Linearising graded manifolds

Our paper, Polarisation of Graded Bundles, with Janusz Grabowski and Mikołaj Rotkiewicz has now been published in SIGMA [1].

In the paper we show that Graded bundles (cf. [2]), which are a particular kind of graded manifold (cf. [3]), can be `fully linearised’ or `polarised’. That is, given any graded bundle of degree k, we can associate with it in a functorial way a k-fold vector bundle – we call this the full linearisation functor. In the paper [1], we fully characterise this functor. Hopefully, this notion will prove fruitful in applications as k-fold vector bundles are nice objects that that various equivalent ways of describing them.

Graded Bundles
Graded bundles are particular examples of polynomial bundles: that is we have a fibre bundle whose are \(\mathbb{R}^{N}\) and the admissible changes of local coordinates are polynomial. A little more specifically, a graded bundle $F$, is a polynomial bundle for which the base coordinates are assigned a weight of zero, while the fibre coordinates are assigned a weight in \(\mathbb{N} \setminus 0\). Moreover we require that admissible changes of local coordinates respect the weight. The degree of a graded bundle is the highest weight that we assign to the fibre coordinates.

Any graded bundle admits a series of affine fibrations
\(F = F_k \rightarrow F_{k-1} \rightarrow \cdots \rightarrow F_{1} \rightarrow F_{0} =M\),
which is locally given by projecting out the higher weight coordinates.

For example, a graded bundle of degree 2 admits local coordinates \((x, y ,z)\) of weight 0,1, and 2 respectively. Changes of coordinates are then, `symbolically’
\(x’ = x'(x)\),
\(y’ = y T(x)\),
\(z’ = z G(x) + \frac{1}{2} y y H(x)\),
which clearly preserve the weight.

We then have a series of fibrations
\(F_2 \rightarrow F_1 \rightarrow M\),
given (locally) by
\((x,y,z) \mapsto (x,y) \mapsto (x)\).

Linearisation
The basic idea of the full linearisation is quite simple – I won’t go into details here. Recall the notion of polarisation of a homogeneous polynomial. The idea is that one adjoins new variables in order to produce a multi-linear form from a homogeneous polynomial. The original polynomial can be recovered by examining the diagonal.

As graded bundles are polynomial bundles, and the changes of local coordinates respect the weight, we too can apply this idea to fully linearise a graded bundle. That is, we can enlarge the manifold by including more and more coordinates in the correct way as to linearise the changes of coordinates. In this way we obtain a k-fold vector bundle, and the original graded bundle, which we take to be of degree k.

So, how do we decide on these extra coordinates? The method is to differentiate, reduce and project. That is we should apply the tangent functor as many times as is needed and then look for a substructure thereof. So, let us look at the degree 2 case, which is simple enough to see what is going on. In particular we only need to differentiate once, but you can quickly convince yourself that for higher degrees we just repeat the procedure.

The tangent bundle \( T F_2\) – which we consider the tangent bundle as a double graded bundle – admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)} \; \underbrace{\dot{x}}_{(0,1)}, \; \underbrace{\dot{y}}_{(1,1)} ,\; \underbrace{\dot{z}}_{(2,1)})\)

The changes of coordinates for the ‘dotted’ coordinates are inherited from the changes of coordinates on \(F_2\),
\(\dot{x}’ = \dot{x}\frac{\partial x’}{\partial x}\),
\( \dot{y}’ = \dot{y}T(x) + y \dot{x} \frac{\partial T}{\partial x}\),
\(\dot{z}’ = \dot{z}G(x) + z \dot{x}\frac{\partial G}{\partial x} + y \dot{y}H(x) + \frac{1}{2}y y \dot{x}\frac{\partial H}{\partial x}\).
Thus we have differentiated.

Clearly we can restrict to the vertical bundle while still respecting the assignment of weights – one inherited from \(F_2\) and the other comes from the vector bundle structure of a tangent bundle. In fact, what we need to do is shift the first weight by minus the second weight. Technically, this means that we no longer are dealing with graded bundles, the coordinate \(\dot{x}\) will be of bi-weight (-1,1). However, the amazing thing here is that we can set this coordinate to zero – as we should do when looking at the vertical bundle – and remain in the category of graded bundles. That is, not only is setting \(\dot{x}=0\) well-defined, you see this from the coordinate transformations; but also this keeps us in the right category. We have preformed a reduction of the (shifted) tangent bundle.

