggi nella societa e nella tecnica. Edizioni di Communita. pp. 188-221 (1970) ([[MontagueEnglishFormalLanguage.pdf:file]])

* {#Montague70b} [[Richard Montague]], _Universal Grammar_, Theoria 36 (3):373--398 (1970) ([doi:10.1111/j.1755-2567.1970.tb00434.x](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1755-2567.1970.tb00434.x))

* {#Montague73} [[Richard Montague]], _The Proper Treatment of Quantification in Ordinary English_, Approaches to Natural Language (1973) ([[MontagueQuantification.pdf:file]])

* R. H. Thomason (ed.), _Formal Philosophy: Selected Papers of Richard Montague_ , Yale UP New Haven 1974.

## Related entries

* [[Montague semantics]]

* [[generalized quantifier]]

* [[Alfred Tarski]]

* [[Solomon Feferman]]

* [[reflection principle]]

* [[compositionality]]

category:people

[[!redirects R. Montague]]


Richard Palais

* [webpage](http://vmm.math.uci.edu/)

## Selected writings

On [[topological G-spaces]], [[classifying spaces]], the [[equivariant Tietze extension theorem]], ...:

* {#Palais60} [[Richard Palais]], *The classification of $G$-spaces*, Memoirs of the AMS **36**, 1960  ([ISBN:978-0-8218-9979-3](https://bookstore.ams.org/memo-1-36) [pdf](http://vmm.math.uci.edu/PalaisPapers/ClassificationOfG-Spaces.pdf), [[Palais_ClassificationOfGSpaces.pdf:file]])

On the [[slice theorem]]:

* {#Palais61} [[Richard Palais]], _Slices and equivariant embeddings_, chapter VIII in: [[Armand Borel]] (ed.), _Seminar on Transformation Groups_, Annals of Mathematics Studies 46, Princeton University Press 1960 ([jstor:j.ctt1bd6jxd](https://www.jstor.org/stable/j.ctt1bd6jxd))

* [[Richard Palais]], _On the Existence of Slices for Actions of Non-Compact Lie Groups_, Annals of Mathematics
Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 ([jstor:1970335](https://www.jstor.org/stable/1970335), [doi:10.2307/1970335](https://doi.org/10.2307/1970335), [pdf](http://vmm.math.uci.edu/ExistenceOfSlices.pdf))

On [[homotopy theory]] of [[infinite-dimensional manifolds]], such as the [[absolute neighbourhood retract]]-property of [[paracompact topological space|paracompact]] [[Banach manifolds]]:

* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

category: people

[[!redirects Richard S. Palais]]


Richard Pink

* [webpage](http://www.math.ethz.ch/~pink/)

category: people


Richard S. Varga

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

## Selected writings

On the [[Gershgorin circle theorem]]:

* [[Richard S. Varga]], *Gershgorin and His Circles*, Springer-Verlag, Berlin, 2004 ([doi:10.1007/978-3-642-17798-9](https://link.springer.com/book/10.1007/978-3-642-17798-9))

category: people

[[!redirects Richard Varga]]


Richard Schoen

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

## related $n$Lab entries

* [[positive energy theorem]]

category: people


Richard Sharpe

* [webpage](https://www2.warwick.ac.uk/fac/sci/maths/people/staff/richard_sharp/)

category: people


Richard Stanley

__Richard P. Stanley__ is a mathematician specializing in [[combinatorics]].

* [personal page](http://www-math.mit.edu/~rstan/), [publications](http://www-math.mit.edu/~rstan/pubs/)

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

## Selected writings

On plane [[partitions]], [[symmetric functions]], [[Schur functions]]:

* [[Richard Stanley]], *Theory and application of plane partitions 1*, Studies in Applied Math. **50** 2 (1971), 167-188 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-1.pdf), [[StanleyPlanePartitions1.pdf:file]])

and introducing the [[hook-content formula]]:

* [[Richard Stanley]], *Theory and application of plane partitions 2*, Studies in Applied Math. **50** 3 (1971), 259-279 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf), [[StanleyPlanePartitions2.pdf:file]])

On [[enumerative combinatorics|enumerative]] [[combinatorics]]:

* {#Stanley11} [[Richard Stanley]], *[Enumerative combinatorics](http://www-math.mit.edu/~rstan/ec/)* -- [Volume 1](http://www-math.mit.edu/~rstan/ec/ec1/), Wadsworth & Brooks/Cole Mathematics Series book series 1986, 1997 ([doi:10.1007/978-1-4615-9763-6](https://link.springer.com/book/10.1007/978-1-4615-9763-6)) reprinted in: Cambridge Studies in Advanced Mathematics, Cambridge University Press 2011 ([ISBN:9781107602625](https://www.cambridge.org/us/academic/subjects/mathematics/discrete-mathematics-information-theory-and-coding/enumerative-combinatorics-volume-1-2nd-edition?format=PB&isbn=9781107602625), [pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf))

specifically on [[symmetric functions]]:

* {#Stanley99} [[Richard Stanley]], *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

## Related $n$Lab entries

* [[hook content formula]]

* [[Catalan number]]

* [[incidence algebra]]

* [[order polynomial]]

* [[zeta polynomial]]

* [[Gian-Carlo Rota]]

category: people

[[!redirects Richard P. Stanley]]
[[!redirects R. P. Stanley]]


Richard Statman

* [webpage](http://www.math.cmu.edu/math/faculty/statman.html)

## Related entries

* [[Structural Complexity Of Proofs]]
* [[lambda calculus]]


Richard Steiner

* [web page](http://www.maths.gla.ac.uk/~rjs/)

category: people
[[!redirects R. Steiner]]


Richard Stong

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=30952)

## Selected wrigings

On [[modular tensor categories]]:

* [[Eric Rowell]], [[Richard Stong]], [[Zhenghan Wang]], *On classification of modular tensor categories*, Comm. Math. Phys. 292 (2009) no. 2, 343--389 ([arXiv:0712.1377](https://arxiv.org/abs/0712.1377))

category: people


Richard Swan

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

## Selected writings

On [[group actions on spheres]]:

* [[Richard G. Swan]], *Periodic Resolutions for Finite Groups*, Annals of Mathematics Second Series **72** 2 (1960) 267-291 [[doi:10.2307/1970135](https://doi.org/10.2307/1970135)]

## Related entries

* [[Serre-Swan theorem]]

* [[algebraic K-theory]]

category: people

[[!redirects Richard G. Swan]]


Richard Szabo

* [website](http://www.ma.hw.ac.uk/maths/People/Frontpages/richard.html)

## Selected writings

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

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

Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number  ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466), [doi:10.4310/ATMP.2019.v23.n8.a5](https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a5))

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_ ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

On [[Donaldson-Thomas theory]] and [[Hilbert schemes]]:

* Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], _Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory_, Nucl. Phys. B809: 452-518, 2009 ([arXiv:0803.4188](https://arxiv.org/abs/0803.4188))

On the [[first-order formulation of gravity]] via [[L-infinity algebras|$L_\infty$-algebras]]:

* Marija Dimitrijević Ćirić, [[Grigorios Giotopoulos]], Voja Radovanović, [[Richard J. Szabo]], *$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity*, J. Math. Phys., **61** 112502 (2020) [[arXiv:2003.06173](https://arxiv.org/abs/2003.06173), [doi:10.1063/5.0011344](https://doi.org/10.1063/5.0011344)]

On the [[type II geometry]] ([[doubled geometry]]) of [[para-Hermitian manifolds]]:

* Vincenzo Emilio Marotta, [[Richard J. Szabo]], *Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality*,  Reviews in Mathematical Physics **33** 09 (2021)  2150031 [[arXiv:1910.09997](https://arxiv.org/abs/1910.09997), [doi:10.1142/S0129055X21500318](https://doi.org/10.1142/S0129055X21500318)]

[[!redirects Richard J. Szabo]]

category: people


Richard Taylor

* Wikipedia

On the [[modularity theorem]]:

* {#TaylorWiles95} [[Richard Taylor]], [[Andrew Wiles]], *Ring-Theoretic Properties of Certain Hecke Algebras*, Annals of Mathematics Second Series **141** 3 (1995) 553-572 $[$[doi:10.2307/2118560 ](https://doi.org/10.2307/2118560 )$]$

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

## Related entries

* [[modularity theorem]]

* [[Galois representation]]

category: people


Richard Thomas

* [webpage](http://www.imperial.ac.uk/people/richard.thomas)

## Selected writings

On [[braid cobordism]] and [[triangulated categories]]:

