On a variant of rhodonea curves

Rhodonea curves or rose curves are plots of a polar equation of the form
\(r = \cos(k \theta)\).

If we specialise to equations with

\(k= \frac{n}{d}\)

for n and d integers (>0), then we have plots of the form below. In the table n runs across and d down

Now, just for fun I considered a slight variant of this given by

\(r = \cos( k \theta) – k\)

The plots are as follows

For another variant I considered

\(r = \cos( k \theta) – k^{-1}\)

I am not sure there is anything mathematically deep here, I just like the images and classify this as some basic mathematical art.

On the physics of chocolate

Researchers at Technische Universität München, Germany, have reported that molecular dynamics can be used to gain new insights into the chocolate conching [1].

Chocolate conching is the stage of manufacturing where aromatic sensation, texture and mouthfeel are developed.

This work seems to be the first to attempt to properly understand the role of lecithins in chocolate production.

Physics, helping to build a tasty more palatable world.

[1] M Kindlein, M Greiner, E Elts and H Briesen, Interactions between phospholipid head groups and a sucrose crystal surface at the cocoa butter interface, 2015 J. Phys. D: Appl. Phys. 48 384002.

Chocolate physics: how modelling could improve mouthfeel, IOP website.

An interview with Prof. Christopher Lintott

Prof. Christopher Lintott of Oxford University, winner of this years Kelvin Medal and Prize from the Institute of Physics and regular on the BBC’s Sky at Night agreed to answer a few questions.

Science and Popularisation

1. What first got you involved in science, and in particular astronomy?

I was a small kid who loved looking through telescopes – first of all a neighbour’s small reflector, then a larger telescope at school. I loved the idea that we could understand what’s happening in space despite being stuck on the surface of a small insignificant planet – and that there was lots left for us still to find out.

2. What was your first telescope?

The same one I have now, a 6” reflector. It’s nothing fancy – it doesn’t even have a motor – but it allows me to explore the sky. I’m a great fan of astrophotography, but I spend too much time looking at my computer as it is. When I’m observing I want the photons to be hitting my eyeballs!

3. How did you get involved in the BBC’s Sky at Night?

I’d been doing some science writing and got invited to be a guest on the show. From there, I was lucky enough to be part of the team and I gradually did more and more. I think Sky at Night’s a wonderful show, with the chance to explore so many fascinating aspects of our relationship with the Universe.

4. What is ‘Citizen Science’?

It’s a modern term for an old idea, which is that anyone can participate in the scientific process. These days, we use the term to cover the kind of projects we build on Zooniverse.org – projects which allows professional astronomers and volunteers together to comb through the vast stores of data which modern surveys produce.

5. Which medium do you think is the most effective at popularising science?

It depends what you’re trying to do, but one of the things that I think we need to remember is that we can’t rely on people choosing to seek out scientific content. A large proportion of the public have been put off science through experiences at school, or through a lack of confidence, and we need to find ways to reach them. In the old days, that meant big budget TV shows, but now that audience is fragmenting we need to find new ways for people to stumble across science. As an example, the Adler Planetarium in Chicago runs a Telescopes in the City program in which they take scopes (and astronomers) to random locations, surprising people with the sky. I think that kind of experience can be life-changing.

6. What, in your opinion, should be the ultimate goal of science popularisation?

I’m not sure it’s an ultimate goal, but there are lots of people who I believe would enjoy following science as it happens, and maybe even participating. I want that crowd to feel like they’re part of the journey, rather than just consumers of pre-packaged scientific results. We just reported on the New Horizons encounter with Pluto, which threw up all sorts of wonderful surprises. Someone said to me that they hadn’t realised that scientists smile when they say they don’t know something – I’d like more people to participate in the joy of not knowing.


1. Can you say a few words about your research?

These days I’m interested in galaxies – in particular, we’re trying to find out why some galaxies form stars and why some appear to have shut down. Most of this work is done with the wonderful data provided by Galaxy Zoo volunteers.

