Publishing negative results

May 28th, 2015
 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.

John Nash killed in car crash

May 25th, 2015

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.

The closing talk: Geometry of Jets and Fields

May 16th, 2015
 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.

My poster won!

May 15th, 2015
 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.

LMS popular lectures

April 21st, 2015
 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:

• Professor Martin Hairer, FRS – University of Warwick
• Professor Ben Green, FRS – University of Oxford
• Dr Ruth King – University of St Andrews
• Dr Hannah Fry – University College London

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

Mechanics on graded bundles

March 26th, 2015
 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.

du Sautoy asks “can anyone be a maths genius?”

March 22nd, 2015
 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.

Can anyone be a maths genius? BBC iWonder

The original review of general relativity

March 20th, 2015
 It has now been 99 years, to the day (20/03/2015) since Einstein published his original summary of general relativity [1].

Before that he had published some incomplete works that have the wrong field equation, but the key ideas were in place by 1914. The core idea is that space-time is dynamical and interacts with the matter and energy.

It is hard to believe that this theory of gravity has stood the test of time so well. We know for various reasons that general relativity cannot be the complete picture, but nature just refuses to give us hints on what could be the more complete theory.

Reference
[1] A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 354 (7), 1916, 769-822.

Two quotes on the philosophy of mathematics

March 13th, 2015
 I gave a talk the other day based on our recent work on graded bundles in the category of Lie groupoids. Anyway, as part of the motivation I drew the audiences attention to two quotes…

“Mathematics is written for mathematicians.” Copernicus

“For the things of this world cannot be made known without a knowledge of mathematics.” Roger Bacon

They show the two different sides of mathematics; mathematics motivated by mathematics and mathematics motivated by applications. I think one should to some extent sit in the middle here, but ultimately it is nice when mathematics has something to to with the real world, even if that connection is somewhat loose.

My real motivation for these two specific quotes was that Copernicus was Polish and Bacon English!

Weighted Lie groupoids

February 24th, 2015
 In collaboration with K. Grabowska and J. Grabowski, we have examined the finite versions of weighted algebroids which we christened ‘weighted Lie groupoids’.

Groupoids capture the notion of a symmetry that cannot be captured by groups alone. Very loosely, a groupoid is a group for which you cannot compose all the elements, a given element can only be composed with certain others. In a group you can compose everything.

Groups in the category of smooth manifolds are known as Lie groups and similarly groupoids in the category of smooth manifolds are Lie groupoids.

It is well-known every Lie groupoid can be ‘differentiated’ to obtain a Lie algebroid, in complete analogy with the Lie groups and Lie algebras. The ‘integration’ is a little more complicated and not all Lie algebroids can be globally integrated to a Lie groupoid. Recall that for Lie algebroids we can always integrate them to a Lie group.

Previously we defined the notion of a weighted Lie algebroids (and applied this to mechanics) as a Lie algebroid with a compatible grading. A little more technically we have Lie algebroids in the category of graded bundles. The question of what such things integrate to is addressed in our latest paper [1].

Lie groupoids in the category of graded bundles
The question we looked at was not quite the integration of weighted Lie algebroids as Lie algebroids, but rather what extra structure do the associated Lie groupoids inherit?

We show that a very natural definition of a weighted Lie groupoid follows as a Lie groupoid with a compatible homogeneity structure, that is a smooth action of the multiplicative monoid of reals. Via the work of Grabowski and Rotkiewicz [2] we know that any homogeneity structure leads to a N-gradation of the manifold in question; and so what they call a graded bundle.

The only question was what should this compatibility condition between the groupoid structure and the homogeneity structure be? It turns out that, rather naturally, that the condition is that the action of the homogeneity structure for a given real number be a morphism of Lie groupoids. Thus, we can think of a weighted Lie groupoid as a Lie groupoid in the category of graded bundles.

I will remark that weighted Lie groupoids are a nice higher order generalisation of VB-groupoids, which are Lie groupoids in the category of vector bundles. These objects have been the subject of recent papers exploring the Lie theory and application to the theory of Lie groupoid representations. I direct the interested reader to the references listed in the preprint for details.

Some further structures
Following our intuition here that weighted versions of our favourite geometric objects are just those objects with a compatible homogeneity structure in [1] we also studied weighted Poisson-Lie groupoids, weighted Lie bi-algebras and weighted Courant algebroids. The classical theory here seems to be pushed through to the weighted case with relative ease.

Contact and Jacobi groupoids
The notion of a weighted symplectic groupoid is clear: it is just a weighted Poisson groupoid with an invertible Poisson, thus symplectic, structure. By replacing the homogeneity structure, i.e. an action of the monoid of multiplicative reals, with a smooth action of its subgroup of real numbers without zero one obtains a principal $\mathbb{R}^{\times}$-bundle in the category of symplectic (in general Poisson) groupoids. Following the ideas of [3] this will give us the ‘proper’ definition of a contact (Jacobi) groupoid. We will shortly be presenting details of this, so watch this space.

References
[1] Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, Graded bundles in the category of Lie groupoids, arXiv:1502.06092

[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), no. 1, 21–36

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