area enclosed by a circle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea ##

The [[area]] enclosed by a [[circle]] as discussed in [[Euclidean geometry]].

## Definition/proposition and proofs

Strictly speaking, we are talking about the area of the [[disk]] whose [[boundary]] is the circle; however, the average person usually identifies the interior of a geometric shape with its boundary.

+-- {: .num_prop}
###### Proposition
Depending on which [[circle constant]] you use, given a radius $r$ of a circle $\mathcal{C}$ in the [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, the area of a circle is expressed either as $A(r) = \frac{1}{2} \tau r^2$ or as $A(r) = \pi r^2$.
=--

### Proof by double integration

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

Given any [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \times [0, r] \to \mathbb{R}^2$ defined as
$$\overrightarrow{r}(\rho, \theta) \coloneqq \rho \cos(\theta) \hat{i} + \rho \sin(\theta) \hat{j}$$

Then the area of $\mathcal{C}$ is given by the following [[double integral]]:

$$A(r) = \int_{0}^{r} \int_{0}^{\tau} \vert \overrightarrow{r}(\rho, \theta) \vert d \theta d \rho$$

which evaluates to

$$A(r) = \int_{0}^{r} \int_{0}^{\tau} \vert \rho \cos(\theta) \hat{i} + \rho \sin(\theta) \hat{j} \vert d \theta d \rho = \int_{0}^{r} \int_{0}^{\tau} \rho((\cos(\theta))^2 + (\sin(\theta))^2) d \theta d \rho = \int_{0}^{r} \int_{0}^{\tau} \rho d \theta d \rho = \int_{0}^{r} \tau \rho d \rho = \frac{1}{2} \tau r$$
=--

### Proof by areal velocity

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

Given any [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. There is an [[geometric algebra]] $\mathbb{G}^2$ on the vector space defined by the equations $\hat{i}^2 = 1$, $\hat{j}^2 = 1$, and $\hat{i} \hat{j} = -\hat{j} \hat{i}$.

The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
$$\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}$$

Then the area of $\mathcal{C}$ is given by integrating the magnitude of the [[areal velocity]]:

$$A(r) = \int_{0}^{\tau} \left|\frac{\overrightarrow{r}(\theta) \wedge \overrightarrow{v}(\theta)}{2}\right| d \theta$$

where $a \wedge b$ is the wedge product of two multivectors $a$ and $b$ and $\overrightarrow{v}$ is the [[velocity]] of a point in $\mathcal{C}$. This expression evaluates to

$$A(r) = \int_{0}^{\tau} \left|\frac{(r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}) \wedge \partial_\theta (r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j})}{2}\right| d \theta$$
$$A(r) = \int_{0}^{\tau} \left|\frac{(r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}) \wedge (-r \sin(\theta) \hat{i} + r \cos(\theta) \hat{j})}{2}\right| d \theta$$
$$A(r) = \int_{0}^{\tau} \left|\frac{(r (\cos(\theta))^2 \hat{i} \hat{j} + r (\sin(\theta))^2 \hat{i} \hat{j})}{2}\right| d \theta$$
$$A(r) = \int_{0}^{\tau} \left|\frac{r \hat{i} \hat{j}}{2}\right| d \theta = \int_{0}^{\tau} \frac{r}{2} d \theta = \frac{1}{2} \tau r$$
=--

### Proof by action functionals

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

Given any [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
$$\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}$$

Then the area of $\mathcal{C}$ is given by the [[action functional]] of the parameterized curve:

$$A(r) = \frac{1}{2} \int_{0}^{\tau} {\vert \overrightarrow{r}(\theta) \vert}^2 d \theta$$

which evaluates to

$$A(r) = \frac{1}{2} \int_{0}^{\tau} {\vert r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j} \vert}^2 d \theta = \frac{1}{2} \int_{0}^{\tau} (r((\cos(\theta))^2 + (\sin(\theta))^2))^2 d \theta = \frac{1}{2} \int_{0}^{\tau} r^2 d \theta = \frac{1}{2} \tau r$$
=--

### Proof by limits of regular polygons

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

The area of a [[regular polygon]] $\mathcal{P}_n$ with $n$ sides and [[circumradius]] $r$ is given by the sequence of functions $P:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$A_\mathcal{P}(n)(r) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2 n}\right)$$

which [[embedding|embeds]] in the $\mathbb{R}_+$-[[action]] $A_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R})$, defined as

$$A_\mathcal{P}^\prime(n)(r) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2n}\right)$$

The [[limit of a function|limit]] of $A_\mathcal{P}^\prime$ as $n$ goes to infinity is the area of a circle with radius $r$:

$$A(r) = \lim_{n \to \infty} A_\mathcal{P}^\prime(n)(r) = \lim_{n \to \infty} \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{n}\right) = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\sin(\tau m)}{m} = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\partial_m \sin(\tau m)}{\partial_m m} = \frac{1}{2} r^2 \lim_{m \to 0} \frac{\tau \cos(\tau m)}{1} = \frac{1}{2} \tau r^2$$
=--

## See also

* [[circle]]

* [[unit circle]]

* [[area]]

* [[circumference of a circle]]

* [[Euclidean space]]

[[!redirects area of a circle]]


areal velocity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea ##

The [[bivector]] whose magnitude equals the rate of change at which [[area]] is swept out by a [[point]]/[[particle]] as it moves along a [[curve]].

## Definition

Given an $n$-[[dimension|dimensional]] [[Euclidean space]] $\mathbb{R}^n$, one could select an [[orthonormal basis]] on $\mathbb{R}^n$ by postulating an origin $0$ and a function $\hat{i}:[1, n] \to \mathbb{R}^n$ such that for all $m, p \in [1, n]$ the pair of vectors $\hat{i}_m$ and $\hat{i}_p$ is mutually [[orthonormal]]. There is an [[geometric algebra]] $\mathbb{G}^n$ on $\mathbb{R}^n$ defined by the equations $\hat{i}_m^2 = 1$ for all $m \in [1, n]$, and $\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m$ for all $m, p \in [1, n]$.

A [[smooth curve]] $\mathcal{C}$ in $\mathbb{R}^n$ could be parameterized by a [[smooth function]] $\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n$. Then the **areal velocity**, **sector velocity**, or **sectorial velocity** of a point in $\mathcal{C}$ in $\mathbb{R}^n$ is given by the bivector-valued function $A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2$

$$A(t) = \frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}$$

where $a \wedge b$ is the wedge product of two vectors $a$ and $b$, and $\overrightarrow{v}$ is the [[velocity]].

## In 3 dimensions

In 3 dimensions, the vector areal velocity $\overrightarrow{a}:\mathbb{R} \to \langle \mathbb{G}^n \rangle_1$ is the [[Hodge dual]] of the areal velocity, which is the product of the pseudoscalar
$$I = \prod_{i:[1, n]} \hat{i}_i$$
with the areal velocity:
$$\overrightarrow{a}(t) = I A(t) = I\left(\frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}\right) = \frac{I(\overrightarrow{r}(t) \wedge \overrightarrow{v}(t))}{2} = \frac{\overrightarrow{r}(t) \times \overrightarrow{v}(t)}{2}$$

## Conservation of areal velocity

Conservation of areal velocity is the same as the conservation of [[angular momentum]].

## See also

* [[area]], [[velocity]]

* [[area of a circle]]

* [[Euclidean space]]

* [[geometric algebra]]

* [[classical mechanics]]

* [[angular momentum]]

* [[torque]]

## References

See also:

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

[[!redirects areal velocities]]
[[!redirects sector velocity]]
[[!redirects sector velocities]]
[[!redirects sectorial velocity]]
[[!redirects sectorial velocities]]


Arend

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Arend_ is a [[proof assistant]] system for [[homotopy type theory]] with native support for [[higher inductive types]] and some [[cubical type theory]].

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References

* [Arend home page](https://arend-lang.github.io/)


Arend Heyting

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

## Selected writings

On [[intuitionistic logic]]:

* {#Heyting1930} [[Arend Heyting]], *Die formalen Regeln der intuitionistischen Logik. I, II, III.* Sitzungsberichte der Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1930) 42-56, 57-71, 158-169

  abridged reprint in:

  Karel Berka, Lothar Kreiser (eds.), *Logik-Texte*, De Gruyter (1986) 188-192 [[doi:10.1515/9783112645826](https://doi.org/10.1515/9783112645826)]

* [[Arend Heyting]], *Die intuitionistische Grundlegung der Mathematik*, Erkenntnis **2** (1931) 106-115 [[jsotr:20011630](https://www.jstor.org/stable/20011630), [pdf](http://www.psiquadrat.de/downloads/heyting1931.pdf)]

* [[Arend Heyting]], *Bemerkungen zu dem Aufsatz von Herrn Freudenthal "Zur intuitionistischen Deutung logischer Formeln"*, Comp. Math. **4** (1937) 117-118 [[doi:CM_1937__4__117_0](http://www.numdam.org/item/?id=CM_1937__4__117_0)]

and making explicit the [[BHK interpretation]]:

* {#Heyting56} [[Arend Heyting]], *Intuitionism: An introduction*, Studies in Logic and the Foundations of Mathematics, North-Holland (1956, 1971) [[ISBN:978-0720422399]()]

## Related entries

* [[Heyting algebra]]

* [[intuitionistic logic]]

category: people

[[!redirects Heyting]]


Arf invariant

## Related concepts

* [[Arf-Kervaire invariant]]

* [[Arf-Kervaire invariant problem]]

## References

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


Arf-Kervaire invariant problem

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

#Contents#
* table of contents
{:toc}
## Idea

The **Arf-Kervaire invariant problem** asks whether certain hypothetical elements $\theta_j \in \pi^s_{2^{j+1} - 2}$ of the stable [[homotopy groups of spheres]] exist. In fact the definition of these elements works in all dimensions but due to a theorem of Browder they do not exist in dimensions $d\neq 2^{j+1} - 2$.

Prior to ([Hill-Hopkins-Ravenel 09](#HillHopkinsRavenel09)), all that was known rested on the explicit construction of such elements for $j=1,...,5$ (so in dimensions 2,6,14,30 and 62). HHR established that these elements do not exist for $j \gt 6$, so the only dimension in which existence remains unknown in 126 (ie $j=6$). The proof is by construction of a 256-periodic [[spectrum]] $\Omega$ and a [[spectral sequence]] for it that can detect the elements $\theta_j$ as elements of $\pi_*(\Omega)$. HHR then show that $\pi_n(\Omega)=0$ for $-4\lt n\lt 0$, which by the periodicity, implies that the images of $\theta_j$ must be elements of the trivial group, and hence are themselves trivial.

## Key properties of the $C_8$ fixed point spectrum $\Xi$

We write $\Xi$ to mean the spectrum "$\Omega$" discussed in [Hill-Hopkins-Ravenel 09](#HillHopkinsRavenel09).

### Detection theorem
It has an [[Adams-Novikov spectral sequence]] in which the image of each $\theta_j$ is non-trivial. This means if $\theta_j\in \pi_*(\mathbb{S})$ exists then it can be seen in $\pi_*(\Xi)$.

### Periodicity Theorem
The spectrum is 256-periodic, as in $\Omega^{256}\Xi \simeq \Xi$.

### Gap Theorem
We have $\pi_k(\Xi) = 0$ for $-4 \lt k \lt 0$. Its proof uses the [[slice spectral sequence]].

### Result

Suppose $\theta_7 \in \pi_{254}(\mathbb{S})$ exists then the detection theorem implies that it has a non-trivial image in $\pi_{254}(\Xi)$. But by the periodicity and gap theorems we see that $\pi_{254}(\Xi)$ is trivial. The argument for $j \ge 7$ is similar since $|\theta_j| = 2^{j+1} \equiv -2 \mod 256$.

