Bell's inequality

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#### Measure and probability theory
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#### Quantum systems
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#Contents#
* table of contents
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## Idea

In [[quantum physics]]/[[quantum information theory]], What came to be called *Bell's inequality* ([Bell 1964](#Bell64)) is an [[inequality]] satisfied by the three pairwise [[correlation functions]] between three [[random variables]] defined on one and the same classical [[probability space]]. As such, it is an elementary statement about classical probability theory which as been argued ([Pitowsky 1989a](#Pitowsky89a)) to have been known already to [[Boole -- The Laws of Thought|Boole (1854)]].

The point of the argument by [Bell 1964](#Bell64) was to highlight that when taking these three random variables to be the results of [[quantum measurements]] of the [[spin]] of an [[electron]] along three pairwise non-orthogonal axes (as in the [[Stern-Gerlach experiment]]) then [[quantum theory]] predicts that this inequality is *violated* -- implying that there is no single classical [[probability space]] (called a *[[hidden variable theory|hidden variable]]* in the context of [[interpretations of quantum mechanics]]) on which these three [[quantum measurement]]-results are jointly [[random variables]].

A number of [[experiments]] have sought to check Bell's inequalities in [[quantum physics]] ("[[Bell tests]]") and all claim to have verified that it is indeed violated in nature (see [Aspect 2015](#Aspect15)), as predicted by [[quantum theory]].

Bell's inequality has been and is receiving an enormous amount of attention, first in discussions of [[interpretations of quantum mechanics]], but more recently and more concretely also in the context of [[quantum information theory]].

## Statement
 {#Statement}

### Original formulation
 {#OriginalFormulation}

> The following is fairly verbatim recap of the original argument in [Bell 1964](#Bell64). For a streamlined re-statement see further [below](#CompactReformulation).

Let us denote the result _A_ of a measurement that is determined by a unit vector, $\vec{a}$, and some parameter $\lambda$ as $A(\vec{a},\lambda)=\pm 1$ where we further suppose that the outcome of the measurement is either +1 or -1.  Likewise, we may do the same for the result _B_ of a second measurement, i.e. $B(\vec{b},\lambda)$.  We further make the vital assumption that the result _B_ does not depend on $\vec{a}$ and likewise _A_ does not depend on $\vec{b}$.

Before proceeding, we should note that $\lambda$ here plays the role of a "hidden" parameter or variable.  We say it is "hidden" because its precise nature is not known.  However, it is still a very real parameter with a probability distribution $\rho(\lambda)$.  The expectation value of the product of the two measurements is

\[
  \label{AnExpectationValue}
  P(\vec{a},\vec{b})=\int d\lambda\rho(\lambda)A(\vec{a},\lambda)B(\vec{b},\lambda).
\]

Because $\rho$ is a normalized probability distribution,

$$
  \int d\lambda \rho(\lambda) = 1
$$

and because $A(\vec{a},\lambda)=\pm 1$ and $B(\vec{b},\lambda)=\pm 1$, P cannot be less than -1.  It can be equal to -1 at $\vec{a}=\vec{b}$ only if $A(\vec{a},\lambda)=\pm 1 = -B(\vec{a},\lambda)=\pm 1$ except at a set of points $\lambda$ of zero probability.  Thus we can write (eq:AnExpectationValue) as

\[
  P(\vec{a},\vec{b})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda).
\]

If we introduce a third unit vector $\vec{c}$ we can find the difference between the correlation of $\vec{a}$ to the two other unit vectors,

\[
  \label{DifferenceOfCorrelations}
  P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)].
\]

Rearranging this we may write (eq:DifferenceOfCorrelations) as

\[
  P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1].
\]

Given the limitations we have placed on the value of _A_, we may write

\[
  |P(\vec{a},\vec{b})-P(\vec{a},\vec{c})| \le \int d\lambda\rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)].
\]

But the second term on the right is simply $P(\vec{b},\vec{c})$ and thus

\[
  1 + P(\vec{b},\vec{c}) \ge |P(\vec{a},\vec{b})-P(\vec{a},\vec{c})|
\]

which is the original form of Bell's inequality.  Note that this may be written in terms of correlation coefficients,

\[
  1 + C(b,c) \ge |C(a,b)-C(a,c)|
\]

where _a_, _b_, and _c_ are now settings on the measurement apparatus.

#### Quantum mechanical violations

The original derivation of Bell's inequalities involved the use of a [[Stern-Gerlach device]] that measures spin along an axis.  Suppose $\sigma_{1}$ and $\sigma_{2}$ are spins.  The result, _A_, of measuring $\sigma_{1}\cdot\vec{a}$ is then interpreted as being entirely determined by $\vec{a}$ and $\lambda$.  Likewise for _B_ and $\sigma_{2}\cdot\vec{b}$.  It is also important to remember that the result _B_ does not depend on $\vec{a}$ and likewise _A_ does not depend on $\vec{b}$.

For a singlet state (that is a state with total spin of zero), the quantum mechanical expectation value of measurements along two different axes (see the Wigner derivation below for a more intuitive explanation of the physical nature of this) is

\[
  \langle\sigma_{1}\cdot\vec{a},\sigma_{2}\cdot\vec{b}\rangle = - \vec{a}\cdot\vec{b}.
\]

In theory this ought to equal $P(\vec{a},\vec{b})$ but in practice it does not.  It is important to remember that we are using _classical_ reasoning throughout our derivations of the various forms of Bell's inequalities.

The setup envisioned here consists of pairs of spin-1/2 particles produced in singlet states that then each pass through separate Stern-Gerlach (SG) devices.  Since they are in singlet states, if we measured the first particle of a pair to be aligned with a given axis, say $\vec{a}$, then the second should be measured to be anti-aligned with that same axis, giving a total spin of zero.

In practice we are dealing with _beams_ of particles and thus we can never be absolutely certain that correlated pairs are measured simultaneously and so we ultimately are making statistical predictions.  Nevertheless, in a given sample consisting of a large-enough number of randomly distributed spin-1/2 particles, we can be certain that, for example, a definite number are aligned with an axis $\vec{a}$ while a definite number are aligned with an axis $\vec{b}$.

Now take an individual particle and suppose that, for this particle, if we measured $\sigma\cdot\vec{a}$ we would obtain a +1 with certainty (meaning it is aligned with $\vec{a}$) but if we instead chose to measure $\sigma\cdot\vec{b}$ we would obtain a -1 with certainty (meaning it is anti-aligned with $\vec{b}$).  Notationally we refer to such a particle as belonging to type $(\vec{a}+,\vec{b}-)$.  Clearly for a given pair of particles in a singlet state, if particle 1 is of type $(\vec{a}+,\vec{b}-)$, then particle 2 must be of type $(\vec{a}-,\vec{b}+)$.

#### Locality

For beams of correlated particles measuring along only two axes, we should expect to get a roughly evenly balanced distribution of types as follows:

$$
\array{
\text{ Particle 1 } & & \text{ Particle 2 } \\
(\vec{a}+,\vec{b}-) & \leftrightarrow & (\vec{a}-,\vec{b}+) \\
(\vec{a}+,\vec{b}+) & \leftrightarrow & (\vec{a}-,\vec{b}-) \\
(\vec{a}-,\vec{b}-) & \leftrightarrow & (\vec{a}+,\vec{b}+) \\
(\vec{a}-,\vec{b}+) & \leftrightarrow & (\vec{a}+,\vec{b}-)
}
$$

There is a very important assumption implied here.  Suppose a particular pair belongs to the first grouping, that is if an observer _A_ decides to measure the spin along $\vec{a}$ for particle 1, he or she _necessarily_ obtains a plus sign (corresponding to it being aligned with $\vec{a}$) _regardless_ of any measurement observer _B_ may make on particle 2.  This is the principle of locality: _A_'s result is predetermined independently of _B_'s choice of what to measure.

#### Wigner's derivation

Now suppose we introduce a third axis, $\vec{c}$, so that we can have, for example, particles of type $(\vec{a}+,\vec{b}+,\vec{c}-)$ corresponding to being aligned if measured on $\vec{a}$ and $\vec{b}$ and anti-aligned on $\vec{c}$.  Further let us "count" the pairs that fall into the various groupings and label the populations as follows:

$$
\array{
\text{ Population } & \text{ Particle 1 } & \text{ Particle 2 } \\
N_{1} & (\vec{a}+,\vec{b}+, \vec{c}+) & (\vec{a}-,\vec{b}-,\vec{c}-) \\
N_{2} & (\vec{a}+,\vec{b}+, \vec{c}-) & (\vec{a}-,\vec{b}-,\vec{c}+) \\
N_{3} & (\vec{a}+,\vec{b}-, \vec{c}+) & (\vec{a}-,\vec{b}+,\vec{c}-) \\
N_{4} & (\vec{a}+,\vec{b}-, \vec{c}-) & (\vec{a}-,\vec{b}+,\vec{c}+) \\
N_{5} & (\vec{a}-,\vec{b}+, \vec{c}+) & (\vec{a}+,\vec{b}-,\vec{c}-) \\
N_{6} & (\vec{a}-,\vec{b}+, \vec{c}-) & (\vec{a}+,\vec{b}-,\vec{c}+) \\
N_{7} & (\vec{a}-,\vec{b}-, \vec{c}+) & (\vec{a}+,\vec{b}+,\vec{c}-) \\
N_{8} & (\vec{a}-,\vec{b}-, \vec{c}-) & (\vec{a}+,\vec{b}+,\vec{c}+)
}
$$

Let's suppose that observer _A_ finds particle 1 is aligned with $\vec{a}$, i.e. $\vec{a}+$, and that observer _B_ finds particle 2 is aligned with $\vec{b}$, i.e. $\vec{b}+$.  From the above table it is clear that the pair belong to either population 3 or 4.  Note that because $N_{i}$ is positive semi-definite we must be able to construct relations like, for instance,

\[
  \label{APositivityCondition}
  N_{3} + N_{4} \le (N_{3} + N_{7}) + (N_{4} + N_{2}).
\]

Now let $P(\vec{a}+;\vec{b}+)$ be the probability that, in a random selection, _A_ finds particle 1 to be $\vec{a}+$ and _B_ finds particle 2 to be $\vec{b}+$.  In terms of populations, we have

\[
  \label{Populations1}
  P(\vec{a}+;\vec{b}+) = \frac{(N_{3} + N_{4})}{\sum_{i}^{8}N_{i}}.
\]

Similarly we have

\[
  \label{Populations2}
  P(\vec{a}+;\vec{c}+) = \frac{(N_{2} + N_{4})}{\sum_{i}^{8}N_{i}}
\]

and

\[
  \label{Populations3}
  P(\vec{c}+;\vec{b}+) = \frac{(N_{3} + N_{7})}{\sum_{i}^{8}N_{i}}.
\]

The positivity condition (eq:APositivityCondition) then becomes

\[
  \label{WignerFormOfBellInequality}
  P(\vec{a}+;\vec{b}+) \le P(\vec{a}+;\vec{c}+) + P(\vec{c}+;\vec{b}+).
\]

This is Wigner's form of Bell's inequality.