2. Which one of your papers are you most proud of, and why?

The discovery paper for the Voorwerp – I had to learn a lot to write it, and we had the most tremendous battles with the referee, but it came out well in the end. Plus it’s about a wonderful object, and consisted of what I thought astronomers did when I was a kid. Find an interesting object, and point telescopes at it until you know what it is…

Chris Lintott’s homepage at Oxford
BBC Sky at Night

On contact and Jacobi geometry

I have placed a preprint on the arXiv Remarks on contact and Jacobi geometry, which is joint work with K. Grabowska and J. Grabowski.

In the preprint we explain how the proper framework of contact and Jacobi geometry is that of \(\mathbb{R}^{\times}\)-principal bundles equipped with homogeneous Poisson structures. Note that in our approach homogeneity is with respect to a principal bundle structure and not just a vector field. This framework allows a drastic simplification of many standard results in Jacobi geometry while simultaneously generalising them to the case of non-trivial line bundles. Moreover, based on what we learned from our previous work, it became clear that this framework gives a very natural and general definition of contact and Jacobi groupoids.

The key concepts of the preprint are Kirillov manifolds and Kirillov algebroids, i.e. homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. Among other results we

  • describe the structure of Lie groupoids with a compatible principal G-bundle structure
  • present the `integrating objects’ for Kirillov algebroids
  • define contact groupoids, and show that any contact groupoid has a canonical realisation as a contact subgroupoid of the latter

Our motivation
The main motivation for this work was to put some order and further geometric understanding into the subject of contact and Jacobi geometry. We take the ‘poissonisation’ as the true starting definition of a ‘Jacobi structure’ and accept all the consequences of that choice. Importantly, once phrased in the correct way, that is in terms of \(\mathbb{R}^{\times}\)-principal bundles and their actions, the true nature of Jacobi geometry as a specialisation and not a generalisation of Poisson geometry becomes clear.

Non-trivial line bundles makes it easier?
Almost oxymoronically, passing to structures on non-trivial line bundles and then the language of \(\mathbb{R}^{\times}\)-principal bundles really does simplify the overall understanding.

This is particularly evident for contact and Jacobi groupoids where insisting on working with a trivialisation leads to unnecessary complications.

In conclusion
We hope that this work will really convince people that contact and Jacobi geometry need not be as complicated as it is often presented in the literature. Quite often the constructions become very ‘computational’ and ‘algebraic’, and in doing so the underlying geometry is obscured. In this work we really try to stick to geometry and avoid algebraic computations.

One nation science

The UK Science Minister Jo Johnson wants to reshape the funding of science and make sure that the funds are distributed wider.

Currently the ‘Golden Triangle’, Oxford, Cambridge and London’s UCL, Imperial and King’s receive 35% of the total £2.7bn annual funding.

I think that spreading the money could be a good idea, though it has to be done carefully and objectively.

Science minister signals shift towards ‘one nation science’ BBC News

The 2nd Conference of the Polish Society on Relativity

I will be attending the 2nd conference of the Polish Society of Relativity which will celebrate 100 years of general relativity.

The conference is in Warsaw and will be held over the period 23-28 November 2015.

The invited speakers include George Ellis, Roy Kerr, Roger Penrose and Kip Thorne. I am a little excited about this.

Registration is now open and you can follow the link below to find out more.

Polskie Towarzystwo Relatywistyczne

Homotopy versions of Jacobi structures

I have placed a preprint on the arXiv ‘Jacobi structures up to homotopy’ (arXiv:1507.00454 [math.DG]) which is joint work with Alfonso G. Tortorella (a PhD student from Universita degli Studi di Firenze, Italy). Our motivation for the work comes from the increasing presence of higher Poisson and Schouten structures in mathematical physics.

In the preprint we ask and answer the question of ‘how to equip sections of (even) line bundles over a supermanifold with the structure of an L-algebra’.

It turns out that the most conceptionally simple way to do this is to adopt the philosophy of [1] and study homogeneous higher Poisson geometry. In essence, we take the ‘higher Poissonisation’ of a ‘homotopy Kirillov structure’ as the starting definition. In this way we go around trying to carefully define homotopy Kirillov structure in the so-called ‘intrinsic set-up’; which would be quite complicated for non-trivial line bundles. We define a higher Kirillov manifold as a principal ℜx-bundle equipped with a homogeneous higher Poisson structure.

