All posts by ajb

Engineering graduates 'taking unskilled jobs'

Nearly a quarter of UK engineering graduates are working in non-graduate jobs or unskilled work such as waiting and shop work, a report suggests.

–BBC News Reporter Katherine Sellgren

A study by researchers Birmingham University seems to go against what people in business are always telling us;

 

“The shortage of science, technology, engineering and maths graduates is an issue for businesses”

–Susan Anderson

I do not understand what people are talking about when they claim there is a shortage of engineers, scientist, mathematicians, computer scientists, and so on.  This is just not true. What is true is that a large proportion of these highly skilled people have to work in “non-scientific” jobs as there is a shortage of jobs relevant to their degrees.

In Physics World (February 2011,  20p ) Jim Grozier basically  complained that banking and finance was taking too many PhD qualified scientists away from science while not directly paying for their training. I agree with the sentiment,  but the point he misses is that there are not enough jobs in science, at Universities or other Labs, to keep these people working in science. Jim himself is one of the lucky ones, he is now based at UCL working in experimental particle physics.  (I vaguely knew Jim when he was PhD student and I a masters student at Sussex. )

It is astonishing, in the light of claims of science graduate shortages, that so few new graduates go into related employment”

–Professor Emma Smith

 

This is why we must all support the Science is Vital campaign.  Much of what the report says also applies to scientists and mathematicians.

To read more about the report by the University of Birmingham have a look at the BBC news report.

 

Science is Vital

From the Science is Vital campaign.

Science Careers: final call for evidence

Following the meeting with Minister of State for Universities and Science, David Willetts, about the ailing state of science careers in the UK, we want to solicit your feedback for the report he requested from us.

Your evidence can be submitted online via this page.

I urge everyone in the UK who has anything to do with science to add their evidence. This means undergraduate students, spouses  and people who have been forced to leave science as well as postdocs, lecturers and professors.

A simple QS and odd Jacobi manifold

Let us quickly recall what I mean by a QS and an odd Jacobi manifold.

Definition A supermanifold equipped with a Schouten structure S and a homological vector field Q such that

\(\{ S, \mathcal{Q} \} =0 \),

where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be a QS-manifold.

This definition allows us to write everything in terms of an odd function quadratic in momenta and an odd function linear in momenta, ie. functions on the total space of the cotangent bundle of our supermanifold. The bracket in the above is the canonical Poisson bracket.  (The example I will give will make this clearer.)

Definition A supermanifold equipped with an almost Schouten structure  S and a homological vector field Q such that

\(\{ S, S \} ={-} 2 \mathcal{Q} S\),

\(\{ S,\mathcal{Q} \} =0\),

where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be an  odd Jacobi manifold.

Both these species of supermanifold are very similar.  QS-manifolds have a genuine Schouten structure, that is an odd function quadratic in momenta such that it Poisson self-commutes and Poisson commutes with the symbol of the homological vector field.  An  odd Jacobi manifold consists of an almost Schouten structure that has a very specific Poisson self-commutator and Poisson commutes with the symbol of the homological vector field.

On to our example…

Consider the supermanifold \(\mathbb{R}^{1|1}\), which we equip with local coordinates \((t, \xi)\). Here \(t\)  is the commuting coordinate and  \(\xi \) is the anticommuting coordinate. This supermanifold comes equipped with a canonical Schouten structure

\(S = {-}\pi p\),

where we employ fibre coordinates \((p, \pi)\) on the cotangent bundle.  As the above structure does not contain conjugate variables is it cleat that

\(\{S,S \}=0\).

We can go a little further than this as we also have a canonical homological vector field, which indeed gives rise to a symbol that Poisson commutes with the Schouten structure:

\(\mathcal{Q} = {-}\pi\).

So \(\mathbb{R}^{1|1}\) is a QS-manifold, canonically.  The associated Schouten bracket is given by

\([f,g]_{S} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}\),

for all \(f,g \in C^{\infty}(\mathbb{R}^{1|1})\).