## Related concepts

* [[Kervaire invariant]]

* [[Hopf invariant one problem]]

## References

A solution of the problem in the negative, except for one outstanding dimension (namely 126), using methods of [[equivariant stable homotopy theory]]:

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics **184** 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)]

* {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)]

* {#HillHopkinsRavenel} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_, Current Developments in Mathematics, 2010: 1-44 (2011) ([[HHRKervaire.pdf:file]], [doi:10.4310/CDM.2010.v2010.n1.a1](https://dx.doi.org/10.4310/CDM.2010.v2010.n1.a1))

On the [[equivariant stable homotopy theory]] involved:

* [Hill-Hopkins-Ravenel 09, Appendix](#HillHopkinsRavenel09)

* [HillHopkinsRavenel, section 4](#HillHopkinsRavenel)

More resources are collected at

* [[Douglas Ravenel]], _[A solution to the Arf-Kervaire invariant problem](http://www.math.rochester.edu/people/faculty/doug/kervaire.html)_, web resources 2009

* {#HillHopkinsRavenelIntroduction} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire invariant problem in algebraic topology: introduction_ (2016) ([pdf](http://math.ucla.edu/~mikehill/Research/CDMHistory.pdf))

[[!redirects Kervaire invariant one problem]]
[[!redirects Kervaire invariant problem]]


argument shift method

Argument shift method is a method to construct commutative subalgebras of the symmetric algebra on a Lie algebra with respect to the Poisson-Lie bracket. It is of importance in the study of [[integrable system]]s.

* A. S. Mishchenko, A. T. Fomenko, _Euler equations on finite-dimensional Lie groups_, Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 42 (1978), no. 2, 386--415; _Integrability of Euler equations on semisimple Lie algebras_, in: Proc. Seminars on Vector and Tensor Analysis [in Russian], no. 19, Moskov. Gos. Univ., Moscow, 1979, 3--94
* S. V. Manakov, _Remark on integrability of Euler equations of the dynamics of an n-dimensional rigid body_, Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 10 (1976), no. 4, 93--94
* L. G. Rybnikov, _The argument shift method and the Gaudin model_, Funct. Anal. Appl. 40, 188--199 (2006)
* V. V. Shuvalov, _On limits of Mishchenko--Fomenko subalgebras in Poisson algebras of semisimple Lie algebras_, Funct. Anal. Appl. 36, 298–305 (2002) [doi](https://doi.org/10.1023/A:1021713927119)
* Y. Ikeda, _Quasidifferential operator and quantum argument shift method_, Theor Math Phys 212, 918--924 (2022) [doi](https://doi.org/10.1134/S0040577922070030)


Ari M. Turner

## Selected writings

On [[symmetry-protected topological order]]:

* {#PBTO09} [[Frank Pollmann]], [[Erez Berg]], [[Ari M. Turner]], [[Masaki Oshikawa]], *Symmetry protection of topological order in one-dimensional quantum spin systems*, Phys. Rev. B **85** 075125 (2012) $[$[arXiv:0909.4059](https://arxiv.org/abs/0909.4059), [doi:10.1103/PhysRevB.85.075125](https://doi.org/10.1103/PhysRevB.85.075125)$]$

On characterizing [[anyon]] [[braiding]] / [[modular transformations]] on [[topological order|topologically ordered]] [[ground states]] by analysis of ([[topological entanglement entropy|topological]]) [[entanglement entropy]] of subregions:

* [[Yi Zhang]], [[Tarun Grover]], [[Ari M. Turner]], [[Masaki Oshikawa]], [[Ashvin Vishwanath]], *Quasiparticle statistics and braiding from ground-state entanglement*, Phys. Rev. B **85** (2012) 235151 $[$[doi:10.1103/PhysRevB.85.235151](https://doi.org/10.1103/PhysRevB.85.235151)$]$

On [[topological semi-metals]]:

* [[Ari M. Turner]], [[Ashvin Vishwanath]], Part I of: *Beyond Band Insulators: Topology of Semi-metals and Interacting Phases*, in: *Topological Insulators*, Contemporary Concepts of Condensed Matter Science **6** (2013) 293-324 $[$[arXiv:1301.0330](https://arxiv.org/abs/1301.0330), [ISBN:978-0-444-63314-9](https://www.sciencedirect.com/bookseries/contemporary-concepts-of-condensed-matter-science/vol/6/suppl/C)$]$

category: people

[[!redirects Ari Turner]]


Aristotle

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

## Selected writings

On [[physics]]:

* {#Aristotle} [[Aristotle]], *Physics*, in: Jonathan Barnes (ed.), *The Complete Works of Aristotle* Vol 1 (1984) [[pdf](https://sites.unimi.it/zucchi/NuoviFile/Barnes%20%20-%20Physics.pdf), [ISBN:9780691016504](https://press.princeton.edu/books/hardcover/9780691016504/complete-works-of-aristotle-volume-1)]

## Related entries

* [[Metaphysics (Aristotle)]]

* [[syllogism]]

* [[potentiality and actuality]]

* [[Science of Logic]]

category: people


arithmetic

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

#Contents#
* table of contents
{:toc}

## Idea

Arithmetic (Greek ἀριθμός: number) is, roughly speaking, the study of [[numbers]] in their various forms, and the structure and properties of the operations defined on them, including at least [[addition]] and [[multiplication]], and sometimes also [[subtraction]], [[division]], and [[exponentiation]].

Notions of "[[number]]" are very broad and not at all easy to encapsulate. There are [[natural numbers]], [[integers]], [[rational numbers]], [[real numbers]], and [[complex numbers]], [[quaternions]] (Hamiltonian numbers), [[octonions]] (Cayley numbers). There are [[dual numbers]] and [[hyperbolic numbers]],  and [[geometric algebra]]s
(geometric numbers). There are [[cyclic group#Ring structure|integers modulo $n$]]. There are [[algebraic numbers]] and [[algebraic integers]], and individual [[fields]] of such ([[number fields]]). There are $p$-[[adic numbers]]. Then there are [[cardinal numbers]], [[ordinal numbers]], and [[surreal numbers]]. There are even [[closed intervals]] and [[open intervals]]. For each one of these one can (and does!) speak of its arithmetic.

This article will provide links to other articles in which these various cases are discussed.

* [[number theory]]

  * [[Peano arithmetic]]

  * [[second-order arithmetic]]

  * [[arithmetic geometry]], [[arithmetic scheme]], [[arithmetic variety]], [[arithmetic curve]], [[arithmetic genus]], [[arithmetic Chern-Simons theory]]

  * [[arithmetic topology]]

  * [[arithmetic pretopos]]

  * [[modular arithmetic]]

* [[transfinite arithmetic]]

  * [[cardinal arithmetic]]

  * [[ordinal arithmetic]]

## References

See also

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

[[!redirects arithmetics]]


arithmetic and noncommutative geometry

A relation between [[noncommutative geometry]] and [[arithmetic]] (= number theory) has been explored much in the work of [[Alain Connes]] and his collaborators, especially Marcolli and Consani.

Some surveys include

* P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics __16__, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030)
* [[Matilde Marcolli]], _Lectures on arithmetic noncommutative geometry_, [math/0409520](http://arxiv.org/abs/math/0409520)

There is also another line of thought in the work Manin-Marcolli on the relation of [[Arakelov geometry]] and noncommutative geometry. Arakelov's geometry is of course, motivated by number theory.

Independetly, one should also notice that the noncommutative geometry over non-archimedean fields is relevant for [[homological mirror symmetry]] as explored in the works of [[Maxim Kontsevich]] and [[Yan Soibelman]].

[[!redirects noncommutative geometry and arithmetic]]
[[!redirects arithmetic noncommutative geometry]]


arithmetic Chern-Simons theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**Arithmetic Chern-Simons theory** names the attempt to apply constructions from [[Chern-Simons theory]] to the field of [[arithmetic]], in view of patterns in the [[function field analogy]]. In particular, the papers ([Kim1](#Kim1), [Kim2](#Kim2)) apply ideas of [[Dijkgraaf-Witten theory]] on 2+1 dimensional [[topological quantum field theory]] to [[arithmetic curves]], that is, the [[ring spectrum|spectra]] of rings of [[algebraic integer|integers]] in [[algebraic number fields]].

This theory pursues the surprising analogies between 3-dimensional topology and number theory, where knots embedded in a 3-manifold behave like prime ideals in a ring of algebraic integers, known as [[arithmetic topology]].

## Related concepts

* [[number theory and physics]]

* [[arithmetic gauge theory]]

* [[p-adic physics]]

  * [[p-adic string theory]]

  * [[p-adic AdS/CFT correspondence]]

## References

* {#Kim1} [[Minhyong Kim]], *Arithmetic Chern-Simons Theory I*, in: *Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants*, Proceedings in Mathematics & Statistics **330**,  Spinger (2020) [[arXiv:1510.05818](http://arxiv.org/abs/1510.05818), [doi:10.1007/978-3-030-51795-3_8](https://doi.org/10.1007/978-3-030-51795-3_8)]

* {#Kim2} Hee-Joong Chung, Dohyeong Kim, [[Minhyong Kim]], Jeehoon Park, Hwajong Yoo, _Arithmetic Chern-Simons Theory II_, ([arXiv:1609.03012](http://arxiv.org/abs/1609.03012))

* Frauke M. Bleher, Ted Chinburg, Ralph Greenberg, Mahesh Kakde, George Pappas, Martin J. Taylor, _Unramified arithmetic Chern-Simons invariants_, ([arXiv:1705.07110](https://arxiv.org/abs/1705.07110))

* Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, George Pappas, Jeehoon Park, Hwajong Yoo, _Abelian arithmetic Chern-Simons theory and arithmetic linking numbers_, ([arXiv:1706.03336](https://arxiv.org/abs/1706.03336))

* Hikaru Hirano, _On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields_, ([arXiv:1911.12964](https://arxiv.org/abs/1911.12964))

* Jungin Lee, Jeehoon Park, _Arithmetic Chern-Simons theory with real places_, ([arXiv:1905.13610](https://arxiv.org/abs/1905.13610))

An introductory talk"

* [[Minhyong Kim]], _Arithmetic topological quantum field theory?_, ([slides](https://simonsfoundation.s3.amazonaws.com/share/mps/conferences/2017_Conference_on_Number_Theory_Geometry_Moonshine_and_Strings/Kim.pdf))

Introduction in  the broader context of [[arithmetic gauge theory]]:

* {#Kim18} [[Minhyong Kim]], *Arithmetic Gauge Theory: A Brief Introduction*,  Modern Physics Letters A **33** 29 (2018) 1830012 [[arxiv:1712.07602](https://arxiv.org/abs/1712.07602), [doi:10.1142/S0217732318300124](https://doi.org/10.1142/S0217732318300124)]


arithmetic Chow group

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

#Contents#
* table of contents
{:toc}

## Idea

Arithmetic Chow groups are refinements of ordinary [[Chow groups]] analogous to how [[ordinary differential cohomology]] refines [[ordinary cohomology]].

Let $X$ be an [[arithmetic variety]], that is: a [[quasi-projective variety|quasi-projective]] [[flat scheme|flat]] [[regular scheme]] over an
[[arithmetic ring]]. In ([Gillet-Soule](#GilletSoule)) the _arithmetic Chow groups_ of $X$, denoted $\hat CH^p(X)$, are defined as groups whose elements are [[equivalence classes]] of
pairs consisting of a codimension $p$ subvariety of $X$ together with a [[Green current]] for it. Later, in ([Burgos Gil 97](#BurgosGil97)), an alternative definition was given in terms of a [[Deligne complex]] of [[differential forms]] with logarithmic
singularities along infinity, that computes a version of [[ordinary differential cohomology]] groups.

When $X$ is [[proper scheme|proper]], the two definitions are [[natural isomorphism|naturally isomorphic]].