* {#KhovanovThomas07} [[Mikhail Khovanov]], [[Richard Thomas]], *Braid cobordisms, triangulated categories, and flag varieties*, Homology, Homotopy and Applications **9**(2007) 19-94 [[arXiv:math/0609335](https://arxiv.org/abs/math/0609335), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-9/issue-2/Braid-cobordisms-triangulated-categories-and-flag-varieties/hha/1201127331.full)]

category: people

[[!redirects R. P. Thomas]]


Richard Tolman

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

category: people


Richard V. Kadison

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

## Selected writings

Introducing the [[Kadison-Singer problem]]:

* [[Richard V. Kadison]], [[Isadore Singer]], _Extensions of pure states_, American Journal of Mathematics, 81(2):383–400, 1959 ([jstor:2372748](https://www.jstor.org/stable/2372748))

On [[operator algebra]]:

* [[Richard V. Kadison]], [[John R. Ringrose]], _Fundamentals of the theory of operator algebras_,

  * Vol I *Elementary Theory*, Graduate Studies in Mathematics **15**, AMS 1997 ([ISBN:978-0-8218-0819-1](https://bookstore.ams.org/gsm-15), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0888.46039&format=complete))

  * Vol II *Advanced Theory*, Graduate Studies in Mathematics **16**, AMS 1997 ([ISBN:978-0-8218-0820-7](https://bookstore.ams.org/gsm-16))

  * Vol III, *Special Topics, Elementary Theory -- An Exercise Approach*, Springer 1991 ([doi:10.1007/978-1-4612-3212-4](https://link.springer.com/book/10.1007/978-1-4612-3212-4), )

category: people

[[!redirects Richard Kadison]]


Richard Williamson

* [[richardwilliamson:HomePage|nlab sub-wiki]]

* [website](http://rwilliamson-mathematics.info/)

* [thesis](http://rwilliamson-mathematics.info/cylindrical_model_structures/cylindrical_model_structures.html)
## Related $n$Lab entries

* [[cylinder functor]]

* [[homotopy group (of an ∞-stack)]]

* [[motivic cohomology]]

* [[A1-homotopy theory]]

category: people


Richard Woodard

* [webpage](http://www.phys.ufl.edu/faculty/woodard.shtml)

## related $n$Lab entries

* [[quantum gravity]]

* [[de Sitter spacetime]]

category: people


Richman premetric space

> This entry is about the family of binary relations indexed by the non-negative rational numbers defined by Fred Richman in [Real numbers and other completions](http://math.fau.edu/richman/docs/RealNums.pdf). For the family of binary relations indexed by the positive rational numbers defined by Auke Booij in [Analysis in univalent type theory](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), see [[Booij premetric space]].

***

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## Idea ##

A more general concept of [[metric space]] by Fred Richman. While Fred Richman simply called these structures "[[premetric spaces]]", there are multiple notions of premetric spaces in the mathematical literature, so we shall refer to these as Richman premetric spaces.

## Definition ##

A __Richman premetric space__ is a [[set]] $S$ with a ternary [[relation]] $(-)\sim_{(-)}(-)\colon S \times \mathbb{Q}_{\geq 0} \times S \to \Omega$, where $\mathbb{Q}_{\geq 0}$ represent the non-negative [[rational numbers]] in $\mathbb{Q}$ and $\Omega$ is the set of [[truth values]], such that

* for all $x \in S$ and $y \in S$, $(x = y) \iff (x \sim_0 y)$

* for all $x \in S$ and $y \in S$, there exists $q \in \mathbb{Q}_{\geq 0}$ such that $x \sim_q y$

* for all $x \in S$, $y \in S$, $q \in \mathbb{Q}_{\geq 0}$, and $r \in (q, \infty)$, where $(q, \infty)$ is the set of all non-negative rational numbers strictly greater than $q$, then $(x \sim_r y) \iff (x \sim_q y)$

* for all $x \in S$, $y \in S$, $z \in S$, $q \in \mathbb{Q}_{\geq 0}$, and $r \in \mathbb{Q}_{\geq 0}$, if $x \sim_q y$ and $y \sim_r z$, then $x \sim_{q + r} z$.

## Properties ##

Assuming [[excluded middle]], every Richman premetric space is a [[metric space]]. Without excluded middle, however, every Richman premetric space is a "metric space" which is valued in the lower Dedekind real numbers, rather than the two-sided Dedekind real numbers.

