moduli stack of elliptic curves

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###Context###
#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### Higher geometry
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[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
A **moduli stack of elliptic curves** is, as the name suggests, a [[moduli stack]] which classifies [[elliptic curves]]. Hence it is a [[stack]] $\mathcal{M}_{ell}$ such that for $X$ any other suitable [[space]], the [[groupoid]] of maps $X \to \mathcal{M}_{ell}$ and [[homotopies]] between them is [[equivalence of groupoids|equivalent]] to that of $X$-parameterized [[elliptic curves]] with equivalences between these. (There are some variants of $\mathcal{M}_{ell}$ corresponding to the choice of which singularities and degeneracies of elliptic curves are taken into account.)

This is formalized in [[algebraic geometry]], hence $X$ here is a [[scheme]] over the [[integers]] in general. The moduli stack always has a tautological construction as a "[[sheaf]] of [[groupoids]]" (a [[stack]], whence the name) over the [[site]] of [[affine schemes]], given by sending any $Spec(R)$ to the groupoid of suitable [[elliptic curves]] over $Spec(R)$. For concrete computations it typically helps to know that the moduli stack of elliptic curbes is [[representable functor|represented]] by a [[geometric stack]], [[Isbell duality|dually]] given by a [[Hopf algebroid]].

The moduli stack $\mathcal{M}_{ell}$ has a [[Deligne-Mumford compactification|compactification]] $\mathcal{M}_{\overline{ell}}$ obtained by adding the [[nodal cubic curve]], and often (but not always) this compactified version is the default meaning of "moduli stack of elliptic curves". Adding also the [[cuspidal cubic curve]] and hence all [[cubic curves]] produces the full moduli stack $\mathcal{M}_{cub}$ of cubic curves, inside which $\mathcal{M}_{ell}$ sits as the locus of non-singular curves.

Since an [[elliptic curve]] is a [[arithmetic genus|genus-1]] [[algebraic curve]] with a marked point (the neutral element of the [[group]] structure), $\mathcal{M}_{ell}$ is equivalently the [[moduli stack of algebraic curves]] for genus $g = 1$ with $n = 1$ punctures, and as such is often equivalently written

$$
  \mathcal{M}_{1,1} = \mathcal{M}_{ell}
  \,.
$$

A special class of cases which is much simpler than the general case but still of paramount interest is the moduli stack of elliptic curves over the [[complex numbers]], hence of maps $Spec(\mathbb{C}) \to \mathcal{M}_{ell}$. These are just [[complex manifold|complex]] [[tori]]/[[Riemann surfaces]] of [[genus of a surface|genus]] 1 which may be identified with quotients of the [[complex plane]] by a framed [[lattice]] well-defined up to [[Möbius transformations]], and so in this case the moduli stack of elliptic curves is just the [[homotopy quotient]] (the [[orbifold]] quotient) of the [[upper half plane]] by the [[action]] of the [[modular group]]. (This is equivalently the [[moduli space of curves]] $\mathcal{M}_{(1,1)}$ which in turn is a quotient of the [[Teichmüller space]] $\mathcal{T}_{(1,1)}$.) This case is considered below in

* _[Description over the complex numbers](#DescriptionOverTheComplexNumbers)_.

Below that is the

* _[Description over general schemes](#DescriptionOverGeneralSchemes)_.

as an [[algebraic stack]] $\mathcal{M}_{ell}$. This is still not the most refined description: by the [[Goerss-Hopkins-Miller theorem]] the assignment to an [[elliptic curve]] of its [[elliptic spectrum]] lifts the ordinary [[structure sheaf]] $\mathcal{O}$ of $\mathcal{M}_{ell}$ to a higher structure sheaf $\mathcal{O}^{top}$ of [[E-∞ rings]] in a way that makes $(\mathcal{M}_{ell}, \mathcal{O}^{op})$ a [[spectral Deligne-Mumford stack]]. The [[global sections]] of this structure sheaf yield the [[spectrum]] [[tmf]] of [[topological modular forms]]:

* _[As a derived scheme in E-infinity geometry](#InEInfinityGeometry)_

See also _[[A Survey of Elliptic Cohomology - elliptic curves]]_ for more.

## Description over the complex numbers
 {#DescriptionOverTheComplexNumbers}

An [[elliptic curve]] $E \to Spec(\mathbb{C})$ over the [[complex numbers]] is determined, up to non-canonical [[isomorphism]], by its [[j-invariant]]

$$
  j(E) \in \mathbb{C}
  \,.
$$

Here every [[complex number]] appears as a value, and therefore the moduli space of elliptic cuves a priori is not [[compact topological space|compact]].

A [[compactification]] of the moduli space is obtained by including also elliptic curves with [[nodal singularity]].

### Upper half plane

The [[upper half plane]] $\mathfrak{h}$ is in [[bijection]] with framed [[lattices]] in the [[complex plane]] $\mathbb{C}$, which in turn is in bijection with [[isomorphism]] classes of framed elliptic curves over $\mathbb{C}$

$$
  \mathfrak{h}
  \simeq
  \{framed\;lattices\;in\;\mathbb{C}\}
  \simeq
  \{framed\;elliptic\;curves\;over\;\mathbb{C}\}/_\sim
$$

and we have

$$
  \{elliptic\;curves\;over\;\mathbb{C}\}_\sim
  \simeq
  \mathfrak{h}/{SL_2(\mathbb{Z})}
$$

where the [[special linear group]] over the [[integers]]

$$
  SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}$$

[[action|acts]] as the [[modular group]] by [[Möbius transformations]]

$$
  \tau \mapsto \frac{a \tau + b}{c \tau + d}
  \,.
$$

### The naive moduli space and its problems

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

Write

$$
  M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})
$$

for the plain [[quotient]] of the [[upper half plane]] by the above group action.

=--

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

A **holomorphic family of [[elliptic curves]]** over a [[complex manifold]] $T$ is

* a [[holomorphic function]] $\pi : X \to T$

* together with a [[section]] $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an [[elliptic curve]] (using the first definition above).

=--

For every family

$$
  \array{
    X \\ \downarrow^{\mathrlap{\pi}} \\ T
  }
$$

we would like to have $F \to M_{1,1}$ such that there is a [[pullback]]

$$
  \array{
    X \simeq \phi^* F &\longrightarrow& F
    \\
    \downarrow & & \downarrow
    \\
    T &\stackrel{\phi}{\longrightarrow}& M_{1,1}
  }
$$

where

$$
  \phi:   t \mapsto [X_t, s(t)]
$$

such that

* $\phi : T \to M_{1,1}$ is a [[holomorphic map]]

* every [[holomorphic map]] $T \to M_{1,1}$ corresponds to a family over $T$;

* there is a universal family over $M_{1,1}$

This is _impossible_ . One can construct explicit counterexamples. These counterexamples involve [[elliptic curves]] with nontrivial [[automorphism]]s.

For instance

$$
  \{
    (x,y,z) \in \mathbb{P}^2 \times X :
    y^2 = x(x-1)(x-\lambda)
  \}
  \to
  X := \mathbb{P}^1 - \{0,1,\infty\}
$$

> but see the discussion at [[moduli space]] for a discussion of the statement "it's the automorphisms that prevent the [[moduli space]] from existing"

### Moduli space of framed elliptic curves

consider

$$
  \mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}
$$

given by

$$
  (m,n) :
  (z,\tau)
  \mapsto
  (z + m \tau + n, \tau)
$$

Then consider the family

$$
  \array{
     E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h}
    \\ \downarrow \\ \mathfrak{h}
  }
$$

is a family of [[elliptic curve]]s over $\mathfrak{h}$

and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with

$$
  \Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau
$$

is a family of framed elliptic curves.

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

The space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a [[fine moduli space]] for [[framed elliptic curves]].

=--

Consider any map $\phi : T \to \mathfrak{h}$

with pullback of the universal family

$$
  \array{
     X \stackrel{?}{\to} \phi^* E &\to & E
    \\
    \downarrow && \downarrow
    \\
    T &\stackrel{\phi}{\to}&
    \mathfrak{h}
  }
$$

**claim** for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose [[differential form|1-forms]] $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.

Notice that _locally_ every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$).  So

$$
  \array{
    && \mathfrak{h}
    \\
    && \downarrow^{SL_2(\mathbb{Z})}
    \\
    M_{1,1} &\stackrel{Id}{\to}&
    M_{1,1}
  }
$$

at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$

isn't locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.

### Moduli stack/orbifold of elliptic curves

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

Consider the global [[quotient stack]] [[orbifold]]

$$
  \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
$$

of the upper half plane by the [[action]] of the [[special linear group]]
over the [[integers]].

=--

This is the [[moduli stack]] of elliptic curves.

### Compactified moduli stack

Consider the complex analytic parameterization over the [[annulus]]

$$
  \{q \in \mathbb{C} | 0 \lt {\vert q \vert} \lt 1 \}
$$

of elliptic curves

$$
  E_q \coloneqq \mathbb{C}^*/q^{\mathbb{Z}}
  \,.
$$

This has an extension to the origin, where $E_0$ is a [[nodal curve]]. Algebraically, in a [[formal neighbourhood]] of the origin, hence over
$Spec(\mathbb{Z}[ [q] ])$, this is the [[Tate curve]].

e.g. ([Lurie, section 4.3](#Lurie)).

## Description over general schemes
 {#DescriptionOverGeneralSchemes}

For $S$ a [[scheme]], a [[cubic curve]] over $S$ is a scheme $p \colon X \to S$ over $S$ equipped with a [[section]] $e \colon S \to X$ and such that [[Zariski topology|Zariski locally]] on $S$, $X$ is given by an [[equation]] in $\mathbb{P}_S^2$ of the form

$$
  y^2 + a_1 x y = x^3 + a_2 x^2 + a_4 x + a_6
$$

such that $e \colon S \to X$ is the line at infinity.

Euivalently this says that $p$ is a [[proper morphism|proper]] [[flat morphism]] with a section contained in the [[smooth locus]] whose [[fibers]] are geometrically integral curves of [[arithmetic genus]] one.

Write $\mathcal{M}_{cub}$ for the [[moduli stack]] of such [[cubic curves]]. Then the moduli stack of elliptic curves is the non-vanishing locus of the discriminant  $\Delta \in H^0(\mathcal{M}_{cub}, \omega^{12})$

$$
  \mathcal{M}_{ell} \hookrightarrow \mathcal{M}_{cub}
  \longrightarrow \mathcal{M}_{FG}.
$$

See at _[[elliptic curve]]_ for details. (A textbook account is in [Silverman 09, III](#Silverman09), a review with an  eye towards [[tmf]] is in [Mathew, section 3](#Mathew)).