Interestingly, we can also consider these structures as being odd Jacobi. Explicitly one can calculate the Poisson self-commutator of the Schouten structure and arrive at

\(\{ S, S\} = {-} 2 \left(  {-} \pi\right)\left(  {-}\pi p\right)\),

which is of course zero as \(\pi^{2}=0\). But also notice that this defines an odd Jacobi structure! We then can assign an odd Jacobi bracket as

\([f,g]_{J} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}{-}({-}1)^{\widetilde{f}}\left(  \frac{\partial f}{\partial \xi}\right)g {-}f\left(  \frac{\partial g}{\partial \xi}\right) \).

 

The Schouten bracket satisfies a strict Leibniz rule as where the odd Jacobi bracket does not, we have an “anomaly” term in the derivation property. Both satisfy the appropriate graded version of the Jacobi identity.

 

Interestingly, the Schouten structure on \(\mathbb{R}^{1|1}\) is in fact non-degenerate so we have an odd symplectic supermanifold. One can also consider \(\mathbb{R}^{1|1}\) as an even contact manifold, but I will delay talking about that for now.

One could of course “compactify” \(\mathbb{R}\) and consider the supercircle \(\mathbb{S}^{1|1}\), and this naturally also can be considered as QS and odd Jacobi. Again we have a natural contact structure here and this has been studied in relation to super versions of the Schwarzian derivative. This is really another story…

More details can be found in an older post of mine here. A preprint about odd Jacobi structures can be found on the arXiv here.

 

 

Contact structures and supersymmetric mechanics

Contact structures and supersymmetric mechanics

Andrew James Bruce

Abstract
We establish a relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with supersymmetric mechanics.

arXiv:1108.5291v1 [math-ph]

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In the preprint above I show that aspects of  d=1, N=2 supersymmetric quasi-classical mechanics in the superspace formulation can be understood in terms of  a contact structure on the supermanifold \(R^{1|2}\).

 

In particular if we pick local coordinates \((t, \theta, \bar{\theta})\) then the super contact structure is given by

 

\(\alpha = dt + i \left(  d \bar{\theta}\theta + \bar{\theta} d \theta  \right)\),
which is a Grassmann odd one form. One could motivate the study of such a one form as a “superisation” of the contact form on \(R^{3}\).

 

Associated with any odd one form that is nowhere vanishing is a hyperplane distribution of codimension (1|0). That is we have a subspace of the tangent bundle that contains one less even vector field in its (local) basis as compared to the  tangent bundle.  This is why we should refer to the above structure as an even (pre-)contact structure.

 

The hyperplane distribution associated with the super contact structure is spanned by two odd vector fields. These odd vector fields are exactly the SUSY covariant derivatives. More over we do have a genuine contact structure as the exterior derivative of the super contact form is non-degenerate on the hyperplane distribution. For more details see the preprint.

 

Generalising contact structures  on manifolds to  supermanifolds appears fairly straight forward. We have the non-classical case of odd contact structures to also handle, here the hyperplane distribution is of corank (0|1), i.e. one less odd vector field. There is also a subtly when defining kernels and contactomorphisms as we will have to take care with nilpotent objects.

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Comments on the preprint will be very much appreciated.

 

________________________________

Update A third revised version has now been submitted. 08/02/2012

Theories in physics

In physics the word theory is used synonymously with mathematical model or mathematical framework. The theory is a mathematical construction   that can be used to describe physical phenomena.  A theory should, at least  in principle be falsifiable, that is make predictions that can be tested.

People who are not trained in physics take theory to mean either  “hypothetical” or loosely an  “idea”.  One may hear “but it is only a theory”, which takes the physics use of the word theory out of context.

A theory, in the sense of modern physics must by definition be phrased in mathematics. We need something to mathematically manipulate and calculate things that can be tested against observation.  Without the mathematical framework it is hard to judge if an “idea” has any merit or not.

Often by theory physicists may have something  a little more specific in mind, they often mean a specified action or Lagrangian.  Most of physics can be stated in terms of actions and so it usually makes sense to start there.  Again the action or equivalently the Lagrangian are mathematical notions.

 

 

 

Astronomy Vs Astrology

Even today people confuse astronomy and astrology.  It is not hard to see why when almost every newspaper has a horoscope and  lots of adverts for astrology phone lines.  Lets set the record straight.