## See also ##

* [[premetric space]]

* [[Booij premetric space]]

* [[metric space]]

## References ##

* [[Fred Richman]], *Real numbers and other completions*, Mathematical Logic Quarterly **54** 1 (2008) 98-108 [[doi:10.1002/malq.200710024](https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024)]

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[[!redirects Richman premetrics]]
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[[!redirects Richman premetric spaces]]


Rick Miranda

* [webpage](http://www.math.colostate.edu/~miranda/)

## related $n$Lab entries

* [[elliptic curve]], [[elliptic fibration]]

category: people


Riemann curvature

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## Definition

For $(X,g)$ a [[Riemannian manifold]] let $\nabla_{g}$ be the corresponding [[Levi-Civita connection]]. The **Riemann curvature** $R_g$ of $(X,g)$ is the [[curvature]] $F_{\nabla_g}$ of $\nabla_g$:

$$
  R_g := F_{\nabla_g}
  \,.
$$

## Properties

(...)

## Related concepts

[[!include curvature in Riemannian geometry -- table]]

* [[torsion of a metric connection]]

* [[Weyl tensor]]

* [[Poincare Lie algebra]]

* [[Ricci tensor]]

* [[gravity]]

* [[higher order curvature corrections]]

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Riemann existence theorem

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## Statement

For a nonsingular [[algebraic variety]] $X$ over the [[complex numbers]], the [[functor]] $Y \mapsto Y^{an}$ which sends complex  [[algebraic varieties]] to their [[complex analytic topology]] establishes an [[equivalence of categories]] between the corresponding [[étale sites]] $X_{et} \simeq X^{an}_{et}$.

## Related concepts

* [[comparison theorem (étale cohomology)]]

## References

* [[James Milne]], section 21 of _[[Lectures on Étale Cohomology]]_

* _Riemann's existence theorem_ ([pdf](http://www.math.uci.edu/~mfried/booklist-ret/chpret4-firsthalf.pdf))

A related result in model theory:

* Rahim Moosa ,_A nonstandard Riemann existence theorem_, Trans. Amer. Math. Soc. 356(5):1781–1797 (2004) [doi](https://doi.org/10.1090/S0002-9947-04-03559-7)

> We study elementary extensions of compact complex spaces and deduce that every complete type of dimension $1$ is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.

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Riemann hypothesis

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## Idea

The **Riemann hypothesis** or **Riemann conjecture** is the famous unproved statement that all nontrivial zeros of the [[Riemann zeta function]] are on the vertical line $Re(z)=1/2$ in the [[complex plane]].

Analogues of the Riemann hypothesis can be considered for many analogues of [[zeta functions]] and [[L-functions]], here one speaks of **generalized Riemann hypotheses**.

An important special case over [[finite fields]] is called the **Riemann--Weil conjecture** and was proved by [[Deligne]] building on earlier ideas of [[Andre Weil|Weil]] and  [[Grothendieck]]. Grothendieck however expected a more natural proof using the (hypothetical) theory of [[motives]].

## Related concepts

* [[Weil conjecture]]

* [[Selberg trace formula]]

* [[Riemann hypothesis and physics]]

## References
 {#General}

### General

* [[Terence Tao]], _[The Riemann hypothesis in various settings](http://terrytao.wordpress.com/2013/07/19/the-riemann-hypothesis-in-various-settings/)_

See also

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

### Proof strategies

The suggestion that the Riemann hypothesis might have a proof that is an analogue of Weil's proof for [[arithmetic curves]] over [[finite fields]] $\mathbb{F}_q$ but generalized to the [[field with one element]] is due to

* {#Manin95} [[Yuri Manin]], _Lectures on zeta functions and motives (according to Deninger and Kurokawa)_ Asterisque, (228):4, 121-163, 1995. Columbia University Number Theory Seminar.

* Wikipedia, _[Generalized Riemann hypothesis](http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis)_.

Identification of the zeros of the [[Riemann zeta function]] with the spectrum of [[p-adic string theory]] is due to

* {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013))

### Computer-checks
 {#ReferencesComputerChecks}

[[experimental mathematics|Experimental checks]] of the Riemann hypothesis with computers:

* R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, _On the Zeros of the Riemann Zeta Function in the Critical Strip. II_, Mathematics of Computation Vol. 39, No. 160 (Oct., 1982), pp. 681-688 ([doi:10.2307/2007345 ](https://www.jstor.org/stable/2007345))

* A. M. Odlyzko, _The $10^{22}$-nd zero of the Riemann zeta function_, in Lapidus et al. (eds.) proceedings of AMS special session on _Dynamical, Spectral, and Arithmetic Zeta Functions_, 1999 ([pdf](http://www.dtc.umn.edu/~odlyzko/doc/zeta.10to22.pdf))