#### Violations and geometry

As we mentioned before, we have used purely classical reasoning to derive the two forms of Bell's inequality that we have thusfar encountered.  Recall that the context within which the above were derived was the Stern-Gerlach experiment are we are measuring along axes of the magnetic field.  As such, there are angles between these various axes.  Thus the quantum mechanically-derived probabilities corresponding to (eq:Populations1), (eq:Populations2), and (eq:Populations3) are

$$
  P(\vec{a}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right),
$$

$$
  P(\vec{a}+;\vec{c}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right),
$$

and

$$
  P(\vec{c}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right),
$$

respectively.  Bell's inequality, (eq:WignerFormOfBellInequality), then becomes

\[
  \label{AnotherFormOfBellInequality}
  \frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right) \le \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right) + \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right).
\]

From a geometric point of view, this inequality is not always possible.  For example, suppose, for simplicity that $\vec{a}$, $\vec{b}$, and $\vec{c}$ lie in a plane and suppose that $\vec{c}$ bisects $\vec{a}$ and $\vec{b}$, i.e.

$$
\array{
  \theta_{ab} = 2\theta & \text{ and } & \theta_{ac}=\theta_{cb}=\theta.
}
$$

Then (eq:AnotherFormOfBellInequality) is violated for $0 \lt \theta \lt \frac{\pi}{2}$.  For example, if $\theta = \frac{\pi}{4}$, (eq:AnotherFormOfBellInequality) would become $0.500 \le 0.292$ which is absurd!

### Compact reformulation
 {#CompactReformulation}

A transparent and compact way to derive the actual [[inequality]] of [Bell 1964](#Bell64) (adjusting the original argument only slightly for mathematical elegance) is reviewed in [Khrennikov 2008, §10.1](#Khrennikov08), which we broadly follow:

\begin{proposition}
Given

1. a [[probability space]] $(\Lambda, d\rho)$ with

1. three [[random variables]] taking values in $\{\pm 1\}$ (regarded inside the [[real numbers]]):

   \[
     \label{TheRandomVariables}
     S_i
       \;\colon\;
     X \longrightarrow \{\pm 1\}
     \hookrightarrow
     \mathbb{R}
     \,,
     \;\;\;\;
     i\,\in\, \{1,2,3\}
   \]

then the [[correlation functions]]

\[
  \label{TheCorrelations}
  \langle S_{i} \, S_j\rangle
  \;\coloneqq\;
  \int_{\Lambda}
    \;
    S_i(\lambda)
    \,
    S_j(\lambda)
  \;
  d\rho(\lambda)
\]

satisfy this [[inequality]]:

\[
  \label{TheInequality}
  \big\vert
    \langle S_1 S_2\rangle
    -
    \langle S_3 S_2\rangle
  \big\vert
  \;\leq\;
  1 - \langle S_1 S_3\rangle
  \,.
\]

(where $\left\vert-\right\vert$ denotes the [[absolute value]])
\end{proposition}
\begin{proof}
  Recall that the [[expectation value]] of a random variable $P \,\colon\, \Lambda \longrightarrow \mathbb{R}$ is given by its [[Lebesgue integral]] against the [[probability measure]]:
  $$
    \langle P \rangle
    \;\coloneqq\;
    \int_\Lambda P(\lambda) \, d\rho(\lambda)
    \,,
  $$
  and that $d\rho$ being a [[probability measure]] implies the normalization
  \[
    \label{NormalizationOfProbability}
    \langle 1 \rangle
    \;\equiv\;
    \int_\Lambda 1 \, d\rho(\lambda)
    \;=\;
    1
    \,.
  \]

Moreover, the assumption (eq:TheRandomVariables) that the random variables $S_i$ take values in $\{\pm 1\}$
immediately implies for all $i,j \,in\, \{1,2,3\}$ that
  \[
    \label{IdempotencyOfTheRandomVariables}
    \big( S_i \cdot S_i \big) \,=\, 1
    \,,
    \;\;\;\;
    \text{i.e.}
    \;\;\;
    \underset{\lambda \,\in\, \Lambda}{\forall}
    S_i(\lambda) \, S_i(\lambda)
    \,=\,
    (\pm 1)^2
    \,=\,
    1
    \,.
  \]

Together this implies -- by repeatedly using the [[Cauchy-Schwarz inequality]] -- the bounds:

$$
  \big\vert
    \langle S_i \rangle
  \big\vert
  \;\leq\;
  1
  \,,
  \;\;\;\;\;\;\;
  \big\vert
    \langle S_i S_j\rangle
  \big\vert
  \;\leq\;
  1
$$

and thus, in particular:

\[
  \label{BoundOnCorrelations}
  \big\vert
  \langle
    P \, S_i \, S_j
  \rangle
  \big\vert
  \;\leq\;
  \big\vert
  \langle
    P
  \rangle
  \big\vert
  \,,
\]

for any [[random variable]] $P \,\colon\, \Lambda \to \mathbb{R}$.

Using these (evident) ingredients, we directly compute as follows
$$
  \begin{array}{ll}
  \big\vert
    \langle S_1 S_2\rangle
    -
    \langle S_3 S_2\rangle
  \big\vert
  &
  \\
  \;=\;
  \Big\vert
    \int_{\Lambda}
      S_1(\lambda) \, S_2(\lambda)
      \,
    d\rho(\lambda)
    -
    \int_{\Lambda}
      S_3(\lambda) \, S_2(\lambda)
      \,
    d\rho(\lambda)
  \Big\vert
  &
  \text{by}\;\text{(eq:TheCorrelations)}
  \\
  \;=\;
  \Big\vert
    \int_{\Lambda}
      \big(
        S_1(\lambda)
        -
        S_3(\lambda)
      \big)
      \,
      S_2(\lambda)
      \,
    d\rho(\lambda)
  \Big\vert
  &
  \text{by linearity of the integral}
  \\
  \;=\;
  \Big\vert
    \int_{\Lambda}
      \big(
        1
        -
        S_1(\lambda) \, S_3(\lambda)
      \big)
      S_1(\lambda) \, S_2(\lambda)
      \,
    d\rho(\lambda)
  \Big\vert
  &
  \text{by}\;\text{(eq:IdempotencyOfTheRandomVariables)}
  \\
  \;\leq\;
  \Big\vert
    \int_{\Lambda}
      \big(
        1
        -
        S_1(\lambda) \, S_3(\lambda)
      \big)
      \,
    d\rho(\lambda)
  \Big\vert
  &
  \text{by}\;\text{(eq:BoundOnCorrelations)}
  \\
  \;=\;
  1 - \langle S_1 S_3\rangle
  &
  \text{by}\;\text{(eq:NormalizationOfProbability)}\;\text{and}\; \text{(eq:TheCorrelations)}
  \end{array}
$$

This is the inequality (eq:TheInequality).
\end{proof}

## Related concepts

* [[Einstein-Podolsky-Rosen paradox]]

* [[Grothendieck inequality]]

* [[quantum probability]]

* [[interpretation of quantum mechanics]]

* [[quantum information theory]]

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Wigner theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References

### General

The original article:

* {#Bell64} [[John Bell]], *On the Einstein Podolsky Rosen paradox*, Physics **1** 195 (1964) [[doi:10.1103/PhysicsPhysiqueFizika.1.195](https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195), [pdf](http://www.drchinese.com/David/Bell_Compact.pdf)]

Review:

* [[John F. Clauser]], [[Abner Shimony]], *Bell's theorem. Experimental tests and implications*, Rep. Prog. Phys. **41** (1978) 1881 [[doi:10.1088/0034-4885/41/12/002](https://iopscience.iop.org/article/10.1088/0034-4885/41/12/002)]

* {#Kuperberg05} [[Greg Kuperberg]], section 1.6.2 of: _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

* [[Valter Moretti]], Thm. 4.49 of: *Fundamental Mathematical Structures of Quantum Theory*, Springer (2019) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]

and on a background of [[quantum logic]]:

* Laura Molenaar, *Quantum logic and the EPR paradox*, Delft (2014) [[uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b](http://resolver.tudelft.nl/uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b)]

Further on experimental verification:

* {#Aspect15} [[Alain Aspect]], *[Closing the Door on Einstein and Bohr’s Quantum Debate](https://physics.aps.org/articles/v8/123)*, Physics **8** 123 (2015)

Relation to the [[Kochen-Specker theorem]]:

* [[Leandro Aolita]], Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, *Fully nonlocal quantum correlations*, Phys. Rev. A **85** 032107 (2012) [[arXiv:1105.3598](https://arxiv.org/abs/1105.3598), [doi:10.1103/PhysRevA.85.032107](https://doi.org/10.1103/PhysRevA.85.032107)]

See also:

* Wikipedia, _[Bell's theorem](http://en.wikipedia.org/wiki/Bell_inequality)_

* Wikipedia, *[Bell test](https://en.wikipedia.org/wiki/Bell_test)*

* Wikipedia, *[Leggett-Garg inequality](https://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality)*

* Stanford Encyclopedia of Philosophy, _Bell's theorem_ ([url](http://plato.stanford.edu/entries/bell-theorem/))

In relation to the [[Grothendieck inequality]]:

* [[Boris S. Tsirelson]], *Quantum analogues of the Bell inequalities. The case of two spatially separated domains*, Journal of Soviet Mathematics **36** (1987) 557–570 [[doi:10.1007/BF01663472](https://doi.org/10.1007/BF01663472)]

* [[Boris S. Tsirelson]], *Some results and problems on quantum Bell-type inequalities* Hadronic Journal Supplement **8** 4 (1993) 329-345 [[pdf](https://www.tau.ac.il/~tsirel/download/hadron.pdf), [[Tsirelson-QuantumBellType.pdf:file]] [web](https://www.tau.ac.il/~tsirel/download/hadron.html)]

  > (but see the erratum [here](https://www.tau.ac.il/~tsirel/Research/bellopalg/main.html))

* Wikipedia, *[Tsirelson's bound](https://en.wikipedia.org/wiki/Tsirelson%27s_bound)*

### In quantum field theory

In the generality of [[quantum field theory]]:

On Bell inequalities in [[particle physics]] and possible relation to the [[weak gravity conjecture]]:

* Aninda Sinha, Ahmadullah Zahed, *Bell inequalities in 2-2 scattering* [[arXiv:2212.10213](https://arxiv.org/abs/2212.10213)]

On [[BRST complex|BRST invariant]] Bell inequality in [[gauge field theory]]:

* David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Giovani Peruzzo, Silvio P. Sorella: *BRST invariant formulation of the Bell-CHSH inequality in gauge field theories* [[arXiv:2304.01028](https://arxiv.org/abs/2304.01028)]

### Probabilistic opposition
  {#ReferencesProbabilisticOpposition}

Identification of Bell's inequalities with much older inequalities in classical [[probability theory]], due to [[George Boole]]'s *[[Boole -- The Laws of Thought|The Laws of Thought]]*, was pointed out by (among others, called the "probabilistic opposition" in [Khrennikov 2007, p. 3](#Khrennikov07)) by:

* {#Pitowsky89a} [[Itamar Pitowsky]], *From George Boole To John Bell — The Origins of Bell’s Inequality*, in: *Bell’s Theorem, Quantum Theory and Conceptions of the Universe*, Fundamental Theories of Physics **37** Springer (1989) [[doi:10.1007/978-94-017-0849-4_6](https://doi.org/10.1007/978-94-017-0849-4_6)]

* {#Pitowsky89b} [[Itamar Pitowsky]], *Quantum Probability -- Quantum Logic*, Lecture Notes in Physics **321**, Springer (1989) [[doi:10.1007/BFb0021186](https://doi.org/10.1007/BFb0021186)]

* [[Luigi Accardi]], *The Probabilistic Roots of the Quantum Mechanical Paradoxes*, in: *The Wave-Particle Dualism*,  Fundamental Theories of Physics **3** Springer (1984) [[doi:10.1007/978-94-009-6286-6_16](https://doi.org/10.1007/978-94-009-6286-6_16)]

reviewed in:

* Elemer E Rosinger, *George Boole and the Bell inequalities* [[arXiv:quant-ph/0406004](https://arxiv.org/abs/quant-ph/0406004)]

* {#Khrennikov07} [[Andrei Khrennikov]], *Bell's inequality: Physics meets Probability* [[arXiv:0709.3909](https://arxiv.org/abs/0709.3909)]

* {#Khrennikov08} [[Andrei Khrennikov]], *Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?*, Entropy **10** 2 (2008) 19-32 [[doi:10.3390/entropy-e10020019](https://doi.org/10.3390/entropy-e10020019)]

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Bell's theorem > history

see *[[Bell's inequality]]*


Belle experiment

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Belle experiment_ is an [[experiment]] with focus on the [[physics]] of [[B-mesons]] (similar to the [[LHCb experiment]], see also the [[BaBar experiment]]).