 

Astronomy the scientific study of  celestial bodies, for example the Sun, planets, starts, comets etc.  The science is based on observation of the  celestial bodies and the application of physical laws to such bodies.  Mathematics and physics are essential in astronomy.

 

Astrology the belief that the position of  celestial bodies influences the personality and human affairs. It is based on superstition and no physical mechanisms have been established. The superstition does not apply the scientific method and in no way follows modern scientific principles.

In short, astronomy and the closely related astrophysics and cosmology add to the human understanding of nature and our place in the Universe.  Astrology is a superstition that people exploit to make money.

It is of course true that the origins of astronomy lie in astrology. Careful observations and recording of data was necessary in order to write astrological charts.  One could equally argue that chemistry owes  a lot to alchemy.  But we have come a long way in our thinking and philosophy.  Astronomy and chemistry are sciences.

Please do not confuse the two, it is rather insulting to all astronomers!


The fundamental misunderstanding of calculus

We all know the fundamental theorems of calculus, if not check Wikipedia.  I now want to  demonstrate what has been called the fundamental misunderstanding of calculus.

Let us consider the two dimensional plane and equip it with coordinates \((x,y)\).  Associated with this choice of coordinates are  the partial derivatives

\(\left( \frac{\partial}{\partial x} , \frac{\partial}{\partial y} \right)\).

You can think about these in terms of the tangent sheaf etc. if so desired, but we will keep things quite simple.

Now let us consider a change of coordinates. We will be quite specific here for illustration purposes

\(x \rightarrow \bar{x} = x +y\),

\(y \rightarrow \bar{y} = y\).

Now think about how these effect the partial derivatives. This is really just a simple change of variables.  Let me now state  the fundamental misunderstanding of  calculus in a way suited to our example:

Misunderstanding: Despite coordinate x changing the partial derivative with respect to x remains unchanged. Despite the coordinate y remaining unchanged the partial derivative with respect to y changes.

This may seem at first counter intuitive, but is correct. Let us prove it.

Note hat we can invert the change of coordinate for x very simply

\(x = \bar{x} {-}\bar{y} \),

using the fact that y does not change. Then one needs to use the chain rule,

\(\frac{\partial}{\partial \bar{x}}  = \frac{\partial x}{\partial \bar{x}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{x}}\frac{\partial}{\partial y}   =    \frac{\partial}{\partial x}\),

\(\frac{\partial}{\partial \bar{y}}  = \frac{\partial x}{\partial \bar{y}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{y}}\frac{\partial}{\partial y}   =    \frac{\partial}{\partial y} {-} \frac{\partial}{\partial x} \).

There we are. Despite our initial gut feeling that that the partial derivative wrt y should remain unchanged we see that it is in fact the partial derivative wrt x that is unchanged.  This can course some confusion the first time you see it,  and hence the nomenclature the fundamental misunderstanding of calculus.

I apologise for forgetting who first named the misunderstanding.

 

A must read for all you Guinness drinkers

W. T. Lee and M. G. Devereux [1] review the bubble formation processes in carbonated drinks,  like fizzy pop or champagne and compare this with heavy stout drinks.  Stout beers have lots of dissolved nitrogen and this makes the physics slightly different to the carbonated drinks.  Specifically, although the same mechanisms apply the time scales are very different.  Stouts will not spontaneously form a head of foam. This means that a widget or similar needs to be added to cans in order to  aid the nucleation of gas bubbles.

In the paper both the mathematical and experimental issues are discussed.  I recommend you use the paper as a conversation starter next time you are wasted on stout at your local ale house.

Bottoms up!

References

[1] W. T. Lee and M. G. Devereux. Foaming in stout beers.  arXiv:1105.2263v1 [physics.chem-ph]

Higher Lie-Schouten brackets

I thought it would be interesting to point out a geometric construction related to  \(L_{\infty}\)-algebras.  (See earlier post here) Recall that given a Lie algebra \((\mathfrak{g}, [,] )\) one can associate on the dual vector space a linear Poisson structure known as the Lie-Poisson bracket.  So, as a  manifold \((\mathfrak{g}^{*}, \{, \}) \) is a Poisson manifold.  It is convenient to  replace the “classical” language of linear and replace this with a graded condition. That is, if we associate weight one to the coordinates on \(\mathfrak{g}^{*} \) then the Lie-Poisson bracket is of weight minus one.