## Related concepts

* [[Chow group]], [[Arakelov geometry]]

## References

### General

Arithmetic intersection theory was introduced in

* Henri Gillet, Christoph Soulé, _Arithmetic intersection theory_ IHES Preprint (1988)
 {#GilletSoule}

Generalization are discussed in

* J. I. Burgos Gil, J. Kramer, U. Kühn, _Cohomological arithmetic Chow rings_ ([arXiv:math/0404122v2](http://arxiv.org/abs/math/0404122v2))
 {#BurgosGilKramerKuehn}

* J. I. Burgos Gil, _Higher arithmetic Chow groups_ ([pdf](http://www.crm.es/Publications/10/Pr925.pdf))

### Relation to differential cohomology
 {#ReferencesRelationToDifferentialCohomology}

Articles that discuss the relation of arithmetic Chow groups to [[ordinary differential cohomology]] include

* Henri Gillet, Christoph Soulé, _Arithmetic Chow groups and differential characters_ in [[Rick Jardine]] (ed.) _Algebraic K-theory: Connections with Geometry and Topology_, Springer (1989)

* J. I. Burgos Gil, _Arithmetic Chow rings_, Ph.D. thesis, University of Barcelona, (1994).

* J. I. Burgos Gil, _Arithmetic Chow rings and Deligne-Beilinson cohomology_, J. Alg. Geom. 6 (1997), 335–377.
 {#BurgosGil97}

[[!redirects arithmetic Chow groups]]
[[!redirects Arithmetic Chow groups]]


arithmetic cohesion -- table

**[[cohesion]] in [[E-∞ arithmetic geometry]]**:

| [[cohesion]] [[modality]]  | symbol |  interpretation  |
|---|---|---|
| [[flat modality]] | $\flat$ | [[formal completion]] at |
| [[shape modality]] | $ʃ$ | [[torsion approximation]] |
| [[dR-shape modality]]  | $ʃ_{dR}$ | [[localization]] away |
| [[dR-flat modality]] | $\flat_{dR}$  | [[adic residual]] |

the **[[differential cohomology hexagon]]/[[arithmetic fracture squares]]**:

$$
  \array{
    &&  localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual
    \\
    & \nearrow & & \searrow & & \nearrow && \searrow
    \\
    \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X  && && X &&  && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow
    \\
    && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation
  }
  \,,
$$


arithmetic cryptography

[[!redirects Arithmetic cryptography]]

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

#Contents#
* table of contents
{:toc}

## Idea

Arithmetic cryptography is the developing subject that describes [public key](http://en.wikipedia.org/wiki/Public-key_cryptography) [[cryptography]] systems based on the use of arithmetic geometry of schemes (or [[global analytic spaces]]) over $\mathbb{Z}$.

There are two well known examples of such systems:

1. The first and most used one, which is very efficient **because** it is conformal to the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle), is based on the fact that it is very difficult (from the computational viewpoint) to factorize a natural number into a product of two big prime numbers.

1. The second one is based on the [discrete logarithm](http://en.wikipedia.org/wiki/Discrete_logarithm) problem on elliptic curves (or more generally abelian varieties) over finite fields. It has the advantage on the first algorithm of allowing to make shorter keys, without compromising security (NSA has generalized the use of [elliptic curve cryptography](http://en.wikipedia.org/wiki/Elliptic_curve_cryptography) algorithm both for commercial and classified use, as explained on the wikipedia page on elliptic curve cryptography).

The basic idea of arithmetic cryptography is to use a finite family $X$ of polynomials with integer coefficients $P_1,\dots,P_m\in \mathbb{Z}[X_1,\dots,X_n]$ (or more generally a quasi-projective scheme $X$ of finite type over $\mathbb{Z}$, or even maybe a [[global analytic space]] $X$ over a convenient Banach ring), encoded in a finite number of integers (the coefficients and degrees of the corresponding polynomials), together with some additional data (such as a way to cut a part of the associated motive) to define a [public key cryptosystem](http://en.wikipedia.org/wiki/Public-key_cryptography).

Some computational aspects of general motives have been investigated in the case of motives of modular forms by Bass Edixhoven and Jean-Marc Couveignes, using étale cohomological methods. Kedlaya and Lauder-Wan also studied the Dwork approach from a computational viewpoint.

## Necessary tools

Since the space $X$ to be used is given by a non-linear equation, it is not directly adapted to the use of computational methods. One will thus need to extract linear invariants from $X$, by the use of a kind of 'differential calculus' (in Quillen's sense, i.e., using a convenient [[tangent (infinity,1)-category]]) and/or the definition of convenient 'cohomological invariants.

It seems that étale cohomological methods, that were originally used by Grothendieck and Deligne (and more recently, Laumon) to prove the full Weil conjectures, are not so easy to implement on a computer (see however the book of Edixhoven and Couveignes). It seems that "[[p-adic]] methods", based on [[p-adic differential equation|p-adic differential calculus]] and Fourier transform, and now completely developed by Berthelot, Lestum, Caro and Kedlaya (p-adic proof of the Weil-conjectures) are better adapted to computations (e.g., of L-functions over finite fields, i.e., characteristic polynomials of frobenius acting on cohomology).

It thus looks like an important project to develop "[[p-adic]] methods" in [[global analytic geometry]], starting with the definition of two types of cohomology theories for [[global analytic spaces]]: an [[absolute cohomology]] theory (related to the Chern character from K-theory to negative cyclic cohomology) and a [[geometric cohomology]] theory (with characteristic $0$ coefficients, e.g., in the full ring $\mathbb{A}$) of adèles). It may also be interesting to develop a notion of [[global Fourier transform]].

One of these cohomology theories (the absolute one) should give a natural way to study various conjectures on [[special values of L-functions]] and the other (the geometric one) should give a natural way to study the position of zeroes and poles of L-functions by spectral methods.

The [[geometric cohomology theory]] should also be (since it has characteristic $0$ coefficients) related to the theory of motives and [[global automorphic representations]], and thus give more generally a spectral way of studying zeroes and poles of the L-function of the rational motive of an [[arithmetic variety]].

## Aims

The aim of arithmetic cryptography is to define a good [[geometric cohomology]] theory for [[global analytic spaces]] based on analytic methods and differential calculus that would allow the definition of [public key cryptography](http://en.wikipedia.org/wiki/Public-key_cryptography) systems based on the datum of a [[global analytic space]] $X$ and of (say) a part $M$ of the associated (maybe absolute) rational motive $M(X)$. The definition of such methods would not give any new attacks to the previous ones, but may give a bigger class of public keys, that may allow the use of shorter ones. However, it is not yet clear that such a general approach will be conformal to the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle). In particular, as explained by Edixhoven in a preprint, it seems to be quite a hard computational task to determine if two cohomology classes are equal, at least in the $\ell$-adic setting.

## Possible constraints on the theory

The constraints on such a theory would be the following:

1. get back the classical discrete logarithm elliptic public key cryptography method when one starts from an elliptic curve over a finite field (or maybe a corresponding weak formal scheme over the ring of p-typical Witt vectors).

2. get back the original prime factorization public key cryptography method when one starts from $\mathbb{Z}$.

3. It must be clearly stressed here that the proper theoretical setting for such a general theory may be very hard to develop, since it should involve the definition of a proper cohomology for the Riemann zeta function, that would allow a spectral proof of its functional equation (as in Tate's thesis) `and` of the [[Riemann hypothesis]]. The full project thus looks like a very far reaching one, since there is, up to now, no precise idea of a way of constructing such a [[geometric cohomology]] theory (even if ideas on the constraints that it should fulfill are widespread in the mathematical litterature, e.g., in Deninger's work, in the [[field with one element]] litterature, in [[Langlands program]] and in the study of [[Weil-étale cohomology]]). Moreover, the use of adelic coefficients instead of complex ones seems necessary to treat the $p$-adic and archimedean cohomologies on equal footing.

4. One may use the classical GRH conjectures (without proving them) to get the estimates analogous to the classical estimate of the number of primes smaller than a given prime. From this point of view, it seems that representation theoretic approaches may be more interesting.

## Possible methods

It is not at all clear that the following propositions will be conformal to the KISS principle. They may also be attacked by quantum computing methods, by higher dimensional generalizations of Shor's algorithm, but one may hope that allowing the use of arbitrary arithmetic schemes may make the quantum computing methods more difficult to apply and/or implement in general. This also means that these propositions may be very hard to implement in practice.

A starting point for crash-testing the compatibility of higher dimensional arithmetic cryptography with the [KISS principle](http://en.wikipedia.org/wiki/KISS_principle) may be to test it in the finite characteristic case (with p-adic methods, say derived analytic spaces over $\mathbb{Z}_p$, for computational purposes). First remark that the discrete logarithm problem for an elliptic curve over $\mathbb{F}_p$ may be understood as a discrete logarithm problem in the first (étale torsion) cohomology group of the curve, given by torsion points. One may try to generalize this problem to the higher dimensional situation by giving an algebraic cohomology class $[c]$, and computing $[d]=n.[c]$. The public key is given by the pair $([c],[d])$ and the message is the number $n$. As pointed out by Edixhoven in a preprint, computing a cohomology class may be a very hard computational task, so that this idea may not be conformal to the KISS principle.

One may also devise another "product type" approach to the definition of a public key: given two prime cohomology classes $[c]$ and $[d]$ (classes of irreductible subvarieties of a given codimension), compute their product class $[e]=[c].[d]$ in the cohomology ring. One may try, given $[e]$, to find back $[c]$ and $[d]$, and this may be a very hard problem.

## Related references

Computing zeta functions of arithmetic schemes, David Harvey, [arXiv](http://arxiv.org/abs/1402.3439)


arithmetic curve

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

#Contents#
* table of contents
{:toc}
## Idea

A [[curve]] in [[arithmetic geometry]], hence an [[arithmetic scheme]] of suitable [[dimension]] 1 etc.

## Properties

### Function field analogy

[[!include function field analogy -- table]]

[[!redirects arithmetic curves]]


arithmetic D-module

#Contents#
* table of contents
{:toc}

## Idea

The theory of _arithmetic D-modules_ was primarily developped by Berthelot to better understand the [[functor|functoriality]] properties of [[rigid cohomology]]. It gives a theory of coefficients for the [[cohomology]] of [[quasi-projective variety|quasi-projective algebraic varieties]] over finite fields that are stable by the [[six operations|six Grothendieck operations]], after Kedlaya and Caro. This allows a purely p-adic proof of Deligne's Weil II theorem, that generalized the
[[Riemann hypothesis]] over finite fields to the category of coefficients for cohomology (i.e., motivic sheaves).

## Related concepts

* [[D-module]]

* [[coherent D-module]], [[holonomic D-module]]

## References

* Pierre Berthelot: [D-modules arithmetiques I, II](http://perso.univ-rennes1.fr/pierre.berthelot/).

* Kiran S. Kedlaya, [Semistable reduction for overconvergent F-isocrystals I, II, III](http://arxiv.org/find/all/1/AND+au:+kedlaya+ti:+AND+semistable+reduction/0/1/0/all/0/1?skip=0&query_id=99b522850f00b5aa).

* Daniel Caro: [Stability of holonomicity over quasi-projective varieties](http://arxiv.org/abs/0810.0304)

[[!redirects arithmetic D-modules]]


arithmetic differential geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
**Arithmetic differential geometry** is an approach to [[arithmetic]] which looks to find analogs of constructions in [[differential geometry]], for example, [[arithmetic jet spaces]]. It has been developed primarily by [[Alexandru Buium]].

According to this approach, the classical derivatives of differential geometry are replaced by [[p-derivations]] for prime $p$, such as the [[Fermat quotient]]

$$
\delta_p: \mathbb{Z} \to \mathbb{Z}, a \mapsto \delta_p a = \frac{a - a^p}{p}.
$$

##Comparison with Borger's absolute geometry

Buium explains that when working with a single prime his approach is consistent with [[Borger's absolute geometry]], which is described as "an algebraization of our analytic theory" ([Buium 17, p. 24](#Buium17)). However, in the case of multiple primes Borger requires [[Frobenius lifts]] to commute, and this diverges from the non-vanishing 'curvature' Buium derives from non-commuting lifts. For him, the ("manifold" corresponding to) the integers, $\mathbb{Z}$, is "intrinsically curved".