After the discovery of the [[Higgs field]] at the two major other detectors of [[LHC]], these [[B-meson]] [[experiments]] currently stand out as seeing hints for physics beyond the [[standard model of particle physics]] with considerable [[statistical significance]]: the _[[flavour anomalies]]_ (see [Adamczyk 19](#Adamczyk19))

## Related concepts

* related [[experiments]]

  * [[LHCb]]

  * [[BaBar experiment]]

* observations

  * [[flavour anomaly]]

  * [[V_cb puzzle]]

  * [[pentaquark]]

  * [[XYZ particle]]

* [[precision experiment]]

## References

### General

* [Belle homepage](https://belle.kek.jp/)

* [Belle II homepage](https://www.belle2.org/)

* C. Z. Yuan (for the [[Belle II Collaboration]]), _The Belle II Experiment at the SuperKEKB_ ([arXiv:1208.3813](https://arxiv.org/abs/1208.3813))

* [[Belle collaboration]], _Physics Achievements from the Belle Experiment_ ([arXiv:1212.5342](https://arxiv.org/abs/1212.5342))

* S.I. Eidelman, A.V. Nefediev, P.N. Pakhlov, V.I. Zhukova, _Superfactory of bottomed hadrons Belle II_ ([arXiv:2012.05147](https://arxiv.org/abs/2012.05147))

See also

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

* Wikipedia, _[Belle II experiment](https://en.wikipedia.org/wiki/Belle_II_experiment)_

More on [[Belle II]]:

* *Belle II Executive Summary* [[arXiv:2203.10203](https://arxiv.org/abs/2203.10203)]

* *Snowmass White Paper: Belle II physics reach and plans for the next decade and beyond* [[arXiv:2207.06307](https://arxiv.org/abs/2207.06307)]

  > Belle II is an experiment operating at the [[intensity frontier]].

### On flavour anomalies

[[flavour anomalies]]:

* [[Belle collaboration]], _Measurement of the $\tau$ lepton polarization and $R(D^\ast)$ in the decay $\bar B \to D^\ast \tau^- \bar \nu_\tau$ with one-prong hadronic τ decays at Belle_ ([arXiv:1709.00129](https://arxiv.org/abs/1709.00129))

* {#Adamczyk19} Karol Adamczyk, _Semitauonic B decays at Belle/Belle II_, Proceedings of the [10th International Workshop on the CKM Unitarity Triangle](https://ckm2018.physi.uni-heidelberg.de/) (CKM 2018), Heidelberg, Germany, September 17-21, 2018 ([arXiv:1901.06380](https://arxiv.org/abs/1901.06380))

* [[Belle collaboration]], _Measurement of the $D^{\ast -}$ polarization in the decay $B^0 \to D^{\ast -} \tau^+ \nu_\tau$_ ([arXiv:1903.03102](https://arxiv.org/abs/1903.03102))

* {#Caria19} Giacomo Caria on behalf of the [[Belle collaboration]], _Measurement of $R(D)$ and $R(D^\ast)$ with a semileptonic tag at Belle_, Moriond, EW  22/03/2019 ([pdf](http://moriond.in2p3.fr/2019/EW/slides/6_Friday/3_YSF/1_gcaria_moriond2019.pdf))

* [[Belle Collaboration]], _Test of lepton flavor universality in $B \to K^\ast \ell^+ \ell^-$ decays at Belle_ ([arXiv:1904.02440](https://arxiv.org/abs/1904.02440))

* {#Belle1910} [[Belle collaboration]], _Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^\ast)$ with a semileptonic tagging method_, Phys. Rev. Lett. 124, 161803 (2020)
([arXiv:1910.05864](https://arxiv.org/abs/1910.05864), [doi:10.1103/PhysRevLett.124.161803](https://doi.org/10.1103/PhysRevLett.124.161803))

Outlook for [[Belle II]] to confirm the [[flavour anomalies]], if they are real:

* Tobias Huber, Tobias Hurth, Jack Jenkins, Enrico Lunghi, Qin Qin, K.Keri Vos, _Phenomenology of inclusive $\bar B \to X_s \ell^+ \ell^-$ for the Belle II era_ ([arXiv:2007.04191](https://arxiv.org/abs/2007.04191))

> With $[$ ... $]$ [[Belle II]], we are at the brink of a new era in [[quark]] [[flavour physics]] $[...]$ we present a phenomenological study of  the potential for [[Belle II]] to reveal possible new physics in the inclusive  decay channel.

> (p. 15:) We see that [[flavour anomalies|anomalies]] in the exclusive sector can be confirmed at the [[statistical significance|5σ]] level  by inclusive measurements if the true values of $C_{9,10}^{\mu NP}$ are at the current best-fit point of the exclusive fits

> (p. 16:)  After including $B_s \to \mu^+ \mu_-$, the reach in the $[C_9^{\mu NP}, C_{10}^{\mu NP}]$ plane improves considerably.  Exclusive [[flavour anomalies|anomalies]] could be confirmed at the [[statistical significance|6σ]] level.

> (p. 20:) Should  the  true  value  of $C_9^{NP}$ and $C_{10}^{NP}$ be at  the current best-fit points of the global fits, an analysis of inclusive $\bar B_ \to X_s \ell^+ \ell^-$ at [[Belle II]] with $50 ab^{-1}$ of data will exclude the SM point $C_9^{NP} = C_{10}^{NP} = 0$ at the level of $\sim 5$[[statistical significance|σ]].  This again underlines the necessity of a full angular analysis of $\bar B_ \to X_s \ell^+ \ell^-$ at [[Belle II]].

[[!redirects Belle]]
[[!redirects Belle II]]

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[[!redirects Belle II collaboration]]
[[!redirects Belle II collaborations]]

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Ben Allanach

* [webpage](http://www.damtp.cam.ac.uk/people/b.allanach/)

## Selected writings

On [[flavour anomalies]]:

* {#Allanach19} [[Ben Allanach]], _Finding Z's responsible for $R_{K^{(\ast)}}$_, talk at [Moriond 2019](http://moriond.in2p3.fr/2019/EW/) ([pdf](http://moriond.in2p3.fr/2019/EW/slides/6_Friday/2_afternoon/5_Allanach.pdf))

* {#AllanachGripaiosYou17} [[Ben Allanach]], Ben Gripaios, Tevong You, _The Case for Future Hadron Colliders From $B \to K^{(\ast)}\mu^+ \mu^-$ Decays_, JHEP03(2018)021 ([arXiv:1710.06363](https://arxiv.org/abs/1710.06363))

* [[Ben Allanach]], _$U(1)_{B_3-L_2}$ Explanation of the Neutral Current $B$−Anomalies_ ([arXiv:2009.02197](https://arxiv.org/abs/2009.02197))

category: people


Ben Brown

PhD student at The University of Edinburgh.

[Personal Homepage](https://bencwbrown.co.uk/)

category: people


Ben Goertzel

__Ben Goertzel__ is a prominent researcher in [[machine learning]]/artificial intelligence predominantly interested in achieving AGI (artificial general intelligence, the term he actually coined in print).

* Ben Goertzel, _Reflective metagraph rewriting as a foundation for an AGI "Language of Thought"_, [arXiv:2112.08272](https://arxiv.org/abs/2112.08272)

category: people


Ben Greenman

[home page](http://ccs.neu.edu/home/types)


Ben Gripaios

* [webpage](https://www.hep.phy.cam.ac.uk/~gripaios/)

## Selected writings

On [[particle physics]]:

* James Dodd, [[Ben Gripaios]], _The Ideas of Particle Physics_, Cambridge University Press 2020 ([ISBN:9781108727402](https://www.cambridge.org/gb/academic/subjects/physics/particle-physics-and-nuclear-physics/ideas-particle-physics-4th-edition?format=PB), [doi:10.1017/9781108616270]( https://doi.org/10.1017/9781108616270))

On [[equivariant ordinary differential cohomology]] and [[action functionals]] for [[topological field theories]] (such as [[Chern-Simons theory]]):

* Joe Davighi, [[Ben Gripaios]], [[Oscar Randal-Williams]], _Differential cohomology and topological actions in physics_ ([arXiv:2011.05768](https://arxiv.org/abs/2011.05768))

category: people


Ben Heidenreich

* [webpage](https://www.physics.umass.edu/people/ben-heidenreich)

## Selected writings

On the [[weak gravity conjecture]]:

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_, JHEP02(2016)140 ([arXiv:1509.06374](https://arxiv.org/abs/1509.06374))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))

* [[Ben Heidenreich]], Matteo Lotito, *Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String* [[arXiv:2401.14449](https://arxiv.org/abs/2401.14449)]

On [[swampland conjectures]]:

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Emergence and the Swampland Conjectures_ ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698))

category: people


Ben Knudsen

* [webpage](http://www.math.northwestern.edu/~knudsen/)

## Selected writings

On [[configuration spaces of points]] in [[algebraic topology]]:

* [[Ben Knudsen]], _Configuration spaces in algebraic topology_ ([arXiv:1803.11165](https://arxiv.org/abs/1803.11165))

On the [[Morava E-theory]] of [[configuration spaces of points]]:

* [[Lukas Brantner]], [[Jeremy Hahn]], [[Ben Knudsen]], *The Lubin-Tate Theory of Configuration Spaces: I* ([arXiv:1908.11321](https://arxiv.org/abs/1908.11321))

## Eelated $n$Lab entries

* [[factorization homology]]

category: people


Ben Moon

### About

Name: I publish as 'Benjamin', but typically go by 'Ben'.

* [Homepage](https://www.cs.kent.ac.uk/people/rpg/bgm4/)

* [GitHub](https://github.com/GuiltyDolphin)

## Selected writings

On [[dependent linear types]] and [[graded modalities]]:

* [[Benjamin Moon]], [[Harley Eades III]], [[Dominic Orchard]], *Graded Modal Dependent Type Theory*. In: N. Yoshida (ed.) *Programming Languages and Systems ESOP 2021*, Lecture Notes in Computer Science **12648**, Springer (2021) 462-490 [[doi:10.1007/978-3-030-72019-3_17](https://doi.org/10.1007/978-3-030-72019-3_17), [arxiv:2010.13163](https://arxiv.org/abs/2010.13163)]

## Related entries

* [[type theory]]

category:people

[[!redirects Benjamin Moon]]


Ben Walter

* [webpage](http://www.math.purdue.edu/~walterb/)

## Selected writings

On [[Lie coalgebras]], [[rational homotopy theory]] and [[Hopf invariants]] via [[homotopy Whitehead integrals]] ("[[functional cup products]]"):

* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883  ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))

## Related $n$Lab entries

* [[model structure on dg-coalgebras]]

* [[model structure on dg-Lie algebras]]

category: people


Ben Webster

Is a professor at Northeastern University.