We also study the notion of a higher Kirillov algebroid, which is essentially a higher Kirillov manifold with an addition compatible regular action of ℜ. This additional action encodes a vector bundle structure [2].

Interestingly, from the structure of a higher Kirillov algebroid we derive a line bundle equipped with a ‘higher representation’ of an associated L-algebroid. This is very similar to the classical case of Jacobi algebroids and is a geometric realisation of Sh Lie-Rinehart representations as define by Vitagliano [4]. As a special example we see that the line bundle underlying a higher Kirillov manifold comes equipped with a higher representation of the first jet vector bundle of the said line bundle (which is naturally an L-algebroid).

Finally we present the higher BV-algebra associated with a higher Kirillov manifold following the ideas of Vaisman [3].

[1] J. Grabowski, Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013) 27-58.

[2] J. Grabowski & M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305

[3] I. Vaisman, Annales Polonici Mathematici (2000) Volume: 73, Issue: 3, page 275-290.

[4] L. Vitagliano, Representations of Homotopy Lie-Rinehart Algebras, Math. Proc. Camb. Phil. Soc. 158 (2015) 155-191.

Riemannian Lie algebroids and harmonic maps

I have placed a preprint on the arXiv ‘Killing sections and sigma models with Lie algebroid targets’ (arXiv:1506.07738 [math.DG]). In the paper I recall the notion of a Riemannian Lie algebroid, collect the basic theory and proceed to define Killing sections.

Lie algebroids are a generalisation of the tangent bundle of a manifold. The mantra here is that whatever you can do on a tangent bundle you can do on a Lie algebroid. This includes developing a theory of Riemannian geometry on them.

The notion of a Riemannian metric on a Lie algebroid is just that of a metric on the underlying vector bundle. There is no compatibility condition or anything like that. So, as all vector bundles can be equipped with metrics, all Lie algebroids can be given a metric. The interesting fact is that the Lie algebroid structure allows you to build the theory of Riemannian geometry in exactly the same way as you would on a standard Riemannian manifold. A Lie algebroid with a metric are known as Riemannian Lie algebroids.

In particular we have a good notion of torsion (which is generally missing) and have the notion of a Levi-Civita connection. Moreover, we have the fundamental theorem that says that such a connection is uniquely defined by metric compatibility and vanishing torsion, just as we have in the classical case. All the formula generalise directly with little fuss.

This all begs the question of developing general relativity on a Lie algebroid. Indeed one can formulate the Einstein field equations in this context, see [1]. The geometry here is clear and very neat, the applications to the theory of gravity are less clear.

Killing sections
Something I noticed that was generally missing in the literature was the notion of a Killing section of a Riemannian Lie algebroid. Such a section is a natural generalisation of a Killing field on a Riemannian manifold; they represent infinitesimal isometries. In the paper I show how the basic idea generalises to Riemannian Lie algebroids giving the notion of a Killing section. Moreover, I show how the common ways of expressing the notion of a Killing field directly generalise to Lie algebroids.

Sigma models and harmonic maps
With the above technology in place, I then look at the theory of sigma models that have a Riemannian Lie algebroid as their target. I took the work of Martinez [2] on classical field theory on Lie algebroids, and applied it to this class of theories. The basic idea is that the fields of such a theory are Lie algebroid morphisms from the tangent bundle of our source manifold to a Lie algebroid target. Equipping both the source and target with a metric allows us to build a model in exactly the same way as a standard sigma model on the space of maps between two Riemannian manifolds. The critical points of the Lie algebroid sigma model are seen to be a generalisation of harmonic maps.

I show, as expected, that the infinitesimal internal symmetries of the Lie algebroid sigma model are described by the Lie algebra of Killing section.

After thoughts
Non-linear sigma models represent a large class of models that have found applications in high energy physics, string theory and condensed matter physics. From a mathematical perspective, sigma models provide a strong link between differential geometry and field theory. In this work, I do not attempt to find such applications of the Lie algebroid sigma model, I focus on the differential geometry. However, studying such models seems very natural and hopefully useful.

[1] M. Anastasiei & M. Girtu, Einstein equations in Lie algebroids, Sci. Stud. Res. Ser. Math. Inform. 24 (2014), no. 1, 5-16.