## Related pages

* [[arithmetic jet space]]

* [[Joyal delta-ring]]

* [[Borger's absolute geometry]]

* [[D-geometry]]

* [[arithmetic geometry]]

## References

* {#Buium17} [[Alexandru Buium]], _Foundations of arithmetic differential geometry_, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, ([AMS](http://bookstore.ams.org/surv-222/), [Preface and Introduction](http://www.math.unm.edu/~buium/foundationsintro.pdf)).

* [[Alexandru Buium]], _Arithmetic differential geometry_, May 19, 2017, [talk slides](http://www.math.unm.edu/~buium/adgtalk.pdf)

[[!redirects arithmetic derivative]]
[[!redirects arithmetic derivatives]]


arithmetic gauge theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Arithmetic gauge theory_ is a theory proposed by [Kim 2018 ](#Kim18) to look at [[Galois representations]] with an action of another group (the [[gauge group]]) as being analogous to [[gauge fields]] in [[gauge theory]]. It is an approach to Diophantine problems such as the effective [[Mordell conjecture]] and is influenced by the theory of the [[Selmer group]] and Chabauty's method.

## Definitions

Let $K$ be a field of characteristic zero and let $G_{K}$ be its [[absolute Galois group]]. A _gauge group over $K$_ is a [[topological group]] $U$ with a continuous action of $G_{K}$.

A _$U$-gauge field_, or _$U$-principal bundle_, is a topological space $P$ with compatible continuous left $G_{K}$-action and simply transitive continuous right $U$ action.

## Examples

This section follows section 8 of [Kim 2018](#Kim18).

Let $V$ be a variety over $\mathbb{Q}$ equipped with a base point $b\in V(\mathbb{Q})$. Our motivation is determining the set $V(\mathbb{Q})$ of rational points of $V$. For our gauge group $U$ we let

$$U=\pi_{1}(\overline{V},b)_{\mathbb{Q}},$$

the $\mathbb{Q}_{p}$-pro-unipotent fundamental group of $V$. For our gauge field $P$ we let

$$P(x)=\pi_{1}(\overline{V},b,x),$$

the $U$-torsor of pro-unipotent paths from $b$ to $x$.

Let $S$ be a finite set of primes. Let $H_{f}^{1}(\mathbb{Z}_{S},U)$ be the set of $U$-torsors over $\mathbb{Q}$ which are unramified outside $S$ and crystalline at $p$ (see [[p-adic Hodge theory]] for the meaning of crystalline). Similarly let $H^{1}(\mathbb{Q}_{p},U)$ be the set of $U$-torsors over $\mathbb{Q}_{p}$. We have a localization map

$$H_{f}^{1}(\mathbb{Z}_{S},U)\to\prod^{'}H^{1}(\mathbb{Q}_{v},U)$$

where the restricted product on the right means that all but finitely many of the components are unramified, and at $p$ the corresponding component is crystalline.

Recall that we are interested in $V(\mathbb{Q})$, the set of rational points of $V$. Now we have a map $A:V(\mathbb{Q})\to H_{f}^{1}(\mathbb{Z}_{S},U)$ given by

$$x\mapsto P(x).$$

where $P(x)=\pi_{1}(\overline{V},b,x)$ as above. This map fits into the diagram

$$
  \array{& V(\mathbb{Q}) & \rightarrow & V(\mathbb{Q}_{p}) & \\
          A & \downarrow &&\downarrow & A_{p} \\
          & H_{f}^{1}(\mathbb{Z}_{S},U) & \underset{loc_{p}}\rightarrow& H_{f}^{1}(\mathbb{Q}_{p},U) &
}$$

Kim conjectures the following:

\begin{conjecture}
Suppose $V$ is a smooth projective curve of genus $g\geq 2$. Then
$$V(\mathbb{Q})=A_{p}^{-1}(Im(loc_{p})).$$
\end{conjecture}

## Relation to L-functions

## Related ideas

* [[gauge theory]]

* [[Mordell conjecture]]

* [[arithmetic Chern-Simons theory]]

## References

* {#Kim18} [[Minhyong Kim]], *Arithmetic Gauge Theory: A Brief Introduction*,  Modern Physics Letters A **33** 29 (2018) 1830012 [[arxiv:1712.07602](https://arxiv.org/abs/1712.07602), [doi:10.1142/S0217732318300124](https://doi.org/10.1142/S0217732318300124)]

* [[Minhyong Kim]], _Recent Progress on the Effective Mordell Problem_, lecture at the Sydney Mathematical Research Institute [YouTube](https://www.youtube.com/watch?v=NDQ_aO7QDEU)

* [[Minhyong Kim]], _Foundations of nonabelian Chabauty_, lectures at the Arizona Winter School 2020 [Slides1](https://swc-math.github.io/aws/2020/2020KimNotes1.pdf), [Slides2] (http://swc-alpha.math.arizona.edu/video/2020/2020KimLecture2Slides.pdf) [Slides3] (https://swc-math.github.io/aws/2020/2020KimNotes3.pdf) [Slides4] (https://swc-math.github.io/aws/2020/2020KimNotes4.pdf)

[[!redirects arithmetic gauge theories]]


arithmetic genus

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the concept of [[genus of a surface]] from [[Riemann surfaces]]/[[complex curves]] to [[algebraic varieties]], hence to [[algebraic curves]].

Further base-chaning to [[arithmetic curves]] there is the _[[genus of a number field]]_ (in the sense of the [[function field analogy]]).

## Definition

(...)

essentially the [[Euler characteristic]] $\chi(\mathcal{O}_{\Sigma})$ of the [[structure sheaf]]:

$$
  g_\Sigma = - (\chi(\mathcal{O}_\Sigma) - 1)
$$

(...)

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## References

* {#Neukirch92} [[Jürgen Neukirch]], _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_,  Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))

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

* _[How to see the genus](http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/)_

[[!redirects genus of a curve]]

[[!redirects genus of an algebraic curve]]


arithmetic geometry

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

#Contents#
* table of contents
{:toc}

## Idea

Arithmetic geometry is a branch of [[algebraic geometry]] studying [[schemes]] (usually of [[morphism of finite type|finite type]]) over the [[spectrum (geometry)|spectrum]] [[Spec(Z)]] of the [[commutative ring]] of [[integers]]. More generally, [[algebraic geometry]] over non-[[algebraically closed fields]] or fields of [[positive characteristic]] is also referred to as "arithmetic algebraic geometry".

Since an [[affine variety]] in this context is given by solutions to [[Diophantine equations]], this is also called _Diophantine geometry_.

An archetypical application of arithmetic geometry is the study of [[elliptic curves]] over the [[integers]] and the [[rational numbers]].

For [[number theory|number theoretic]] purposes, i.e. in actual [[arithmetic]]; usually one complements this with some data "at the prime at infinity" leading to a more modern notion of an _[[arithmetic scheme]]_ (cf. [[Arakelov geometry]]).

The refinement to [[higher geometry]] is [[E-infinity geometry]] ([[spectral geometry]]).

## Properties

### Base over $\mathbb{F}_1$

Arithmetic geometry naturally has as [[base topos]] the topos over [[F1]] in the sense of [[Borger's absolute geometry]], which gives an [[essential geometric morphism]] of [[etale toposes]]

$$
  Et(Spec(\mathbb{Z}))  \longrightarrow Et(Spec(\mathbb{F}_1))
  \,.
$$

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[arithmetic variety]], [[arithmetic scheme]]

  * [[arithmetic curve]]

* [[arithmetic Chow group]]

* [[analytic geometry]]

* [[Weil-étale topology for arithmetic schemes]]

* [[absolute cohomology]]

* [[function field analogy]]

* [[Weil conjecture on Tamagawa numbers]]

* [[Borger's absolute geometry]]

* [[Mordell conjecture]]

* [[Iwasawa-Tate theory]]

* [[arithmetic jet space]]

* [[arithmetic differential geometry]]

* [[adelic integration]]

* [[shtuka]]

* [[higher arithmetic geometry]]

* [[E-∞ arithmetic geometry]]

* [[arithmetic topology]]

* [[Frobenioid]]

* [[p-adic physics]]

  * [[p-adic string theory]]

  * [[p-adic AdS/CFT correspondence]]

## References

* Dino Lorenzini, _An Invitation to Arithmetic Geometry_ (Graduate Studies in Mathematics, Vol 9) GSM/9

An almost entirely self-contained introduction and (according to Werner Kleinert) "the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form (after Grothendieck's [[EGA]])":

* Qing Liu, _Algebraic Geometry and Arithmetic Curves_, Oxford Graduate Texts in Mathematics __6__, 2002/2006, 600pp.

Lecture notes include

* {#Sutherland13} [[Andrew Sutherland]], _Introduction to Arithmetic Geometry_, 2013 ([web](http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/index.htm))

* [[Christophe Soulé|C. Soulé]], D. Abramovich, J. F. Burnol, J. K. Kramer, _Lectures on Arakelov Geometry_, Cambridge Studies in Advanced Mathematics __33__, 188 pp.

and with an eye towards [[anabelian geometry]]:

* {#MK09} [[Minhyong Kim]], _Galois Theory and Diophantine geometry_, 2009 ([pdf](http://people.maths.ox.ac.uk/kimm/papers/cambridgews.pdf))

Further resources:

* Wikipedia: *[Glossary of arithmetic and Diophantine geometry](http://en.wikipedia.org/wiki/Glossary_of_arithmetic_and_Diophantine_geometry),

* Wikipedi, *[Arakelov geometry](http://en.wikipedia.org/wiki/Arakelov_theory)*
* _Arakelov geometry preprint arxiv_, list of [links](http://people.math.jussieu.fr/~vmaillot/Arakelov)

* conferences in arithmetic geometry, at [[Kiran Kedlaya]]'s [wiki](http://scripts.mit.edu/~kedlaya/wiki/index.php?title=Conferences_in_Arithmetic_Geometry)

Some relation to [[modular tensor categories]]:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

[[!redirects arithmetic geometries]]

[[!redirects Diophantine geometry]]
[[!redirects Diophantine geometries]]


arithmetic geometry - contents

**[[number theory]]**

* [[arithmetic]]

* [[arithmetic geometry]], [[arithmetic topology]]

* [[higher arithmetic geometry]], [[E-∞ arithmetic geometry]]

**[[number]]**

* [[natural number]], [[integer number]], [[rational number]], [[real number]], [[irrational number]], [[complex number]], [[quaternion]], [[octonion]], [[adic number]], [[cardinal number]], [[ordinal number]], [[surreal number]]

**[[arithmetic]]**

* [[Peano arithmetic]], [[second-order arithmetic]]

* [[transfinite arithmetic]], [[cardinal arithmetic]], [[ordinal arithmetic]]

* [[prime field]], [[p-adic integer]], [[p-adic rational number]], [[p-adic complex number]]

**[[arithmetic geometry]]**, [[function field analogy]]

* [[arithmetic scheme]]

  * [[arithmetic curve]], [[elliptic curve]]

  * [[arithmetic genus]]

* [[arithmetic Chern-Simons theory]]

* [[arithmetic Chow group]]

* [[Weil-étale topology for arithmetic schemes]]

* [[absolute cohomology]]

* [[Weil conjecture on Tamagawa numbers]]

* [[Borger's absolute geometry]]

* [[Iwasawa-Tate theory]]

  * [[arithmetic jet space]]

* [[adelic integration]]

* [[shtuka]]

* [[Frobenioid]]

**[[Arakelov geometry]]**

* [[arithmetic Riemann-Roch theorem]]

* [[differential algebraic K-theory]]


arithmetic geometry -- contents

[[arithmetic]]/[[number theory]], [[geometry]]

**[[arithmetic geometry]]**, **[[E-∞ geometry]]**

* [[Arakelov geometry]]

* [[arithmetic scheme]]

* [[arithmetic Chow group]]


arithmetic jet space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An analog of [[jet spaces]] in [[arithmetic geometry]].

## Definition
 {#Definition}

Notice that the [[p-adic integers]] $\mathbb{Z}_p$ are (by the discussion at [p-adic integer -- as formal power series](p-adic+integer#AsFormalNeighbourhoodOfPrime))  the analog in [[arithmetic geometry]] of a [[formal power series]] ring (around the point $p \in$ [[Spec(Z)]]), hence their [[formal spectrum]] $Spf(\mathbb{Z}_p)$ is an incarnation in [[arithmetic geometry]] of an abstract [[formal disk]].

