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The paper ‘Higher Order Mechanics on Graded Bundles’ has been chosen for ‘Publisher’s Pick’ by the Journal of Physics A. |
As part of this, my collaborators and I have given a short interview about the motivation and reasoning behind the paper. You can read the interview at the link below.
Link
Interview with Andrew Bruce, Katarzyna Grabowska and Janusz Grabowski
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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.
References
[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.
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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:
As a warning, I will not be too technical here, but will use some standard language from quantum field theory.
Renormalisable
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).
Recently the journal New Negatives in Plant Science, was launched with the aim of publishing negative, unexpected or controversial results in the field plant biology this.
This journal is aimed at plant science, but I have always thought that some kind of journal in mathematics that presents results that are ‘close but no cigar’ could be useful; for example one could present results of things that at first look should work, but do not. (Everybody’s note book is full of such things!) However, no-one would want to publish results that are not correct. The only way I can see to turn this around is to develop ‘no-go theorems’.
By ‘no-go theorems’ I mean clear mathematical reason why something the community expected to work does not. Such theorems are usually to be found in theoretical physics, but they can appear in pure mathematics also.
Such concrete statements are of course published in standard journals. Examples that spring to my mind are the Weinberg–Witten theorem, Coleman–Mandula theorem and the no-cloning theorem. Plenty of other examples exist.
US mathematician John Nash was killed along side his wife in a taxi crash in New Jersey. Nash is best known for his work in game theory which lead him to be awarded the 1994 Nobel Prize for Economics. He is also more popularly known for the man who inspired the film ‘A Beautiful Mind’.
Nash battled with schizophrenia along side mathematics.
I know Nash’s work in differential geometry; his famous theorem states ‘every Riemannian manifold can be isometrically embedded into some Euclidean space’.
Our thoughts are with his friends and family.
Link
‘Beautiful Mind’ mathematician John Nash killed in crash
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I gave the final talk at the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski. The reason was because I won the poster competition. As Prof. Grabowski had on the opening day discussed our applications in geometric mechanics, I discussed some more mathematical ideas around this. |
In particular I sketched our theory of weighted Lie algebroids and weighted Lie groupoids. Importantly, I gave our guiding principal which states that `compatibility with grading means the action of the homogeneity structure is a morphism in the category you are interested in’. For sure, so far that principal seems to be working.
You can find the slides here. You can also find these slides and others via the conference homepage.
I think, or I should say hope, that the talk was well received. It was an honour and a pleasure to give a talk at the conference in his honour.
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The poster that I presented in the conference ‘Geometry of Jets and Fields‘ in honour of Prof. Grabowski has won the competition. The prize is to give the closing talk! |
The poster is based on my joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” which appears as 2015 J. Phys. A: Math. Theor. 48 205203.
The basic rule that I followed is that ‘less is more’. I tried to only sketch the basic ideas and give the important example. I noticed that my poster is quite informal in the sense that I present no theorems or similar, I just sketch our application of graded bundles and weighted Lie algebroids to mechanics in the Lagrangian picture.
You can find the wining poster here.
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The London Mathematical Society (LMS) Popular Lectures present exciting topics in mathematics and its applications to a wide audience. Because the LMS is 150 years old this year they are having 4 lectures this year instead of the usual 2. |
This years speakers are:
The lectures will be held in London, Birmingham, Leeds and Glasgow.
The topics seem to be catered to the general populous, I won’t expect the opening line to be “Let E be a quasicoherent sheaf of modules on X…”
For more details follow the link below
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My joint paper with K. Grabowska and J. Grabowski entitled “Higher order mechanics on graded bundles” has now been accepted for publication in Journal of Physics A: Mathematical and Theoretical. The arXiv version is arXiv:1412.2719 [math-ph]. |
I am very happy about this as it is my first joint paper to be published. The paper presents some novel and interesting ideas on how to geometrically formulate higher order mechanics, hopefully our expected applications will be realised.
One interesting possible application, as pointed out by one of the referees, is computational anatomy; this is the quantitative analysis of variability of biological shape. There has been some applications of higher derivative mechanics via optimal control theory to this discipline [1].
We were not thinking of such applications in the biomedical sciences when writing this paper. For me, the main motivation for higher order mechanics is as a toy model for higher order field theories and these arise as effective field theories in various contexts. It is amazing that these ideas may find some use in ‘more down to Earth’ applications. However, we will have to wait and see just how the applications pan out.
You can read more about the preprint in an earlier blog entry.
References
[1] F. Gay-Balmaz, D. Holm, D.M. Meier, T.S. Ratiu & F. Vialard, Invariant higher-order variational problems, Comm. Math. Phys. 309(2), (2012), 413-458.
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Prof. du Sautoy asks this very question. |
How many times have you heard someone say ”I can’t do maths”? Chances are you’ve said it yourself.
du Sautoy talking to the BBC
In all honesty I find myself thinking the above at least twice a day.
Genes or hard work
I am not an expert in how genes play a role in our intelligence, but for sure they do. That said, no-one is born an expert in mathematics and it takes a lot of hard work. Like everything in life, becoming proficient in mathematics to the level you set yourself is about perseverance and the willingness to struggle with things until you have mastered them.
Link
Can anyone be a maths genius? BBC iWonder