[2] E. Martinez, Classical field theory on Lie algebroids: variational aspects, J. Phys. A: Math. Gen. 38 (2005) 7145.

Quantum gravity

The subject of a quantum theory of gravity is interesting, technical and very difficult. However, there are three basic principles that we expect such a theory to obey.

Creating a full quantum theory of gravity seems to be out of our reach right now. String theory comes close, but the full theory here is not understood. Loop quantum gravity also offers a good picture, but again technicalities spoil achieving the goal.

I am no expert in quantum gravity, but I thought it maybe interesting to outline three basic ‘rules’. The full quantum theory of gravity should be:

  1.  Renormalisable (maybe not perturbatively) or finite.
  2. Background independent.
  3. Reducible to general relativity (plus small corrections) in a sensible classical limit.

As a warning, I will not be too technical here, but will use some standard language from quantum field theory.

The standard methods of quantum field theory are to expand the theory about some fixed configuration, usually the vacuum, and consider small fluctuations about this reference configuration. However, in doing so some techniques are needed to remove the appearance of infinite values of things you would like to measure in the lab. These methods are collective known as ‘perturbative renormalisation’. For example, we know that the quantum theory of electrodynamics can be handled properly using these methods.

However, general relativity as described by Einstein is not amenable to methods of perturbative renormalisation. Well, this is true if we want a full theory. What one can do is consider quantum general relativity as an effective theory. That is we accept that at some energy scale the theory will breakdown, but as long as we are not at that scale the theory is okay. By adding a ‘cut-off’ we can understand quantum general relativity using Feynman diagrams to ‘one-loop’ and calculate graviton scattering amplitudes and so on.

Interestingly, there is some evidence that general relativity or something close to it is nonperturbatively renormalisable; this is known as asymptotic safety. With no details, the idea is that quantum general relativity is not ‘sick’ and well-defined, just not as a perturbative theory like quantum electrodynamics. This is fascinating as it means that a proper quantum theory of gravity may not be a theory of gravitons after all! Recall that small ripples in the electromagnetic field are quantised and understood to be photons. Maybe it is not really possible to describe quantum gravity in a similar way where small ripples in space-time are quantised.

Alternatively, a full theory of quantum gravity could be finite. That is we can employ perturbative methods, but do not need renormalisation techniques. Amazingly, we know of supersymmetric Yang-Mills theories that are finite. Moreover, superstring theory is also finite (I am unsure as to how rigours the proof are here, but the string community generally accept this as fact). It maybe possible that the full theory of quantum gravity is finite from the start. This suggests that looking at supersymmetric theories of gravity is a good idea, but by no means the only thing one can think about.

In short, any full quantum theory of gravity must allow us to calculate things we can hope to measure.

Background independence
This means that the theory should not depend on any chosen background geometric fields. In particular, this is taken to mean that the theory should not require some chosen background metric.

String theory as it stands fails on this. However, string theory is usually employed using perturbation theory and so some classical background is chosen, often 10-d flat space-time.

Loop quantum gravity seems better in this respect, but it has other problems.

In short, any full quantum theory of gravity should not require us to fix the geometry (and maybe topology) from the start.

Reduce to general relativity
General relativity has been so successful in describing classical gravitational phenomena. It is tested to some huge degree of accuracy and so far no deviations from it’s predictions have been found. General relativity is a good theory within the expected domains of validity.

Thus, any quantum theory of gravity must in some classical limit reduce to general relativity, up to small corrections. These quantum corrections must be small enough as not to be seen already in astrophysics and cosmology.

If a quantum theory of gravity cannot be shown to reduce to general relativity in some limits (there maybe several ways of doing this) then we cannot be sure that we really have a quantum theory of gravity.

Today we know that string theory gives us general relativity + small corrections. In essence this is because the spectra of closed string theory contains a spin-2 boson, via rather general arguments we know that this has to be the graviton and the field equations are essentially the Einstein field equations. (Remember this is all in perturbation theory).

Recovering general relativity from loop quantum gravity has yet to be done. This I would say is a sticking point right now.

In short, any full quantum theory of gravity must reproduce the phenomena of general relativity is some classical limit(s).

Random thoughts on mathematics, physics and more…