Faster Than the Speed of Light?

Professor Marcus du Sautoy offered a rather sobering view of the results of OPERA on the BBC last night. The BBC iPlayer version is available to view  until the 31st October, follow this link.

Until now I have resisted posting anything about the superluminal speed of neutrinos as measured by OPERA. There are plenty of blogs about this.  All I really want to say is that Marcus du Sautoy does a great job in approaching the topic from a science/maths point of view and does not “fan the flames” of the media hype.  His programme, in my opinion combats some of the hysteria and plain rubbish out there.

Please take the time to watch it.


du Sautoy’s Oxford homepage

Wikpedia article about du Sautoy


Number theory and numerology

In a similar way to the historical link between  astronomy and astrology the subjects of number theory and numerology are also  linked.  The very early impetus for number theory was numerology.  The Pythagorean  school (500BC) were interested in the philosophical and mystical properties of numbers. Plato was influenced by this and mentions numerology in his works, notably The Republic (380BC).  Judaism ,  Christianity and Islam all have elements of numerology.

Number theory itself is probably older than this and goes back  almost to counting and simple arithmetic in  prehistory.

In the same way as astronomy developed into a science, so did number theory.

Definition Number theory is the branch of mathematics that deals with the study of numbers, usually the integers, rational numbers ,  prime numbers etc.

Definition Numerology is the study of the supposed relationship  between numbers, counting and everyday life.

There is another kind of numerology that is the study of numerical coincidences. This happens a lot in physics, where a series of apparent coincidences  can occur between various rations of physical constants or physical observables.

The famous example of this is Dirac’s large number hypothesis which enforces a ration between the cosmic scale and the scale of fundamental forces. Dirac’s hypothesis predicts that  Newton’s constant is varying in time. There has been some work in understanding the implications of physical constants changing in time.

Although Dirac’s hypothesis is the most famous, it was Eddington and Weyl who first noticed such numerical coincidences.

The trouble is that this cannot really be called science.  Physics is all about mathematical models that can be used to explain physical phenomena.  Noticing numerical coincidences by itself does not really add to our understanding of nature.  One would like to explain the  coincidences clearly and mathematically within some theory.  Generally these coincides are interesting, but it is not clear how they are fundamental. Of course, this is apart from those that are really just due to our choices in units etc.

Number theory also has some intersection with physics. Recently there has been some considerable crossover between   arithmetic and algebraic geometry  and string theory (via modular forms largely). I will have to postpone talking about this.


Suggestions for giving talks

We all have to give talks as part of our work from time to time.  In fact giving talks is  very important. Let’s face the truth, people won’t read your papers.  People working on very similar and related things might. Placing a preprint on the arXiv helps,  but generically people won’t get past the title and if you are lucky the abstract. Giving talks at conferences and seminars is the only way to get people to notice your work and importantly YOU.

My general advice (and I have no great insight) is simple,

  1. Select very modest goals and maybe 3 or 4 key points.  Most of the audience will not be experts in your precise field, unless you are at a specialist conference.  For departmental seminars you will have to take great care in what you say and how you say it. You want people to learn something and also see that you are great at communicating.
  2. Open with setting the context of your work.  People need to know how it all fits in. They want to know if what you have done relates to their work.
  3. Be slack with unimportant details.  You can remove terms in equations that play no significant role. Just use words like “+ small terms”.  You can suppress indices.  Don’t be frightened of using analogies or examples to get a point across.  The ethos does  have to be exactly correct,  just close enough to what you are really doing, but you must say this.
  4. Never run overtime. The audience will hate you for this. It is disrespectful to keep going past your allotted time.  It can also show that you do not take talking seriously and have not though out your talk at all.

Geroch way back in 1973 wrote some notes called “Suggestions For Giving Talks”.  He gives plenty of sound advice for giving talks. The notes are available on the arXiv  here.  I suggest anyone giving a scientific talk take a look at it.  I know it has helped me.


Other things online I have found useful and full of good advice include

There is much advice online, so have a quick ” google”.

To improve you talking skills one should do do two things

  1. Give as many talks as possible.
  2. Attend as many talks as possible.

Only with practice will you improve.  You can also take inspiration from good speakers and avoid imitating the bad ones. We have all been to talks by very well respected people, only to be disappointed by their presentation.  Learn from the masters, both good and bad.


Finally, if you have to give a talk soon, good luck.


First Order Differential Operators

I thought I would share some interesting things about first order differential operators, acting on functions on a supermanifold. One can reduce the theory to operators on manifolds by simply dropping the sign factors and ignoring the parity.