* [website](https://uwaterloo.ca/scholar/b2webste/home)
* [Secret Blogging Seminar](http://sbseminar.wordpress.com/).

## Selected writings

On [[Hilbert schemes]] as [[Coulomb branches]]:

* [[Ben Webster]], _Coherent sheaves on Hilbert schemes through the Coulomb lens_, 2018 ([[WebsterHilbertScheme18.pdf:file]])

##Expository posts:##

###Algebraic geometry###
[Interpreting the Hecke Algebra II: the sheafification ](http://sbseminar.wordpress.com/2009/04/09/interpreting-the-hecke-algebra-ii-the-sheafification/)

[Geometry and triply graded knot homology](http://sbseminar.wordpress.com/2009/04/09/geometry-and-triply-graded-knot-homology/)

[The Grothendieck trace formula as categorification, I: the category and the comparison theorem ](http://sbseminar.wordpress.com/2009/02/12/the-grothendieck-trace-formula-as-categorification-the-category-and-the-comparison-theorem/)

[About that field isomorphism...](http://sbseminar.wordpress.com/2009/02/15/about-that-field-isomorphism/)

###Knot theory###

[How to get an algebra from a knot invariant](http://sbseminar.wordpress.com/2009/04/13/how-to-get-an-algebra-from-a-knot-invariant/)

[Hochschild homology](http://sbseminar.wordpress.com/2007/07/22/hochschild-homology/)

[Soergel bimodules](http://sbseminar.wordpress.com/2007/07/23/soergel-bimodules/)

[The geometry of Soergel bimodules](http://sbseminar.wordpress.com/2007/08/23/the-geometry-of-soergel-bimodules/)

###Finite dimensional algebras###

[Gale duality and linear programing](http://sbseminar.wordpress.com/2008/04/06/gale-duality-and-linear-programing/)

[Gale and Koszul duality, together at last](http://sbseminar.wordpress.com/2008/07/14/gale-and-koszul-duality-together-at-last/)

[Koszul algebras and Koszul duality](http://sbseminar.wordpress.com/2007/11/01/koszul-algebras-and-koszul-duality/)

###Linear equivalence###

[Laplacian spectra and linearly equivalent G-sets](http://sbseminar.wordpress.com/2007/09/19/laplacian-spectra-and-linearly-equivalent-g-sets/)

[Zeta function relations and linearly equivalent group actions](http://sbseminar.wordpress.com/2007/08/29/zeta-function-relations-and-linearly-equivalent-group-actions/)

###Social choice theory###
[Why Arrow's theorem is a scam](http://sbseminar.wordpress.com/2008/02/17/why-arrows-theorem-is-a-scam/)

[Stable marriages](http://sbseminar.wordpress.com/2008/01/21/stable-marriages/)

category: people
[[!redirects BWebster:Ben Webster]]


Ben Williams

* [webpage](http://www.math.ubc.ca/~tbjw/)

category: people


Benedict Gross

[website](http://www.math.harvard.edu/~gross/)

category: people


Benedikt Ahrens

* [webpage](https://benediktahrens.net/)

## Selected writings

On the [[UniMath project]]:

* [[Benedikt Ahrens]], _UniMath: its origins, present, and future_ (2017) [[pdf](https://unimath.github.io/bham2017/UniMath_origins-present-future.pdf), [[Ahrens-UniMathHistory.pdf:file]]]

On [[2-dimensional type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, August 2022, No. 12, Pages 1–14, ([doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334))

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Michael Shulman]], [[Dimitris Tsementzis]], *The Univalence Principle* ([arXiv:2102.06275](https://arxiv.org/abs/2102.06275))

On [[univalence]] in [[homotopy type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent foundations and the equivalence principle*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

On [[2-dimensional type theory]]:

* [[Benedikt Ahrens]], [[Paige Randall North]], [[Niels van der Weide]], *Semantics for two-dimensional type theory*, LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 12 (2022) 1–14, [[doi:10.1145/3531130.3533334](https://doi.org/10.1145/3531130.3533334)]

On [[univalent bicategory|univalent]] [[bicategories]] in [[homotopy type theory]]:

* [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], _Bicategories in Univalent Foundations_, Mathematical Structures in Computer Science **31** 10 (2022) 1232-1269 [[arXiv:1903.01152](https://arxiv.org/abs/1903.01152), [doi:10.1017/S0960129522000032](https://doi.org/10.1017/S0960129522000032)&rbrack

On [[monoidal category|monoidal]] [[univalent categories]]:

* [[Kobe Wullaert]], [[Ralph Matthes]], [[Benedikt Ahrens]], *Univalent Monoidal Categories* [[arXiv:2212.03146](https://arxiv.org/abs/2212.03146)]

## Related entries

* [[internal category in homotopy type theory]]

* [[UniMath project]]

category: people


Benedikt Fluhr

* [Homepage](https://bfluhr.com/)


Bengt Nilsson

* [webpage](http://fy.chalmers.se/~tfebn/)

## Selected writings

On [[Kaluza-Klein compactification]] in [[supergravity]]:

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Kaluza-Klein supergravity_, Physics Reports Volume 130, Issues 1–2, January 1986, Pages 1-142 ([spire:229417](https://inspirehep.net/record/229417), doi:10.1016/0370-1573(86)90163-8)

On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", then also called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (doi:10.1016/0550-3213(87)90397-X)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

On [[DBI-action]] and [[Green-Schwarz action functional]] for [[D-branes]]:

* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_,  Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))

* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

category: people


Beni Yoshida

* [institute page](https://perimeterinstitute.ca/people/beni-yoshida)

## Selected writings

An observation on classical [[error correcting codes]] preconceiving aspects of [[holographic tensor network]] [[quantum error correcting codes]]:

* [[Beni Yoshida]], *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275))

Introducing the [[HaPPY code]] (a [[quantum error correcting code]] whose generating [[tensor network]] exhibits [[holographic entanglement entropy]]):

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

On [[AdS/CFT in condensed matter physics]]:

* Mike Blake, Yingfei Gu, [[Sean A. Hartnoll]], Hong Liu, [[Andrew Lucas]], Krishna Rajagopal, [[Brian Swingle]], [[Beni Yoshida]], *Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes* [[arXiv:2203.04718](https://arxiv.org/abs/2203.04718)]

category: people


Benjamin Antieau

* [website](http://www.math.washington.edu/~bantieau/)

## Selected writings

On [[prismatic cohomology]] applied to [[algebraic K-theory]]:

* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)[

category: people


Benjamin Assel

* [webpage](https://www.perimeterinstitute.ca/people/benjamin-assel)

## Selected writings

On [[Coulomb branch]] [[singularities]] in 3d [[super QCD]]:

* [[Benjamin Assel]], [[Stefano Cremonesi]], _The Infrared Physics of Bad Theories_, SciPost Phys. 3, 024 (2017) ([arXiv1707.03403](https://arxiv.org/abs/1707.03403))

category: people


Benjamin Balsam

* [webpage](http://www.math.sunysb.edu/~balsam/)

## related $n$Lab entry

* [[Turaev-Viro model]]

category: people


Benjamin Blander

#related $n$Lab entries#

* [[model structure on simplicial presheaves]]

category: people


Benjamin Grinstein

* [webpage](http://leewick.ucsd.edu/~ben/)

## Selected writings

On [[flavor physics]]:

* [[Benjamin Grinstein]], _TASI-2013 Lectures on Flavor Physics_ ([arXiv:1501.05283](https://arxiv.org/abs/1501.05283))

On [[flavor physics]] in view of the [[flavour anomalies]] seen at [[LHCb]] and other experiments

* [[Benjamin Grinstein]], _A path to flavor_, talk at _[Implications of LHCb measurement and future prospects](https://indico.cern.ch/event/769902/)_ CERN 2019 ([pdf](https://indico.cern.ch/event/769902/contributions/3582540/attachments/1926501/3193107/Grinstein-high-res.pdf), [[GrinsteinFlavor2019.pdf:file]], [indico:3582540](https://indico.cern.ch/event/769902/contributions/3582540))

category: people

[[!redirects Ben Grinstein]]


Benjamin Hennion

* [webpage](http://guests.mpim-bonn.mpg.de/hennion/)

## related $n$Lab entries

* [[tangent complex]]

* [[formal neighbourhood of the diagonal]]

category: people

[[!redirects Hennion]]


Benjamin M. Mann

[[!redirects B. M. Mann]]

## Selected writings

On the [[moduli space of Yang-Mills monopoles]] and [[rational maps]] to [[complex projective spaces]]:

* [[Charles P. Boyer]], [[B. M. Mann]], *Monopoles, non-linear $\sigma$-models, and two-fold loop spaces*,  Commun. Math. Phys. 115, 571–594 (1988) ([arXiv:10.1007/BF01224128](https://doi.org/10.1007/BF01224128))

On [[homotopy type of spaces of rational maps]]:

* [[Fred Cohen]], [[Ralph Cohen]], [[B. M. Mann]], [[R. J. Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

*  [[Benjamin M. Mann]], [[R. James Milgram]],  *Some spaces of holomorphic maps to complex Grassmann manifolds*, J. Differential Geom. 33(2): 301-324 (1991) ([doi:10.4310/jdg/1214446318](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-2/Some-spaces-of-holomorphic-maps-to-complex-Grassmann/10.4310/jdg/1214446318.full))

* [[Charles P. Boyer]], [[B. M. Mann]], [[J. C. Hurtubise]], [[R. J. Milgram]], *The topology of the space of rational maps into generalized flag manifolds*, Acta Mathematica. 1994;173(1):61-101 ([doi:10.1007/BF02392569](https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full))

category: people


Benjamin Pierce

* [website](http://www.cis.upenn.edu/~bcpierce/)

## Selected writings

On [[subtypes]] (with an early discussion of what came to be called *[[lenses (in computer science)]]* motivated by [[object-oriented programming]]):

* {#HofmannPierce96} [[Martin Hofmann]], [[Benjamin Pierce]], *Positive Subtyping*, Information and Computation **126** 1 (1996) 11-33 [[doi:10.1006/inco.1996.0031](https://doi.org/10.1006/inco.1996.0031)]

Introducing the terminology of [[lenses (in computer science)]]:

* Aaron Bohannon, [[Benjamin C. Pierce]], Jeffrey A. Vaughan, *Relational lenses: a language for updatable views*, Proceedings of Principles of Database Systems (PODS) (2006) 338-347 [[doi:10.1145/1142351.1142399](https://doi.org/10.1145/1142351.1142399), [pdf](https://www.cis.upenn.edu/~bcpierce/papers/dblenses-pods.pdf)]

* {#FosterGreenwaldMoorePierceSchmitt07} J. N. Foster, M. B. Greenwald, J. T. Moore, [[Benjamin C. Pierce]], A. Schmitt, _Combinators for bidirectional tree transformations: A linguistic approach to the view-update problem_, ACM Transactions on Programming Languages and Systems **29** 3 (2007) 17-es [[doi:x10.1145/1232420.1232424](https://doi.org/10.1145/1232420.1232424)]

category: people

[[!redirects Benjamin C. Pierce]]


Benjamin Schumacher

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

## Selected writings

Early discussion of [[quantum computation]] and introducing the terminology *[[q-bit]]*:

* [[Benjamin Schumacher]], *Quantum coding*, Phys. Rev. A **51** (1995) 2738 $[$[doi:10.1103/PhysRevA.51.2738](https://doi.org/10.1103/PhysRevA.51.2738)$]$

On [[quantum systems]] and [[quantum information theory]]:

* [[Benjamin Schumacher]], [[Michael Westmoreland]], *Quantum Processes, Systems, and Information*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511814006](https://doi.org/10.1017/CBO9780511814006)]

category: people


Benjamin Steinberg

[[!redirects Ben Steinberg]]
[[!redirects Ben Steinberg]]
[[!redirects B. Steinberg]]

* [Homepage](http://www.sci.ccny.cuny.edu/~benjamin/)

* [Blog](http://bensteinberg.wordpress.com/author/bsteinbg/)

category:people


Benjamin-Ono equation

* thesis by Alexandre Landry [pdf](http://archimede.bibl.ulaval.ca/archimede/fichiers/27192/27192.pdf)

related entries: [[integrable model]]


Benno van den Berg

* [website](https://staff.fnwi.uva.nl/b.vandenberg3/)

## Selected writings

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

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

On a [[topos]] for [[continuous logic]]:

* Daniel Figueroa, [[Benno van den Berg]], _A topos for continuous logic_ ([arXiv:2107.10543](https://arxiv.org/abs/2107.10543))

On [[objective type theory]]:

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

category: people


Benny Lautrup

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

## related $n$Lab entries

* [[Nakanishi-Lautrup field]], [[antighost field]]

category: people


Beno Eckmann

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

## Selected writings

On [[internalization]], [[H-spaces]], [[monoid objects]], [[group objects]] in [[algebraic topology]]/[[homotopy theory]] and introducing the [[Eckmann-Hilton argument]]:

* [[Beno Eckmann]], [[Peter Hilton]], Theorem 1.12: *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221
([doi:10.4064/fm-50-2-207-221](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III primitive categories_,  Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _Seminar on triples and categorical homology theory_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

[[!redirects B. Eckmann]]
[[!redirects Eckmann]]
category:people


Benoit Fresse

__[Benoit Fresse](http://math.univ-lille1.fr/~fresse/)__ is a mathematician at Lille. One of his interests concerns the theory of [[operads]].

## Selected writings:

On [[Lie theory]] of [[formal groups]] over an [[operad]]:

* _Lie theory of formal groups over an operad_, J. Alg. __202__, 455--511, 1998, [doi](http://dx.doi.org/10.1006/jabr.1997.7280)

* _Modules over operads and functors_, Springer LNM __1967__, 2009, x+308 pp. [MR2010e:18009](http://www.ams.org/mathscinet-getitem?mr=2494775)

On the [[homotopy theory]] of [[operads]] and [[Grothendieck-Teichmüller groups]]:

* {#FresseHOGTG} [[Benoit Fresse]], *Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2)*, Mathematical Surveys and Monographs **217** (2017) [[ISBN:978-1-4704-3480-9](https://bookstore.ams.org/surv-217/), [website](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/), [pdf (draft of Part I)](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.648.9110&rep=rep1&type=pdf)]

  > Fresse lists two main aims of the book project: 1. the development of a [[rational homotopy theory]] for operads, and 2. to explain the relationship between operads and [[Grothendieck-Teichmüller groups]].

On the [[bar-cobar construction]] for [[dg-Hopf algebras]]:

* [[Benoit Fresse]], _The universal Hopf operads of the bar construction_ ([arXiv:math/0701245](https://arxiv.org/abs/math/0701245))

category: people


Benoit Jubin

My web page is [here](http://math.berkeley.edu/~jubin/).

category: people

[[!redirects Benoît Jubin]]


Benoît Valiron

* [webpage](https://www.monoidal.net/)

* [InSpire page](https://inspirehep.net/authors/2012244)

## Selected writings
Early consideration of [[quantum programming languages]] as [[quantum lambda-calculus]] invoking [[linear type theory|linear types]]:

* [[Benoît Valiron]], *A functional programming language for quantum computation with classical control*, MSc thesis, University of Ottawa (2004) [[doi:10.20381/ruor-18372](http://dx.doi.org/10.20381/ruor-18372), [pdf](https://ruor.uottawa.ca/bitstream/10393/26790/1/MR01625.PDF)]

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 [[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)]

* {#SelingerValiron09} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *Semantic Techniques in Quantum Computation*, Cambridge University Press (2009) 135-172 [[doi:10.1017/CBO9781139193313.005](https://doi.org/10.1017/CBO9781139193313.005), [pdf](https://www.mscs.dal.ca/~selinger/papers/qlambdabook.pdf)]

* Pablo Arrighi, Alejandro Díaz-Caro, [[Benoît Valiron]], *A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus*, EPTCS **88** (2012) 1-15 [[arXiv:1012.4032](https://arxiv.org/abs/1012.4032), [doi:10.4204/EPTCS.88.1](https://doi.org/10.4204/EPTCS.88.1)]

> [[Selinger (2016)](Quipper#Selinger16):] When the [QPL workshop series](https://www.mathstat.dal.ca/~selinger/qpl/) was first founded, it was called "*Quantum Programming Languages*". One year I wasn't participating, and while I wasn't looking they changed the name to "*Quantum Physics and Logic*" --- same acronym!

> Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing $[$[Selinger & Valiron 2004](#SelingerValiron04)$]$, but the sad thing is: In those days nobody really cared. $[...]$

> Now it's 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. $[...]$.

> So now people are working on quantum programming languages *again*.

Implementation as an [[embedded programming language]] in [[Haskell]]:

* Richard Eisenberg, [[Benoît Valiron]], Steve Zdancewic, *Typechecking Linear Data: Quantum Computation in Haskell* [[[EisenbergValironZdancewic-LinearData.pdf:file]]]

Introducing the [[functional programming language|functional]] [[quantum programming language]] [[Quipper]]:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices 48(6):333-342, 2013 ([arXiv:1304.3390](https://arxiv.org/abs/1304.3390))

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 ([arXiv:1304.5485](https://arxiv.org/abs/1304.5485))

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

On [[linear algebra]] ([[finite-dimensional vector spaces]] over [[finite fields]]) as [[categorical semantics]] for [[linear logic]]/[[linear type theory]]:

* [[Benoît Valiron]], [[Steve Zdancewic]], *Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi*, in: *Proc. of ICTAC'14*, Lecture Notes in Computer Science **8687**, Springer (2014) 442-459 [[arXiv:1406.1310](https://arxiv.org/abs/1406.1310), [doi:10.1007/978-3-319-10882-7_26](https://doi.org/10.1007/978-3-319-10882-7_26), slides:[pdf](https://www.cs.bham.ac.uk/~drg/bll/steve.pdf), [[Zdancewic-LinearLogicAlgebra.pdf:file]]]

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the [[Quipper]]-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

On [[dynamic lifting]] (of [[quantum measurement]]-results) in [[quantum programming]]/[[quantum computing]]:

* [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]

* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]

See also:

* Kostia Chardonnet, Marc de Visme, Benoît Valiron, Renaud Vilmart, *The Many-Worlds Calculus* [[arXiv:2206.10234](https://arxiv.org/abs/2206.10234)]

## Related entries

* [[software verification]]

category: people

[[!redirects Benoit Valiron]]


Benson Farb

* [personal page](http://www.math.uchicago.edu/~farb/)

* [institute page](https://mathematics.uchicago.edu/people/profile/benson-farb/)

* [Wikipedia page](https://en.wikipedia.org/wiki/Benson_Farb)

## Selected writings

Introducing the notion of *[[FI-modules]]* ([[FinInj|$FinSet_{inj}$]]-[[representations]]):

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], [[Rohit Nagpal]], *FI-modules over Noetherian rings*, Geom. Topol. **18** (2014) 2951-2984 [[arXiv:1210.1854](https://arxiv.org/abs/1210.1854), [doi:10.2140/gt.2014.18.2951](https://doi.org/10.2140/gt.2014.18.2951)]

and the related notion of [[representation stability]]:

* {#ChurchFarb13} [[Thomas Church]], [[Benson Farb]], *Representation theory and homological stability*, Advances in Mathematics
**245** (2013) 250-314 [[doi:10.1016/j.aim.2013.06.016](https://doi.org/10.1016/j.aim.2013.06.016)]

* [[Benson Farb]], *Representation Stability* [[arXiv:1404.4065](https://arxiv.org/abs/1404.4065)]

category: people

[[!redirects Benson S. Farb]]
[[!redirects Benson Stanley Farb]]


Beppe Metere

My web page is [here](https://gmetere.github.io/).

category: people

## Selected writings

* A. S. Cigoli, S. Mantovani, [[G. Metere]], *On pseudofunctors sending groups to 2-groups*, Mediterr. J. Math. 20, 17 (2023). [[arXived version](https://arxiv.org/abs/2112.08695)]

* A. S. Cigoli, S. Mantovani, [[G. Metere]], [[E. M. Vitale]], *Fibered aspects of Yoneda's regular span*, Advances in Mathematics, 360 (2020). [[arXived version](https://arxiv.org/abs/1806.02376)]

* S. Mantovani, [[G. Metere]], *Normalities and commutators*, Journal of Algebra 324 (2010), no. 9. [[arXived version](https://arxiv.org/abs/1002.1372)]

[[!redirects G. Metere]]

[[!redirects Giuseppe Metere]]


Beren Sanders

* [homepage](http://www.math.ucla.edu/~beren/)

category:people


Berezin integral

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{:toc}

## Idea

A _Berezinian integral_ is an [[integration over supermanifolds|integral over a supermanifold]] that is  a [[superpoint]] $\mathbb{R}^{0|q}$.

For the infinite-dimensional version see [[fermionic path integral]].

## Related entries

* [[integration]], [[integral]]

  [[integration over supermanifolds]]

* [[superpoint]], [[Grassmann algebra]]

* [[determinant]]

* [[superdeterminant]]/[[Berezinian]]

* [[Ausdehnungslehre]]

## References

The concept is originally due to [[Felix Berezin]].

