chord diagram

> This entry is about standard round chord diagrams. For [[linear chord diagrams]] or [[horizontal chord diagrams]] or [[Jacobi diagrams]] or [[Sullivan chord diagrams]] see there.

***

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

\tableofcontents

## Idea
 {#Idea}

A _chord diagram_ is a finite trivalent undirected [[graph]] with an embedded [[orientation|oriented]] [[circle]] and all [[vertices]] on that circle, regarded modulo cyclic identifications, if any.

A typical chord diagram looks like this:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Equivalently this is a pairing (by chords) of all elements in a [[cyclic order]] (the boundary vertices).

Chord diagrams are a basic object of study in [[combinatorics]] with remarkably many applications in [[mathematics]] and [[physics]], notably in [[knot theory]] and [[Chern-Simons theory]] (where they control [[Vassiliev invariants]]) but more recently also in [[string theory|stringy]] [[quantum gravity]] (see the references [below](#ReferencesWeightSystemsOnChordDiagramsInPhysics)).

If also (trivalent) internal vertices are considered, one speaks of _[[Jacobi diagrams]]_. Under standard [[equivalence relations]] these are actually equivalent to chord diagrams, see [below](#RelationToChernSimonsDiagrams)).

## Definitions

Chord diagrams can be defined both in topological terms, which formalize the graphical intuition, as well as in purely combinatorial terms.
We present only the combinatorial definition here, starting from _[[rooted chord diagrams]]_ and using those to define unrooted chord diagrams, rather than the other way around.

(For topological definitions of both unrooted and [[rooted chord diagrams]], see for example Chapter 6 of [Lando and Zvonkin]({#LandoZvonkin}), or Definition 1.5 of [Bar-Natan 1995](#Bar-Natan1995) for the unrooted case.)

### A combinatorial definition

**Rooted chord diagrams**

If we would label the points of a chord diagram,

[[!include chord diagram > chord diagram]]

then the chords can be seen as describing a matching (or equivalently, an [[involution]]) on the set of marked points which pairs (or equivalently, _swaps_) $A$ with $C$, $B$ with $D$, and $E$ with $F$.

A subtle point is that a chord diagram can have non-trivial [[symmetries]] when considered up to this action of rotation.
For that reason, in many situations it is useful to equip the circle with a distinguished [[basepoint]] (distinct from the marked points), the result being called a _[[rooted chord diagram]]_.
Rooted chord diagrams may be conveniently visualized by "cutting open" the circle at the basepoint.
For example, here we have rooted the chord diagram above at a basepoint lying on the interval AF, and opened up the circle:

[[!include chord diagram > arc diagram latin]]

The marked points of a [[rooted chord diagram]] are intrinsically equipped with a [[linear order]] ($A \lt B \lt C \lt D \lt E \lt F$) rather than a cyclic order.

Finally, it is also at times useful to consider chord diagrams with marked points which are not attached to any chord; if we think of the diagram as representing an involution, then this allows for the possibility of representing involutions with [[fixed points]].

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

A **[[rooted chord diagram]]** of order $n$ is a [[surjective]] [[monotone function]]
$$
  D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
$$
from the $n$-fold [[coproduct]] of the [[walking arrow]] (seen as a 2-element [[poset]]) into the [[linear order]] $[0,2n-1] = \{ 0 \lt \dots \lt 2n-1 \}$.
The symmetric group $S_n$ acts freely on the domain $[0,1] + \dots + [0,1]$ of a rooted chord diagram of order $n$, and we consider two rooted chord diagrams to be isomorphic if they only differ by precomposition with such a permutation action.

=--

Equivalently, a rooted chord diagram of order $n$ is a way of arranging the elements of $[0,2n-1]$ into a collection of (necessarily mutually disjoint) ordered pairs $(D_{10}, D_{11}), \dots, (D_{n0},D_{n1})$, where $D_{i0} \lt D_{i1}$ for each $1 \le i \le n$.

Equivalently, a rooted chord diagram of order $n$ is a [[shuffle]] of $n$ distinct 2-letter words $\bar x x$, $\bar y y$, $\bar z z$, etc., considered up to [[alpha-conversion]]. Under this view, rooted chord diagrams are also sometimes referred to as **double-occurrence words**. (For example, the rooted chord diagram in the illustration above can be encoded as the double-occurrence word $\bar x \bar y x y \bar z z$.)

**Unrooted chord diagrams**

Intuitively, the [[cyclic group]] $C_{2n}$ acts on rooted chord diagrams
$$
  D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
$$
by acting on the codomain $[0,2n-1]$ via the map $\tau_n = x \mapsto x-1\,(\mod{2n})$.
This is not quite well-typed, however, since the composite function $\tau_n \circ D$ is not quite order-preserving.
It is easiest to define the action of $C_{2n}$ in terms of the view of a rooted chord diagram $D$ as a collection of ordered pairs $(D_{10}, D_{11}), \dots, (D_{n0},D_{n1})$.
The action of $\tau_n$ is defined on each pair by
$$
 \tau_n(x,y) = \begin{cases}(x-1,y-1) & x\gt 0 \\ (y-1,2n-1) & x = 0 \end{cases}
$$
and this extends to an action on rooted chord diagrams by acting independently on the pairs.

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

An **unrooted chord diagram** of order $n$ is a rooted chord diagram of order $n$, considered up to the action of $\tau_n$.

=--

### 4T relation and weight systems
 {#4TRelationAndWeightSystems}

Often one is interested in chord diagrams only modulo the [[4T relation]]:

For $R \in $ [[CRing]] a [[commutative ring]], let $R\langle \mathcal{D}^c \rangle$ denote the $R$-[[linear span]] of the [[set]] $\mathcal{D}^c$ of [[chord diagrams]].

Then one traditionally writes

\[
  \label{QuotientSpaceBy4T}
  \mathcal{A}^c
    \;\coloneqq\;
  R\langle \mathcal{D}^c \rangle/4T
\]

for the [[quotient spaces]] of the [[linear span]] of [[chord diagrams]] by the [[4T relations]]:

Hence:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

A [[linear function]]

$$
  w \;\colon\; \mathcal{A}^c \longrightarrow R
  \,,
$$

-- hence a plain $R$-valued [[function]] on the [[set]] $\mathcal{D}^c$ of [[chord diagrams]] which is [[invariant]] under the [[4T relation]] --

is called a _[[framed weight system]]_. See there for more.

### Trace from horizontal chord diagrams
 {#TraceFromHorizontalChordDiagrams}

Given a [[horizontal chord diagram]] on $n$ strands and given any choice of [[cyclic permutation]] of $n$ elements, the [[trace of horizontal to round chord diagrams]] is the [[round chord diagram]] obtained by gluing the ends of the strands according to the cyclic permutation, and retaining the chords in the evident way.

The following graphics shows basic examples of the trace operation for [[cyclic permutation]] of strands one step to the right:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

This defines a [[function]]

$$
  tr \;\colon\; \mathcal{C}^{pb} \longrightarrow \mathcal{C}^c
$$

from the [[set]] of [[horizontal chord diagrams]] to the set of [[round chord diagrams]].