Two standard versions of [[Hopf algebroids]] representing $\mathcal{M}_{\overline{ell}}$ as a [[geometric stack]] are usefully reviewed in ([Mathew, section 4](#Mathew)).

## As a derived scheme in $E_\infty$-geometry
 {#InEInfinityGeometry}

By the [[Goerss-Hopkins-Miller theorem]] the [[structure sheaf]] $\mathcal{O}$ of the moduli stack of elliptic curves lifts to a sheaf $\mathcal{O}^{top}$ of [[E-∞ rings]] which over a given [[elliptic curve]] is the corresponding [[elliptic spectrum]].

By ([[A Survey of Elliptic Cohomology|Lurie (Survey), theorem 4.1]]), this yields a  [[spectral Deligne-Mumford stack]] refinement

$$
  \mathcal{M}_{ell}^{der} \coloneqq  (\mathcal{M}_{ell}, \mathcal{O}^{top})
$$

which is the moduli stack of [[derived elliptic curves]], in that there is a [[natural equivalence]] in [[E-∞ rings]] $A$ of the form

$$
  Hom(Spec(A), \mathcal{M}_{ell}^{der})
  \simeq
  E(A)
  \,,
$$

where on the left we have maps of [[structured (∞,1)-toposes]] and on the right the [[∞-groupoid]] of [[derived elliptic curves]] over $A$.

This is based on the representability theorem ([[A Survey of Elliptic Cohomology|Lurie (Survey), prop. 4.1]], [[Representability Theorems|Lurie (Representability)]]).

In this derived picture the compactified dericed moduli space is obtained by gluing in the [[spectrum]] of [[Tate K-theory]] $KO[ [q] ] \simeq KU[ [q] ]/\mathbb{Z}_2$ by forming the [[homotopy pushout]]

$$
  \array{
    Spec(K((q))) &\longrightarrow& \mathcal{M}_{ell}^{der}
    \\
    \downarrow && \downarrow
    \\
    Spec(K[ [q] ]) &\longrightarrow& \mathcal{M}_{\overline{ell}}^{der}
  }
  \,.
$$

([[A Survey of Elliptic Cohomology|Lurie(Survey), p. 33]]). Again, the underlying ordinary [[Deligne-Mumford stack]] is the ordinary $\mathcal{M}_{\overline{\ell}}$.

## Properties

### Finite covers

The [[moduli space]] $\mathcal{M}_{ell}[n]$ of [[elliptic curves with level-n structure]] (for some $n \in \mathbb{N}$) provides a finite [[covering]] of $\mathcal{M}_{ell}$ (similarly for the compactifications).

(Over the [[complex numbers]] this is the _[[modular curve]]_).

### Cohomology

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

$$
  H_1(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12\mathbb{Z}
$$

$$
  H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0
$$

$$
  H^2(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12 \mathbb{Z}
$$

$$
  H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q})
  \simeq
  H_\bullet(M_{1,1}, \mathbb{Q})
$$

and similarly for [[integral cohomology]]

$$
  \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
$$

$$
  Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}
$$

=--

### Euler characteristic
 {#EulerCharacteristic}

The [[orbifold Euler characteristic]] of the moduli space of complex elliptic curves is given by the [[special values of L-functions|special value]] of the [[Riemann zeta function]] at $s= -1$

$$
  \chi(\mathcal{M}_{1,1}) = \zeta(-1) = - \frac{1}{12}
  \,.
$$

This is a special case of the result in ([Zagier-Harer 86](#ZagierHarer86)) discussed at _[[moduli space of curves]]_. See also the first page here: [[EulerCharacteristicOfSpaceOfCurves.pdf:file]].

## Related concepts

* [[modular form]]

* [[elliptic cohomology]]

[[!include moduli spaces -- contents]]

[[!include moduli stack of curves -- table]]

## References

Introductory lecture notes on the moduli space of elliptic curves over the [[complex numbers]] include

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

* {#IntroOrbifolds} section 4 of _Introduction to Orbifolds_ ([[IntroductionToOrbifolds.pdf:file]])

Accounts of the general case include

* Nicholas M. Katz, [[Barry Mazur]], _Arithmetic moduli of elliptic curves_, Annals of Mathematics Studies_, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR MR772569 (86i:11024)

* {#Silverman09} [[Joseph Silverman]], _The arithmetic of elliptic curves_, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)

Lecture notes/talk notes reviewing this include

* James Parson, _Moduli of elliptic curves_  ([pdf](http://math.stanford.edu/~conrad/vigregroup/vigre03/moduli.pdf))

* {#Mathew} [[Akhil Mathew]], section 3 of _The homotopy groups of $TMF$_ ([pdf](http://math.mit.edu/~sglasman/tmfhomotopy.pdf))

* [[Andre Henriques]], _The moduli stack of elliptic curves_   ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter04/henriques.pdf)) in _Topological modular forms_ Talbot workshop 2007 ([web](http://math.mit.edu/conferences/talbot/2007/tmfproc/))

For more of the general picture in view of [[elliptic cohomology]] and [[tmf]] see also

* {#Lurie} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

The [[orbifold Euler characteristic]] of the moduli space of curves was originally computed in

* {#ZagierHarer86} [[Don Zagier]], John Harer, _The Euler characteristic of the moduli space of curves_, Inventiones mathematicae (1986) Volume: 85, page 457-486 ([EUDML](https://eudml.org/doc/143377))

Reviews of the orbifold Euler characteristic computation include

* _Mathematical ideas and notions in quantum field theory -- 5. The Euler characteristic of the moduli space of curves_ ([[EulerCharacteristicOfSpaceOfCurves.pdf:file]])

[[!redirects moduli space of elliptic curves]]


moduli stack of formal groups

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli stack]] $\mathcal{M}_{FG}$ of all [[formal groups]]. Often meant are 1-dimensional commutative formal groups-

## Definition

Let $L = \pi_\bullet MU$ be the [[Lazard ring]].

Write $G^+$ for the [[group scheme]] given on a [[ring]] $R$ by

$$
  G^+(R) \coloneqq \{g\in R[ [x] ] \vert g(t) = b_1 t + b_2 t^2 + \cdots \; with\; b_1 \in R^\times \}
  \,.
$$

There is a canonical [[action]] of $G^+$ on $Spec(L)$. The [[quotient stack]] of this action is the moduli stack of (1d commutative) formal groups

$$
  \mathcal{M}_{fg} = (Spec(L))//G^+
  \,.
$$

(e.g. [Lurie, lecture 11, def. 2](#LurieLect11))

## Properties

### Chromatic height stratification

The moduli stack of formal groups $\mathcal{M}_{FG}$ admits a natural [[stratification]] whose open [[strata]] are labeled by a [[natural number]] called the _[[height of a formal group|height of formal groups]]_.

The [[complex oriented cohomology theories]] associated to these [[formal groups]] by the [[Landweber exact functor theorem]] accordingly also inherit such an integer label, called _[[chromatic filtration]]_. Studying this is the topic of [[chromatic homotopy theory]].

[[!include Lurie spectral sequences -- table]]

### Morava stabilizer group

Write $\overline{\mathbb{F}_{\mathrm{p}}}$ for the [[algebraic closure]] of $\mathbb{F}_p$.

The [[stratum]] $\mathcal{M}_{FG}^n$ can be identified with the [[homotopy quotient]] $Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}$, where the [[group]] $\mathbb{G}$ is the _Morava stabilizer group_.

This is ([Lurie 10, lect. 19, prop. 1](#LurieLect19)) See also the beginning of [Lurie 10, lect 21](#LurieLect21).

### Deformation theory

The [[deformation theory]] around these [[strata]] is [[Lubin-Tate theory]].

### Relation to moduli of elliptic curves and tori

Inside the moduli stack of formal groups sit, in that order, that of [[cubic curves]], the [[moduli stack of elliptic curves]], the [[moduli stack of tori]].

## Related concepts

* [[Spec(S)]]

[[!include moduli spaces -- contents]]

[[!include moduli stack of curves -- table]]

## References

* [[Niko Naumann]], _Comodule categories and the geometry of the stack of formal groups_, Advances in Mathematics **215** (2007) pp 569-600, doi:[10.1016/j.aim.2007.04.007](http://dx.doi.org/10.1016/j.aim.2007.04.007), arXiv:[math/0503308](http://arxiv.org/abs/math/0503308).

* Brian D. Smithling, _On the moduli stack of commutative, 1-parameter formal Lie groups_ ([arXiv:0708.3326](http://arxiv.org/abs/0708.3326))

* {#LurieLect11} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 11 _Formal groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture11.pdf))

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 14 _Classification of formal groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture14.pdf))

* {#LurieLect19} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 19 _Morava stabilizer groups_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture19.pdf))

* {#LurieLect21} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 21 _Lubin-Tate theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf))

* [[Jeroen van der Meer]], _A Stack-Theoretic Perspective on $\mathrm{KU}$-Local Stable Homotopy Theory_, Chapter 3 _The Moduli Stack of Formal Groups_. 2019 ([pdf](https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/2018-2019/jvd-meer-thesisroot.pdf))

On [[quasicoherent sheaves]] over $\mathcal{M}_{fg}$:

* [[Paul Goerss]], _Realizing Families of Landweber Exact Homology Theories_ ([arXiv:0905.1319](http://arxiv.org/abs/0905.1319))

* [[Paul Goerss]], _Quasi-coherent sheaves on the moduli stack of formal groups_ ([arXiv:0802.0996](http://arxiv.org/abs/0802.0996))

[[!redirects moduli stacks of formal groups]]

[[!redirects height filtration]]

[[!redirects moduli stack of formal group laws]]

[[!redirects moduli space of formal groups]]
[[!redirects moduli spaces of formal groups]]


moduli stack of L-parameters

#Contents#
* table of contents
{:toc}

## Idea

The *moduli stack of L-parameters* is an [[algebraic stack|algebraic]] [[moduli stack]] that parametrizes [[L-parameters]], which generalize the [[Weil-Deligne representations]] for the [[local Langlands correspondence]] when the [[reductive group]] is not the [[general linear group]] $GL_n$.

In the "classical" $\ell$-adic case of the local Langlands correspondence, this moduli stack was constructed independently by Dat-Helm-Kurinczuk-Moss ([DHKM20](#DHKM20)), Fargues-Scholze ([FarguesScholze21](#FarguesScholze21)), and Zhu ([Zhu20](#Zhu20)). For the p-adic local Langlands correspondence, the relevant stack has to be the moduli stack of etale $(\varphi,\Gamma)$-modules (also known as the [[Emerton-Gee stack]]), constructed by Emerton-Gee ([EmertonGee19](#EmertonGee19)).