Therefore in the sense of [[synthetic differential geometry]] the $p$-[[formal neighbourhood]] of any [[arithmetic scheme]] $X$ around a global point $x \colon Spec(\mathbb{Z}) \to X$ is the space of lifts

$$
  \array{
    Spf(\mathbb{Z}_p)  && \stackrel{\hat x}{\longrightarrow}&& X
    \\
    & \searrow &&  \swarrow
    \\
    && Spec(\mathbb{Z})
  }
  \,.
$$

Moreover the map that sends an commutative ring, hence an [[arithmetic variety]], to its $p$-formal power series in this sense is the construction of the [[ring of Witt vectors]] ($p$-typical Witt vectors if one fixes one prime, and "big Witt vectors" if one considers all at once) - see e.g. [Hartl 06, section 1.1](#Hartl06).

The following definition says essentially this, but further sends the resulting space to [[F1]]-geometry in the sense of [[Borger's absolute geometry]]:

For $X= Spec(R)$ an [[affine scheme]] over [[Spec(Z)]] (hence the formal dual of a [[ring]]), then the _arithmetic jet space_ of $X$ at [[prime]] $p$ is $(W_n)_\ast$ applied to the $p$-adic completion of $X$, where $(W_n)_\ast$ is the [[ring of Witt vectors]]-construction, the [[direct image]] of [[Borger's absolute geometry]] $Et(Spec(\mathbb{Z})) \to Et(Spec(\mathbb{F}_1))$.

The definition is originally due to ([Buium 96, section 2](#Buium96), [Buium 05, section 3.1](#Buium05)), reviewed in ([Buium 13, 1.2.3](#Buium13)) as part of his [[arithmetic differential geometry]] program. The above formulation is in ([Borger 10, (12.8.2)](#Borger10)). Buium and Borger have also defined the notion of an arithmetic jet space for a finite set of primes in ([BB09](#BB09)).

## Related concepts

* [[arithmetic differential geometry]]

## References

The original articles are

* {#Buium96} [[Alexandru Buium]], _Geometry of $p$-jets. Duke Math. J., 82(2):349–367, 1996. ([Euclid](http://projecteuclid.org/euclid.dmj/1077245037))

* {#Buium05} [[Alexandru Buium]], _Arithmetic differential equations_, volume 118 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. ([pdf](http://www.math.unm.edu/~buium/prebook.pdf))

Introduction and survey is in

* {#Buium13} [[Alexandru Buium]], _Differential calculus with integers_ ([arXiv:1308.5194](http://arxiv.org/abs/1308.5194), [slightly differing pdf](http://www.math.unm.edu/~buium/statupdated.pdf))

* [[Alexandru Buium]], _Lectures on arithmetic differential equations_ ([pdf](http://www.lorentzcenter.nl/lc/web/2009/342/presentations/lectures%20A.%20Buium.pdf))

* [[Alexandru Buium]], _Foundations of arithmetic differential geometry_, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, ([AMS](http://bookstore.ams.org/surv-222/), [Preface and Introduction](http://www.math.unm.edu/~buium/foundationsintro.pdf)).

Discussion in the context of the [[function field analogy]] is in

* {#Hartl06} [[Urs Hartl]], _A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic_ ([arXiv:math/0607182](http://arxiv.org/abs/math/0607182))

Discussion in the context of [[Borger's absolute geometry]] over [[F1]] is in

* {#Borger10} [[James Borger]],  _The basic geometry of Witt vectors, II: Spaces_ ([arXiv:1006.0092](http://arxiv.org/abs/1006.0092))

See also

* [[Alexandru Buium]], Taylor Dupuy, _Arithmetic differential equations on $GL_n$, I: differential cocycles_ ([arXiv:1308.0748](http://arxiv.org/abs/1308.0748))

* {#BB09} [[James Borger]], [[Alexandru Buium]], _Differential forms on arithmetic jet spaces_, ([arXiv:0908.2512](https://arxiv.org/abs/0908.2512))

[[!redirects arithmetic jet spaces]]

[[!redirects arithmetic differential equation]]
[[!redirects arithmetic differential equations]]


arithmetic pretopos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An **arithmetic pretopos** is a [[pretopos]] $C$ with a _parameterized [[natural numbers object]]_. A **list-arithmetic pretopos** is a pretopos with all _parameterized list objects_ ([Maietti 10, 2.6](#Maietti10)).

Using the (equivalent) definition given in [Cockett 1990](#Cockett90), a **parameterized list object** is a [[W-type]] for the [[polynomial functor]] $B+ A\times(-)\colon C\to C$. This definition makes sense since a pretopos has [[finite products]] and [[disjoint coproducts]] (here denoted "$+$").

Discussion via its [[internal language]], which is a [[dependent type theory]]... ([Maietti 05](#Maietti05), [Maietti 10, p.6](#Maietti10)).

Maietti ([05](#Maetti05),[10](#Maetti10)) proposed that list-arithmetic pretoposes serve as the **arithmetic universes** that [[André Joyal]] (cf. [Joyal 05](#Joyal05)) once suggested to use for discussion of [[incompleteness theorems]] (cf. [van Dijk/Oldenziel 2020](#VanDijkOldenziel)); they are used directly as the definition of arithmetic universes e.g. in ([Maietti-Vickers 2012](#MaiettiVickers12)).

## Related entries

* [[free monoid]]

* [[Gödel's incompleteness theorem]]

* [[Lawvere's fixed point theorem]]

* [[classifying topos for the theory of objects]]

## References

* {#Cockett90} [[Robin Cockett]], _List-arithmetic distributive categories: Locoi_, JPAA **66** no.1 (1990) pp.1-29.

* {#Cockett97} [[Robin Cockett]], _Finite objects in a locos_, JPAA **116** (1997) pp.169-183.

* {#vanDijkOldenziel20} Joost van Dijk, Alexander Gietelink Oldenziel, _Gödel's Incompleteness after Joyal_, arXiv:2004.10482 (2020). ([abstract](https://arxiv.org/abs/2004.10482))

* {#Joyal05} [[André Joyal]], _The Gödel incompleteness theorem, a categorical approach_, (abstract) Amiens 2005, Cah. Top. Géom. Diff. Cat. **46** no.3 (2005) p.202. ([numdam](http://www.numdam.org/item/CTGDC_2005__46_3_163_0))

* {#Maietti05a} [[Maria Maietti]], _Reflection Into Models of Finite Decidable FP-sketches in an Arithmetic Universe_, Electronic Notes in Theoretical Computer Science
**122** (2005) 105-126 [[doi:10.1016/j.entcs.2004.06.054](https://doi.org/10.1016/j.entcs.2004.06.054)]

* {#Maietti05} [[Maria Maietti]], _Modular correspondence between dependent type theories and categories including pretopoi and topoi_, Mathematical Structures in Computer Science **15**  6 (2005) 1089-1149 [[doi:10.1017/S0960129505004962](https://doi.org/10.1017/S0960129505004962), [pdf](https://www.math.unipd.it/~maietti/papers/tumscs.pdf)]

* {#Maietti10} [[Maria E. Maietti]], _Joyal's arithmetic universe as list-arithmetic pretopos_, TAC **24** 3 (2010) 39-83 [[tac:24-03](http://www.tac.mta.ca/tac/volumes/24/3/24-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/24/3/24-03.pdf)]

* {#MaettiVickers12} [[Maria E. Maietti]], [[Steve Vickers]], _An induction principle for consequence in arithmetic universes_, JPAA **216** (2012) pp.2049-2067. [[doi:10.1016/j.jpaa.2012.02.040](https://doi.org/10.1016/j.jpaa.2012.02.040), [pdf](http://www.math.unipd.it/~maietti/papers/aumv12.pdf)]

* [[Paul Taylor]], _Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe_, Electronic Notes in Theoretical Computer Science **122** (2005) 247-296 [[doi:10.1016/j.entcs.2004.06.059](https://doi.org/10.1016/j.entcs.2004.06.059)]

* {#Vickers16} [[Steve Vickers]], _Sketches for arithmetic universes_, arXiv:1608.0159 (2016) &lbrack[arXiv:1608.01559](http://arxiv.org/abs/1608.01559)]

* {#Vickers17} [[Steve Vickers]], _Arithmetic universes and classifying toposes_ [[arXiv:1701.04611](https://arxiv.org/abs/1701.04611)]

[[!redirects arithmetic pretoposes]]
[[!redirects arithmetic pretopoi]]

[[!redirects list-arithmetic pretopos]]
[[!redirects list-arithmetic pretoposes]]
[[!redirects list-arithmetic pretopoi]]

[[!redirects arithmetic universe]]
[[!redirects arithmetic universes]]


arithmetic Riemann-Roch theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[Grothendieck-Riemann-Roch theorem]] to [[Arakelov geometry]].

## References

* [[Henri Gillet]], Damian Rössler, [[Christophe Soulé]], _An arithmetic Riemann-Roch theorem in higher
degrees_ (2007) [pdf](http://www.ihes.fr/~soule/arrfin.pdf)

[[!redirects arithmetic Grothendieck-Riemann-Roch theorem]]


arithmetic scheme

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

#Contents#
* table of contents
{:toc}

## Idea

An _arithmetic scheme_ is a [[scheme]] in [[arithmetic geometry]], hence a scheme over [[Isbell duality|formal dual]]/[[spectrum of a commutative ring|spectrum]] $Spec \mathbb{Z}$ of the [[ring]] of  [[integers]].

Typically one takes an arithmetic scheme to be ([[regular scheme|regular]]) [[separated scheme|separated]] and of [[scheme of finite type|of finite type]].

## Properties

### Differential K-theory

A [[differential algebraic K-theory]] over arithmetic schemes is considered in ([Bunke-Tamme 12](#BunkeTamme12)).

## Related concepts

* [[arithmetic curve]]

## References

* {#BunkeTamme12} [[Ulrich Bunke]], [[Georg Tamme]], _Regulators and cycle maps in higher-dimensional differential algebraic K-theory_ ([arXiv:1209.6451](http://arxiv.org/abs/1209.6451))

[[!redirects arithmetic schemes]]


arithmetic topology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

## Contents
* table of contents
{:toc}

## Idea

**Arithmetic topology** is a theory describing some surprising [[analogies]] between [[low-dimensional topology|3-dimensional topology]] and [[number theory]] ([[arithmetic]]), where [[knots]] embedded in a [[3-manifold]] behave like [[prime ideals]] in a [[ring of algebraic integers]]. See also at _[Spec(Z) -- As a 3d space containing knots](Spec%28Z%29#As3dSpaceContainingKnots)_.  More broadly, the scope of arithmetic topology is now taken to include the intersection of [[arithmetic geometry]],  [[algebraic topology]] and [[low-dimensional topology]] (see [GGW20](#GGW20)).

Under the original analogy, the [[3-sphere]], $S^3$ corresponds to the ring of [[rational numbers]] $\mathbb{Q}$, or rather (the closure of) $spec(\mathcal{O}_{\mathbb{Q}})$ (i.e., $spec(\mathbb{Z})$), since the 3-sphere has no non-trivial ([[branched cover|unbranched]]) covers while $\mathbb{Q}$ has no non-trivial [[unramified extensions]]. The [[linking number]] between two embedded knots in the 3-sphere then corresponds to the [[Legendre symbol]] between two primes in the ordinary integers.

The so-called _M^2KR dictionary_ (Mazur-Morishita-Kapranov-Reznikov) relates terms from each side of the analogy (see sec 2.2 of [Sikora](#Sikora)).