## Properties

### Representation as involutions

Every rooted chord diagram
$$
  D : \underset{n\,\text{times}}{\underbrace{[0,1] + \dots + [0,1]}} \twoheadrightarrow [0,2n-1]
$$
uniquely determines a fixed point free involution
$$
  i_D : \{0,\dots,2n-1\} \to \{0,\dots,2n-1\}
$$
swapping the elements of each chord,
$$
  i_D(x) = \begin{cases}y & (x,y) \in D \\ y & (y,x) \in D\end{cases}
$$
and conversely, any such involution uniquely determines a rooted chord diagram up to isomorphism.
In particular, there are a double-factorial number
$$
 (2n-1)!! = (2n-1)(2n-3)\cdots 1
$$
of distinct isomorphism classes of rooted chord diagrams of order $n$.

### Symmetries

Every rooted chord diagram uniquely determines a chord diagram simply by forgetting the basepoint.
Conversely, an unrooted chord diagram of order $n$ can be rooted in up to $2n$ different ways (corresponding to the $2n$ intervals between the marked points), but might also have fewer rootings in the case of symmetries.

### Relation to Jacobi diagrams
 {#RelationToChernSimonsDiagrams}

[[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Gauss diagrams of ordinary (virtual) knots

Any [[knot diagram]] can be represented faithfully by a certain kind of chord diagram with some extra structure, known as a _Gauss diagram_ [(Polyak and Viro 1994)](#PV94).

Recall that classically, a knot is defined as an embedding of the circle $S^1$ into $\mathbb{R}^3$, which can be represented two-dimensionally by projecting onto the plane $\mathbb{R}^2$ and marking each double point as either an over-crossing or an under-crossing.
One obtains a chord diagram from this knot diagram by considering the original immersing circle and connecting the [[preimages]] of each double point by a chord.
Then, information about crossings can be represented by orienting the chords from over-crossing to under-crossing, while information about the local [[writhe]] of each crossing (in the case of a [[framed knot]]) may be encoded by assigning each chord an additional positive or negative sign.
Finally, fixing a [[base point]] on the original knot gives rise to a Gauss diagram whose underlying chord diagram is naturally rooted.

Once again, this geometric object can be abstracted into a purely combinatorial one, called the _Gauss code_ of the knot (diagram).
To construct the Gauss code, begin by assigning arbitrary labels to the crossings, then run the length of the knot (starting at some arbitrary base point) and output the name of each crossing as you visit it, together with a [[bit]] (or two bits in the case of a framed knot) specifying whether you are at an over-crossing or an under-crossing (of positive or negative writhe).
The underlying (rooted) chord diagram of the knot is thus encoded as a double-occurrence word.

Polyak and Viro called these "Gauss diagrams" after [[Carl Gauss]], who apparently studied the question of which chord diagrams arise from immersions of circles (see the chapter titled "Gauss is back: curves in the plane" of Ghys's _[A singular mathematical promenade](#Ghys16)_).
This question also played a foundational role in [[virtual knot theory]] — indeed, according to [Kauffman (1999)](#Kauffman1999), the "fundamental combinatorial motivation" for the definition of virtual knots was the idea that it should be possible to interpret an _arbitrary_ Gauss diagram as encoding a knot, now no longer seen as an embedding of $S^1$ into $\mathbb{R}^3$, but into a thickened surface of arbitrary genus.

## Chord diagrams of singular knots

By an analogous mechanism, any [[singular knot]] $K$ with $n$ simple double points (i.e., points where the knot intersects itself transversally) gives rise to a chord diagram $c(K)$ with $n$ chords, by connecting the preimages of these double points.
This chord diagram $c(K)$ of course does not faithfully represent the original knot $K$ (e.g., it does not include any information about over-crossings and under-crossings).
However, such chord diagrams nonetheless play an important role in the theory of [[Vassiliev invariants]], by the theorem that the value of a Vassiliev invariant $v$ of degree $\le n$ on any singular knot $K$ with $n$ simple double points depends only on the chord diagram $c(K)$.

## Related concepts

[[!include chord diagrams and weight systems -- table]]

* [[Motzkin number]]

* [[combinatorial map]]

## References

### General

Early consideration of round chord diagrams (as "linked diagrams"):

* Paul R. Stein, *On a class of linked diagrams, I. Enumeration*, Journal of Combinatorial Theory, Series A Volume 24, Issue 3, May 1978, Pages 357-366 (doi:10.1016/0097-3165(78)90065-1)

* Paul R. Stein, C. J. Everett, *On a class of linked diagrams II. Asymptotics*, Discrete Mathematics Volume 21, Issue 3, 1978, Pages 309-318 (doi:10.1016/0012-365X(78)90162-0)

Original discussion of chord diagrams in the context of [[Vassiliev invariants]]:

* {#BarNatan95} [[Dror Bar-Natan]], _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

Textbook accounts:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of:  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* [[David Jackson]], [[Iain Moffat]], Section 11 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

Lecture notes:

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

For chord diagrams and arc diagrams from the perspective of [[combinatorics]], see:

* Jacques Touchard, _Sur un problème de configurations et sur les fractions continues_ Canadian Journal of Mathematics 4 (1952), 2-25. ([doi](http://dx.doi.org/10.4153/CJM-1952-001-8))

* Philippe Flajolet and Marc Noy, _Analytic combinatorics of chord diagrams_, INRIA technical report 3914, March 2000. ([pdf](http://algo.inria.fr/flajolet/Publications/FlNo00.pdf))

* A. Khruzin, _Enumeration of chord diagrams_ arXiv:math/0008209, August 2000. ([arxiv](http://arxiv.org/abs/math/0008209))

For the relationship to Gauss diagrams in classical and [[virtual knot theory]], see:

* {#PV94} Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, _International Mathematics Research Notes_ 1994:11. [pdf](http://www.math.stonybrook.edu/~oleg/math/papers/1994-Polyak-Viro.pdf)

* {#Kauffman1999} [[Louis Kauffman]], Virtual Knot Theory, _European Journal of Combinatorics_ (1999) 20, 663-691. [pdf](http://homepages.math.uic.edu/~kauffman/VKT.pdf)

* {#Viro05} Oleg Viro, Virtual Links, Orientations of Chord Diagrams and Khovanov Homology, Proceedings of 12th Gökova Geometry-Topology Conference, pp. 184–209, 2005. [pdf](http://www.pdmi.ras.ru/~olegviro/ggt05-viro.pdf)

* [Gauss codes](http://katlas.org/wiki/Gauss_Codes) at the Knot Atlas.

For the connection to [[Vassiliev invariants]] of [[singular knots]], see:

* {#LandoZvonkin} [[Sergei K. Lando]] and [[Alexander K. Zvonkin]], _Graphs on Surfaces and Their Applications_, Springer, 2004. (Chapter 6.)

* {#Bar-Natan1995} [[Dror Bar-Natan]], On the Vassiliev knot invariants, _Topology_ 34 (1995), 423-472. ([html](http://www.math.toronto.edu/~drorbn/papers/OnVassiliev/))

* [[Simon Willerton]], Vassiliev invariants and the [[Hopf algebra]] of chord diagrams, Math. Proc. Camb. Phil. Soc. (1996), 119, 55.