* {#Gourdon04} Xavier Gourdon, _The $10^{13}$ first zeros of the Riemann Zeta function, and zeros computation at very large height_, 2004 ([pdf](http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf), [webpage](http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html))

* David J. Platt, _Isolating some non-trivial zeros of zeta_, Mathematics of Computation 86 (2017), 2449-2467 ([doi:10.1090/mcom/3198 ](https://doi.org/10.1090/mcom/3198 ))

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Riemann hypothesis and physics

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## Idea

There are several ways in which the _[[Riemann hypothesis]]_ has or might have an incarnation in or leave an imprint on [[physics]], specifically in [[quantum physics]].

### Via string theory

By the general discussion at _[[zeta function]]_ and at _[[functional determinant]]_ zeta functions are closely related to 1-loop [[vacuum amplitudes]] and to [[vacuum energy]] in [[quantum field theory]] in general and in [[string theory]] in particular.

Concretely, the [[Rankin-Selberg-Zagier method]] implies that the [[partition function]] of the [[superstring]] asymptotes for small [[proper time]] to a constant times a converging oscillatory term (reminiscent of the [[explicit formulae for L-functions]]) whose [[frequencies]] are proportional to the imaginary values of the zeros of the [[Riemann zeta function]] ([ACER 11](#ACER11)).

Similarly there the [[Veneziano amplitude]] of the string has an expression in terms of the [[Riemann zeta function]] ([HJM 15](#HJM15)).

### Via noncommutative geometry

In other parts of the literature there is the desire to interpret the Riemann zeta function instead as a [[partition function]] of a [[quantum mechanical system]]. (Notice that the 1-loop [[vacuum amplitude]] mentioned before is instead a [[Mellin transform]] of the partition function.) The main example here is maybe the [[Bost-Connes system]].

## Related entries

* [[number theory and physics]]

* [[Selberg trace formula]]

* [[random matrix theory]]

## References

### Via string theory

via [[string theory]]:

* Sergio Cacciatori, Matteo Cardella, _Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis_ ([arXiv:1007.3717](http://arxiv.org/abs/1007.3717))

  "the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory"

* {#ACER11} Carlo Angelantonj, Matteo Cardella, [[Shmuel Elitzur]], [[Eliezer Rabinovici]], *Vacuum stability, string density of states and the Riemann zeta function*, JHEP **1102** 024 (2011) [[arXiv:1012.5091](http://arxiv.org/abs/1012.5091)]

* Sergio L. Cacciatori, Matteo A. Cardella, _Uniformization, Unipotent Flows and the Riemann Hypothesis_ ([arXiv:1102.1201](http://arxiv.org/abs/1102.1201))

* {#HJM15} Yang-Hui He, Vishnu Jejjala, Djordje Minic, _From Veneziano to Riemann: A String Theory Statement of the Riemann Hypothesis_ ([arXiv:1501.01975](http://arxiv.org/abs/1501.01975))

### Via scattering amplitudes

via [[scattering amplitudes]] in [[perturbative quantum field theory]]:

* Grant N. Remmen, *Amplitudes and the Riemann Zeta Function* ([arXiv:2108.07820](https://arxiv.org/abs/2108.07820))

### Via noncommutative geometry

via [[noncommutative geometry]]:

* [[Matthew Watkins]], _[the nontrivial Riemann zeta zeros interpreted as a spectrum of energy levels](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics1.htm)_

* [[Alain Connes]], _Noncommutative geometry and the Riemann zeta-function_ ([pdf](http://www.alainconnes.org/docs/imufinal.pdf))

* [[Masoud Khalkhali]], _What is new with Connes' approach to the Riemann hypothesis?_ [pdf](http://www.math.uwo.ca/~masoud/files/TehProg.pdf)

via [[random matrix theory]]:

* Nina Snaith, _Random matrix theory and zeta functions_, 2000 ([pdf](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/snaith-thesis.pdf))


Riemann integration

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## Idea

A method of [[integration]].

## Related concepts

* [[Lebesgue integration]]

* [[Jordan content]]

* [[Darboux integration]]

## References

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

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Riemann normal coordinates

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## Idea

Around every point of a [[Riemannian manifold]] there is a [[coordinate system]] such that the [[Levi-Civita connection]] of the metric pulled back to these coordinates vanishes _at that point_. (Notice that the [[Riemann curvature]] will not in general vanish even at that point).