The corresponding notion in the [[geometric Langlands correspondence]] is the [[moduli stack]] of [[local systems]] (recall -- e.g. from [Deligne 1970](#Deligne70) -- that local systems on a curve (or any space) give [[linear representations]] of its [[fundamental group]], hence are akin to [[Galois representations]]).

## References

* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 October 2018, Pages 299-357 ([arXiv:1407.6494](https://arxiv.org/abs/1407.6494), [doi:10.1016/j.jalgebra.2018.06.012](https://doi.org/10.1016/j.jalgebra.2018.06.012))

* [[Ernst-Wilhelm Zink]], with [[Allan J. Silberger]], *Langlands classification for L-parameters* ([pdf](https://www.math.hu-berlin.de/~zyska/zink/Lvortrag1.pdf))

* {#DHKM20} Jean Francois Dat, David Helm, Robert Kurinczuk, and Gilbert Moss, _Moduli of Langlands Parameters_ ([arXiv:2009.06708](https://arxiv.org/abs/2009.06708))

* {#Zhu20} [[Xinwen Zhu]], _Coherent sheaves on the stack of Langlands parameters_ ([arXiv:2008.02998](https://arxiv.org/abs/2008.02998))

* {#FarguesScholze21} [[Laurent Fargues]], [[Peter Scholze]], _Geometrization of the local Langlands correspondence_ ([arXiv:2102.13459](https://arxiv.org/abs/2102.13459))

* {#EmertonGee19} [[Matthew Emerton]], [[Toby Gee]], _Moduli stacks of etale (phi,Gamma) modules and the existence of crystalline lifts_ ([arXiv:1908.07185](https://arxiv.org/abs/1908.07185))

[[!redirects moduli stacks of L-parameters]]


moduli stack of tori

#Contents#
* table of contents
{:toc}

## Idea

Inside the compactified [[moduli stack of elliptic curves]], at the [[cusp]] point corresponding to the [[nodal cubic curve]] sits the _moduli stack of [one dimensional tori](http://ncatlab.org/nlab/show/torus#InAlgebraicGeometry)_ $\mathcal{M}_{\mathbb{G}_m}$ ([Lawson-Naumann 12, def. A.1, A.3](#LawsonNaumann12)). This is equivalent to the [[quotient stack]] of the point by the [[group of order 2]]

$$
  \mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2
$$

([Lawson-Naumann 12, prop. A.4](#LawsonNaumann12)). Here the $\mathbb{Z}_2$-action is the inversion involution on [[abelian groups]].

Using the [[Goerss-Hopkins-Miller theorem]] this stack carries an [[E-∞ ring]]-valued [[structure sheaf]] $\mathcal{O}^{top}$ ([Lawson-Naumann 12, theorem A.5](#LawsonNaumann12)); and by the above equivalence this is a single [[E-∞ ring]] equipped with a $\mathbb{Z}_2$-[[∞-action]]. This is [[KU]] with its involution induced by [[complex conjugation]], hence essentially is $KR$.

Accordingly, the [[global sections]] of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-[[homotopy fixed points]] of this action, hence is [[KO]]. This is further amplified in ([Mathew 13, section 3](#Mathew13)) and ([Mathew, section 2](#Mathew)).

As suggested there and by the main ([Lawson-Naumann 12, theorem 1.2](#LawsonNaumann12)) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of [[tmf]] built as the global sections of the similarly derived [[moduli stack of elliptic curves]].

## Related concepts

[[!include moduli spaces -- contents]]

## References
Discussion of $KU$ with its $\mathbb{Z}_2$-action as the [[E-∞ ring]]-valued [[structure sheaf]] of the moduli stack of tori is due to

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], appendix of _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_ ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

which is reviewed and amplified further in

* {#Mathew13} [[Akhil Mathew]], section 3 of _The homology of $tmf$ ([arXiv:1305.6100](http://arxiv.org/abs/1305.6100))

* {#Mathew} Akhil Mathew, section 2 of _The homotopy groups of $TMF$_, talk notes ([pdf](http://math.mit.edu/~sglasman/tmfhomotopy.pdf))

[[!redirects moduli stack of 1-dimensional tori]]


modulus of continuity

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

\tableofcontents

## Definition

Given [[metric spaces]] $(X, \delta_X)$ and $(Y, \delta_Y)$,

* given an element $x:A$, a **local modulus of continuity** for a function $f:X \to Y$ at $x:A$ is a [[monotonic function|monotonic]] [[continuous function]] on the extended non-negative [[real numbers]] $\omega_x \colon [0, \infty] \to [0, \infty]$ whose [[limit of a sequence|limit]] at $0$ is equal to $0$, such that for all elements $y:X$, $d_Y(f(x), f(y)) \leq w_x(d_X(x, y))$.

* a **global modulus of continuity** for a function $f:X \to Y$ is a monotonic continuous function on the extended non-negative real numbers $\omega:[0, \infty] \to [0, \infty]$ whose limit at $0$ is equal to $0$, such that for all elements $x:X$ and $y:X$, $d_Y(f(x), f(y)) \leq w(d_X(x, y))$.

## Related concepts

* [[pointwise continuous function]]

* [[uniformly continuous function]]

## References

* Wikipedia, [Modulus of continuity](https://en.wikipedia.org/wiki/Modulus_of_continuity)

[[!redirects modulus of continuity]]
[[!redirects moduli of continuity]]

[[!redirects local modulus of continuity]]
[[!redirects local moduli of continuity]]

[[!redirects global modulus of continuity]]
[[!redirects global moduli of continuity]]


modulus of convergence

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

#Contents#
* table of contents
{:toc}

## Definition ##

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$$

be the positive rational numbers. Let $S$ be a [[Booij premetric space]], and let $x:I \to S$ be a [[net]] with index set $I$. A __$I$-modulus of convergence__ is a function $M \in {\mathbb{Q}_{+}} \to I$ such that for all positive rational numbers $\epsilon$ and all indices $i \in I$ and $j \in I$, if $i \geq M(\epsilon)$ and $j \geq M(\epsilon)$, then $x_i \sim_{\epsilon} x_j$.

The composition $x \circ M$ of a net $x$ and a modulus of convergence $M$ is also a [[net]].

## See also ##

* [[limit of a net]]
* [[Booij premetric space]]
* [[modulated Cauchy complete space]]
* [[Cauchy structure]]
* [[Cauchy approximation]]
* [[HoTT book real numbers]]

## References ##

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)


modus ponens

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

# Contents
* table of contents
{: toc}

## Idea

In [[formal logic]], _modus ponens_ is the [[elimination rule]] for the [[logical connective]] $\to$ (forming [[conditional statements]]).

On more general [[types]] in [[type theory]] this is [[function application]].

Even in Hilbert-style logic where there are almost no [[rule of inference|rules of inference]] (besides the many [[axioms]]), there is usually a rule of modus ponens.  (In particular, modus ponens is the *only* non-axiom rule of inference in [[David Hilbert|Hilbert]]\'s version of the [[propositional calculus]].)

## Statements

Modus ponens is the [[rule of inference]] which says that from the [[sequents]]

$$
  \vdash \psi
$$

and

$$
  \vdash \psi \to \phi
$$

asserting (respectively) the [[judgement]] that a [[proposition]] $\psi$ is [[true]] and the [[judgement]] that the [[conditional statement]] $\psi \to \phi$ is also true, the sequent

$$
  \vdash \phi
$$

(asserting the judgement that the proposition $\phi$ is true) may be [[deduction|deduced]].

Depending on what sort of sequents are allowed in the [[sequent calculus]] that one is working with, there may be additional [[context]] on either side of the sequents.  So rather generally, from

$$
  \Gamma \vdash \Delta,\; \psi,\; \Theta
$$

and

$$
  \Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta
,$$

we may deduce

$$
  \Gamma \vdash \Delta,\; \phi,\; \Theta
;$$

this rule may be summarized as

$$
  \frac{ \Gamma \vdash \Delta,\; \psi,\; \Theta ;\;\;\;
         \Gamma \vdash \Delta,\; (\psi \to \phi),\; \Theta }
       { \Gamma \vdash \Delta,\; \phi,\; \Theta }
.$$

In [[linear logic]], we must distinguish between the additive conditional $\to$, whose elimination rule is as above, and the multiplicative conditional $\multimap$, whose elimination rule is

$$
  \frac{ \Gamma \vdash \Delta,\; \psi,\; (\psi \multimap \phi),\; \Theta }
       { \Gamma \vdash \Delta,\; \phi,\; \Theta }
.$$

In [[noncommutative logic]], we would further distinguish this last version from

$$
  \frac{ \Gamma \vdash \Delta,\; (\psi \multimap^{op} \phi),\; \psi,\; \Theta }
       { \Gamma \vdash \Delta,\; \phi,\; \Theta }
.$$

## Properties

The converse to modus ponens, which is the [[introduction rule]] for conditional statements, is much less commonly admitted directly as a rule of inference, but its validity is typically a theorem, the _[[deduction theorem]]_; see also the discussion at _[[metalanguage]]_.

The [[categorical semantics]] of modus ponens in the form

$$
  \psi,\; (\psi \to \phi) \;\vdash \; \phi
$$

is the [[evaluation map]] of the ambient ([[locally cartesian closed category|locally]]) [[cartesian closed category]].

[[!redirects modus ponens]]


Moebius transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Let $k$ be a [[field]], and let $\mathbb{P}^1(k)$ be the [[projective line]] over $k$. A _Möbius transformation_ (also called a _homography_, a _linear fractional transformation_, or a _fractional linear transformation_) is a [[function]] $f: \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ defined by the rule

$$
  f(x)
  =
  \frac{a x + b}{c x + d}
$$

where $a, b, c, d \in k$ and $a d - b c \in k^\times$ (the [[group of units]]). Möbius transformations form a [[group]] under [[composition]], [[isomorphic]] to the [[projective linear group]]

$$
  PGL_2(k)
  \coloneqq
  GL_2(k)/\{\lambda I: \lambda \in k^\times\}
  \,.
$$

If, as in the case $k = \mathbb{C}$ ([[complex numbers]]), each element of $k^\times$ has a [[square root]], then this group is identified with the [[projective special linear group]]

$$
  PSL_2(k)
  \coloneqq
  SL_2(k)/\pm I
  \,.
$$

Alternatively, a fractional linear transformation can be considered as synonymous with an [[automorphism]] of the [[field]] of [[rational functions]] $k(x)$ as a field over $k$ of [[transcendence degree]] 1.

In [[complex analysis]] (which is the usual context when one speaks of Möbius transformations; otherwise one usually calls them by some combination of "linear" and "fractional"), Möbius transformations are precisely the [[biholomorphisms]] of the [[Riemann sphere]], hence exactly its [[bijection|bijective]] [[conformal transformations]].