Exposition of the standard lore includes

* [[Andreas Wipf]], _Chapter 10: Berezin integral_ ([pdf](https://www.tpi.uni-jena.de/qfphysics/homepage/wipf/lectures/pfad/pfad10.pdf))

* [[Urs Schreiber]], [_Integration over supermanifolds_](http://www.math.uni-hamburg.de/home/schreiber/sin.pdf)

Discussion in relation to the [[Hodge star operator]]:

* [[Leonardo Castellani]], [[Roberto Catenacci]], [[Pietro Antonio Grassi]], _The Hodge Operator Revisited_ ([arXiv:1511.05105](https://arxiv.org/abs/1511.05105))

A [[general abstract]] discussion in terms of [[D-module]] theory is in

* [[Frédéric Paugam]], _Homotopical Poisson Reduction of gauge theories_ ([pdf](http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf))

[[!redirects Berezin integrals]]

[[!redirects Berezinian integral]]
[[!redirects Berezinian integrals]]

[[!redirects Berezin integration]]


Berger's theorem

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## Statement

Berger's theorem says that if a [[manifold]] $X$ is

* [[simply connected topological space|simply connected]]

* neither locally a product nor a [[symmetric space]]

then the possible special holonomy groups are the following

[[!include special holonomy table]]

## References

Original article:

* {#Berger} [[Marcel Berger]], _Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes_, Bull. Soc. Math. France 83 (1955) ([doi:10.24033/bsmf.1464](https://doi.org/10.24033/bsmf.1464))

* Carlos Olmos, _A Geometric Proof of the Berger Holonomy Theorem_, Annals of Mathematics Second Series, Vol. 161, No. 1 (Jan., 2005), pp. 579-588 (10 pages) ([jstor:3597350](https://www.jstor.org/stable/3597350))

See also

* Wikipedia, _[Holonomy -- The Berger classification](https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification)_

[[!redirects Berger theorem]]


Bergfinnur Durhuus

* [personal page](https://web.math.ku.dk/~durhuus/)

* [institute page](https://www.math.ku.dk/english/staff/?pure=en/persons/23254)

* [inspire page](https://inspirehep.net/authors/1011165)

## Selected writings

On [[linear operators]] on [[Hilbert spaces]]:

* [[Bergfinnur Durhuus]], *Operators on Hilbert Space* [[pdf](https://web.math.ku.dk/~durhuus/MatFys/MatFys4.pdf), [[Durhuus-OperatorsOnHilbertSpace.pdf:file]]]

category: people


Berkovich space

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## Idea

__Berkovich analytic spaces__ are a version of [[analytic space]]s over [[nonarchimedean fields]]. Unlike the _rigid analytic spaces_ (see [[rigid analytic geometry]]) of Tate, which are locally defined via [[maximal spectra]] of Tate algebras glued via the [[Grothendieck topology|Grothendieck]] [[G-topology]], the Berkovich analytic spaces are actual [[topological space]] equipped with a cover by [[affinoid domains]] via the [[analytic spectrum]] construction, due to [[Vladimir Berkovich]]. This spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, [[Huber's adic spaces]] which may also contain valuations of higher order).

For more background see _[[analytic geometry]]_.

## Definition of Berkovich analytic spaces

Let $k$ be a [[non-archimedean field]].

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

Given $n \in \mathbb{N}$  and positive elements $\{r_1, \cdots, r_n \in k\}$, consider the sub-[[power series]] [[associative algebra|algebra]] over $k$ of those series which [[convergence|converge]] inside the radii $k_i$, i.e. the algebra defined by

$$
  \{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
  :=
  \left\{
    \sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to \infty} {\vert a_\nu \vert} r^\nu = 0
  \right\}
  \,.
$$

This is a commutative [[Banach algebra]] over $k$ with [[norm]] ${\Vert f \Vert} = max {\vert a_\nu\vert} r^\nu$.

A **$k$-[[affinoid algebra]]** is a commutative Banach $k$-algebra $A$ for which there exists $n$ and $\{r_i\}$ as above and an [[epimorphism]]

$$
  \{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
  \to
  A
$$

such that the [[norm]] on $A$ is the [[quotient norm]].

If one can choose here $r_i = 1$ for all $i$ then $A$ is called **strictly $k$-affinoid**.

The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-[[affinoid algebras]] and bounded [[homomorphisms]] between them.

=--

Via the [[analytic spectrum]] $Spec_{an}$ there is a [[topological space]] associated with any $k$-affinoid space. Often this underlying topological space is referred to as _the analytic space_.

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

An **[[affinoid domain]]** in an [[affinoid space]] $X = Spec_{an} A$ is a [[closed subset]] $V \subset X$ such that there is a [[homomorphism]] of $k$-affinoid spaces

$$
  \phi : Spec_{an} A_V \to X
$$

for some $A_V$, whose [[image]] is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

=--

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

A **$k$-analytic space** is a [[locally Hausdorff topological space|locally Hausdorff]] [[topological space]] $X$ equipped with an [[atlas]] by $k$-[[affinoid domains]] and [[affinoid domain embeddings]], such that their underlying [[analytic spectra]] [[topological spaces]] form a [[quasinet|net]] of [[compact subsets]] on $X$.

=--

([Berkovich 09, def. 3.1.4](#Berkovich09))

## Properties

### Cohomology
 {#Cohomology}

Under some mild conditions, the algebraic and the analytic [[étale cohomology]] of Berkovich spaces agree. ([Berkovich 95](#Berkovich95))

The underlying [[topological space]] $X^{an}$ given by the [[Berkovich analytic spectrum]] has as [[singular cohomology]] the [[weight filtration|weight 0]]-cohomology of $X$ ([Berkovich 09](#Berkovich09)).

See also MO discussion [here](http://mathoverflow.net/a/171653/381).

### Local contractibility
 {#LocalContractibility}

A [[complex analytic manifold]] and a _smooth_ [[complex analytic space]] is locally isomorphic to a [[polydisk]] and hence is trivially a [[locally contractible space]]. But over a [[non-archimedean field]] analytic spaces no longer need to be locally isomorphic to polydisks (but $p$-adic polydisks are still contractible ([Berkovich 90](#Berkovich90))). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.

Assume that the [[valuation]] on the ground field $k$ is nontrivial.

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

A $k$-analytic space $X$ is called _locally embeddable in a smooth space_ if each point of $X$ has an [[open neighbourhood]] [[isomorphism|isomorphic]] to a strictly $k$-analytic domain in smooth $k$-analytic space.

=--

+-- {: .num_theorem}
###### Theorem

Every $k$-analytic space which is locally embeddable in a smooth space, def. \ref{LocallyEmbeddableInASmoothSpace}, is a [[locally contractible space]].

More precisely, every point of a locally smooth $k$-analytic space has an open neighbourhood $U$ which is contractible, and which is a [[union]] $U = \cup_{i = 1}^\infty U_i$ of analytic domains.

=--

The local contractibility is [Berkovich (1999), theorem 9.1](#BerkovichContractible). The refined statment in terms of inductive systems of analytic domains is in [Berkovich (2004)](#BerkovichContractibleII).

## Examples

* [[analytic affine line]]

## Applications

* The proof of the [[local Langlands conjecture]]
for $GL_n$ by Harris–Taylor uses [[étale cohomology]] on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.

## Related concepts

* [[non-commutative analytic space]]

* [[p-adic geometry]]

* [[G-topology]]

* [[Huber space]], [[perfectoid space]]

* [[global analytic geometry]]

## References

### Introductions and reviews

A nice survey is in

* {#LeStum12} [[Bernard Le Stum]], _One century of $p$-adic geometry -- From Hensel to Berkovich and beyond_, talk notes, June 2012 ([pdf](http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf))

A good introduction to the general idea is at the beginning of

* {#Payne13} [[Sam Payne]], _Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry_ ([arXiv:1309.4403](http://arxiv.org/abs/1309.4403))

Basic notions are listed in

* M. Temkin, _Non-archimedean analytic spaces_ ([pdf slides](http://www.math.huji.ac.il/~temkin/lectures/Non-Archimedean_Analytic_Spaces.pdf))

A review of basic definitions and facts about affinoid and rigid $k$-analytic spaces can be found in

* Gaëtan Chenevier, _lecture 5_ ([pdf](http://www.math.polytechnique.fr/~chenevier/coursIHP/chenevier_lecture5.pdf))

See also the references at [[rigid analytic geometry]].

A review of definitions and results on  $k$-analytic spaces is in

* {#Berkovich98} [[Vladimir Berkovich]], _$p$-Adic analytic spaces_, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/ICM98_1998.pdf))

A more detailed set of lecture notes along these lines is

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

Introductory exposition of the Berkovich [[analytic spectrum]] is

* {#Poineau07} [[Jérôme Poineau]], _Global analytic geometry_, pages 20-23 in EMS newsletter September 2007 ([pdf](http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf))

* [[Frédéric Paugam]], section 2.1.4 of_Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))

A exposition of examples of Berkovich spectra is in

* [[Scott Carnahan]], _Berkovich spaces I_ ([web](http://sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i/))

### Original articles

* [[Vladimir Berkovich]], _Spectral theory and analytic geometry over non-Archimedean fields_, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.
  {#Berkovich90}

* [[Vladimir Berkovich]], _Étale cohomology for non-Archimedean analytic spaces_, Publ. Math. IHES 78 (1993), 5-161.

Discussion of Berkovich09cohomology of Berkovich analytic spaces includes

* {#Berkovich95} [[Vladimir Berkovich]], _On the comparison theorem for étale cohomology of non-Archimedean analytic spaces._ Israel Journal of Mathematics 92.1-3 (1995): 45-59.

* {#Berkovich09} [[Vladimir Berkovich]], _A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures_, Algebra, Arithmetic, and Geometry. Birkhäuser Boston, 2009. 49-67.

Discussion of local contractibility of smooth $k$-analytic spaces is in

* {#BerkovichContractible} [[Vladimir Berkovich]], _Smooth $p$-adic analytic spaces are locally contractible_, Invent. Math. 137 1-84 (1999) ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Inven_1999_137_contra.pdf))

* {#BerkovichContractibleII} [[Vladimir Berkovich]], _Smooth p-adic analytic spaces are locally contractible. II_, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Dworkvol_2004_contraII.pdf))

and more generally in

* {#HrushovskiLoeser10} [[Ehud Hrushovski]], [[François Loeser]], _Non-archimedean tame topology and stably dominated types_ ([arXiv:1009.0252](http://arxiv.org/abs/1009.0252))

* [[Ehud Hrushovski]], [[François Loeser]], [[Bjorn Poonen]], _Berkovich spaces embed in Euclidean spaces_ ([arXiv:1210.6485](http://arxiv.org/abs/1210.6485))

### Relation to other topics

On the relation to [[buildings]]:

* Annette Werner, _Buildings and Berkovich Spaces_ ([pdf](http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/werner/talks/dmvmuench10.pdf))

Relation to [[integration]] theory

* [[Vladimir Berkovich]], _Integration of 1-forms on $p$-adic analytic spaces_, Princeton University Press,

Aspects of the [[homotopy theory]]/[[étale homotopy]] of analytic spaces are discussed in

* [[Aise Johan de Jong]], _Étale fundamental groups of non-archimedean analytic spaces_, Mathematica, 97 no. 1-2 (1995), p. 89-118  ([numdam](http://www.numdam.org/item?id=CM_1995__97_1-2_89_0))

Relation to [[formal schemes]]:

* Jérôme Poineau, [MO comment](http://mathoverflow.net/a/138577/381)

Discussion of Berkovich analytic geometry as [[algebraic geometry]] in the general sense of [[Bertrand Toën]] and [[Gabriele Vezzosi]] is in

* [[Oren Ben-Bassat]], [[Kobi Kremnizer]], _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

[[!redirects Berkovich analytic space]]
[[!redirects Berkovich analytic spaces]]
[[!redirects Berkovich spaces]]
[[!redirects p-adic analytic space]]
[[!redirects p-adic analytic spaces]]


Berkovits superstring

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#Contents#
* table of contents
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## Idea

The formulation of the [[sigma-model]] 2d [[CFT]]s that are used in (super) [[string theory]] had originally been of two types:

1. In the [[RNS-string]] formulation the worldsheet is a [[supermanifold]], while the target space is an ordinary [[manifold]].

   The advantage of this formulation is that its [[quantization]] is comparatively tractable: it is a 2-dimensional superconformal field theory, coming with a super [[vertex operator algebra]].