In the context of [[general relativity]] this reflects aspects of the [[equivalence princile (physics)]].

In the sense of [[integrability of G-structures]], Riemann normal coordinates exhibit the first-order integrability of [[orthogonal structure]], see at _[integrability of G-structures -- Examples -- Orthogonal structure](integrability+of+G-structures#ExamplesOrthogonalStructure)_.

## Related concepts

* [[general covariance]]

* [[Walker coordinates]]

## References

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

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Riemann series theorem

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## Idea

The set $\Sigma_\mathbb{N} \coloneqq \mathbb{N} \cong \mathbb{N}$ is the [[symmetric group]] of the [[natural numbers]], whose elements $p \in \Sigma_\mathbb{N}$ are the [[permutations]] of the set of natural numbers.

The **Riemann series theorem** or the **Riemann rearrangement theorem** states that given a [[sequence]] of [[real numbers]] $a:\mathbb{N} \to \mathbb{R}$, if the [[series]] $\sum_{n=0}^{\infty} a_n$ is [[conditionally convergent]], then

* for all real numbers $M \in \mathbb{R}$, there exists a [[permutation]] $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges to $M$.

* there exists a [[permutation]] $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ diverges.

## Construction of the real numbers

The Riemann series theorem could be used to construct the [[real numbers]]. Since the Riemann series theorem holds for any conditionally convergent series, it suffices to use the conditionally convergent series defined by the sequence of [[rational numbers]] $a:\mathbb{N} \to \mathbb{Q}$ where $a_{2 n} = \frac{1}{n}$ and $a_{2 n + 1} = -\frac{1}{n}$. Then we define the real numbers as a quotient set $\mathbb{R} \coloneqq C(\Sigma_\mathbb{N})/\sim$ of the subset of the symmetric group of the natural numbers $C(\Sigma_\mathbb{N}) \subseteq \Sigma_\mathbb{N}$ for which the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges for $p \in \Sigma_\mathbb{N}$; i.e. for which the sequence of partial sums is a [[Cauchy sequence]]. We say that permutations $p \in C(\Sigma_\mathbb{N})$ and $q \in C(\Sigma_\mathbb{N})$ are similar $p \sim q$ if their corresponding series have the same limit. This is similar to the construction of the [[real numbers]] as the quotient set of all Cauchy sequences of rational numbers.

## In constructive mathematics

The Riemann series theorem cannot be proved in [[constructive mathematics]]. In general, given a sequence of rational numbers $a:\mathbb{N} \to \mathbb{Q}$ such that the series $\sum_{n=0}^{\infty} a_n$ is [[conditionally convergent]], any permutation of the indices $p \in \Sigma_\mathbb{N}$ such that the series $\sum_{n=0}^{\infty} a_{p(n)}$ converges would only converges to the [[Cantor real numbers]], since the sequence of partial sums of a series of rational numbers is a sequence of rational numbers, and the limit of any Cauchy sequence of rational numbers is only a [[Cantor real number]]. In the absence of [[excluded middle]] or [[countable choice]], one cannot prove that any [[sequentially Cauchy complete]] [[Archimedean ordered field]], such as the [[Dedekind real numbers]], is equivalent to the [[Cantor real numbers]].

## See also

* [[conditional convergence]]

## References

* Wikipedia, *[Riemann series theorem](https://en.wikipedia.org/wiki/Riemann_series_theorem)*


Riemann sphere

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## Idea

The 2-[[sphere]] with its canonical structure of a [[complex manifold]] is called the _[[Riemann sphere]]_.

As such this is [[complex projective space]] $\mathbb{C}P^1$.

## Properties

The [[biholomorphisms]], i.e. the bijective [[conformal transformations]] from the Riemann sphere to itself are the [[Möbius transformations]].

## Related concepts

* [[geometric quantization of the 2-sphere]].

* [[celestial sphere]]

* [[branched cover of Riemann sphere]]

* [[stereographic projection]]

* [[complex projective space]]

  * [[complex projective plane]]

  * [[complex projective 3-space]]

## References

See also

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

On the [[homotopy type]] of the [[space of maps|space of]] [[rational functions]] from the [[Riemann sphere]] to itself (related to the [[moduli space of monopoles]] in $\mathbb{R}^3$ and to the [[configuration space of points]] in $\mathbb{R}^2$):

* [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

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