The Lie Poisson bracket is very important in deformation quantisation (both formal and C*-algebraic). There are some nice theorems and results that I should point to at some later date.

Now, it is also known that one has an odd version of this known as the Lie-Schouten brackets on \(\Pi \mathfrak{g}^{*}\). The key difference is the shift in the Grassmann parity of the “linear” coordinates.  Note that this all carries over to Lie super algebras with no problem.  I will drop the prefix super from now on…

 

So, let us look at the situation for \(L_{\infty}\)-algebras. We understand these either as a series of higher order brackets on a vector space  \(U\) that satisfies a higher order generalsiation of the Jacobi identities or more conveniently we can understand all this in terms of a homological vector field on the formal manifold \(\Pi U\).

Definition An \(L_{\infty}\)-algebra is a vector space \(V = \Pi U\) together with a homological vector field \(Q = (Q^{\delta} + \xi^{\alpha} Q_{\alpha}^{\delta} + \frac{1}{2!} \xi^{\alpha} \xi^{\beta} Q_{\beta \alpha}^{\delta} + \frac{1}{3!} \xi^{\alpha} \xi^{\beta} \xi^{\gamma} Q_{\gamma \beta \alpha}^{\delta} + \cdots) \frac{\partial}{\partial \xi^{\delta}}\),

where we have picked coordinates on \(\Pi U\)  \(\{  \xi^{\alpha}\}\). Note that these coordinates are odd as compared to the coordinates on \(U\). Thus we assign the Grassmann parity \(\widetilde{\xi^{\alpha}} = \widetilde{\alpha} + 1\)  Note that \(Q\) is odd and that if we restrict to the quadratic part then we are back to Lie algebras.

I will simply state the result, rather than derive it.

Proposition Let \((\Pi U, Q)\) be an \(L_{\infty}\)-algebra. Then the formal manifold \(\Pi U^{*}\) has a homotopy Schouten algebra structure.

Let us pick local coordinates \(\{ \eta_{\alpha}\}\) on \(\Pi U^{*}\). Furthermore, we consider this as a graded manifold and attach a weight of one to each coordinate.  A general function,  a  “multivector” has the form

\(X = \stackrel{0}{X} + X^{\alpha} \eta_{\alpha} + \frac{1}{2!}X^{\alpha \beta}\eta_{\beta} \eta_{\alpha} + \cdots \)

The higher Lie-Schouten brackets are given by

\((X_{1}, X_{2}, \cdots, X_{r}) = \pm Q_{\alpha_{r}\cdots  \alpha_{1} }^{\beta}\eta_{\beta}\frac{\partial X_{1}}{\partial \eta_{\alpha_{1}}} \cdots   \frac{\partial X_{1}}{\partial \eta_{\alpha_{r}}}\),

being slack with an overall sign.  Note that with respect to the natural weight the n-bracket has weight (1-n). Thus not unexpectedly, restricting to n=2 gives an odd bracket of weight minus one: up to conventions this is the Lie-Schouten bracket of a Lie algebra.

The above collection of brackets forms an \(L_{\infty}\)-algebra in the “odd super” conventions that satisfies a derivation rule of the product of “multivectors”. Thus the nomenclature homotopy Schouten algebra and higher Lie-Schouten bracket.

A similar statement holds in terms of a  homotopy Poisson algebra on \(U^{*}\). Here the brackets as skewsymmetric and of  even/odd Grassmann parity for even/odd number of arguments.  (I rather the odd conventions overall).

Now this is quite a new construction and the technical exploration of this nice geometric construction awaits to be explored. How much of the geometric theory associated with Lie algebras and Lie groups carries over to \(L_{\infty}\)-algebras and \(\infty\)-groups is an open question.

Details can be found in Andrew James Bruce ” From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures”, Reports on Mathematical Physics Vol. 67, (2011), No. 2  (also on the arXiv).

 

Also see earlier post here on Lie infinity algebroids.