   The disadvantage is that, as one says, "target space supersymmetry is not manifest": as described at [[string theory]], every 2d CFT gives rise to an effective background [[quantum field theory]] (that quantum field theory whose [[S-matrix]] approximates the [[S-matrix]] obtained by evaluating the given CFT correlators on all possible surfaces and summing over the conformal [[moduli space]] and all genera). One can identify explcitly in the [[vertex operator algebra]] of the 2d CFT those operators that correspond to the fields of this effective background field theory (the _vertex operators_). One finds that these fields are those of a theory of [[supergravity]] and [[super Yang-Mills theory]]. So in particular there is a [[super Lie algebra]] acting on them. But this "supersymmetry" of the effective target space theory appears like an accident if one unwraps all this: there is no structural one-line argument (known) that would guarantee that the effective background quantum field theory of a 2d super [[CFT]] is itself supersymmetric.

1. In the [[Green-Schwarz superstring]] [[sigma-model]] formulation, the [[worldsheet]] is an ordinary [[manifold]], but target space is a [[supermanifold]].

   This has the immediate advantage that target space supersymmetry is now "manifest": as described at [[supergravity]] it is just the super-[[diffeomorphism]] invariance of the theory, and the [[sigma-models]] in question all manifestly have this property.

   The big disadvantage is, that it is not known how to [[quantization|quantize]] this system. The reason is that the standard procedure for quantization shows that the GS-type sigma models are [[constrained mechanics|constrained]] systems with second-class constraints. Little to nothing is known how to deal with that.

Berkovits was motivated by the desire to find a formulation of the superstring sigma model CFT that would combine the advantages of both the RNS formulation and the Green-Schwarz formulation. The goal was to have a CFT that looked like a free field theory (as the RNS string does for a flat Minkowski background, but as the GS string does not) if one just looked at a small part of the range of its fields, but which was globally constrained: the spinorial worldsheet fields here are maps into a space of [[pure spinors]] that locally lools like a Euclidean space, but globally has a cone geometry (...details...).

Berkovits originally wrote down some more or less ad-hoc expressions. Later it was understood that what described is a _[[sheaf]] of vertex operator algebras_ (in general, not a [[sheaf]] but a [[stack]], in fact a [[gerbe]]) on target space: to each contractible open patch of target space is associate the [[vertex operator algebra]] of a free [[sigma-model]] CFT whose fields take values _just_ in that patch, such that on overlaps these vertex operator algebras glue in some way.

This at once made the previously mathemtically rather unjustified approach make close contact with the developing theory of [[chiral de Rham complex]], which is one of the most-studied examples of sheaves of vertex operator algebras.

## Related concepts

* [[superstring]]

* [[string scattering amplitude]]

## References

### General

The original articles:

(...)

Relation to the [[RNS string]]:

* [[Nathan Berkovits]], *Manifest Spacetime Supersymmetry and the Superstring* ([arXiv:2106.04448](https://arxiv.org/abs/2106.04448))

Relation to the [[Green-Schwarz superstring]]:

* [[Max Guillen]], *Green-Schwarz and Pure Spinor Formulations of Chiral Strings* ([arXiv:2108.11724](https://arxiv.org/abs/2108.11724))

Review:

* [[Nathan Berkovits]], [[Carlos R. Mafra]], *Pure spinor formulation of the superstring and its applications*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.10510](https://arxiv.org/abs/2210.10510)]

Discussion of [[string scattering amplitudes]]:

* [[Carlos R. Mafra]], [[Oliver Schlotterer]], *Tree-level amplitudes from the pure spinor superstring* [[arXiv:2210.14241](https://arxiv.org/abs/2210.14241)]

Relation to $D=5$ [[holomorphic Chern-Simons theory]]:

* [[Nathan Berkovits]], *$D=5$ Holomorphic Chern-Simons and the Pure Spinor Superstring* [[arXiv:2211.06731](https://arxiv.org/abs/2211.06731)]

Relation to [[semi-topological 4d Chern-Simons theory]]:

* [[Nathan Berkovits]], Rodrigo S. Pitombo, *4D Chern-Simons and the pure spinor $AdS_5 \times S^5$ superstring* [[arXiv:2401.03976](https://arxiv.org/abs/2401.03976)]

### Sheaf ov VOAs

A good reference that explains the sheaf of vertex operator algebra perspective on the Berkovits superstring is

* [[Nikita Nekrasov]], _Lectures on curved $\beta$-$\gamma$ systems, pure spinors, and anomalies_ ([arXiv](http://arxiv.org/abs/hep-th/0511008))

The standard reference on the closely related mathematical theory of the chiral deRham complex is

* [[Vassily Gorbounov]], Fyodor Malikov, [[Vadim Schechtman]], _Gerbes of chiral differential operators_  Math. Res. Lett. __7__ (2000),  no. 1, 55--66, [MR2002c:17040](http://www.ams.org/mathscinet-getitem?mr=2002c:17040), [math.AG/9906117](http://arxiv.org/abs/math/9906117); _Gerbes of chiral differential operators. II. Vertex algebroids_, Invent. Math. __155__ (2004), no. 3, 605--680, [MR2005e:17047](http://www.ams.org/mathscinet-getitem?mr=2005e:17047), [math.AG/0003170](http://arxiv.org/abs/math/0003170), [doi](http://dx.doi.org/10.1007/s00222-003-0333-4); _Gerbes of chiral differential operators. III_, in: The orbit method in geometry and physics (Marseille, 2000),  73--100, Progr. Math. __213__, Birkhäuser 2003, [MR2005a:17028](http://www.ams.org/mathscinet-getitem?mr=2005a:17028), [math.AG/0005201](http://arxiv.org/abs/math/0005201), _On chiral differential operators over homogeneous spaces_, Int. J. Math. Math. Sci. __26__ (2001), no. 2, 83--106, [MR2002g:14020](http://www.ams.org/mathscinet-getitem?mr=2002g:14020), [math.AG/0008154](https://arxiv.org/abs/math/0008154), [doi](https://doi.org/10.1155/S0161171201020051)

Here is some blog discussion about this topic:

* [[Jacques Distler]]

  * _[CDO and pure spinors](http://golem.ph.utexas.edu/~distler/blog/archives/000664.html)_

  * _[Nekrasov on pure spinors](http://golem.ph.utexas.edu/~distler/blog/archives/000670.html)_

* [[Urs Schreiber]]

  * _[Nekrasov on pure spinor superstring](http://golem.ph.utexas.edu/string/archives/000661.html)_

    _[Sheaves of CDOs](http://golem.ph.utexas.edu/string/archives/000668.html)_

Nekrasov work referred to above is partly from

* Nathan Berkovits, [[Nikita Nekrasov]], _The character of pure spinors_, Letters in Mathematical Physics (2005) 74:75--109 [doi](https://doi.org/10.1007/s11005-005-0009-7)

[[!redirects Berkovits string]]

[[!redirects pure spinor superstring]]


Bernard Badzioch

* [website](http://www.math.buffalo.edu/Badzioch_Bernard.html)

category: people


Bernard Bolzano

Bernard Bolzano (1789-1848) was a Bohemian mathematician, logician, and philosopher.

In his lifetime he was renowned perhaps mainly for his *Wissenschaftslehre* which puts forth an encompassing theory of science. In mathematics, much of his work became well known only after his death, but he can be regarded as an early proponent of [[epsilontic analysis]] and the introduction of rigor in analysis, as for example in the result known as the [[Bolzano-Weierstrass theorem]] (q.v.).

## References

Collected works are available at

* [Bolzano collection](http://dml.cz/handle/10338.dmlcz/400001) at the Czech Digital Mathematics Library

For biographical and other detail, see

* [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/bolzano/)

* [Bolzano's Logic](https://plato.stanford.edu/entries/bolzano-logic/) in Stanford EP

* [MacTutor biographical sketch](http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bolzano.html)

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


Bernard de Wit

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

## Selected writings

On the regularized [[quantization]] of the [[supermembrane]] (the [[kappa-symmetry|kappa-symmetric]] [[Green-Schwarz sigma-model]] for the [[M2-brane]]) to the [[BFSS matrix model]]:

* {#deWitHoppeNicolai88} [[Bernard de Wit]], [[Jens Hoppe]], [[Hermann Nicolai]], _On the Quantum Mechanics of Supermembranes_, Nucl. Phys. B305 (1988) 545. ([pdf](http://pubman.mpdl.mpg.de/pubman/item/escidoc:153408:1/component/escidoc:153407/353961.pdf), [[deWitHoppeNicolai88.pdf:file]], [spire:261702](http://inspirehep.net/record/261702))

* [[Bernard de Wit]], W. Lüscher, [[Hermann Nicolai]], _The supermembrane is unstable_, Nucl. Phys. B320 (1989) 135 ([spire:266584](http://inspirehep.net/record/266584/), doi:10.1016/0550-3213(89)90214-9)

On [[gauged supergravity]] and introducing the concept of [[tensor hierarchies]]:

* [[Bernard de Wit]], [[Henning Samtleben]], _Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems_, Fortsch. Phys. 53 (2005) 442-449 ([arXiv:hep-th/0501243](https://arxiv.org/abs/hep-th/0501243))

* [[Bernard de Wit]], [[Henning Samtleben]], _The end of the $p$-form hierarchy_, JHEP 0808:015, 2008 ([arXiv:0805.4767](https://arxiv.org/abs/0805.4767))

On [[tensor hierarchies]] in [[gauged supergravity]] and possible relation to [[U-duality]] and [[M-theory]]:

* [[Bernard de Wit]], [[Hermann Nicolai]], [[Henning Samtleben]], _Gauged Supergravities, Tensor Hierarchies, and M-Theory_, JHEP 0802:044, 2008 ([arXiv:0801.1294](https://arxiv.org/abs/0801.1294))

category: people


Bernard Julia

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

## Selected writings

Introducing [[D=11 supergravity]]:

* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. B **76** (1978) 409 [doi:10.1016/0370-2693(78)90894-8]

On [[first-order formulation of gravity|first order formulation]] of [[supergravity]]:

* [[Bernard Julia]], S. Silva, *On first order formulations of supergravities*, JHEP 0001 (2000) 026 [[arXiv:hep-th/9911035](https://arxiv.org/abs/hep-th/9911035), [doi:10.1088/1126-6708/2000/01/026](https://doi.org/10.1088/1126-6708/2000/01/026)]

category: people


Bernard Le Stum

* [webpage](http://perso.univ-rennes1.fr/bernard.le-stum/Bienvenue.html)

## related $n$Lab entries

* [[p-adic geometry]]

* [[non-archimedean analytic geometry]]

category: people


Bernard Leclerc

_Bernard Leclerc_ is a mathematician at Université de Caen. His specialties include [[algebraic combinatorics]] and [[representation theory]].