Often, and particularly when $k$ is the the [[commutative ring]] of [[integers]] $\mathbb{Z}$, one considers a _modular group_ where the coefficients $a, b, c, d$ are assumed to lie in an [[integral domain]] and $a d - b c = 1$. (The [[homotopy quotient]] of the [[upper half-plane]] by the group $PGL_2(\mathbb{Z})$ is the [[moduli stack of elliptic curves]] over the [[complex numbers]].)

## Properties

### Cross-ratios

+-- {: .num_prop}
###### Proposition
The action $PGL_2(k) \times \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ is [[transitive action|3-transitive]], i.e., any triplet of distinct points $(a, b, c)$ may be mapped to
any other triplet of distinct points $(a', b', c')$ by applying a group element.
=--

+-- {: .proof}
###### Proof
It suffices to consider $a' = 0, b' = 1, c' = \infty$ where one applies the transformation $x \mapsto \frac{(x-a)(b-c)}{(x-c)(b-a)}$.
=--

This motivates the following definition: given a 4-tuple $(a, b, c, d)$ of distinct points in $\mathbb{P}^1(k)$, its _cross-ratio_ is

$$\gamma(a, b, c, d) = \frac{(d-a)(b-c)}{(d-c)(b-a)}.$$

It is not hard to see that the group action preserves the cross-ratio, i.e., $\gamma(g \cdot a, g \cdot b, g \cdot c, g \cdot d) = \gamma(a, b, c, d)$. Moreover, the group action is transitive on each cross-ratio-equivalence class of 4-tuples.

In the case $k = \mathbb{C}$ where $\mathbb{P}^1$ is interpreted as the Riemann sphere, it turns out that the cross-ratio of a 4-tuple is a [[real number]] if and only if the four points lie on a [[circle]] (or a line which is a circle passing through $\infty$). Hence Möbius = conformal transformations take circles to circles.

### Action on hyperbolic space

As explained at [[Poincare group]], the group $PSL_2(\mathbb{C})$ can be identified with those [[linear transformations]] of Minkowski space $\mathbb{R}^4$ that preserve the Minkowski [[quadratic form|form]] $Q$, are [[orientation]]-preserving, and take the forward [[light cone]] $\{v = (\vec{x}, t): Q(v) = 0, t \gt 0\}$ to itself. It follows that $PSL_2(\mathbb{C})$ acts on the hyperboloid sheet

$$H^3 = \{v = (\vec{x}, t): Q(v) = 1, t \gt 0\}$$

which is naturally identified with hyperbolic 3-space. There is a Poincaré disk model for $H^3$; consider the disk $D^3$ that is the intersection of the [[future cone]] $\{v = (\vec{x}, t): Q(v) \geq 0, t \gt 0\}$ with the hyperplane $t = 1$. Its interior is an open 3-disk $int(D^3)$ which can be placed in perspective with $H^3$ by considering lines through the origin in $\mathbb{R}^4$: each line that passes through a unique point in $H^3$ passes through a unique point of $int(D^3)$. In this way, $D^3$ is viewed as a natural compactification of $H^3$, and the action of $PSL_2(\mathbb{C})$ on $H^3$ induces an action of $PSL_2(\mathbb{C})$ on $D^3$. The restriction of this action to the boundary $S^2 = \partial D^3$ ("the heavenly sphere") coincides with the action on the Riemann sphere $S^2 = \mathbb{P}^1(\mathbb{C})$.

## Modular group

The _modular group_ $\Gamma$ is the subgroup $PSL_2(\mathbb{Z}) \hookrightarrow PSL_2(\mathbb{C})$ consisting of Möbius transformations with [[integer]] [[coefficients]], hence maps $z \mapsto \frac{a z + b}{c z + d}$ with $a, b, c, d \in \mathbb{Z}$ and $a d - b c = 1$.

The group $PSL_2(\mathbb{R})$ acts on the [[upper half-plane]] $H = \{z \in \mathbb{C}: Im(z) \geq 0\}$ (or rather $H \cup \{\infty\}$ as a [[subspace]] of the [[Riemann sphere]]), by restriction of the action of $PSL_2(\mathbb{C})$ on the Riemann sphere. Indeed, the action of $PSL_2(\mathbb{R})$ takes the real line $\mathbb{R} \cup \{\infty\}$ to itself, and any element $f(z) = \frac{a z + b}{c z + d}$ takes $i$ to $(b + a i)(d - c i)/(c^2 + d^2)$, whose imaginary part $(ad - b c)/(c^2 + d^2) = 1/(c^2 + d^2)$ is positive. By continuity it follows that the action preserves the sign of the imaginary part, hence takes the upper-half plane $H$ to itself.

It is illuminating to think of complex numbers $\tau$ such that $Im(\tau) \gt 0$ as representing [[elliptic curves]] $E$. Indeed, the [[field]] of [[meromorphic functions]] on an elliptic curve (i.e., a complex projective curve $E$ of [[genus]] $1$, or a [[torus]] equipped with a structure of complex analytic 1-manifold) can be identified with a field of doubly periodic [[holomorphic functions]] $\mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ where $L = \langle \omega_1, \omega_2\rangle$ is a fundamental lattice (a discrete cocompact subgroup of the additive [[topological group]] $\mathbb{C}$) attached to $E$. In essence, this field is generated by [[Weierstrass elliptic functions]] $\wp(z), \wp'(z): \mathbb{C}/L \to \mathbb{P}^1(\mathbb{C})$ (here $\wp'$ is the [[derivative]] of $\wp$) which satisfy a [[cubic curve|cubic algebraic relation]]

$$(\wp')^2 = 4\wp^3 - g_2\wp - g_3$$

where the constants $g_2, g_3$ are expressed as certain [[Eisenstein series]] in the fundamental periods $\omega_1, \omega_2$. The $\mathbb{Z}$-[[linear basis]] elements $\omega_1, \omega_2$ of the [[lattice]] may be arranged so that $\tau = \omega_2/\omega_1$ has positive imaginary part. Of course, if there is a [[homothety]] $z \mapsto \lambda z$ that takes a lattice $L$ to a lattice $L'$, then the elliptic curves $E = \mathbb{C}/L$ and $E' = \mathbb{C}/L'$ are analytically isomorphic, so the map

$$\tau \mapsto \mathbb{C}/\langle 1, \tau\rangle$$

gives a surjection from complex numbers with positive imaginary part to isomorphism classes of smooth elliptic curves. Thus we may restrict attention to lattices of the form $L = \langle 1, \tau \rangle$.

Of course, $L$ admits more than one such basis $(1, \tau)$, but for any other $(1, \tau')$ there is a [[linear transformation]] $\gamma \in \Gamma \coloneqq PSL_2(\mathbb{Z})$ such that $\gamma(\tau) = \tau'$. In summary, the [[orbit space]]

$$\{\tau \in \mathbb{C}: Im(\tau) \gt 0\}/\Gamma$$

is a coarse [[moduli space]] for elliptic curves. In this context, one often says that elliptic curves are paramatrized by the $j$-[[j-invariant|invariant]], a certain [[modular form]] $j(\tau)$ defined on the upper half-plane such that $j(\tau) = j(\tau')$ if and only if $\tau' = \gamma \cdot \tau$ for some $\gamma \in \Gamma$.

Of course, in some cases there may be more than one $\gamma \in \Gamma$ that fixes a given $\tau$; this is notably the case when $\tau$ is a fourth root of unity or a sixth root of unity. A more refined approach then is to consider, instead of the coarse orbit space, the (compactified) [[moduli stack]] $(H \cup \{\infty\})//\Gamma$ for elliptic curves, as a central geometric object of interest.

### Abstract structure of modular group

As an abstract [[group]], $\Gamma = PSL_2(\mathbb{Z})$ is a [[free product]] $\mathbb{Z}/(2) \ast \mathbb{Z}/(3)$; explicitly, we may take the generator of order $2$ to be given by the Moebius transformation $z \mapsto -1/z$, and the generator of order $3$ to be given by $z \mapsto (z-1)/z$.

This [[concrete group]] and certain of its subgroups, such as [[congruence subgroups]], are fairly ubiquitous; for example the modular group appears in the theory of [rational tangles](http://ncatlab.org/nlab/show/continued+fraction#rational_tangles) and of  [[combinatorial map|trivalent maps]], and these groups frequently crop up in the theory of buildings (stuff on hyperbolic buildings to be filled in).

It is also worth pointing out that $\Gamma$ is a quotient of the [[braid group]] $B_3$. Consider the standard Artin presentation of $B_3$, with two generators $\sigma_1$, $\sigma_2$ subject to the relation

$$\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2.$$

Then $z \coloneqq (\sigma_1 \sigma_2)^3$ is a [[center|central element]] of $B_3$, and there is a central extension

$$1 \to \mathbb{Z} \stackrel{1 \mapsto z}{\to} B_3 \stackrel{q}{\to} \Gamma \to 1$$

where $q$ is the unique homomorphism mapping $\sigma_1\sigma_2$ to $\lambda z. (z-1)/z$, and $\sigma_1\sigma_2\sigma_1$ to $\lambda z. \frac{-1}{z}$.

## Related concepts

* [[SL(2,Z)]]

* [[conformal group]]

* [[Kleinian group]]

* [[elliptic fibration]]

* [[congruence subgroup]]

* [[level n subgroup]],

* [[modular equivariant elliptic cohomology]]

## References

Named after _[[August Möbius]]_.

See also:

* Wikipedia, _[Projective linear group](https://en.m.wikipedia.org/wiki/Projective_linear_group)_

[[!redirects PSL(2,C)]]

[[!redirects Moebius transformations]]
[[!redirects moebius transformation]]
[[!redirects moebius transformations]]
[[!redirects Möbius transformation]]
[[!redirects Möbius transformations]]
[[!redirects möbius transformation]]
[[!redirects möbius transformations]]
[[!redirects linear fractional transformation]]
[[!redirects linear fractional transformations]]
[[!redirects fractional linear transformation]]
[[!redirects fractional linear transformations]]

[[!redirects Möbius group]]
[[!redirects Moebius group]]

[[!redirects modular group]]
[[!redirects modular groups]]


MOFr

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

> under construction

#Contents#
* table of contents
{:toc}

## Idea

[[cobordism theory]] for unoriented manifolds with stably [[framed manifold|framed]] [[manifold with boundary|boundaries]], thus unifying [[MO]] with [[MFr]].