* Greg Muller, ["Chord Diagrams and Lie Algebras"](https://cornellmath.wordpress.com/2007/12/25/chord-diagrams-and-lie-algebras/), blog post at _The Everything Seminar_, 25 December 2007.

The following book contains an extensive discussion of chord diagrams associated to singular points of analytic curves:

* {#Ghys16} Étienne Ghys, _A Singular Mathematical Promenade_, preliminary version of December 2016. ([arXiv:1612.06373](https://arxiv.org/abs/1612.06373)) ([author pdf](http://perso.ens-lyon.fr/ghys/articles/Promenade.pdf))

On [[Vassiliev invariants]] of [[braid group|braids]] via [[chord diagrams]]:

* [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

Relation to [[spherical curves]]:

* Noboru Ito, _Based chord diagrams of spherical curves_ ([arXiv:2004.12121](https://arxiv.org/abs/2004.12121))

See also:

* Ali Assem Mahmoud, Karen Yeats, _Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions_ ([arXiv:2010.06550](https://arxiv.org/abs/2010.06550))

[[!include weight systems on chord diagrams in physics]]

[[!redirects chord diagrams]]

[[!redirects chord]]
[[!redirects chords]]

[[!redirects Gauss diagram]]
[[!redirects Gauss diagrams]]
[[!redirects Gauss code]]
[[!redirects Gauss codes]]
[[!redirects double-occurrence word]]
[[!redirects double-occurrence words]]

[[!redirects round chord diagram]]
[[!redirects round chord diagrams]]

[[!redirects algebra of chord diagrams]]
[[!redirects algebras of chord diagrams]]


chord diagram > arc diagram

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chord diagram > arc diagram latin

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chord diagram > chord diagram

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chord diagram > chord diagram 2a

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chord diagram > chord diagram 2b

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chord diagram > chord diagram unlabelled

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chord diagrams and weight systems -- table

| [[chord diagrams]] | [[weight systems]] |
|--|--|
| [[linear chord diagrams]],
 [[round chord diagrams]]
 [[Jacobi diagrams]],
[[Sullivan chord diagrams]] | [[Lie algebra weight systems]],
 [[stringy weight system]],
 [[Rozansky-Witten weight systems]] |

\linebreak

| [[knots]] | [[braids]] |
|-----------|------------|
| [[chord diagram]],
 [[Jacobi diagram]] | [[horizontal chord diagram]] |
| [[1T relation|1T]]&[4T relation](4T+relation#ForCircularChordDiagrams) | [[2T relation|2T]]&[4T relation](4T+relation#ForHorizontalChordDiagrams)/
 [[infinitesimal braid relations]] |
| [[weight system]] | [[horizontal weight system]] |
| [[Vassiliev knot invariant]] | [[Vassiliev braid invariant]] |
| [[weight systems are associated graded of Vassiliev invariants]] | [[horizontal weight systems are cohomology of loop space of configuration space]] |


chord diagrams as multi-trace observables in BMN matrix model -- example

### Chord diagrams as multi-trace observables in the BMN matrix model
 {#AsLieWeightSystemsInBMNModel}

The [[supersymmetry|supersymmetric]] [[states]] of the [[BMN matrix model]] are temporally constant complex matrices which are [[complex number|complex]] [[metric Lie representations]] $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of $\mathfrak{g}=$[[su(2)]] (interpreted as [[fuzzy 2-sphere]] [[noncommutative geometries]] of [[giant gravitons]] or equivalently as [[fuzzy funnels]] of [[D0-D2 brane bound states]]).

A [[fuzzy 2-sphere]]-[[rotation]] [[invariant]] [[multi-trace observable]] on these supersymmetric states is hence an expression of the following form:

> (from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]])

Here we are showing the corresponding [[string diagram]]/[[Penrose notation]] for [[metric Lie representations]], which makes manifest that

1. these multi-trace observables are encoded by [[Sullivan chord diagrams]] $D$

1. their value on the supersymmetric states $\mathfrak{su}(2) \otimes V \overset{\rho}{\to}V$ is the [[evaluation]] of the corresponding [[Lie algebra weight system]] $w_{{}_V}$ on $D$.

Or equivalently, if $\widehat D$ is a [[horizontal chord diagram]] whose $\sigma$-[[permutation|permuted]] closure is $D$ (see [here](horizontal+chord+diagram#ClosureToSullivanChordDiagrams)) then the values of the invariant multi-trace observables on the supersymmetric states of the BMN matrix model are the evaluation of $w_{V,\sigma}$ on $\widehat D$, as shown here:

> (from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]])

But since [[all horizontal weight systems are partitioned Lie algebra weight systems]] this way, this identifies supersymmetric states of the BMN matrix model as seen by invariant multi-trace observables as [[horizontal chord diagrams]] evaluated in [[Lie algebra weight systems]].


chord diagrams modulo 4T are Jacobi diagrams modulo STU

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

#Contents#
* table of contents
{:toc}

## Ingredients

Consider

1. the [[set]] $\mathcal{D}^c$ of [[chord diagrams]];

1. the [[set]] $\mathcal{D}^t$ of [[Jacobi diagrams]];

Write

\[
  \label{InjectionOfChordDiagramsIntoJacobiDiagrams}
  \mathcal{D}^c
    \overset{i}{\hookrightarrow}
  \mathcal{D}^t
\]

for the canonical [[injection]], regarding [[chord diagrams]] as [[Jacobi diagrams]] without internal [[vertices]], hence with all [[vertices]] on the circle.

For $R \in $ [[CRing]] a [[commutative ring]], write

1. $R\langle \mathcal{D}^c \rangle$ for the $R$-[[linear span]] of [[chord diagrams]];

1. $R\langle \mathcal{D}^t \rangle$ for the $R$[[linear span]] of [[Jacobi diagrams]]

and finally

\[
  \label{QuotientSpaces}
  \mathcal{A}^c
    \;\coloneqq\;
  R\langle \mathcal{D}^c \rangle/4T
  \,,
  \phantom{AAAA}
  \mathcal{A}^t
    \;\coloneqq\;
  R\langle \mathcal{D}^t\rangle/STU
\]

for the respective [[quotient spaces]] by the [[4T relations]]

and by the [[STU relations]]

respectively.