* [homepage](http://www.math.unicaen.fr/~leclerc)

category:people
[[!redirects B. Leclerc]]


Bernard M.A.G. Piette

* [Institute page](https://www.dur.ac.uk/directory/profile/?id=474)

* [spire page](https://inspirehep.net/authors/1048203)

## Selected writings

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

* [[Nicholas S. Manton]], [[Bernard M.A.G. Piette]], *Understanding Skyrmions using Rational Maps*, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 ([doi:10.1007/978-3-0348-8268-2_27](https://doi.org/10.1007/978-3-0348-8268-2_27), [arXiv:hep-th/0008110](https://arxiv.org/abs/hep-th/0008110))

* W.T. Lin, [[Bernard M.A.G. Piette]], *Skyrmion Vibration Modes within the Rational Map Ansatz*,  	Phys. Rev. D77:125028, 2008 ([arXiv:0804.4786](https://arxiv.org/abs/0804.4786), [doi:10.1103/PhysRevD.77.125028](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.77.125028))

category: people

[[!redirects Bernard Piette]]

[[!redirects B.M.A.G. Piette]]

[[!redirects B. M. A. G. Piette]]

[[!redirects Bernard M. A. G. Piette]]


Bernard Malgrange

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

## Related entries

* [[Koszul-Malgrange theorem]]

* [[holomorphic vector bundle]]

category: people


Bernard S. Kay

* [GoogleScholar page](https://scholar.google.com/citations?user=8M4mKZsAAAAJ&hl=en)

## Selected writings

On [[AQFT on curved spacetimes]]:

* [[Bernard S. Kay]], *Quantum Field Theory in Curved Spacetime*, Encyclopaedia of Mathematical Physics (2023) [[arXiv:2308.14517](https://arxiv.org/abs/2308.14517)]

category: people

[[!redirects Bernard Kay]]


Bernardo Uribe

* [website](https://sites.google.com/site/bernardouribejongbloed/)

## Selected writings

Introducing [[twisted K-theory|twisted]] [[orbifold K-theory]] presented by
[[twisted bundles|twisted]] [[equivariant vector bundles]] (including their form that came to be called called *[[bundle gerbe modules]]*):

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489.  ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x))

On [[equivariant ordinary differential cohomology]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], _Deligne Cohomology for Orbifolds, discrete torsion and B-fields_, in: _Geometric and topological methods for quantum field theory_ Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 ([arXiv:hep-th/0201184](https://arxiv.org/abs/hep-th/0201184), [spire:582101](https://inspirehep.net/literature/582101))

On [[transgression]] and [[higher holonomy]] in [[Deligne cohomology]] (for [[bundle gerbes with connection]]) over [[orbifolds]]:

* [[Ernesto Lupercio]], [[Bernardo Uribe]], *Holonomy for Gerbes over Orbifolds*, J. Geom.Phys. **56** (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), [doi:10.1016/j.geomphys.2005.08.006](https://doi.org/10.1016/j.geomphys.2005.08.006)]

On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:

* [[Edward Becerra]], [[Bernardo Uribe]], _Stringy product on twisted orbifold K-theory for abelian quotients_, Trans. Amer. Math. Soc. 361 (2009), 5781-5803  ([arXiv:0706.3229](https://arxiv.org/abs/0706.3229), [doi:10.1090/S0002-9947-09-04760-6](https://doi.org/10.1090/S0002-9947-09-04760-6))

On the [[topology]] of the infinite [[unitary group]] [[U(ℋ)]] on a [[separable Hilbert space]] ([[operator topology]] and [[compact-open topology]]):

* {#EspinozaUribe} [[Jesus Espinoza]], [[Bernardo Uribe]], _Topological properties of the unitary group_, JP Journal of Geometry and Topology **16** (2014) Issue 1, pp 45-55 ([arXiv:1407.1869](https://arxiv.org/abs/1407.1869), [journal](http://www.pphmj.com/abstract/8730.htm))

and of the corresponding [[projective unitary group]] [[PU(ℋ)]]:

* [[Jesus Espinoza]], [[Bernardo Uribe]], *Topological properties of spaces of projective unitary representations*, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 ([arXiv:1511.06785](https://arxiv.org/abs/1511.06785), [scielo:S0370-39082016000200013](http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013), [doi:10.18257/raccefyn.317](https://doi.org/10.18257/raccefyn.317))

On 3-[[twisted equivariant K-theory]] via the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]:

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

On [[equivariant principal bundles]] and their [[classifying spaces]]:

* [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

On the analog of the [[Atiyah-Hirzebruch spectral sequence]] for [[twisted equivariant K-theory]]:

* [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Bernardo Uribe]], [[Mario Velasquez]], _Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions_, Proceedings of the Edinburgh Mathematical Society **61** 1 (2018) ([arXiv:1307.1003](https://arxiv.org/abs/1307.1003), [doi:10.1017/S0013091517000281](https://doi.org/10.1017/S0013091517000281))

category: people


Bernd Fischer

* [Wikipedia page](http://en.wikipedia.org/wiki/Bernd_Fischer)

## Related $n$Lab entries

* [[Monster group]]

* [[Moonshine]]

category: people


Bernd Schroers

Bernd Schroers is professor of [[mathematics]] at Heriot-Watt in Edinburgh.

* [webpage](http://www.hw.ac.uk/schools/mathematical-computer-sciences/staff-directory/bernd-schroers.htm)

## Selected writings

On [[Yang-Mills monopoles]] for [[gauge group]] [[SU(3)]]:

* F. A. Bais, [[Bernd Schroers]], *Quantisation of Monopoles with Non-abelian Magnetic Charge*, Nucl. Phys. B512 (1998) 250-294 ([arXiv:hep-th/9708004](https://arxiv.org/abs/hep-th/9708004))

On [[Michael Atiyah]]'s work relating to [[physics]] ([[Yang-Mills theory]], [[Skyrmions]]):

* [[Bernd Schroers]], _Michael Atiyah and Physics: the Later Years_,  ([arXiv:1910.10630](https://arxiv.org/abs/1910.10630)), in: Notices of the AMS, _Memories of Sir Michael Atiyah_ ([pdf](https://www.ams.org/journals/notices/201911/rnoti-p1834.pdf), [[MemoriesOfAtiyah.pdf:file]])

## Related $n$Lab entries

* [[Chern-Simons theory]]

* [[Kaluza-Klein compactification]]

category: people


Bernd Siebert

* [institute page](https://www.math.uni-hamburg.de/home/siebert/)

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

* [GoogleScholar page](https://scholar.google.de/citations?user=kgOhnLYAAAAJ&hl=de)

## Selected writings

Early discussion of [[quantum cohomology rings]]:

* [[Bernd Siebert]], [[Gang Tian]], *On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator*,  Asian Journal of Mathematics **1** 4 (1997) 679 – 695 [[arXiv:alg-geom/9403010](https://arxiv.org/abs/alg-geom/9403010), [doi:10.4310/AJM.1997.v1.n4.a2](https://dx.doi.org/10.4310/AJM.1997.v1.n4.a2)]

category: people


Bernd Sturmfels

Bernd Sturmfels is a professor at Berkeley, California.

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


Bernd Thaller

* [home page](https://imsc.uni-graz.at/thaller/)

* [Wikipedia entry (deutsch)](https://de.wikipedia.org/wiki/Bernd_Thaller)

## Selected writings

* [[Bernd Thaller]], *The Dirac Equation*,  Texts and Monographs in Physics, Springer (1992) ([doi:10.1007/978-3-662-02753-0](https://link.springer.com/book/10.1007/978-3-662-02753-0))

category: people


Bernhard Banaschewski

Bernhard Banaschewski (1926-2022)]

* [McMaster obituary](https://math.mcmaster.ca/bernhard-banaschewski-1926-2022/#:~:text=Bernhard%20Banaschewski%2C%20McKay%20Professor%20(Emeritus,of%20the%20Department%20in%201955.)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=13403)

## Selected writings

* {#Banaschewski_Pultr94} [[B. Banaschewski]], A. Pultr, *Variants of openness*, Appl. Cat. Struc. **2** (1994) 1-21 [[doi:10.1007/BF00873038](https://doi.org/10.1007/BF00873038)]

* {#Banaschewski_Pultr96} [[B. Banaschewski]], A. Pultr, *Booleanization*, Cah. Top. Géom. Diff. Cat. **XXXVII** **1** (1996)  41-60 [[numdam:CTGDC_1996__37_1_41_0](http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0)]

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]

On the [[Boolean algebra]] underlying Spencer Brown's "Laws of form":

* {#Banaschewski77} [[B. Banaschewski]], *On G. Spencer Brown's laws of form*, Notre Dame J. Formal Logic **18** 3 (1977) 507-509 [[doi:10.1305/ndjfl/1093888028](https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/On-G-Spencer-Browns-laws-of-form/10.1305/ndjfl/1093888028.full)]

On [[algebraic closures]] of [[fields]] in the absence of the [[axiom of choice]]:

* {#Banaschewski92} [[Bernhard Banaschewski]], *Algebraic closure without choice*, Mathematical Logic Quarterly, Volume 38, Issue 1, 1992, Pages 383-385, [[doi:10.1002/malq.19920380136](https://doi.org/10.1002/malq.19920380136)]

## Related entries

* [[weakly open subtopos]]

* [[skeletal geometric morphism]]

category:people

[[!redirects B. Banaschewski]]
[[!redirects Banaschewski]]


Bernhard Hanke

* [webpage](https://www.uni-augsburg.de/de/fakultaet/mntf/math/prof/diff/team/bernhard-hanke/)

## Selected writings

On [[equivariant bordism homology theory]]:

* [[Bernhard Hanke]], _Geometric versus homotopy theoretic equivariant bordism_, Math. Ann. 332, 677–696 (2005) ([arXiv:math/0412550](https://arxiv.org/abs/math/0412550), [doi:10.1007/s00208-005-0648-0](https://doi.org/10.1007/s00208-005-0648-0))

category: people


Bernhard Keller

_Bernhard Keller_ is a Swiss-French mathematician.

* [homepage](https://webusers.imj-prg.fr/~bernhard.keller/indexe.html)

* [quelques prépublications et publications](https://webusers.imj-prg.fr/~bernhard.keller/publ/index.html)

## Selected writings

On [[differential graded categories]]:

* [[Bernhard Keller]], _On differential graded categories_,  International Congress of Mathematicians. **II** (2006) 151-190, Eur. Math. Soc., Zürich [[arXiv:math/0601185](http://arxiv.org/abs/math/0601185)]

category: people

[[!redirects B. Keller]]


Bernhard Riemann

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

* [Wikipedia entry (fr)](http://fr.wikipedia.org/wiki/Bernhard_Riemann)

## Seleted writings

Envisioning the modern foundations of [[differential geometry]] via the notion of ([[differentiable manifold|differentiable]], [[smooth manifold|smooth]]) [[manifolds]] and of what camed to be called [[Riemannian geometry]]:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

## Related entries

* [[manifold]], [[smooth manifold]]

* [[Riemannian geometry]], [[Riemannian metric]]

* [[Riemann conjecture]]

* [[Riemann surface]]

category: people

[[!redirects Riemann]]


Bernhard Schölkopf

* [webpage](https://www.is.mpg.de/~bs)

* [Wikipedia entry](https://www.is.mpg.de/~bs)

## Selected writings

On [[kernel methods]] in [[machine learning]]:

* Thomas Hofmann, [[Bernhard Schölkopf]], Alexander J. Smola, *Kernel methods in machine learning*, Annals of Statistics 2008, Vol. 36, No. 3, 1171-1220 ([arXiv:math/0701907](https://arxiv.org/abs/math/0701907))

On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

On some philosophical aspects of [[statistical learning theory]]:

* [[David Corfield]], [[Bernhard Schölkopf]], Vladimir Vapnik, _Falsificationism and statistical learning theory: Comparing the Popper and Vapnik-Chervonenkis dimensions_, 2009, Journal for General Philosophy of Science 40 (1), 51-58 ([doi:10.1007/s10838-009-9091-3](https://doi.org/10.1007/s10838-009-9091-3))

category: people


Bernhard Ömer

* [webpage](http://www.itp.tuwien.ac.at/~oemer/)

## Selected writings

On classical computational control in [[quantum computation]]:

* [[Bernhard Ömer]], *Structured Quantum Programming*, 2003/2009 ([pdf](http://www.itp.tuwien.ac.at/~oemer/doc/structquprog.pdf))

category: people

[[!redirects Bernhard Omer]]