## Definition

Consider the [[cofiber sequence]] of the unit morphism of the [[ring spectrum]] [[MO]]

\[
  \label{AsUnitCofiber}
  \array{
    \mathbb{S}
    &
      \overset{
        1^{M\mathrm{O}}
      }{
        \longrightarrow
      }
    &
    M \mathrm{O}
    \\
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    \\
    \ast
    &\longrightarrow&
    M \mathrm{O}/ \mathbb{S}
  }
\]

The [[homotopy cofiber]]

$$
  M(\mathrm{O},fr)
  \;\coloneqq\;
  M \mathrm{O} / \mathbb{S}
  \,,
$$

has [[stable homotopy groups]] the [[cobordism ring]] of unoriented bordisms with [[stable tangent bundle|stably]] [[framed manifold|framed]] [[manifold with boundary|boundaries]]

\[
  \label{OFrCobordismRing}
  \Omega^{\mathrm{O},fr}_{\bullet}
  \;\coloneqq\;
  \pi_{\bullet}
  \big(
    M\mathrm{O}/\mathbb{S}
  \big)
\]

## Properties

### Boundary morphism to $MFr$
  {#BoundaryMorphism}

The realization (eq:AsUnitCofiber) makes it manifest  that there is a [[cohomology operation]] to [[MFr]] of the form

\[
  \label{BoundaryCohomologyOperation}
  \array{
    M(\mathrm{O},fr)
    \;=
    &
    M \mathrm{O}/\mathbb{S}
    &
    \overset{
      \;\;\;
      \partial
      \;\;\;
    }{\longrightarrow}
    &
    \Sigma
    \mathbb{S}
    & =\;
    \Sigma Mfr
    \\
    \pi_{2d+2}\big( M(\mathrm{O},fr) \big)
    &&
    \longrightarrow
    &&
    \pi_{2d+1}\big( Mfr \big)
  }
  \,.
\]

Namely, $\partial$ is the second next step in the long [[homotopy cofiber]]-sequence starting with $1^{M \mathrm{O}}$. In terms of the [[pasting law]]:

\[
  \label{BoundaryOperationViaPastingLaw}
  \array{
    \mathbb{S}
    &
      \overset{
        1^{M\mathrm{O}}
      }{
        \longrightarrow
      }
    &
    M \mathrm{O}
    &
    \longrightarrow
    &
    \ast
    \\
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    \\
    \ast
    &
    \longrightarrow
    &
    M \mathrm{O}/ \mathbb{S}
    &
      \underset{
        \partial
      }{
        \longrightarrow
      }
    &
    \Sigma \mathbb{S}
  }
\]

### Relation to $M \mathrm{O}$ and $M Fr$

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

The unit morphism of [[MO]] is trivial on [[stable homotopy groups]] in [[positive number|positive]] degree:

$$
  \array{
    \pi_{n+1}
    \big(
      \mathbb{S}
    \big)
    &
    \overset{ 1^{M \mathrm{O}} }{\longrightarrow}
    &
    \pi_{n + 1}
    \big(
      M \mathrm{O}
    \big)
    \\
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    \\
    \Omega^{fr}_{n + 1}
    &\underset{i}{\longrightarrow}&
    \Omega^{\mathrm{O}}_{n + 1}
  }
  \phantom{AAAAAAA}
  n \in \mathbb{N}
$$

=--

([Stong 68, p. 102-103](#Stong68))

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

In [[positive number|positive]] degree, the underling [[abelian groups]] of the [[bordism rings]] for [[MO]], [[MFr]] and $MOFr$ (eq:OFrCobordismRing) sit in [[short exact sequences]] of this form:

\[
  \label{ShortExactSequenceOfUFrBordismRings}
  0
  \to
  \Omega^{\mathrm{O}}_{n+2}
  \overset{i}{\longrightarrow}
  \Omega^{\mathrm{O},fr}_{n+2}
  \overset{\partial}{
    \longrightarrow
  }
  \Omega^{fr}_{n+1}
  \to
  0
  \,,
  \phantom{AAAA}
  n \in \mathbb{N}
  \,,
\]

where $i$ is the evident inclusion, while $\partial$ is the [[boundary]] homomorphism from [above](#BoundaryMorphism).

=--

([Stong 68, p. 102-103](#Stong68))

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

We have the [[long exact sequence of homotopy groups]] obtained from the [[cofiber sequence]] $\mathbb{S} \overset{1^{M\mathrm{O}}}{\longrightarrow} M \mathrm{O} \to M \mathrm{O}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}$ (eq:BoundaryOperationViaPastingLaw), the relevant part of which looks as follows:

\[
  \label{SToMULongExactSequenceOfHomotopyGroups}
  \array{
    \pi_{d+2}
    \big(
      \mathbb{S}
    \big)
    &
    \overset{
      1^{M\mathrm{O}}
    }{
      \longrightarrow
    }
    &
    \pi_{d+2}
    \big(
      M\mathrm{O}
    \big)
    &
    \overset{
    }{\longrightarrow}
    &
    \pi_{d+2}
    \big(
      M\mathrm{O}/\mathbb{S}
    \big)
    &
    \overset{
      \partial
    }{\longrightarrow}
    &
    \pi_{d+1}\big(\mathbb{S}\big)
    &\longrightarrow&
    \pi_{d+1}\big(M\mathrm{O}\big)
    \\
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    \\
    \Omega^{fr}_{d+2}
    &
    \underset{
      \color{green}
      0
    }{
      \longrightarrow
    }
    &
    \Omega^{\mathrm{O}}_{d+2}
    &
    \underset{
      i
    }{\longrightarrow}
    &
    \Omega^{(\mathrm{O},fr)}_{d+2}
    &
    \underset{
      \partial
    }{\longrightarrow}
    &
    \Omega^{fr}_{d + 1}
    &
    \underset{
      \color{green}
      0
    }{\longrightarrow}
    &
    \Omega^{\mathrm{O}}_{d+1}
  }
\]

Here the two outermost morphisms shown are [[zero morphisms]], by Prop. \ref{PositiveDegreeFramedBordismTrivialInUnorientedBordism}, and hence the claim follows.

=--

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#Stong68} [[Robert Stong]], p. 102-106 of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

Analogous discussion for [[MO]]-bordism with [[MSO]]-boundaries:

* G. E. Mitchell, _Bordism of Manifolds with Oriented Boundaries_, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 ([doi:10.2307/2040234](https://doi.org/10.2307/2040234))


Mogens Nielsen

* [institute page](https://pure.au.dk/portal/en/persons/mogens-nielsen(7e58ccfe-511f-4618-81a7-b7bf5b914f42).html)

* [GoogleScholar page](https://scholar.google.com/citations?user=EfAgUqYAAAAJ)

## Selected writings

On [[transition systems]] and other [[models for concurrency]]:

* [[Mogens Nielsen]], [[Glynn Winskel]]: *Models for concurrency*, in: *Handbook of Logic in Computer Science* vol 4, Oxford Univ. Press (1994) 1-148 [[pdf](https://www.cl.cam.ac.uk/~gw104/winskel-nielsen-models-for-concurrency.pdf), [[WinskelNielsen-Concurrency.pdf:file]], [doi:10.5555/218623.218630](https://dl.acm.org/doi/10.5555/218623.218630), [ISBN:9780198537809](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537809?lang=en&cc=at)]

On [[transition systems]], [[bisimulations]] and [[open morphisms]]:

* {#JoyalNielsenWisnkel94} [[André Joyal]], [[Mogens Nielsen]], [[Glynn Winskel]]: *Bisimulation from open maps*, BRICS report series RS-94-7 (1994) [[doi:10.7146/brics.v1i7.21663](https://doi.org/10.7146/brics.v1i7.21663), [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)], Information and Computation **127** 2 (1996) 164-185 [[doi:10.1006/inco.1996.0057](https://doi.org/10.1006/inco.1996.0057)]

category: people


Mohamed Barakat

* [webpage](https://mohamed-barakat.github.io/)

category: people


Mohamed Rameez

* [webpage](https://www.nbi.ku.dk/english/staff/?pure=en/persons/550338)

## Selected writings

On [[inhomogeneous cosmology]] as an explanation for apparent [[dark energy]]:

* {#SarkarEtAl18} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _Evidence for anisotropy of cosmic acceleration_, Astronomy & Astrophysics Letters ([arXiv:1808.04597](https://arxiv.org/abs/1808.04597))

* {#Rameez19} [[Mohamed Rameez]], _On the Real Inhomogeneous Universe and the Weirdness of 'Dark Energy'_ ([Nov. 14, 2019](https://4gravitons.com/2019/11/15/guest-post-on-the-real-inhomogeneous-universe-and-the-weirdness-of-dark-energy/))

* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))

On the evidence for [[dark energy]] in [[cosmology]]:

* {#MRS21} [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], *Do supernovae indicate an accelerating universe?* ([arXiv:2106.03119](https://arxiv.org/abs/2106.03119))

On evidence for [[inhomogeneous cosmology]] (large anisotropies) in observed radio galaxies and quasars:

* {#SHRMS22} Nathan Secrest, Sebastian von Hausegger, [[Mohamed Rameez]], [[Roya Mohayaee]], [[Subir Sarkar]], *A Challenge to the Standard Cosmological Model*,  ApJ Letters (2022) [[arXiv2206.05624](https://arxiv.org/abs/2206.05624)]

category: people


Mohamed Saidi

Mohamed Saidi works at the University of Exeter.  His research interests are in arithmetic geometry.