## The dictionary

### Version of Mazur-Morishita-Kapranov-Reznikov (M^2KR)
 {#VersionOfM^2KR}

1. [[closed manifold|Closed]], [[orientation|orientable]], [[connected topological space|connected]] [[3-manifolds]] correspond to (the closure of) schemes $Spec \mathcal{O}_K$ for number fields $K$.
1. Links correspond to ideals in $\mathcal{O}_K$ and knots correspond to prime ideals (tame in both cases). Knots can be represented by immersions of $S^1$ into $M$, and prime ideals in $\mathcal{O}_K$ can be identified with closed immersions $Spec \mathbb{F} \to Spec \mathcal{O}_K$, where $\mathbb{F}$'s are finite fields. Each link decomposes uniquely as a union of knots and each ideal decomposes uniquely as a product of primes.
1. An algebraic integer corresponds to an embedded surface (possibly with boundary), and the operation $a \to (a)$ corresponds to taking its boundary. Closed embedded surfaces correspond to units in $\mathcal{O}_K$. Ideals of the form $(a)$ represent the identity in $Cl(K)$, and the links of the form $\partial S$ represent the identity in $H_1(M,\mathbb{Z})$.
1. $Cl(K)$ corresponds to the torsion component of first integral homology.
The free component of $H_1(M,\mathbb{Z})$ corresponds to the group of units
in $\mathcal{O}_K$ after removing the torsion (roots of unity).
1. Finite extensions of number fields correspond to finite branched coverings.
1. $S^3$ is supposed to correspond to $\mathbb{Q}$. Notice that $S^3$ has no nontrivial unbranched covers, and similarly $\mathbb{Q}$ has no nontrivial unramified extensions.
1. A Galois extension $L/K$ with Galois group $G$ induces a morphism
$Spec \mathcal{O}_L \to (Spec \mathcal{O}_L)/G = Spec \mathcal{O}_K$. Such maps correspond to the quotient maps $M \to M/G$ induced by orientation preserving actions of finite groups $G$ on 3-manifolds $M$. One can show that $M/G$ is always a 3-manifold and that the maps $M \to M/G$ are branched
coverings.
1. Let $q = p^n$. Consider the cyclotomic extension $\mathbb{Q}(\zeta_q)$. It is ramified only at $p$. These correspond to cyclic branched covers of knots in $S^3$. The union of these as $q$ ranges over all powers of $p$ should correspond to the universal abelian cover of $S^3 \setminus K$. There is a natural action of $\mathbb{Z}$ on the first homology group of the infinite cyclic cover of the knot complement corresponding to the natural action of the $p$-adic integers on the $p$-torsion of $Cl(\mathbb{Q}(\zeta_{p^{\infty}}))$. This concerns the [[Alexander polynomial]] of the knot and [[Iwasawa theory]]. ([Sikora, pp. 5-6](#Sikora), [Koberda08, pp. 32-33](#Koberda08))

Note: Regarding (4), some have argued that $Cl(K)$ should correspond to the full first integral homology group, (see, e.g., [Goundaroulis & Kontogeorgis](#Goundaroulis)).

The correspondence between $\pi^{et}_1(\mathbb{Z} -\{p\})$ and $\pi_1(S^3 \setminus K)$ can be developed to relate the Legendre symbol for two primes to the linking number of two knots, and further to the Rédei symbol for three primes and Milner's triple linking number. Thus we can find a 'Borromean link' of primes, such as $(13, 61, 397)$, where each pair is unlinked.

#### Disanalogies

1. The algebraic translation of the Poincaré Conjecture is false. $\mathbb{Q}$ is not the only number field with no unramified extensions. Nevertheless, $\hat{H}^i(Spec \mathcal{O}_K, \mathbb{Z}/n\mathbb{Z}) = \hat{H}^i(Spec \mathbb{Z}, \mathbb{Z}/n\mathbb{Z}) (i \in \mathbb{Z}, n \geq 2)$ if and only if $\mathcal{O}_K = \mathbb{Z}$.
1. Let $M_1 \to M$ be a covering of 3-manifolds. A knot $K$ in $M$ does not necessarily lift to a knot in $M_1$, while every prime ideal $p \triangleleft \mathcal{O}_K$ gives rise to an ideal $p \mathcal{O}_L$, where $L/K$ is a Galois number field extension. ([Goundaroulis & Kontogeorgis](#Goundaroulis)).

### Version of Deninger

Similar to M^2KR, but with the introduction of a 2-dimensional foliation on the 3-manifold and a flow such that finite primes $p$ correspond to periodic orbits of length $log N p$ and the infinite primes correspond to the fixed points of the flow ([Deninger02](#Deninger02)). (See also the work of [Baptiste Morin](http://www.math.uni-muenster.de/reine/u/baptiste.morin/) on the Weil-étale topos.)

### Version of Reznikov
 {#ReznikovVariant}

Reznikov has modified the dictionary ([Reznikov 00, section 12](#Reznikov00)) so as to associate a number field with what he calls a $3\frac{1}{2}$-manifold, that is a closed three-manifold $M$, bounding a four-manifold $N$, such that the map of fundamental groups $\pi_1(M) \to \pi_1(N)$ is surjective.

##Explanations for the analogy

[[Barry Mazur]] observed that for an affine spectrum $X = Spec(D)$ of the ring of integers $D$ in a number field, the groups $H^n_{et}(X, \mathbb{G}_{m, X})$ vanish (up to 2-torsion) for $n \gt 3$, and is equal to $\mathbb{Q}/\mathbb{Z}$ for $n = 3$, where $\mathbb{G}_{m, X}$ is the étale sheaf on $X$ defined by associating to a connected finite étale covering $Spec(B) \to X$ the multiplicative group $\mathbb{G}_{m, X}(Y) = B^{\times}$.

Also, there is a non-degenerate pairing for any constructible abelian sheaf $M$,

$$
H^r_{et}(X,M^{'}) \times Ext^{3-r}_X(M,\mathbb{G}_{m, X}) \to H^3_{et}(X,\mathbb{G}_{m, X})\simeq \mathbb{Q}/\mathbb{Z},
$$

where $M^{'} = Hom(M, \mathbb{G}_{m, X})$. This resembles Poincaré duality for 3-manifolds.

[[Minhyong Kim]] argues that the normal bundle of an embedding of a circle corresponding to a prime in $Spec(\mathbb{Z})$ is 2-dimensional ([Kim](#Kim)).

[[Baptiste Morin]] claims to provide a unified treatment via equivariant etale cohomology ([Morin06](#Morin06)).

## Related concepts

* [[Alexander polynomial]]

* [[function field analogy]]

* The [[virtually fibered conjecture]] says that every [[closed manifold|closed]], [[irreducible manifold|irreducible]], [[atoroidal 3-manifold|atoroidal]] [[3-manifold]] with infinite [[fundamental group]] has a [[finite cover]] which is a [[surface]] [[fiber bundle]] over the [[circle]].

* [[arithmetic Chern-Simons theory]]

##References
* {#Deninger02} [[Christopher Deninger]], _A note on arithmetic topology and dynamical systems_, ([arxiv:0204274](http://arxiv.org/abs/math/0204274))

* {#Goundaroulis} Dimoklis Goundaroulis, Aristides Kontogeorgis, _On the Principal Ideal Theorem in Arithmetic Topology_, ([talk](http://users.uoa.gr/~kontogar/talks/GkountPSATHA.pdf), [paper](http://arxiv.org/abs/0705.3937))

* {#Kim} Minhyong Kim, [note](http://minhyongkim.files.wordpress.com/2013/05/baez13-12.pdf)

* {#Koberda08} Thomas Koberda, _Class Field Theory and the MKR Dictionary
for Knots_, ([pdf](http://users.math.yale.edu/users/koberda/minorthesis.pdf))

* {#Morin06} [[Baptiste Morin]], _Applications of an Equivariant Etale Cohomology to Arithmetic Topology_, [arxiv:0602064](http://arxiv.org/abs/math/0602064) and  Utilisation d'une cohomologie étale équivariante en topologie arithmétique, Compositio Math. 144 (2008), no. 1, 32-60.

* {#Morishita09} Masanori Morishita, _Analogies between Knots and Primes, 3-Manifolds and Number Rings_, ([arxiv:0904.3399](http://arxiv.org/abs/0904.3399))

* {#Reznikov00} Alexander Reznikov, _Embedded incompressible surfaces and homology of ramified coverings of three-manifolds_, Selecta Math. 6(2000), 1–39

* {#Sikora} Adam Sikora, _Analogies between group actions on 3-manifolds and number fields_, ([arxiv](http://arxiv.org/abs/math/0107210))

* {#KohnoMorishita06} [[Toshitake Kohno]], [[Masanori Morishita]] (eds.), _Primes and Knots_, Contemporary Mathematics, AMS 2006 ([conm:416](http://www.ams.org/bookstore-getitem/item=CONM-416))

* {#Morishita12} [[Masanori Morishita]], _Knots and Primes: An Introduction to Arithmetic Topology_, 2012, Springer, ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))

* {#GGW20} Claudio Gómez-Gonzáles, [[Jesse Wolfson]], _Problems in Arithmetic Topology_ ([arXiv:2012.15434](https://arxiv.org/abs/2012.15434))

[[!redirects MKR dictionary]]
[[!redirects MKR analogy]]


arithmetic variety

## Idea

[[variety]] in [[arithmetic geometry]]

## Related entries

* [[arithmetic scheme]]

* [[arithmetic curve]]

[[!redirects arithmetic varieties]]


arithmetic zeta function

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

#Contents#
* table of contents
{:toc}

## Idea

Under the [[function field analogy]] one understands the [[Riemann zeta function]] and [[Dedekind zeta function]] as associated to [[arithmetic curves]], hence to [[spaces]] in [[arithmetic geometry]] of [[dimension]] 1. As one passes to [[higher dimensional arithmetic geometry]] the corresponding generalization are the _arithmetic zeta functions_.

## References

* Wikipedia, _[Arithmetic zeta function](http://en.wikipedia.org/wiki/Arithmetic_zeta_function)_

* [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))

[[!redirects arithmetic zeta functions]]


arity

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

# N-ary operations
* table of contents
{: toc}

## Definitions

Given a [[natural number]] $n$, an __n-ary operation__ on a [[set]] $S$ is a [[function]]

$$
  \phi
  \;\colon\;
  \big(
    \mathrm{Fin}(n) \to S
  \big)
  \longrightarrow
  S
$$

from the [[function set]] $\mathrm{Fin}(n) \to S$ to $S$ itself, where $\mathrm{Fin}(n)$ is the [[finite set]] with $n$ elements.

The __arity__ of the operation is $n$.

In general, if the natural number $n$ is not specified, these are called **finitary operations**.

Sets equipped with finitary operations are also called **finitary [[magmas]]** (or "finitary groupoids" in older terminology which now clashes with another meaning of *[[groupoid]]*, see at *[[historical notes on quasigroups]]*).

More generally, a finitary operation in a [[multicategory]] is just a [[multimorphism]].

### Arbitrary arity

More generally, one could use an arbitrary [[set]] instead of a [[finite set]]. However, the generalizations are only definable in [[closed monoidal category|closed]] [[multicategories]], rather than any [[multicategory]].

## Properties

Every set $S$ with an $n$-ary operation $\phi$ comes with an [[endomorphism]] called the *$n$-th [[power operation]]*

$$
  \array{
    S
    &
    \overset{\;\;(-)^n\;\;}{\longrightarrow}
    &
    S
    \\
    x
    &\mapsto&
    \phi \circ diag_n(x)
    \,,
  }
$$

where $ S \overset{diag_n}{\longrightarrow} S^n$ is the [[diagonal morphism]].

## See also

* [[magma]]

* [[monoid]]

* [[operation]]

* [[multicategory]]

* [[operad]]

### $n$-ary algebraic structures

* [[n-ary semigroup]]

* [[n-ary quasigroup]]

* [[n-ary group]]

* [[group]]

## References

* Wikipedia, [n-ary group](https://en.wikipedia.org/wiki/N-ary_group)

[[!redirects n-ary operation]]
[[!redirects n-ary operations]]

[[!redirects n-ary magma]]
[[!redirects n-ary magmas]]

[[!redirects finitary operation]]
[[!redirects finitary operations]]

[[!redirects finitary magma]]
[[!redirects finitary magmas]]


arity class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Regular and Exact categories
+-- {: .hide}
[[!include regular and exact categories - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Arity classes
* table of contents
{: toc}

## Idea

An *arity class* is a class of [[cardinalities]] which is suitable to be the collection of [[arity|arities]] for the operations in an [[algebraic theory]].