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Statement
 {#Statement}

The linear extension of the canonical inclusion $\mathbb{D}^c \overset{i}{\hookrightarrow} \mathbb{D}^t$ (eq:InjectionOfChordDiagramsIntoJacobiDiagrams) descends to the quotients (eq:QuotientSpaces) and yields a [[linear isomorphism]]:

$$
  \mathcal{A}^c
  \underoverset{\simeq}{i}{\longrightarrow}
  \mathcal{A}^t
$$

This is due to [Bar-Natan 95, Theorem 6](#BarNatan95). See also [Chmutov-Duzhin-Mostovoy 11, 5.3](#ChmutovDuzhinMostovoy11)

> graphics from [Sati-Schreiber 19]()

The key step of the **proof** is to observe that the [[STU-relations]] imply the [[4T-relations]] as follows:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Application

### Relation to weight systems

The immediate consequence is that the space of [[framed weight systems]] $\mathcal{W}$, which by definition is the [[dual vector space]] to the [[linear span]] of [[chord diagrams]] [[quotient vector space|modulo]] the [[4T-relations]], is equiuvalently also the [[dual vector space]] to the [[linear span]] of [[chord diagrams]] [[quotient vector space|modulo]] the [[4T-relations]]:

$$
  \begin{aligned}
    \mathcal{W}
    & \coloneqq
    (\mathcal{A}^c)^\ast
    \\
    & \simeq
    (\mathcal{A}^t)^\ast
  \end{aligned}
$$

## Related theorems

[[!include facts about chord diagrams and weight systems -- contents]]

## Related concepts

* [[Lie algebra weight system]]

## References

The result is due to

* {#BarNatan95} [[Dror Bar-Natan]], Theorem 6 in: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

Lecture notes

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

Textbook accounts:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 5.3 of  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

[[!redirects chord diagrams modulo 4T are Chern-Simons diagrams modulo STU]]


Chow group

#Contents#
* table of contents
{:toc}

## Idea

The _Chow groups_ of a [[noetherian scheme]] $X$ are the analogs of the [[singular homology]] groups of a [[topological space]].

## Definition

Let $X$ be a [[noetherian scheme]].  One defines the **$k$-th Chow group** of $X$ as the [[quotient]] of the group $Z_k(X)$ of [[algebraic cycles]] of [[dimension]] $k$ by the [[subgroup]] of [[algebraic cycles]] [[rational equivalence|rationally equivalent]] to zero:

$$ CH_k(X) \coloneqq Z_k(X) / \sim_{\rat} $$

The **Chow ring** is the [[graded ring]] which is the [[direct sum]] of the Chow groups, with multiplication being the [[intersection product]].

More generally one can use any [[adequate equivalence relation]] $\sim$ (e.g. $\sim_{num}, \sim_{hom}, \sim_{alg}$) in place of rational equivalence, to get groups

$$ CH^{\sim}_k(X) = Z_k(X) / \sim $$

## Cohomological interpretation

Chow groups appear as the [[cohomology groups]] of [[motivic cohomology]] (see there for details) with [[coefficients]] in suitable [[Eilenberg-MacLane objects]].

## Related concepts

* [[arithmetic Chow group]]
* [[adequate equivalence relation]]
* [[Chow motive]]

## References

Named after [[Wei-Liang Chow]].

The canonical reference is

* [[William Fulton]], _Intersection theory_, 1998.  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2, Berlin, New York: Springer-Verlag

The original references are

* [[Pierre Samuel]], _Rational Equivalence of Arbitrary Cycles_. American Journal of Mathematics, Vol. 78, No. 2 (Apr., 1956), pp. 383-400

* [[Claude Chevalley]], _Les classes d'equivalence rationnelles I_. Séminaire Claude Chevalley, 3 (1958), Exp. No. 2, 14 (on [NUMDAM](http://www.numdam.org/item?id=SCC_1958__3__A2_0))

* [[Claude Chevalley]], _Les classes d'équivalence rationnelle, II_. Séminaire Claude Chevalley, 3 (1958), Exp. No. 3, 18 (on [NUMDAM](http://www.numdam.org/numdam-bin/item?id=SCC_1958__3__A3_0))

The most general treatment can be found in the [[The Stacks Project]]:

* [[Aise Johan de Jong]] et. al., [[The Stacks Project]], chapter 38, _Chow Homology and Chern Classes_ ([URL](https://stacks.math.columbia.edu/tag/02P3))

See also

* Wikipedia, *[Chow group](https://en.wikipedia.org/wiki/Chow_group)*

Informal lecture notes by [[Jacob Murre]]:

* [[Jacob Murre]], _Lectures on algebraic cycles and Chow groups_.  Summer school on Hodge theory and related topics, ICTP, 2010.  [PDF](http://jfresan.files.wordpress.com/2010/11/lectures-murre.pdf)

A concise definition of the notion of Chow group and related concepts is in

* [this pdf](http://www.math.utah.edu/vigre/minicourses/algebra/spiroff3.pdf) from some [Vigre minicours](http://www.math.utah.edu/vigre/minicourses/) (???)

[[!redirects Chow groups]]
[[!redirects Chow ring]]


Chow groups with modulus

## Idea

Chow groups with modulus are a non-[[homotopy invariant]] variant of [[Chow groups]].

In the theory of [[motives]] one studies [[affine line|A^1]]-[[homotopy invariant]] phenomena, like [[higher Chow groups]]/[[motivic cohomology]] and homotopy invariant [[algebraic K-theory]].
It is expected that the various [[categories of motives]] can be embedded into larger ones which also contain non-homotopy invariant phenomena, like (honest) [[algebraic K-theory]].
(Higher) Chow groups with modulus are expected to play the role of [[higher Chow groups]] in these larger categories.

## See also

* [[motivic cohomology]]
* [[higher Chow groups]]

## References

* [[Kay Rülling]], [[Takao Yamazaki]], _Suslin homology of relative curves with modulus_, [arXiv:1505.05922](http://arxiv.org/abs/1505.05922).

* Federico Binda, [[Shuji Saito]], _Relative cycles with moduli and regulator maps_, [arXiv:1212.0385](http://arxiv.org/abs/1412.0385v2).

* [[Moritz Kerz]], [[Shuji Saito]], _Chow group of 0-cycles with modulus and higher dimensional class field theory_, [arXiv:1304.4400](http://arxiv.org/abs/1304.4400).

* [[Amalendu Krishna]], [[Jinhyun Park]], _A module structure and a vanishing theorem for cycles with modulus_, [arXiv:1412.7396](http://arxiv.org/abs/1412.7396).

* [[Amalendu Krishna]], _0-cycles with modulus on surfaces_, [arXiv:1504.03125](http://arxiv.org/abs/1504.03125).

* [[Kay Rülling]], [[Shuji Saito]], _Higher Chow groups with modulus and relative Milnor K-theory_, [arXiv:1504.02669](http://arxiv.org/abs/1504.02669).

* [[Amalendu Krishna]], [[Jinhyun Park]], _A moving lemma for cycles with very ample modulus_, [arXiv:1507.05429](http://arxiv.org/abs/1507.05429).

* [[Wataru Kai]], _A moving lemma for algebraic cycles with modulus and contravariance_, [arXiv:1507.07619](http://arxiv.org/abs/1507.07619).

### Historical precursors

* [[Bruno Kahn]], [[Shuji Saito]], [[Takao Yamakazi]], _Reciprocity sheaves, I_, [arXiv:1402.4201](http://arxiv.org/abs/1402.4201).

* [[Bruno Kahn]], _Reciprocity sheaves_, notes from a talk in "Homotopical methods in algebraic geometry", USC, Los Angeles, May 28 – June 1, 2013, [pdf](http://webusers.imj-prg.fr/~bruno.kahn/preprints/reciprocity-USC.pdf).

* [[Florian Ivorra]], Kay Rülling, _K-groups of reciprocity functors_, [arXiv:1209.1217](http://arxiv.org/abs/1209.1217).