* [Webpage](http://empslocal.ex.ac.uk/people/staff/ms220/Site/Welcome.html)

category:people


Mohammad Ahmady

* [webpage](https://www.mta.ca/Community/Bios/Mohammad_Ahmady/Mohammad_Ahmady/)

## Selected writings

[[!include application of holographic QCD to B-meson physics -- references]]

category: people


Mohammad Garousi

* [webpage](http://garoosi.profcms.um.ac.ir)

## Selected writings

On [[anti D-brane|anti]] [[M2-branes]]:

* [[Mohammad Garousi]], _A proposal for M2-brane-anti-M2-brane action_, Phys. Lett.B686:59-63, 2010 ([arXiv:0809.0381](https://arxiv.org/abs/0809.0381))

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

* [[Mohammad Garousi]], _T-duality of curvature terms in D-brane actions_, JHEP 1002:002, 2010 ([arXiv:0911.0255](https://arxiv.org/abs/0911.0255))

* [[Mohammad Garousi]], _S-duality of D-brane action at order $O(\alpha'{}^2)$_, Phys. Lett. B701:465-470, 2011 ([arXiv:1103.3121](https://arxiv.org/abs/1103.3121))

* Ali Jalali, [[Mohammad Garousi]], _On D-brane action at order $\alpha'{}^2$_, Phys. Rev. D 92, 106004 (2015) ([arXiv:1506.02130](https://arxiv.org/abs/1506.02130))

* [[Mohammad Garousi]], _An off-shell D-brane action at order $\alpha'{}^2$ in flat spacetime_, Phys. Rev. D 93, 066014 (2016) ([arXiv:1511.01676](https://arxiv.org/abs/1511.01676))

On [[higher curvature corrections]] to [[D=11 supergravity]] in [[M-theory]]:

* [[Mohammad Garousi]], _Minimal gauge invariant couplings at order $\ell^6_p$ in M-theory_ ([arXiv:2102.00639](https://arxiv.org/abs/2102.00639))

category: people


Mohammad Sheikh-Jabbari

* [webpage](http://www.ipm.ac.ir/personalinfo.jsp?PeopleCode=IP0300097)

## Selected writings

Discussion of [[pp-wave spacetimes]] as [[Penrose limits]] ([[Inönü-Wigner contractions]]) of [[anti de Sitter spacetime|AdSp]] [[product manifold|x]] [[n-sphere|S^q]] spacetimes and of the corresponding limit of [[AdS-CFT duality]]:

* Darius Sadri, [[Mohammad Sheikh-Jabbari]], _The Plane-Wave/Super Yang-Mills Duality_, Rev. Mod. Phys. 76:853, 2004 ([arXiv:hep-th/0310119](https://arxiv.org/abs/hep-th/0310119))

On the [[BMN matrix model]]:

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

* [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Protected Multiplets of M-Theory on a Plane Wave_, JHEP 0209:021, 2002 ([arXiv:hep-th/0207050](https://arxiv.org/abs/hep-th/0207050))

On [[M5-branes]] in the [[BMN matrix model]]:

* [[Juan Maldacena]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Transverse Fivebranes in Matrix Theory_, JHEP 0301:038, 2003 ([arXiv:hep-th/0211139](https://arxiv.org/abs/hep-th/0211139))

category: people


Mohammed Abouzaid

__Mohammed Abouzaid__ is a geometer ([[symplectic geometry]], [[Fukaya categories]], [[mirror symmetry]]), now a postdoc at MIT and a Clay Institute Research Fellow.

* [webpage](http://math.mit.edu/~abouzaid) at MIT

* Clay institute fellowship [description](http://www.claymath.org/fas/research_fellows/Abouzaid), [research proposal](http://www.claymath.org/fas/research_fellows/Abouzaid/research_statement.pdf)

## Selected writings

On the [[Arnold conjecture]] and [[Morava K-theory]]:

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

[[!redirects M. Abouzaid]]


Mohammed Labbi

* [webpage](https://sites.google.com/site/mllabbi/)

## selected writings

* {#Labbi97} [[Mohammed Labbi]], _Stability under surgeries of the $p$-curvature positivity and manifolds with positive Einstein tensor_, Annals of Global Analysis and Geometry, 15 no 4, 299-312 (1997). ([pdf](http://streaming.ictp.trieste.it/preprints/P/96/097.pdf))

* [[Mohammed Labbi]], _Manifolds with positive second Gauss-Bonnet curvature_, Pacific Journal of Math. Vol. 227, No. 2, 295-310, (2006) ([pdf](http://msp.org/pjm/2006/227-2/pjm-v227-n2-p04-s.pdf))

* Boris Botvinnik, [[Mohammed Labbi]], _Highly connected manifolds of positive $p$-curvature_, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 ([arXiv:1201.1849](http://arxiv.org/abs/1201.1849), [doi:10.1090/S0002-9947-2014-05939-4](https://doi.org/10.1090/S0002-9947-2014-05939-4))

## related $n$Lab entries

* [[p-curvature]]

* [[fivebrane structure]]

category: people


Moira Chas

* [website](http://www.math.sunysb.edu/~moira/)

## related $n$Lab entries:

* [[string topology]]

category: people


Moishezon manifold

Moishezon manifold is a compact [[complex manifold]] $M$ such that each irreducible component has a field of [[meromorphic function]]s of transcendence degree $dim_{\mathbf{C}}M$.

According to a result of Moishezon, a Moishezon manifold has a structure of projective algebraic variety iff it admits a Kahler structure.

There is a somewhat bigger category of Moishezon spaces which happens to be equivalent to [[algebraic space]]s proper over (the spectrum of) $\mathbf{C}$.

Moishezon variety is a compact complex variety that is bimeromorphic to a projective variety.

[[!redirects Moishezon variety]]


Mojtaba Alipour

* [webpage](http://chem.shirazu.ac.ir/?page_id=443)

## related $n$Lab entries

* [[Lamb shift]]

category: people


molecule

## Related concepts

* [[atom (physics)]]

* [[chemistry]], [[chemical element]]

* [[particle]]

* [[Æ code]]

## References

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

[[!redirects molecules]]


moment

>This article is on the concept of moment in probability theory. On some pages 'moment' is being used for [[idempotent (∞,1)-monad]], e.g., see [[negative moment]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $\mu$ a [[probability measure]] on the [[real line]] $\mathbb{R}$, its **$n$th moment** for $n \in \mathbb{N}$ is the [[expectation value]] of $x^n$.

## Properties

* Moments of [[Gaussian probability distributions]] have a combinatorial expression by [[Wick's lemma]].

## Related concepts

* [[expectation value]]

* [[Feynman diagram]]

[[!redirects moments]]


moment map

[[!redirects moment map]]

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

# The momentum map
* table of contents
{: toc}

## Idea

A _momentum map_ is a dual incarnation of a [[Hamiltonian action]] of a [[Lie group]] (or [[Lie algebra]]) on a [[symplectic manifold]].

An [[action]] of a [[Lie group]] $G$ on a [[symplectic manifold]] $X$ by ([[Hamiltonian symplectomorphisms|Hamiltonian]]) [[symplectomorphisms]] corresponds [[infinitesimal space|infinitesimally]] to a Lie algebra [[homomorphism]] from the [[Lie algebra]] $\mathfrak{g}$ to the [[Hamiltonian vector fields]] on $X$. If this lifts to a coherent choice of [[Hamiltonians]], hence to a Lie algebra homomorphism $\mathfrak{g} \to (C^\infty(X), \{-,-\})$ to the [[Poisson bracket]], then, by [[dual vector space|dualization]], this is equivalently a [[Poisson manifold]] homomorphism of the form

$$
  \mu : X \to \mathfrak{g}^*
  \,.
$$

This $\mu$ is called the _momentum map_ (or _moment map_[^name]) of the Hamiltonian action.
[^name]: Moment map is a misnomer and physically incorrect. It is an erroneous translation of the French notion application moment. See [this mathoverflow question](https://mathoverflow.net/q/242468) for the history of the name.

The name derives from the special and historically first case of [[angular momentum]] in the [[rigid body dynamics|dynamics of rigid bodies]], see _[Examples - Angular momentum](#AngularMomentum)_ below.

## Definition

The [Preliminaries](#Preliminaries) below review some basics of [[Hamiltonian vector fields]]. The definition of the momentum map itself is below in _[Hamiltonian action and the momentum map](#HamiltonianActionAndTheMomentumMap)_.

### Preliminaries
 {#Preliminaries}

This section briefly reviews the notion of [[Hamiltonian vector fields]] on a [[symplectic manifold]]

The basic setup is the following: Let $(M,\omega)$ be a [[symplectic manifold]] with a [[Hamiltonian action]] of a [[Lie group]] $G$. In particular that means that there is an [[action]] $\nu\colon G \times M \to M$ via [[symplectomorphism]]s (diffeomorphisms $\nu_g$ such that $\nu_g^*(\omega) = \omega$). A vector field $X$ is [[symplectic vector field|symplectic]] if the corresponding flow preserves (again by pullbacks) $\omega$. The symplectic vector fields form a Lie subalgebra $\chi(M,\omega)$ of the Lie algebra of all smooth vector fields $\chi(M)$ on $M$ with respect to the Lie bracket.

By the [[Cartan homotopy formula]] and closedness $d \omega = 0$

$$
\mathcal{L}_X \omega = d \iota_X \omega
$$

where $\mathcal{L}_X$ denotes the [[Lie derivative]]. Therefore a vector field $X$ is symplectic iff $\iota(X)\omega = d H$ for some function $H\in C^\infty(M)$, usually called Hamiltonian (function) for $X$. Here $X$ is determined by $H$ up to a locally constant function. Such $X = X_H$ is called the __Hamiltonian vector field__ corresponding to $H$.  The Poisson structure on $M$ is the bracket $\{,\}$ on functions may be given by

$$
\{ f, g\} := [X_f,X_g]
$$

where there is a Lie bracket of vector fields on the right hand side.

For $(M,\omega)$ a connected symplectic manifold,  there is an exact sequence of [[Lie algebras]]

$$
  0 \to \mathbf{R}\to (C^\infty(M), \{-,-\}) \to \chi(M,\omega) \to 0
  \,.
$$

See at _[Hamiltonian vector field -- Relation to Poisson bracket](Hamiltonian+vector+field#RelationToFunctions)_.

### Hamiltonian action and momentum map
 {#HamiltonianActionAndTheMomentumMap}

Let $(X, \omega)$ be a [[symplectic manifold]] and let $\mathfrak{g}$ be a [[Lie algebra]]. Write $(C^\infty(X), \{-,-\})$ for the [[Poisson bracket]] Lie algebra underlying the corresponding [[Poisson algebra]].

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

A _[[Hamiltonian action]]_ of $\mathfrak{g}$ on $(X, \omega)$ is a [[Lie algebra]] [[homomorphism]]

$$
  \tilde \mu \;\colon\; \mathfrak{g} \longrightarrow (C^\infty(X), \{-,-\})
  \,.
$$

The corresponding function

$$
  \mu \;\colon\; X \longrightarrow \mathfrak{g}^*
$$

to the [[dual vector space]] of $\mathfrak{g}$, defined by

$$
  \mu \;\colon\; x \mapsto \tilde \mu(-)(x)
$$

is the corresponding **momentum map**.

=--

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

If one writes the evaluation pairing as

$$
  \langle -,-\rangle : \mathfrak{g}^* \otimes \mathfrak{g} \to \mathbb{R}
$$

then the equation characterizing $\mu$ in def. \ref{MomentumMap} reads for all $x \in X$ and $v \in \mathfrak{g}$

$$
  \langle \mu(x), v \rangle = \tilde \mu(v)(x)
  \,.
$$

This is the way it is often written in the literature.

Notice that this in turn means that

$$
  \tilde \mu(v)= \mu^\ast \langle -,v\rangle
  \,.
$$

=--

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

The following are equivalent

1. the linear map underlying $\tilde\mu$ in def. \ref{MomentumMap} is [[Lie algebra]] [[homomorphism]];

1. its dual $\mu$ is a [[Poisson manifold]] [[homomorphism]] with respect to the [[Lie-Poisson structure]] on $\mathfrak{g}^\ast$.

=--

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

This follows by just unwinding the definitions.