## Definition

An **arity class** is a [[class]] $\kappa$ of [[small set|small]] [[cardinalities]] such that

1. $1\in\kappa$.

2. $\kappa$ is closed under indexed sums: if $\lambda\in\kappa$ and     $\alpha: \lambda \to\kappa$, then $\sum_{i\in \lambda} \alpha(i)$ is also in $\kappa$.

3. $\kappa$ is closed under indexed decompositions: if $\lambda\in\kappa$ and $\sum_{i\in \lambda} \alpha(i)\in \kappa$, then each $\alpha(i)$ is also in $\kappa$.

A [[set]] or [[family]] is called **$\kappa$-small** if its cardinality belongs to $\kappa$.  A [[theory]] or other object with a collection of "operations" whose inputs are all $\kappa$-small is called **$\kappa$-ary**.

+-- {: .un_remark}
###### Remark
By [[induction]], the second condition implies closure under iterated indexed sums, in the sense that for any $n\ge 2$, we have
$$\sum_{i_1\in\lambda_1} \; \sum_{i_2\in\lambda_2(i_1)} \cdots
  \sum_{i_{n-1} \in\lambda_{n-1}(i_1,\dots,i_{n-2})} \lambda_n(i_1,\dots,i_{n-1})
$$
is in $\kappa$ if all the $\lambda$'s are.  The first condition may be regarded as the case $n=0$ of this (the case $n=1$ being just "$\lambda\in\kappa$ iff $\lambda\in\kappa$").
=--

+-- {: .un_remark}
###### Remark
An alternative, more category-theoretic, way to state the second and third conditions is that for any [[function]] $f:I\to J$, if ${|J|}\in\kappa$, then ${|I|}\in\kappa$ if and only if all fibers of $f$ are in $\kappa$.
=--

## Examples

* The set $\{1\}$ is an arity class.  A $\{1\}$-ary object is called **unary**.

* The set $\{0,1\}$ is an arity class.  A$\{0,1\}$-ary object is called **subunary**.

* The set $\omega = \mathbb{N} = \{0,1,2,3\dots\}$ is an arity class.  An $\omega$-ary object is called **finitary**.

* For any [[regular cardinal]] $\kappa$, the set of all cardinalities strictly less than $\kappa$ is an arity class, which we abusively denote also by $\kappa$.  The previous example $\omega$ is a special case of this, as is $\{0,1\}$ if we consider $2$ to be a regular cardinal.

* In particular, if $\kappa$ is the "size of the universe" --- e.g., an [[inaccessible cardinal]] for which we have chosen to call sets of cardinality $\lt\kappa$ [[small set|small]], or literally the proper-class cardinality of the [[universe]], depending on how one thinks of it ---, then it is an arity class.  In this case we call $\kappa$-ary objects **infinitary** or $\infty$-ary.

In [[classical mathematics]], these examples in fact exhaust *all* arity classes.  Classically, if $\lambda$ is any cardinal number strictly greater than $1$, then for any cardinal numbers $\mu\le \nu$, we can write $\nu$ as a $\lambda$-indexed sum containing $\mu$.  Hence, if an arity class contains any cardinality $\gt 1$, it must be down-closed, and a down-closed arity class must arise from a regular cardinal.

In [[constructive mathematics]], however, not every arity class besides $\{1\}$ must be downward-closed, and not every downward-closed arity class must arise from a regular cardinal.  Arguably, however, in constructive mathematics one should consider downward-closed arity classes instead of regular cardinals.

## Related pages

* [[algebraic theory]]

* [[∞-ary exact category]], [[∞-ary site]]

* [[arity space]]

## References

* [[Michael Shulman]], "Exact completions and small sheaves".  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)

[[!redirects arity class]]
[[!redirects arity classes]]

[[!redirects unary]]
[[!redirects subunary]]
[[!redirects finitary]]

[[!redirects n-ary]]

[[!redirects κ-ary]]
[[!redirects kappa-ary]]

[[!redirects ∞-ary]]
[[!redirects infinitary]]
[[!redirects infinity-ary]]


arity space

# Arity spaces

* table of contents
{: toc}

## Idea

An **arity space** is a common generalization of [[coherence spaces]], [[finiteness spaces]], and [[totality spaces]] to an arbitrary set of "arities".

+--{: .standout}
This is an original and tentative definition.  In particular, it's not clear whether the allowed sets of arities should be restricted in some way.  Should they be an [[arity class]]?  The only previously studied examples appear to be the cases $\{0,1\}$ (coherence spaces), $\{0,1,2,3,\dots\}$ (finiteness spaces), and $\{1\}$ (totality spaces), which are all arity classes.
=--

## Definition

Let $\kappa$ be a set of [[cardinal numbers]].  Given two [[subsets]] $u,v\subseteq X$ of the same [[set]] $X$, we write $u\perp v$ if $|u\cap v|\in \kappa$.  This relation defines a [[Galois connection]] in the usual way: for $\mathcal{U}\subseteq P(X)$ we have $\mathcal{U}^\perp = \{ v \mid \forall u\in \mathcal{U}. u\perp v \}$.  Since $\perp$ is symmetric, $(-)^\perp$ is self-adjoint on the right.

We define a **$\kappa$-arity space** to be a set $X$ together with a $\mathcal{U}\subseteq P(X)$ that is a fixed point of this Galois connection, $\mathcal{U} = \mathcal{U}^{\perp\perp}$.  We call the sets in $\mathcal{U}$ **$\kappa$-ary** and the sets in $\mathcal{U}^{\perp}$ **co-$\kappa$-ary**.

A **morphism** or **relation** between $\kappa$-arity spaces is a [[relation]] $R: X ⇸ Y$ such that

1. If $u\subseteq X$ is $\kappa$-ary, then $R[u] = \{ y \mid \exists x\in u, R(x,y) \}$ is $\kappa$-ary.
1. If $v\subseteq Y$ is co-$\kappa$-ary, then $R^{-1}[v] = \{ x \mid \exists y\in v, R(x,y) \}$ is co-$\kappa$-ary.

## Examples

* If $\kappa=\{0,1\}$, then a $\kappa$-arity space is precisely a [[coherence space]].

* If $\kappa = \omega = \{0,1,2,3,\dots\}$, then a $\kappa$-arity space is precisely a [[finiteness space]].

* If $\kappa=\{1\}$, then a $\kappa$-arity space is (almost?) precisely a [[totality space]].

## Properties

**Conjecture:** For any $\kappa$, the category of $\kappa$-arity spaces is [[star-autonomous]].

This might follow from constructing it using [[double gluing]] and orthogonality.

## Construction as a Comma Double Category

We can define arity spaces by a variation on the [[double gluing]] construction.

Define a double category $Orth$ of orthogonalities
1. Objects are relations $\bot \subseteq X \times Y$
2. A vertical morphism from $(X_1,Y_1,\perp_1)$ to $(X_2, Y_2, \perp_2)$ exists when $X_1$ and $Y_1$ are orthogonal subsets of $X_2$ and $Y_2$ respectively.
3. A horizontal morphism from $(X_1,Y_1,\perp_1)$ to $(X_2,Y_2,\perp_2)$ is a pair of a function $f_* : X_1 \to X_2$ and $f^* : Y_2 \to Y_1$
4. A square from $f$ to $g$ exists when $f_*$ is the restriction of $g_*$ and similarly for $f^*$ and $g^*$.

Then the (2-)category of arity spaces can be defined as the comma double category (where $Rel$ and $Set$ are viewed as vertically discrete double categories:

\begin{tikzcd}
  Arity(\kappa) \ar[d] \ar[r] \ar[dr,phantom,"\Downarrow"] & Set \times Set^o \ar[d] \\
  Rel \ar[r,"L_\kappa"'] & Orth
\end{tikzcd}

Where $L_\kappa$ maps a set $X$ to the orthogonality $|U \cap V| \leq \kappa$ on $Subset(X) \times Subset(X)$ and a pair of sets $X, Y$ is given the trivial orthogonality $x \perp y = \bot$

[[!redirects arity spaces]]


Arjan Keurentjes

* [webpage](http://www.phys.ens.fr/~arjan/home.html)

## Selected writings

On [[string theory]] [[perturbative string theory vacuum|vacua]] [[KK-compactification|compactified]] on [[orbifolds]] and [[orientifolds]]:

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

On [[U-duality]]:

* [[Arjan Keurentjes]], _The topology of U-duality (sub-)groups_, Class.Quant.Grav. 21 (2004) 1695-1708 ([arXiv:hep-th/0309106](https://arxiv.org/abs/hep-th/0309106))

category: people


Arkaday Tseytlin

* [webpage](http://www.imperial.ac.uk/people/a.tseytlin)

## related $n$Lab entries

* [[AdS-CFT]]

category: people


Arkady Berenstein

Professor of Mathematics in the Mathematics Department of University of Oregon "working in representation theory, quantum groups, Schubert calculus, combinatorics, commutative and noncommutative algebraic geometry".

* [webpage](https://pages.uoregon.edu/arkadiy) at Univ. of Oregon; [papers](https://pages.uoregon.edu/arkadiy/papers.html)
* Arkady Berenstein, [[Andrei Zelevinsky]], _Quantum cluster algebras_, [math.QA/0404446](https://arxiv.org/abs/math/0404446)
* D. Alessandrini, A. Berenstein, [[V. Retakh]], E. Rogozinnikov,  A. Wienhard, _Symplectic groups over noncommutative algebras_ Sel. Math. New Ser. __28__, 82 (2022) [doi](https://doi.org/10.1007/s00029-022-00787-x)
* A. Berenstein, J. Greenstein, _Canonical bases of quantum Schubert cells and their symmetries_, Sel. Math. New Ser. __23__, 2755--2799 (2017) [doi](https://doi.org/10.1007/s00029-017-0316-8)
* Yuri Bazlov, Arkady Berenstein, _Noncommutative Dunkl operators and braided Cherednik algebras_, Selecta Math. (N.S.) __14__ (2009), no. 3-4, 325--372 [pdf](https://pages.uoregon.edu/arkadiy/doubles.pdf), [MR2010k:16044](https://mathscinet.ams.org/mathscinet-getitem?mr=2511188) [doi](https://doi.org/10.1007/s00029-009-0525-x)
* Yuri Bazlov, Arkady Berenstein, _Braided doubles and rational Cherednik algebras_, Adv. Math. __220__ (2009) 1466--1530 [doi](https://doi.org/10.1016/j.aim.2008.11.004)

> We introduce and study a large class of algebras with triangular decomposition which we call __braided doubles__. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of __quasi-Yetter–Drinfeld__ (QYD) __modules__ over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double--this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the __braided Heisenberg double__ attached to the corresponding complex reflection group.

* Arkady Berenstein, Sebastian Zwicknagl, _Braided symmetric and exterior algebras_, Trans. Amer. Math. Soc. 360 (2008) 3429--3472 [doi](https://doi.org/10.1090/S0002-9947-08-04373-0)

category: people


Arkady Tseytlin

Arkady Tseytlin is professor for theoretical [[physics]] at Imperial College London.