* [[Spencer Bloch]], [[Hélène Esnault]], _An additive version of higher Chow groups_, [arXiv:math/0112101](http://arxiv.org/abs/math/0112101).

[[!redirects Chow group with modulus]]
[[!redirects higher Chow group with modulus]]
[[!redirects higher Chow groups with modulus]]
[[!redirects motivic cohomology with modulus]]


Chow homology

## References

* [[The Stacks Project]], _Chow homology_ ([pdf](http://stacks.math.columbia.edu/download/chow.pdf))


Chow motive

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

#Contents#
* table of contents
{:toc}

## Definition

The [[category]] of _effective [[pure motives|pure]] Chow [[motive]]s_ , $Mot_{rat}^{eff}(k)$, is the [[idempotent completion]] of the [[category]] $Corr_{rat}(k)$ whose [[objects]] are [[smooth variety|smooth]] [[projective varieties]] over some [[field]] $k$, and whose [[hom-sets]] are [[Chow groups]] in the [[product]] of two varietes (see for instance [Vishik09, def. 2.1](#Vishik09)).

$$
  Hom_{Corr_{rat}(k)}(Y,X) \coloneqq CH^{dim X}(X \times Y)
  \,.
$$

Hence a [[morphism]] $X \to Y$ in $Mor_{rat}^{eff}(k)$ is an [[equivalence class]] of [[linear combinations]] of [[correspondences]]/[[spans]] of the form

$$
  X \leftarrow \Sigma \rightarrow Y
  \,.
$$

If one furthermore inverts the [[Lefschetz motive]] $\mathbf{L}$ then one obtains the category of _pure Chow motives_

$$
  Mot_{rat}(k) \coloneqq Mot_{rat}(k)[\mathbf{L}^{-1}]
  \,.
$$

This was introduced by [[Grothendieck]]. See e.g. ([Vishik09, p. 6](#Vishik09)), ([Mazza-Voevodsky-Weibel, p. 181](#MazzaVoevodskyWeibel)).

This is a special case of the more general notion of _[[pure motives]]_. See there for more.

## Properties

### Relation to Voevodsky motives

There is a [[full and faithful functor]] from the category of Chow motives into that of [[Voevodsky motives]]:

$$
  Chow^{eff} \hookrightarrow DM^{eff}
  \,.
$$

(e.g [Mazza-Voevodsky-Weibel, prop. 20.1](#MazzaVoevodskyWeibel))

### Relation to noncommutative Chow motives

The relation between Chow motives and [[noncommutative Chow motives]] is recalled as theorem 4.6 in ([Tabuada 11](#Tabuada11)).

This relation is best understood via _[[K-motives]]_, see there.

### Relation to KK-theory

The definition of Chow motives in algebraic geometry is somewhat analogous to the construction of [[KK-theory]] in [[noncommutative topology]]. See at _[KK-theory -- Relation to motives](http://ncatlab.org/nlab/show/KK-theory#AsAnAnalogOfMotives)_.

## Related concepts

* [[pure motive]]

  * **Chow motive**

  * [[numerical motive]]

* [[motivic cohomology]]

* [[category of spans]]

## References

A quick and complete statement of the definition is in

* Alexander Vishik, _Chow groups and motives_, lecture notes 2009 ([pdf](http://alg-geo.epfl.ch/cours/geomthqf_vishik_09/docs/D2.pdf))
 {#Vishik09}

See also

* [[Carlo Mazza]], [[Vladimir Voevodsky]] and [[Charles Weibel]], _Lectures in motivic cohomology_ ([web](http://math.rutgers.edu/~weibel/motiviclectures.html) [pdf](http://www.math.rutgers.edu/~weibel/MVWnotes/prova-hyperlink.pdf))
 {#MazzaVoevodskyWeibel}

A review of [[noncommutative Chow motives]] is in section 4 of

* [[Goncalo Tabuada]], _A guided tour through the garden of noncommutative motives_, ([arxiv1108.3787](http://arxiv.org/abs/1108.3787));
 {#Tabuada11}

Discussion of how the [[triangulated categories of sheaves|derived category]] of a [[scheme]] determines its commutative and [[noncommutative motive|noncommutative]] Chow motive is in

* [[Adeel Khan]], _On derived categories and noncommutative motives of varieties_, [arXiv](http://arxiv.org/abs/1401.7222).

[[!redirects Chow motives]]
[[!redirects category of Chow motives]]


Chow's theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+-- {: .hide}
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Chow's theorem_ states that a [[complex analytic space]] which is given as a [[closed subset]] of a complex [[projective space]] is a complex [[algebraic variety]].

e.g. ([Zhao 13, prop. 5.1](#Zhao13), [Neeman 07, theorem 1.1.3](#Neeman07)).

## Related concepts

* [[analytification]]

* [[comparison theorem (étale cohomology)]]

* [[GAGA]]

## References

Named after:

* [[Wei-Liang Chow]], *On compact complex analytic varieties*, American Journal of Mathematics **71** 4 (1949) 893-914 [[doi:10.2307/2372375](https://doi.org/10.2307/2372375)]

Review:

* {#Zhao13} Yan Zhao, section 5 of _Géométrie algébrique et géométrie analytique_, 2013 ([pdf](http://pub.math.leidenuniv.nl/~jinj/2013/efg/gaga.pdf))

* Evan Warner, _Three approaches to Chow's theorem_ ([pdf](https://www.math.columbia.edu/~warner/notes/AlgGeomSemTalk.pdf))

* {#Neeman07} [[Amnon Neeman]], _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry))


Chris Akers

## Selected writings

On [[holographic Renyi entropy]] and [[quantum error correcting codes]]:

* [[Chris Akers]], [[Pratik Rath]], *Holographic Renyi entropy from quantum error correction*,  J. High Energ. Phys. 2019, 52 (2019) ([arXiv:1811.05171](https://arxiv.org/abs/1811.05171))

On [[holographic entanglement entropy]] and [[quantum error correcting codes]]:

* [[Chris Akers]], [[Geoff Penington]], *Quantum minimal surfaces from quantum error correction* ([arXiv:2109.14618](https://arxiv.org/abs/2109.14618))

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

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

category: people


Chris Beasley

* [webpage](http://www.math.neu.edu/people/profile/chris-beasley)

category: people


Chris Brav

I am a post-doc at the University of Toronto working on derived and non-commutative algebraic geometry.

You may email me at brav?@?math?.?toronto?.?edu, with the ?s removed.