In one direction, suppose that $\tilde \mu$ is a Lie homomorphism. Since the [[Lie-Poisson structure]] is fixed on linear functions on $\mathfrak{g}^\ast$, it is sufficient to check that $\mu^\ast$ preserves the Poisson bracket on these.
Consider hence two Lie algebra elements $v_1, v_2 \in \mathfrak{g}$ regarded as linear functions $\langle -,v_i\rangle$ on $\mathfrak{g}^\ast$. Noticing that on such linear functions the Lie-Poisson structure is given by the Lie bracket we have, using remark \ref{Notation}

$$
  \begin{aligned}
    \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\}
    &=
    \mu^\ast \langle-,[v_1,v_2]\rangle
    \\
    & = \tilde \mu([v_1,v_2])
    \\
    & = \{\tilde\mu(v_1), \tilde\mu(v_2)\}
    \\
    & =
   \left\{
      \mu^\ast \langle -,v_1\rangle, \mu^\ast \langle -,v_2\rangle
   \right\}
  \end{aligned}
$$

and hence $\mu^\ast$ preserves the Poisson brackets.

Conversely, suppose that $\mu$ is a Poisson homomorphism. Then

$$
  \begin{aligned}
    \tilde\mu [v_1,v_2]
    &=
    \mu^\ast \langle -, [v_1,v_2]\rangle
    \\
    & =
    \mu^\ast \{\langle -,v_1\rangle, \langle -,v_2\rangle\}
    \\
    & =
    \left\{
      \mu^\ast \langle -, v_1\rangle,
      \mu^\ast \langle -, v_2\rangle
    \right\}
    \\
    & =
    \left\{
      \tilde\mu(v_1), \tilde\mu(v_2)
    \right\}
  \end{aligned}
$$

and so $\tilde \mu$ is a Lie homomorphism.

=--

## Examples

### Angular momentum
 {#AngularMomentum}
Consider the action of SO(3) on $\mathbb{R}^3$, which induces a Hamiltonian action on $T^*\mathbb{R}^3\cong\mathbb{R}^3\times\mathbb{R}^3$ via
$$(q,p)\xrightarrow{A\in\text{SO(3)}}(A q, p A^{-1})$$
where $q$ is a column vector and $p$ is a row vector. Then the momentum map for this Hamiltonian action is
$$\mu\colon T^*(\mathbb{R}^3)\to \mathfrak{so}(3)^*,\quad \left\langle\mu(q,p),\;\vec\theta\cdot\begin{pmatrix}\Omega_1\\\Omega_2\\\Omega_3\end{pmatrix} \right\rangle\to (\vec{q}\times \vec p)\cdot\vec{\theta}$$
where
$$\Omega_1=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix},\quad\Omega_2=\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix},\quad \Omega_3=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}$$

If we choose $\Omega_1,\Omega_2,\Omega_3$ as an orthonormal basis of $\mathfrak{so}(3)$ and then identify $\mathfrak{so}(3)\cong\mathfrak{so}(3)^*\cong\mathbb{R}^3$, then $\mu(q,p)=\vec q\times\vec p$, which is the angular momentum.

## Properties

### Relation to conserved quantities

The values of the momentum map for each given Lie algebra generator may be regarded as the [[conserved current|conserved currents]] given by a _Hamiltonian [[Noether theorem]]_.

Specifically if $(X,\omega)$ is a [[symplectic manifold]] equipped with a "time evolution" [[Hamiltonian action]] $\mathbb{R} \to \mathfrak{Poisson}(X,\omega)$ given by a [[Hamiltonian]] $H$ and if $\mathfrak{g} \to \mathfrak{Poisson}(X,\omega)$ is some [[Hamiltonian action]] with momentum $\Phi(\xi)$ for $\xi \in \mathfrak{g}$ which preserves the Hamiltonian in that the [[Poisson bracket]] vanishes

$$
  \{\Phi^\xi, H\} = 0
$$

then of course also the time evolution of the momentum vanishes

$$
  \frac{d}{d t} \Phi^\xi = \{H, \Phi^\xi\} = 0
  \,.
$$

See at _[Noether theorem -- In terms of momentum maps/Hamiltonian Noether theorem ](Noether%27s+theorem#HamiltonianNoetherTheorem)_.

### Relation to constrained mechanics
 {#RelationToConstrainedMechanics}

In the context of [[constrained mechanics]] the components of the momentum map (as the Lie algebra argument varies) are called [[first class constraints]]. See _[[symplectic reduction]]_ for more.

## Related concepts

The momentum map is a crucial ingredient in the construction of Marsden--Weinstein [[symplectic quotients]] and in other variants of symplectic reduction.

## References

The concept is originally due to [[Jean-Marie Souriau]].

### General

* [[Victor Guillemin]], [[Yael Karshon]], [[Viktor Ginzburg]], _Moment maps, cobordisms, and Hamiltonian group actions_, Mathematical Surveys and Monographs, volume 98 (2002) 350 pp

Lecture notes and surveys include

* [[Joel W. Robbin]], _The moment map_, an exposition, [pdf](http://www.math.wisc.edu/~robbin/moment.pdf)

* [[Nicole Berline]], [[Michèle Vergne]], _Hamiltonian manifolds and moment maps_ ([pdf](http://www.math.polytechnique.fr/~berline/cours-Fudan.pdf))

Original articles include

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric asymptotics_, AMS (1977) ([online](http://www.ams.org/online_bks/surv14))

* [[Michael Atiyah]], _Convexity and commuting Hamiltonians_, Bull. London
Math. Soc. __14__ (1982), 1-15.

* [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology, Vol 23, No. 1 (1984) ([pdf](https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))
  {#AtiyahBott84}

Further developments are in

* M. Spera, _On a generalized uncertainty principle, [[coherent state]]s and the moment map_, J. of Geometry and Physics __12__ (1993) 165-182, [MR94m:58097](http://www.ams.org/mathscinet-getitem?mr=1237511), doi

* [[Ctirad Klimčik]], [[Pavol Ševera]], _[[T-duality]] and the moment map_, IHES/P/96/70, [hep-th/9610198](https://arxiv.org/abs/hep-th/9610198); _Poisson-Lie T-duality: open strings and D-branes_, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, [hep-th/9512124](https://arxiv.org/abs/hep-th/9512124)

* A. Cannas da Silva, [[Alan Weinstein]], _Geometric models for noncommutative algebras_, Berkeley Math. Lec. Notes Series, AMS 1999, ([pdf](http://math.berkeley.edu/%7Ealanw/Models.pdf))

* Friedrich Knop, _Automorphisms of multiplicity free Hamiltonian manifolds_, [arxiv/1002.4256](https://arxiv.org/abs/1002.4256)

* [[W. Crawley-Boevey]], _Geometry of the moment map for representations of quivers_, Compositio Math. __126__ (2001), no. 3, 257-293 [doi](https:/doi.org/10.1023/A:1017558904030) [MR1834739](https://mathscinet.ams.org/mathscinet-getitem?mr=1834739)

See also

* wikipedia, _[momentum map](http://en.wikipedia.org/wiki/Moment_map)_

Momentum maps in [[higher geometry]], [[schreiber:Higher geometric prequantum theory]], are discussed in

* Martin Callies, [[Yaël Frégier]], [[Chris Rogers]], [[Marco Zambon]], _Homotopy moment maps_, Advances in Mathematics __303__ (2016) 954--1043 ([arXiv:1304.2051](https://arxiv.org/abs/1304.2051))

### Relation to symplectic reduction

Reviews include for instance

* {#FOF} [[José Figueroa-O'Farrill]], p. 26 of  chapter 2+3 of PhD thesis  ([pdf](http://empg.maths.ed.ac.uk/Activities/BRST/Ch2+3PhD.pdf))

### Relation to equivariant cohomology

Relation to [[equivariant cohomology]]:

* [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology vol. 23 No. 1 1984  ([pdf](https://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))

### Generalization: group-valued momentum maps

* [[Anton Alekseev]], Anton Malkin, [[Eckhard Meinrenken]], _Lie group valued moment maps_, J. Differential Geom. Volume 48, Number 3 (1998), 445-495. [euclid](http://projecteuclid.org/euclid.jdg/1214460860), [MR1638045](http://www.ams.org/mathscinet-getitem?mr=1638045)

* [[Eckhard Meinrenken]], _Lectures on group-valued moment maps and Verlinde formulas_, 35 pages, January 2012, [pdf](http://www.math.toronto.edu/mein/research/NotreLectures.pdf)

The relation between momentum maps and [[conserved currents]]/[[Noether's theorem]] is amplied for instance in

* {#Fan} Huijun Fan, _Lecture 8, Moment map and symplectic reduction_ ([pdf](http://www.math.pku.edu.cn/teachers/fanhj/courses/symp-8.pdf))

### In diffeology
 {#ReferencesInDiffeology}

Discussion in [[diffeology]]:

* {#PIZ} [[Patrick Iglesias-Zemmour]], *The moment map in diffeology*, Memoirs of the American Mathematical Society (2010) $[$[web](http://math.huji.ac.il/~piz/Site/The%20Articles/084469EE-D890-4CAA-8A13-1C655A76CF63.html), [ISBN: 978-1-4704-0586-1](https://bookstore.ams.org/memo-207-972)$]$

### In thermodynamics
 {#ReferencesInThermodynamics}

Since momentum maps generalize [[energy]]-functionals, they provide a covariant formulation of [[thermodynamics]]:

* [[Jean-Marie Souriau]], _Thermodynamique et géométrie_,  Lecture Notes in Math. 676 (1978), 369–397 ([scan](http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=197810025))

* {#IglesiasSouriau84} [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]], _Heat, cold and Geometry_, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 ([web](http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm), [pdf](http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf), [doi:978-94-009-7022-9_5](https://doi.org/10.1007/978-94-009-7022-9_5))

* [[Jean-Marie Souriau]], chapter IV "Statistical mechanics" of _Structure of dynamical systems. A symplectic view of physics_ . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

* [[Patrick Iglesias-Zemmour]], _Essai de «thermodynamique rationnelle» des milieux continus_, Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 1,  p. 1-24 ([numdam:AIHPA_1981__34_1_1_0](http://www.numdam.org/item/AIHPA_1981__34_1_1_0))

* [[Jean-Marie Souriau]], _Mécanique statistique, groupes de Lie et cosmologie_, In Colloques int. du CNRS; numéro 237; Aix-en-Provence, France, 1974; pp. 24–28, 59–113. English translation by F. Barbaresco, April, 2020. Available online: https://www.academia.edu/42630654/Statistical_Mechanics_Lie_Group_and_Cosmology_
1_st_part_Symplectic_Model_of_Statistical_Mechanics(access on 20 April 2020)