First order differential operators naturally include vector fields as their homogeneous “top component”.  The lowest order component is left multiplication by a smooth function.   I will attempt to demonstrate that  from an algebraic point of view first order differential operators  are quite natural and in some sense more fundamental that just the vector fields.

Geometrically, vector fields are key as they represent infinitesimal diffeomorphisms and are used to construct Lie derivatives as “geometric variations”.  This is probably why in introductory geometry textbooks first order differential operators are not described.

I do not think anything I am about to say is in fact new.  I assume the reader has some idea what a differential operator is and that they form a Lie algebra under the commutator bracket.  Everything here will be done on supermanifolds.

I won’t present full proofs, hopefully anyone interested can fill in any gaps.  Any serious mistakes then let me know.

Let \(M\) be a supermanifold and let \(C^{\infty}(M)\) denote its algebra of functions.

Definition A differential operator \(D\) is said to be a first order differential operator if and only if

\(\left[  \left[ D,f \right],g \right]1=0\),

for all \(f,g \in C^{\infty}(M)\).

We remark that we have a filtration here rather than a grading (nothing to do with the supermanifold grading) as we include zero order operators here (left multiplication by a function).

Let us denote the vector  space of  first order differential operators as \(\mathcal{D}^{1}(M)\).

Theorem The first order differential operator  \(D \in\mathcal{D}^{1}(M) \) is a vector field if and only if \(D(1)=0\).

Proof Writing out the definition of a first order differential operator gives

\(D(f,g) = D(f)g + (-1)^{\widetilde{D}\widetilde{f}}f D(g)- D(1)fg\),

which reduces to the strict Leibniz rule when \(D(1)=0\).  QED.

Lemma First order differential operators always decompose as

\(D = (D-D(1)) + D(1)\).

The above lemma says that we can write any first order differential operator as the sum of a vector field and a function.

Theorem A first order differential operator \(D\) is a zero order operator if and only if \(D(1) \neq 0\) and

\(\left[  D,f\right]1 = 0\),

for all \(f \in C^{\infty}(M)\).

Proof Writing out the definition of a first order differential operator and using the above Lemma we get

\(\left[  D,f\right]1 =  (D(f) {-} D(1)f) { -} (-1)^{\widetilde{D}\widetilde{f}}f (D {-} D(1)) =0\).

Thus we decompose the condition into the sum of a function and a vector field.  As theses are different they must both vanish separately.  In particular \(D- D(1)\) must be the zero vector. Then \(D = D(1)\) and we have “just” a non-zero function.  QED

We assume that the function is not zero, otherwise we can simply consider it to be the zero vector.  This avoids the obvious “degeneracy”.

Theorem The space of first order differential operators \(D \in\mathcal{D}^{1}(M) \) is a bimodule over \(C^{\infty}(M)\).

Proof Let \(D\) be a first order differential operator and let \(k,l \in C^{\infty}(M)\)  be functions. Then using all the definitions one arrives at

\(kDl = k \left(  (-1)^{\widetilde{l} \widetilde{D}}(D- D(1))   + D(l) \right)\),

which clearly shows that we have a first order differential operator. QED

Please note that this is different to the case of vector fields, they only form a left module. That is \(f \circ X\) is a vector field but \(X \circ  f\) is not.

Theorem The space of first order differential operators is a Lie algebra with respect to the commutator bracket.

Proof Let us assume the basic results for the commutator. That is we take for granted that is forms a Lie algebra. The non-trivial thing is that the space of first order differential operators is closed with respect to the commutator. By the definitons we get

\(\left[ D_{1}, D_{2}  \right] = \left[(D_{1}-D_{1}(1)) , (D_{2} – D_{2}(1))  \right] + (D_{1}-D_{1}(1))(D_{2}(1)){ -} (-1)^{\widetilde{D_{1}} \widetilde{D}_{2}} (D_{2}- D_{2}(1)) (D_{1}(1))\),

which remains a first order differential operator. QED

Note that the above commutator contains the standard Lie bracket between vector fields.  So as one expects vector fields are closed with respect to the commutator.

The commutator bracket between first order differential operators is often known as THE Jacobi bracket.

So in conclusion we see that the first order differential operators have a privileged place in geometry. They form a bimodule over the smooth functions and are closed with respect to the commutator.  No other order differential operators have these properties.

They are also very important from other angles including Jacobi algebroids and related structures like Courant algebroids and generalised geometry. But these remain topics for discussion another day.