* [webpage](http://people.maths.ox.ac.uk/brav/)

## Related $n$Lab entries

* [[Bridgeland stability condition]]

category: people

[[!redirects Christopher Brav]]


Chris D. A. Blair

* [InSpire page](https://inspirehep.net/authors/1253270)

## Selected writings

Generalization of [[Poisson-Lie T-duality]] to [[U-duality]] in [[exceptional generalized geometry]]:

* [[Chris D. A. Blair]], [[Daniel C. Thompson]], Sofia Zhidkova, _Exploring Exceptional Drinfeld Geometries_, JHEP 09 (2020) 151 [[arxiv:2006.12452](https://arxiv.org/abs/2006.12452)]

A review of [[exceptional field theory]]:

* [[David Berman]], [[Chris D. A. Blair]], _The Geometry, Branes and Applications of Exceptional Field Theory_,  International Journal of Modern Physics A **35** 30 (2020) 2030014  [[arXiv:2006.09777](http://arxiv.org/abs/2006.09777), [doi:10.1142/S0217751X20300148](https://doi.org/10.1142/S0217751X20300148)]

category: people

[[!redirects Chris D. Blair]]
[[!redirects Chris Blair]]


Chris Doran

* [institute page](https://www.sid.cam.ac.uk/aboutus/people/cjld1)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Chris_J._L._Doran)

## Selected writings

* {#Doran94} [[Chris Doran]], _Geometric Algebra and its Application to Mathematical Physics_, 1994 ([pdf](http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/DoranThesis.pdf))

* [[Chris Doran]], Anthony Lasenby, _Geometric algebra for physicists_, Cambridge University Press (2003) ([pdf](http://catdir.loc.gov/catdir/samples/cam033/2002035182.pdf))

  (on [[Clifford algebra]] in the guise of _[[geometric algebra]]_ applied to [[mathematical physics]])

## Related $n$Lab entries

* [[geometric algebra]]

category: people


Chris Elliott

* [webpage](http://www.ihes.fr/~celliott/)

## related $n$Lab entries

* [[topological twist]]

* [[topologically twisted D=4 super Yang-Mills theory]]

category: people


Chris Fields

* [research group](https://chrisfieldsresearch.com/)

* [ConstructivistFoundation page](https://constructivist.info/authors/chris-fields)

## Selected writings

On [[quantum biology]], arguing that cellular [[information theory|information processing]] must employ [[quantum coherence]] as a resource for [[reversible computation|reversibility]] in order to maintain a biologically reasonable [[free energy]] budget:

* [[Chris Fields]], [[Michael Levin (biologist)|Michael Levin]], _Metabolic limits on classical information processing by biological cells_ Biosystems **209** (2021) 104513 [[doi:10.1016/j.biosystems.2021.104513](https://doi.org/10.1016/j.biosystems.2021.104513)]

category: people


Chris Grossack

* [website](https://mathdept.ucr.edu/people/chris-grossack)

* [personal website](https://grossack.site)

## Publications

* [[Chris Grossack]], *Extensions of Abelian Automata Groups*, ([arXiv:1903.06997](https://arxiv.org/abs/1903.06997))

## Talks

* [[Chris Grossack]], *Where are the open sets? Comparing HoTT with Classical Topology*, [[Homotopy Type Theory Electronic Seminar Talks]], 6 October 2022, ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Grossack-2022-10-06-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=syftUuf5Hpk))

category: people


Chris Halcrow

* [webpage](https://eps.leeds.ac.uk/staff/5456/dr-chris-halcrow)

## Selected writings

On [[nucleon]]-[[interaction]] modeled via [[Skyrmions]]:

* [[Chris Halcrow]], [[Derek Harland]], _An attractive spin-orbit potential from the Skyrme model_ ([arXiv:2007.01304](https://arxiv.org/abs/2007.01304))

* [[Derek Harland]], [[Chris Halcrow]], _Nucleon-nucleon potential from skyrmion dipole interactions_ ([arXiv:2101.02633](https://arxiv.org/abs/2101.02633))

On [[Skyrmions]]:

* {#GudnasonHalcrow18} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *Vibrational modes of Skyrmions*, Phys. Rev. D **98** 125010 (2018) [[arXiv:1811.00562](https://arxiv.org/abs/1811.00562), [doi:10.1103/PhysRevD.98.125010](https://doi.org/10.1103/PhysRevD.98.125010)]

  *Database of Skyrmion Vibrations* [[www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/](http://www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/)]

* Alberto García Martín-Caro, [[Chris Halcrow]],
*The charge density and neutron skin thickness of Skyrmions* [[https://arxiv.org/abs/2312.04335](https://arxiv.org/abs/2312.04335)]

On [[Skyrmions]] via [[Yang-Mills instantons]]:

* [[Chris Halcrow]], [[Thomas Winyard]], _A consistent two-skyrmion configuration space from instantons_ ([arXiv:2103.15669](https://arxiv.org/abs/2103.15669))

* [[Josh Cork]], [[Chris Halcrow]], *ADHM skyrmions* ([arXiv:2110.15190](https://arxiv.org/abs/2110.15190))

Computer-survey of [[Skyrmion]]-solutions:

* {#GudnasonHalcrow22} [[Sven Bjarke Gudnason]], [[Chris Halcrow]], *A Smörgåsbord of Skyrmions* [[arXiv:2202.01792](https://arxiv.org/abs/2202.01792)]

category: people


Chris Harris

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=168375)

## related $n$Lab entries

* [[Witten genus]]

category: people

[[!redirects Christopher Harris]]


Chris Heunen

Chris Heunen is EPSRC research fellow in [[computer science]] at Oxford.

* [website](http://www.comlab.ox.ac.uk/people/chris.heunen/about.html)

## Selected writings

On [[Bohr toposes]]:

* {#HeunenLandsmanSpitters09} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], *Bohrification of operator algebras and quantum logic*, Synthese **186** 3 (2012) 719-752 [[arXiv:0905.2275](http://arxiv.org/abs/0905.2275), [doi;10.1007/s11229-011-9918-4](https://doi.org/10.1007/s11229-011-9918-4)]

On [[monads]] on [[dagger categories]] with an eye towards [[quantum measurement]] expressed in [[finite quantum mechanics in terms of dagger-compact categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [TAC:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

On the [[category]] of [[matroids]] with *strong maps* between them:

* [[Chris Heunen]], [[Vaia Patta]], *The category of matroids*, Applied Categorical Structures **26** (2018) 205–237 [[arXiv:1512.01390](https://arxiv.org/abs/1512.01390), [doi:10.1007/s10485-017-9490-2](https://doi.org/10.1007/s10485-017-9490-2)]

On [[limits]] in [[dagger categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Limits in dagger categories*, Theory and Applications of Categories **34** 18 (2019) 468-513 [[arXiv:1803.06651](https://arxiv.org/abs/1803.06651), [34-18](http://www.tac.mta.ca/tac/volumes/34/18/34-18abs.html)]

On [[monoidal category]]-theory with an eye towards [[finite quantum mechanics in terms of dagger-compact categories]] and [[quantum computation]]:

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], *Categories for Quantum Theory*, Oxford University Press 2019 ([ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616))

  based on:

  {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], *Lectures on categorical quantum mechanics* (2012) [[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]]

Generalization to [[Hilbert bundles]]:

* [[Chris Heunen]], [[Manuel L. Reyes]], *Frobenius Structures Over Hilbert $C^\ast$-Modules*, Commun. Math. Phys. **361** (2018) 787–824 [[doi:10.1007/s00220-018-3166-0](https://doi.org/10.1007/s00220-018-3166-0), [arXiv:1704.05725](https://arxiv.org/abs/1704.05725)]

On [[Bohr toposes]]:

* [[John Harding]], [[Chris Heunen]], *Topos quantum theory with short posets*, Order **38** 111–125 (2021) [[arXiv:1903.01897](https://arxiv.org/abs/1903.01897), [doi:10.1007/s11083-020-09531-6](https://doi.org/10.1007/s11083-020-09531-6)]

On an [[axiom|axiomatic]] characterization of the [[compact closed dagger category]] [[Hilb]] of [[Hilbert spaces]]:

* [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces*, PNAS **119** 9 (2022) e2117024119 [[arXiv:2109.07418](https://arxiv.org/abs/2109.07418), [doi:10.1073/pnas.2117024119](https://doi.org/10.1073/pnas.2117024119)]

* [[Chris Heunen]], [[Andre Kornell]], [[Nesta van der Schaaf]], *Axioms for the category of Hilbert spaces and linear contractions* [[arXiv:2211.02688](https://arxiv.org/abs/2211.02688)]

## Related entries

* [[quantum logic]]

* [[quantum operation]]

* [[finite quantum mechanics in terms of dagger-compact categories]]

* [[classical structure]]

category: people


Chris Isham

Chris Isham is an Emeritus Professor and Senior Research Investigator at Imperial College, London.

* [website](https://www.imperial.ac.uk/people/c.isham)

* [biography](http://en.wikipedia.org/wiki/Christopher_Isham) (brief)

* Section 3 of: *Chris Isham: mentor, colleague, friend*, by [[Michael Duff]] ([arXiv:2112.13722](https://arxiv.org/abs/2112.13722))

## Selected writings

On [[supersymmetry]] and [[G-structure]] (notably [[Spin(7)-structure]] in [[M-theory on 8-manifolds]]):

* {#IshamPope88} [[Chris Isham]], [[Christopher Pope]], _Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings_, Class. Quant. Grav. 5 (1988) 257 ([spire:251240](http://inspirehep.net/record/251240), [doi:10.1088/0264-9381/5/2/006](https://doi.org/10.1088/0264-9381/5/2/006))

* {#IshamPopeWarner88} [[Chris Isham]], [[Christopher Pope]], [[Nicholas Warner]], _Nowhere-vanishing spinors and triality rotations in 8-manifolds_,  Classical and Quantum Gravity, Volume 5, Number 10, 1988 ([cds:185144](http://cds.cern.ch/record/185144), [doi:10.1088/0264-9381/5/10/009](https://iopscience.iop.org/article/10.1088/0264-9381/5/10/009))

On [[quantum mechanics]]:

* [[Chris Isham]], *Lectures on Quantum Theory -- Mathematical and Structural Foundations*, World Scientific (1995) [[doi:10.1142/p001](https://doi.org/10.1142/p001), [ark:/13960/t4xh7cs99](https://archive.org/details/lecturesonquantu0000isha)]

Proposal that the [[Kochen-Specker theorem]] suggests to understand [[quantum physics]] via the [[internal logic]] of
(what later would be called) a *[[Bohr topos]]*:

* {#ButterfieldHamiltonIsham} [[Jeremy Butterfield]],  John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_, _I. quantum states as generalized valuations_, Internat. J. Theoret. Phys. **37** 11 (1998) 2669-2733 [[MR2000c:81027](http://www.ams.org/mathscinet-getitem?mr=1669557), [doi:10.1023/A:1026680806775](http://dx.doi.org/10.1023/A:1026680806775)]

  _II. conceptual aspects and classical analogues_ Int. J. of Theor. Phys. **38**  3 (1999) 827-859 [[MR2000f:81012](http://www.ams.org/mathscinet-getitem?mr=1697983), [doi:10.1023/A:1026652817988](http://dx.doi.org/10.1023/A:1026652817988)]

  _III. Von Neumann algebras as the base category_, Int. J. of Theor. Phys. **39** 6 (2000) 1413-1436 [[arXiv:quant-ph/9911020](http://arxiv.org/abs/quant-ph/9911020), [MR2001k:81016](http://www.ams.org/mathscinet-getitem?mr=1788498),[doi:10.1023/A:1003667607842](http://dx.doi.org/10.1023/A:1003667607842)]

  _IV. Interval valuations_, Internat. J. Theoret. Phys. __41__ 4 (2002) 613-639 [[MR2003g:81009](http://www.ams.org/mathscinet-getitem?mr=1902067), [doi](http://dx.doi.org/10.1023/A:1015276209768)]

with some review and outlook in

* [[Chris Isham]], [[Jeremy Butterfield]], *Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity*, Found. Phys. **30** (2000) 1707-1735 [[arXiv:gr-qc/9910005](https://arxiv.org/abs/gr-qc/9910005), [doi:10.1023/A:1026406502316](https://doi.org/10.1023/A:1026406502316)]

and then

* {#IshamDoering07} [[Andreas Döring]], [[Chris Isham]], _A Topos Foundation for Theories of Physics_

  *I. Formal Languages for Physics*, J. Math. Phys. **49** (2008) 053515 [[arXiv:quant-ph/0703060](http://arxiv.org/abs/quant-ph/0703060), [doi:10.1063/1.2883740](https://doi.org/10.1063/1.2883740)]

  *II. Daseinisation and the Liberation of Quantum Theory*, J. Math. Phys. **49** (2008) 053516
  [[arXiv:quant-ph/0703062](http://arxiv.org/abs/quant-ph/0703062), [doi:10.1063/1.2883742](https://doi.org/10.1063/1.2883742)]

  *III. The Representation of Physical Quantities With Arrows*, J. Math. Phys. **49** (2008) 053517 [[arXiv:quant-ph/0703064](https://arxiv.org/abs/quant-ph/0703064), [doi:10.1063/1.2883777](https://doi.org/10.1063/1.2883777)]

  *IV. Categories of Systems*, J. Math. Phys. **49** (2008) 053518 [[arXiv:quant-ph/0703066](http://arxiv.org/abs/quant-ph/0703066), [doi:10.1063/1.2883826](https://doi.org/10.1063/1.2883826)]

On [[differential geometry]] in [[mathematical physics]]:

* _[[Modern Differential Geometry for Physicists]]_

## Related $n$Lab entries

* [[Kochen-Specker theorem]]

* [[Bohr topos]]

* [[Timeline of category theory and related mathematics]]

category: people

[[!redirects Christopher Isham]]


Chris Kapulkin

* [webpage](https://www.math.uwo.ca/faculty/kapulkin/)

## Selected writings

On the [[relation between type theory and category theory|categorical semantics]] of [[homotopy type theory]] in [[simplicial sets]]/[[infinity-groupoids|$\ionfty$-groupoids]]:

* {#KapulkinLumsdaine12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The Simplicial Model of Univalent Foundations (after Voevodsky)*, Journal of the European Mathematical Society **23** (2021) 2071–2126  $[$[arXiv:1211.2851](https://arxiv.org/abs/1211.2851), [doi:10.4171/jems/1050](https://doi.org/10.4171/jems/1050)$]$

On the [[model structure for quasi-categories|Joyal-type]] [[model structure for cubical quasi-categories]] on [[cubical sets]] [[connection on a cubical set|with connections]]:

* [[Brandon Doherty]], [[Chris Kapulkin]], [[Zachery Lindsey]], [[Christian Sattler]], _Cubical models of (∞,1)-categories_, Memoirs of the AMS ([accepted 2022](https://www.ams.org/cgi-bin/mstrack/accepted_papers/memo)) [[arXiv:2005.04853](https://arxiv.org/abs/2005.04853)]

On a [[calculus of fractions]] generalized from categories to [[quasi-categories]] ("$(\infty,1)$-calculus of fractions", for *[[localization of an (infinity,1)-category|localization of $(\infty,1)$-categories]]*):

* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]

exposition:

* [[Chris Kapulkin]], *Calculus of Fractions for Quasicategories (Part I)*, [talk at](CQTS#KapulkinOct2023) [[CQTS]] (18 Oct 2023) [video:[YT](https://youtu.be/96ViSKAuApc)]

On the [[model structure on compactly generated topological spaces]] and [[model structure on Delta-generated topological spaces|on Delta-generated topological spaces]], and on a [[model category]] of [[locales]] which makes the [[reflective subcategory|reflection]] of [[sober topological spaces]] a [[Quillen adjunction]] to the sober-restriction of the [[classical model structure on topological spaces]]:

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

## Related entries

* [[internal category in homotopy type theory]]

category: people

[[!redirects Krzysztof Kapulkin]]


Chris Kottke

* [webpage](http://www.math.brown.edu/~ckottke/)

category: people


Chris Mulvey

* [webpage at Sussex University](http://www.maths.sussex.ac.uk/Staff/CJM/)

## Selected writings

On the [[Stone-Weierstrass theorem]] in [[constructive mathematics]]:

* [[B. Banaschewski]], [[C. J. Mulvey]], *A constructive proof of the Stone-Weierstrass theorem*, J. Pure Appl. Algebra __116__ (1997) 25-40 [doi:10.1016/S0022-4049(96)00160-0]

category:people

[[!redirects Christopher J. Mulvey]]
[[!redirects C. J. Mulvey]]


Chris Peters

* [webpage](http://www-fourier.ujf-grenoble.fr/~peters/)

## Related entries

* [[Hodge theory]], [[Hodge structure]]

category: people


Chris Rogers

I'm an assistant professor of mathematics at the [University of Nevada, Reno](http://www.unr.edu/math/).

My research lies on the interface between algebraic topology, geometry, and mathematical physics. I am particularly interested in the role that homotopy theoretic and higher categorical structures play in quantization.

My Ph.D thesis was on [[higher symplectic geometry]], and was supervised by [[John Baez]].

More information (current research projects, publications, etc.) can be found [here](https://sites.google.com/site/chrisrogersmath/home).

## Selected writings

On [[graph complexes]] and the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]]:

* {#DolgushevRogers12} [[Vasily Dolgushev]], [[Christopher Rogers]], _Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction_, In: Mathematical aspects of quantization 583 (2012): 25-145. ([arXiv:1202.2937](https://arxiv.org/abs/1202.2937))

On a [[category of fibrant objects]]-structure [[model structure for L-infinity algebras|for L-infinity algebras]]:

* [[Christopher L. Rogers]], _An explicit model for the homotopy theory of finite type Lie $n$-algebras_, Algebr. Geom. Topol. 20 (2020) 1371-1429 ([arXiv:1809.05999](https://arxiv.org/abs/1809.05999), [doi:10.2140/agt.2020.20.1371](https://doi.org/10.2140/agt.2020.20.1371))

category: people

[[!redirects Chris L. Rogers]]

[[!redirects Christopher Rogers]]
[[!redirects Christopher L. Rogers]]


Chris Schommer-Pries

Chris Schommer-Pries is an assistant professor at the University of Notre Dame.  He studies the interactions of topology, higher category theory, and quantum field theory. Preprints,  papers, and more details can be found on his [website](http://sites.nd.edu/chris-schommer-pries/).

## Slides and talks

* 2d vs 3d TQFT [slides](http://sites.google.com/site/chrisschommerpriesmath/Home/Slides-MFO-6-11-09.pdf?attredirects=0)

* ([[DSSFusionSlides.pdf:file]])

## related $n$Lab entries

* [[cobordism hypothesis]]

* [[FQFT]]

* [[bi-brane]]

* [[canonical transformation]]

* [[string 2-group]]

* [[(infinity,n)-category]]

category: people

[[!redirects Christopher Schommer-Pries]]


Chris Woodward

* [website](http://www.math.rutgers.edu/~ctw/)

## Selected writings

On the [[symplectic category]]:

* [[Katrin Wehrheim]], [[Chris T. Woodward]], _Functoriality for Lagrangian correspondences in Floer theory_, Quantum Topology **1** 2 (2010) 129-170 [[doi:10.4171/qt/4](https://doi.org/10.4171/qt/4), [arXiv:0708.2851](https://arxiv.org/abs/0708.2851)]

## Related entries

* [[symplectic category]]

* [[Lagrangian correspondences and category-valued TFT]]

category: people

[[!redirects Chris T. Woodward]]


Christian Ausoni

* [webpage](http://www.math.univ-paris13.fr/~ausoni/)

## Selected writings

On [[iterated algebraic K-theory]] and the [[red-shift conjecture]]:

* [[Christian Ausoni]], [[John Rognes]], _The chromatic red-shift in algebraic K-theory_, Enseign. Math. (2) 54 (2008), 9-11. ([pdf](http://www.math.univ-paris13.fr/~ausoni/papers/gmconj-AR.pdf), [[AusoniRognesRedShiftInAlgebraicKTheory.pdf:file]], [doi:10.5169/seals-109873](http://doi.org/10.5169/seals-109873))

On the first [[algebraic K-theory]] $K(ku)$ of [[topological K-theory]] ([[iterated algebraic K-theory]]):

* [[Christian Ausoni]], [[John Rognes]], _Algebraic K-theory of topological K-theory_, Acta Math. Volume 188, Number 1 (2002), 1-39 ([euclid:acta/1485891473](https://projecteuclid.org/euclid.acta/1485891473))

* [[Christian Ausoni]], [[John Rognes]], _Rational algebraic K-theory of topological K-theory_, Geom. Topol. 16 (2012) 2037-2065 ([arXiv:0708.2160](https://arxiv.org/abs/0708.2160), [doi:10.2140/gt.2012.16.2037](http://dx.doi.org/10.2140/gt.2012.16.2037))

* [[Christian Ausoni]], _On the algebraic K-theory of the complex K-theory spectrum_, Inventiones mathematicae volume 180, pages 611–668 (2010) ([arXiv:0902.2334](http://arxiv.org/abs/0902.2334), [doi:10.1007/s00222-010-0239-x](https://doi.org/10.1007/s00222-010-0239-x))

Divisibility of a [[gerbe]] on the [[3-sphere]] seen as a [[BDR 2-vector bundle]] is in

* [[Christian Ausoni]], [[Bjørn Ian Dundas]] and [[John Rognes]],  _Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere_, Doc. Math. 13, 795-801 (2008). ([pdf](http://www.math.univ-paris13.fr/~ausoni/papers/ADR08.pdf))

## Related entries

* [[iterated algebraic K-theory]]

* [[red-shift conjecture]]

category: people