Review includes

* {#Marle16} Charles-Michel Marle, _From tools in symplectic and Poisson geometry to Souriau's theories of statistical mechanics and thermodynamics_, Entropy 2016, 18(10), 370 ([arXiv:1608.00103](https://arxiv.org/abs/1608.00103))

* Charles-Michel Marle, _On Gibbs states of mechanical systems with symmetries_, arXiv:2012.00582v2 [math.DG] (2021)

* F. Barbaresco, F. Gay-Balmaz, _Lie group cohomology and (multi)symplectic integrators: new geometric tools for Lie group machine learning based on Souriau geometric statistical mechanics_, Entropy 22 (2020) 498  [link](https://www.mdpi.com/1099-4300/22/5/498)

* Koszul, J.-L., _Introduction to symplectic geometry_, SPRINGER, 2019

* F. Barbaresco (2021) Archetypal Model of Entropy by Poisson Cohomology as Invariant Casimir Function in Coadjoint Representation and Geometric Fourier Heat Equation. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol 12829. Springer

* F. Barbaresco, _Koszul lecture related to geometric and analytic mechanics, Souriau’s Lie group thermodynamics and information geometry_, Info. Geo. 4, 245--262 (2021)

* F. Barbaresco, (2021) Jean-Marie Souriau’s Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum. In: Barbaresco F., Nielsen F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning. SPIGL 2020. Springer Proceedings in Mathematics & Statistics, vol 361. Springer

* CM Marle, _Gibbs states on symplectic manifolds with symmetries_, In: F. Nielsen, F. Barbaresco (eds) Geometric Science of Information. GSI 2021. Lec. Notes in Comp. Sci. 12829. Springer 2021

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moment of inertia

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## Idea

In the [[mechanics]] of [[rigid body dynamics]] in [[Cartesian space]] $\mathbb{R}^n$, the _moment of inertia_ of a [[rigid body]] is the analog of [[mass]] for [[rotational dynamics]].  In [[linear dynamics]], we have the formula
$$ p = m v $$
which says that the [[momentum]] $p$ is proportional to the [[velocity]] $v$.  Similarly, in rotational dynamics, we have the analogous formula
$$ L = I \Omega $$
where $L$ is the [[angular momentum]], $\Omega$ is the [[angular velocity]], and $I$ is the __moment of inertia__.

However, the rotational equation is somewhat more complicated than the linear one: firstly because $L$ and $\Omega$ are not naturally vectors but [[bivectors]]; and secondly because they are not necessarily proportional, so that $I$ cannot be a [[scalar]].  In general, the moment of inertia is a [[linear function]]
$$
  I \colon \wedge^2 \mathbb{R}^n \to \wedge^2 \mathbb{R}^n
$$
so that the above equation becomes simply
$$ L = I(\Omega). $$
This linear function is additionally *symmetric* with respect to the induced [[inner product]] on $\bigwedge^2 \mathbb{R}^n$, so it can be represented in coordinates by a symmetric $\frac{n(n-1)}{2} \times \frac{n(n-1)}{2}$ [[matrix]].

Similarly, differentiating this equation once with respect to [[time]] (and assuming that $I$ is constant as it is for a rigid body), we have
$$ \tau = I \alpha ,$$
relating the total [[torque]] $\tau$ to the [[angular acceleration]] $\alpha$ --- this is the rotational analogue of [[Newton's second law]] $F = m a$ (where $m$ must be constant).

## In low dimensions

In low dimensions, the situation can be (and usually is) simplified.

* In two [[dimensions]], [[bivectors]] form a one-dimensional [[vector space]], so that the moment of inertia is simply a scalar.

* In three dimensions, bivectors form a three-dimensional [[vector space]], so that the moment of inertia can be represented by a symmetric $3 \times 3$ matrix.  Additionally, in three dimensions, there is an [[isomorphism]] between [[bivectors]] and [[vectors]] (once we choose an [[orientation]] to go with our inner product); so that [[angular velocity]] and [[momentum]] can be (and usually are) identified with [[vectors]], and the moment of inertia with a symmetric rank-2 [[tensor]].

## In Hamiltonian dynamics

In terms of the discussion at [[Hamiltonian dynamics on Lie groups]], the [[rigid body dynamics]] in $\mathbb{R}^n$ is given by Hamiltonian motion on the [[special orthogonal group]] $SO(n)$. It is defined by any [[left invariant]] [[Riemannian metric]]

$$
  \langle -,-\rangle \in Sym^2_{C^\infty(G)} \Gamma(T G)
$$

hence a bilinear non-degenarate form on the [[Lie algebra]] $\mathfrak{so}(n)$ (not necessarily the [[Killing form]]).

This bilinear form is the __moment of inertia__. (For instance [AbrahamMarsden, section 4.6](#AbrahamMarsden).)

## In terms of mass density

If a [[rigid body]] has [[mass density]] $\rho$, then its angular momentum is defined in terms of $\Omega$ by the $n$-dimensional integral
$$ L = \int \rho \vec{x} \wedge (\vec{x} \cdot \Omega) \,d^n x $$
over all space, where $\vec{x}$ is the vector from the origin to the point of integration, $\cdot$ denotes the [[interior product]] of a vector with a bivector (yielding a vector), and $\wedge$ denotes the [[exterior product]] of two vectors (yielding a bivector).

When $\Omega$ is the same everywhere (as for a rigid body), then we may view this as a function from $\Omega$ to $L$; this function is the __moment of inertia__.

## Related pages

* [[Hamiltonian dynamics on Lie groups]]

  * [[rigid body dynamics]]

    * [[angular velocity]]

    * [[angular momentum]]

## References

A classical textbook discussion is for instance section 4.6 of

* [[Ralph Abraham]], [[Jerrold Marsden]], _[[Foundations of Mechanics]]_
 {#AbrahamMarsden}

A pedestrian discussion of _moment of inertia_ in terms of [[bivector]]s that applies in any [[dimension]] of [[space]]([[spacetime]]) is around page 74 of

* Chris Doran, Anthony Lasenby, _Geometric Algebra for Physicists_ Cambridge University Press

or around page 56 of

* Chris Doran, Anthony Lasenby, _Physical applications of geometric algebra_ ([pdf](https://dspace.ist.utl.pt/bitstream/2295/52599/1/Lectures_on_Geometric_Algebra.pdf#page=56))

and around slide 6 in

* Anthony Lasenby, Chris Doran and Robert Lasenby, _Rigid Body Dynamics and Conformal Geometric Algebra_ ([pdf](http://staff.science.uva.nl/~leo/agacse2010/talks_world/Lasenby.pdf))

These authors amplify the canonical embedding of bivectors into the [[Clifford algebra]], which they call "[[Geometric Algebra]]".


momentum

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> this entry needs attention

Momentum is, according to [[Isaac Newton|Newton]], the 'amount of motion' in an object.  For particles it is  proportional to the object\'s [[mass]] and to its [[velocity]], and thus measured in units of mass times velocity. For photons it is proportional to the frequency of the photon.

In [[Lagrangian mechanics]] (including relativistic and quantum versions), to every (generalized) coordinate $q_i$ one associates a (generalized) momentum
$$
p_i = \frac{\partial L}{\partial \dot{q}^i}
$$
where $L$ is the [[Lagrangian]] of the system.  More generally, we can say that $p_i$ is the coordinate of $\mathrm{d}q^i$ in the [[action functional|action form]].

If $L$ takes the form $\frac{1}{2} \sum_i m_i \dot{q}_i^2 - U(q_1,\ldots,q_n)$, then we have $p_i = m_i \dot{q}_i$.  Thus, the __linear momentum__ of a point particle in Newtonian mechanics is traditionally *defined* as the intrinsic [[mass]] $m$ times the [[velocity]] $\vec{\dot{q}}$.  However, this breaks down in some situations:

*  Relativistically, the velocity factor is more complicated (tending to $\infty$ as velocity tends to the [[speed of light]]).  However, it is possible to keep $p = m v$ using a velocity-dependent relativistic mass (especially since the mass may depend on other dynamical considerations).
*  In the presence of velocity-dependent forces (notably those produced by a [[magnetic field]]), there is an additional position-dependent term.
*  In nonlinear coordinates (such as for [[angular momentum]]), one not only needs to use a nonlinear velocity (such as [[angular velocity]]) but also replace mass with something more complicated (such as [[moment of inertia]]) which is often position-dependent.

In [[Hamiltonian mechanics]], the momentum coordinates are half of the basic coordinates on [[phase space]].  If phase space is derived as the [[cotangent bundle]] of a [[configuration space]], then the momentum coordinates are the new coordinates on the cotangent bundle, dual to the original coordinates on configuration space; we have the [[symplectic form]] $\sum_i p_i \mathrm{d}q^i$.  However, Hamiltonian mechanics cares only about phase space as a [[symplectic manifold]] with a [[Hamiltonian]]; there is no inherent distinction between position and momentum coordinates, and you can even make a [[canonical coordinate transformation|canonical transformation]] that swaps them (up to a minus sign).

In [[quantum mechanics]], the [[canonical quantization]] process formally replaces momentum by $-i \hbar \frac{\partial}{\partial x},$ where $x$ is position.  (Equivalently, in [[momentum space]], canonical quantization replaces $x$ by $i \hbar \frac{\partial}{\partial p}.)$  In this situation, position and momentum [[canonical commutation relation|fail to commute]].

There is a generalization of momentum in symplectic geometry, so called [[moment map]].

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Mon(∞,1)Cat

#Contents#
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## Idea

The [[(∞,2)-category]] of [[monoidal (∞,1)-categories]].

## Variants

### $\mathcal{O}$-monoidal $(\infty,1)$-category

A [[monoidal (∞,1)-category]] $(C, \otimes)$ is equivalently a [[model structure for dendroidal Cartesian fibrations|coCartesian fibration of (∞,1)-operads]] over [[Assoc]].

A [[symmetric monoidal (∞,1)-category]] $(C, \otimes)$ is equivalently a [[model structure for dendroidal Cartesian fibrations|coCartesian fibration of (∞,1)-operads]] over [[Comm]].

Accordingly, for $\mathcal{O}$ any [[(∞,1)-operad]], a coCartesian fibration of $(\infty,1)$-operads over $\mathcal{O}$ may be called an _$\mathcal{O}$-monoidal $(\infty,1)$-category_.

$$
  O Mon(\infty,1)Cat := coCart_{\mathcal{O}}
  \,.
$$

## Properties

### Model category presentations

see _[[table - models for (infinity,1)-operads]]_

category: category

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Mona Merling

* [personal page](https://www2.math.upenn.edu/~mmerling/)

## Selected writings

On [[classifying spaces]]/[[universal principal bundles]] for [[equivariant principal bundles]]:

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

category: people