* [webpage](http://www3.imperial.ac.uk/people/a.tseytlin)

## Selected writings

Introducing the [[DBI-action]] for [[D-branes]]:

* [[Efim Fradkin]], [[Arkady Tseytlin]], _Non-linear electrodynamics from quantized strings_, Physics Letters B Volume 163, Issues 1–4, 21 November 1985 (doi:10.1016/0370-2693(85)90205-9)

Review:

* [[Arkady Tseytlin]], _Born-Infeld action, supersymmetry and string theory_, in: [[Mikhail Shifman]] (ed.) _[[The many faces of the superworld]]_, pp. 417-452, World Scientific (2000) ([arXiv:hep-th/9908105](https://arxiv.org/abs/hep-th/9908105), [doi:10.1142/9789812793850_0025](https://doi.org/10.1142/9789812793850_0025))

On [[higher curvature corrections]] to the (abelian) [[DBI-action]] for (single) [[D-branes]]:

* {#AndreevTseytlin88} O. D. Andreev, [[Arkady Tseytlin]], _Partition-function representation for the open superstring effective action:: Cancellation of Möbius infinites and derivative corrections to Born-Infeld lagrangian_, Nuclear Physics B Volume 311, Issue 1, 19 December 1988, Pages 205-252 (doi:10.1016/0550-3213(88)90148-4)

On the [[3-brane in 6d]] as [[brane intersection|self-intersection]] of [[M5-branes]]:

* [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl. Phys. B475 (1996) 149 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), doi:10.1016/0550-3213(96)00328-8)

On [[single trace operators]]/[[BMN operators]] in [[D=4 N=4 super Yang-Mills theory]] identified as [[integrable system|integrable]] [[spin chains]] with respect to the [[dilatation operator]], and the correspondence of their spectrum with the [[classical field theory|classical]] [[Green-Schwarz superstring]] on [[anti de Sitter spacetime|AdS5]] under the [[AdS/CFT correspondence]]:

* {#BeisertFrolovStaudacherTseytlin03} [[Niklas Beisert]], [[Sergey Frolov]], [[Matthias Staudacher]], [[Arkady Tseytlin]], _Precision Spectroscopy of AdS/CFT_, JHEP 0310:037, 2003 ([arXiv:hep-th/0308117](https://arxiv.org/abs/hep-th/0308117))

On defect branes in [[D=4 N=4 SYM]] via [[AdS/CFT]]:

* Hongliang Jiang, [[Arkady A. Tseytlin]], *On co-dimension 2 defect anomalies in $\mathcal{N}=4$ SYM and $(2,0)$ theory via brane probes in AdS/CFT* [[arXiv:2402.07881](https://arxiv.org/abs/2402.07881)]

## Related  entries

* [[worldline formalism]], [[beta function]]

* [[Dirac-Born-Infeld action]]

* [[AdS-CFT]]

category: people

[[!redirects A. Tseytlin]]
[[!redirects Arkady A. Tseytlin]]


Arkady Vainshtein

* [webpage](https://www.physics.umn.edu/people/vainshte.html)

## Selected writings

On the [[anomalous magnetic moment]] of the [[muon]]:

* Kirill Melnikov, [[Arkady Vainshtein]], _Theory of the Muon Anomalous Magnetic Moment_, Springer Tracts in Modern Physics 216, 2006

On [[quantum field theory]] and [[string theory]]:

* [[Mikhail Shifman]], [[Arkady Vainshtein]], [[John Wheater]] (eds.),

  _[[From Fields to Strings -- Circumnavigating Theoretical Physics]]_

  Ian Kogan Memorial Collection (In 3 Volumes)

  World Scientific 2005

  [doi:10.1142/5621](https://doi.org/10.1142/5621)

category: people


Arkady Vaintrob

* [webpage](http://pages.uoregon.edu/vaintrob/)

## Selected writings

On ([[Lie algebra weight system|Lie algebra-]])[[weight systems]] on [[chord diagrams]]:

* [[Arkady Vaintrob]], _Vassiliev knot invariants and Lie S-algebras_, Mathematical Research Letters1, 579–595 (1994) ([pdf](https://pdfs.semanticscholar.org/bdc3/ac1d8da476245e2408e481a70b115b3e9aab.pdf))

* [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197))

More on [[Lie algebra weight systems]] arising from [[super Lie algebras]]:

* {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014))

category: people

[[!redirects A. Vaintrob]]
[[!redirects A. Y. Vaintrob]]
[[!redirects A. Yu. Vaintrob]]


Armand Borel

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

category: people


Armin Frei

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=18372)

## Selected writings

On [[monads]] (old term: "triples"), and their [[monad transformations]] in relation to their [[Eilenberg-Moore categories]]:

* {#Frei69} [[Armin Frei]], *Some remarks on triples*, Mathematische Zeitschrift **109** (1969) 269–272 [[doi:10.1007/BF01110118](https://doi.org/10.1007/BF01110118)]

category: people


Arnaud Beauville

__Arnaud Beauville__ is an algebraic geometer and Professeur émérite at Université de Nice. In addition to the study of various "classical" problems of algebraic geometry, he applied complex algebraic geometry to [[integrable model]]s, [[conformal field theory]], study of moduli spaces etc.

* [webpage](http://math.unice.fr/~beauvill), [publications](http://math.unice.fr/~beauvill/bibli.html)

## Selected writings

On [[holomorphic symplectic geometry]]:

* [[Arnaud Beauville]], _Holomorphic symplectic geometry_, 2011 ([pdf](https://math.unice.fr/~beauvill/conf/Lisbon.pdf), [[BeauvilleHolomorphicSymplectic.pdf:file]])

On [[compact hyperkähler manifolds]] as [[Hilbert schemes of points]] of [[K3-surfaces]]:

* {#Beauville83} [[Arnaud Beauville]], _Variétés Kähleriennes dont la premiere classe de Chern est nulle_, Jour.
Diff. Geom. 18 (1983), 755–782 ([euclid.jdg/1214438181](https://projecteuclid.org/euclid.jdg/1214438181))

category: people

[[!redirects A. Beauville]]


Arnaud Spiwack

* [webpage](http://assert-false.science/arnaud/)

## Selected writings

On [[homological algebra]] in [[constructive mathematics]] via [[type theory]]:

* {#CoquandSpiwack} [[Thierry Coquand]], [[Arnaud Spiwack]], *Towards constructive homological algebra in type theory*, in: *Towards Mechanized Mathematical Assistants. MKM Calculemus 2007*,  Lecture Notes in Computer Science **4573** Springer (2007) [[doi:10.1007/978-3-540-73086-6_4](https://doi.org/10.1007/978-3-540-73086-6_4), [pdf](https://hal.inria.fr/inria-00432525/document)]

On notions of [[finite sets]] in [[constructive mathematics]]:

* [[Arnaud Spiwack]], [[Thierry Coquand]], *Constructively Finite?*, in: *Contribuciones cientifícas en honor de Mirian Andrés Gómez*, Universidad de La Rioja (2010) 217-230 [ISBN:978-84-96487-50-5, [inria-00503917](https://inria.hal.science/inria-00503917)]

On a [[dependent linear type theory]]-version of [[system L]]:

* {#Spiwack14} [[Arnaud Spiwack]], _A dissection of L_ (2014) [[pdf](http://assert-false.science/arnaud/papers/A%20dissection%20of%20L.pdf)]

category: people


Arnav Tripathy

* [webpage](https://www.math.harvard.edu/people/tripathy-arnav/)

## Selected writings

Systematic construction of [[Ricci tensor|Ricci flat]] [[Riemannian metrics]] on [[K3]] [[orbifolds]]:

* [[Shamit Kachru]], [[Arnav Tripathy]], [[Max Zimet]], _K3 metrics_ ([arXiv:2006.02435](https://arxiv.org/abs/2006.02435))

* [[Arnav Tripathy]], [[Max Zimet]], _A plethora of K3 metrics_ ([arXiv:2010.12581](https://arxiv.org/abs/2010.12581))

category: people


Arne Ostvaer

__Paul Arne Østvær__ is a Norwegian mathematician specialized in abstract [[homotopy theory]]. Paul Arne used Quillen [[model categories]] to study questions in [[algebraic topology]], [[operator algebras]] and [[algebraic geometry]]. In algebraic geometry this includes $A^1$-homotopy theory, [[motive]]s, and even [[noncommutative motive]]s. In operator algebras he introduced a Quillen model structure on certain category of cubical presheaves on a category of operator algebras.

* [web](http://folk.uio.no/paularne)

* _Homotopy theory of $C^*$-algebras_, Frontiers in Mathematics,
Springer Basel, 2010, ([arxiv/0812.0154](http://arxiv.org/abs/0812.0154), [pdf](http://folk.uio.no/paularne/file0.pdf))

  (on the [[model structure on operator algebras]])

* Markus Spitzweck, Paul Arne Østvær, _Motivic twisted K-theory_, [arxiv/1008.4915](http://arxiv.org/abs/1008.4915)

[[!redirects Paul Arne Østvær]]
[[!redirects Paul Arne Ostvaer]]
[[!redirects Arne Østvær]]
[[!redirects Paul Østvær]]


Arne Strøm

* [webpage](https://www.sv.uio.no/econ/english/people/aca/arnest/)

## Selected writings

Early discussion and results on [[Hurewicz cofibrations]] and [[NDR pairs]]:

* [[Arne Strøm]], _Note on cofibrations_,  Math. Scand.  **19** (1966) 11-14 ([jstor:24490229](https://www.jstor.org/stable/24490229), [dml:165952](https://eudml.org/doc/165952), MR0211403)

* [[Arne Strøm]], _Note on cofibrations II_,  Math. Scand.  **22** (1968) 130--142  ([jstor:24489730](https://www.jstor.org/stable/24489730), [dml:166037](https://eudml.org/doc/166037), MR0243525)

Introducing the [[Strøm model structure]] [[model structure on topological spaces|on topological spaces]] and discussion of the [[classical homotopy category]]:

* [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])

Introducing the

* {#Strom72} [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])

## Related $n$Lab entries

* [[Strøm model structure]]

* [[Strøm's theorem]]

[[!redirects Arne Strom]]


Arnold conjecture

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[symplectic geometry]]

## Literature

Review and introduction:

* Roman Golovko, *On variants of the Arnold conjecture*, Archivum Mathematicum, Tomus 56 (2020), 277–28 ([pdf](https://www.emis.de/journals/AM/20-5/Golovko_2020.pdf), [doi:10.5817/AM2020-5-277](http://dx.doi.org/10.5817/AM2020-5-277))

Via [[algebraic topology]]/[[stable homotopy theory]] and specifically in relation to [[Morava K-theory]]:

* [[Mohammed Abouzaid]], [[Andrew J. Blumberg]], _Arnold Conjecture and Morava K-theory_, ([arXiv:2103.01507](https://arxiv.org/abs/2103.01507))


Arnold Neumaier

[Arnold Neumaier](
http://arnold-neumaier.at)

I am interested in the foundations of mathematics
(building an automatic mathematical research system),
the foundations of theoretical physics (interpretation
of quantum mechanics and statistical physics),
and in applied mathematics of all sorts (from protein folding
to cattle breeding to uncertainty modeling in space system design).

I wrote books on topics in finite geometry, numerical analysis, and Lie algebras (different ones for each of
these subjects).

## Selected writings

On the foundations and [[interpretation of quantum mechanics|interpretation]] of [[quantum physics]]:

* [[Arnold Neumaier]], *Introduction to [[coherent product|coherent spaces]]*, ([arXiv:1804.01402](https://arxiv.org/abs/1804.01402))

* [[Arnold Neumaier]], *Coherent Quantum Physics: A Reinterpretation of the Tradition*, 1st edition, De Gruyter (2019) [ISBN:9783110667295, [doi:10.1515/9783110667387](https://doi.org/10.1515/9783110667387)]

On [[generalized quantum measurement]] via [[POVMs]] (and [[Born's rule]]):

* [[Arnold Neumaier]], *Born's rule and measurement* [[arXiv:1912.09906](https://arxiv.org/abs/1912.09906)]

category: people


Arnold Nordsieck

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

## related $n$Lab entries

* [[infrared divergence]], [[quantum electrodynamics]]

category: people


Arnold Shapiro

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

## Selected writings

Introducing the [[Atiyah-Bott-Shapiro orientation]] [[MSpin]]$\to$[[KO]] and [[MSpin^c|MSpinc]]$\to$[[KU]]:

* {#AtiyahBottShapiro64} [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]], _Clifford modules_, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, [pdf](http://dell5.ma.utexas.edu/users/dafr/Index/ABS.pdf))

category: people


Arnold Sommerfeld

One of the founding fathers of [[quantum mechanics]].

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

## Related entries

* [[Bohr-Sommerfeld quantization]]

category: people