Thus we arrive at a double graded bundle \(VF_2\) which admits local coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)} ,\; \underbrace{z}_{(2,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
and the obvious admissible changes thereof.

Now, observe that we have the degree of \(z\) as (2,0), which is the coordinate with the highest first component of the bi-weight. Thus, as we have the structure of a graded bundle, we can project to a graded bundle of one lower degree \(\pi : VF_2 \rightarrow l(F_2)\). The resulting double vector bundle is what we will call the linearisation of \(F_2\).

So we have constructed a manifold with coordinates
\((\underbrace{x}_{(0,0)}, \; \underbrace{y}_{(1,0)}, \; \underbrace{\dot{y}}_{(0,1)} ,\; \underbrace{\dot{z}}_{(1,1)})\),
with changes of coordinates
\(x’ = x'(x)\),
\(y’ = y T(x)\)
\( \dot{y}’ = \dot{y}T(x)\),
\(\dot{z}’ = \dot{z}G(x) + y \dot{y}H(x)\).

Then, by comparison with the changes of local coordinates on \(F_2\) you see that we have a canonical embedding of the original graded bundle in its linearisation as a ‘diagonal’
\(\iota : F_2 \rightarrow l(F_2)\),
by setting \(\dot{y} = y\) and \(\dot{z} = 2 z\).

References
[1] Andrew James Bruce, Janusz Grabowski and Mikołaj Rotkiewicz, Polarisation of Graded Bundles, SIGMA 12 (2016), 106, 30 pages.

[2] Janusz Grabowski and Mikołaj Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36.

[3] Th.Th. Voronov, Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168.

HISTRUCT — Workshop on higher structures

There will be a workshop on Leibniz algebras and other higher structures at the University of Luxembourg December 13–16, 2016. For details check the announcement below.

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HISTRUCT — Workshop on higher structures

When: 13–16 December 2016

Where: University of Luxembourg-campus Kirchberg, Luxembourg, LUXEMBOURG

Website: http://math.uni.lu/leibniz/

Aim and scope
The purpose of this workshop is to bring together mathematicians working on Leibniz algebras and other higher structures.

Confirmed speakers include:
Olivier ELCHINGER (University of Luxembourg)
Yaël FRÉGIER (Université d’Artois)
Xevi GUITART (Universitat de Barcelona)
Honglei LANG (Max Planck Institute for Mathematics)
Camille LAURENT-GENGOUX (University of Lorraine)
Zhangju LIU (Peking University)
Mykola MATVIICHUK (University of Toronto)
Sergei MERKULOV (University of Luxembourg)
Norbert PONCIN (University of Luxembourg)
Florian SCHÄTZ (University of Luxembourg)
Martin SCHLICHENMAIER (University of Luxembourg)
Boris SHOIKET (Antwerp University)
Mathieu STIENON (Pennsylvania State University, USA)
Ping XU (Pennsylvania State University, USA)

Registration : http://math.uni.lu/leibniz/reg.html
The deadline for registration is the 2nd of December 2016.

Research Project
– This conference is funded in the frame of the OPEN Scheme of the Fonds National de la Recherche Luxembourg (FNR) with the project QUANTMOD O13/5707106 and
– Partial funding by the Mathematics Research Unit is acknowledged.

Please feel free to circulate this announcement around you!

The organizers:
Martin Schlichenmaier (Luxembourg)
Ping Xu (Penn State, USA)
Olivier Elchinger (Luxembourg)

Trouble at the maths department at Leicester

This email made its way to me.

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Twenty four members of the Department of Mathematics at the University of Leicester – the great majority of the members of the department – have been informed that their post is at risk of redundancy, and will have to reapply for their positions by the end of September. Only eighteen of those applying will be re-appointed (and some of those have been changed to purely teaching positions). This is supposedly because of a financial crisis at the University, though the union disputes this claim. It should be noted that there is no formal tenure in the UK, but such mass redundancies are highly unusual.

You can add your name to the online petition against this unusual
attempt at:

http://www2.le.ac.uk/institution/unions/ucu/news/no-redundancies-no-confidence

In this case it would be helpful to mention in the comments section that your signature is in support of the Mathematics Department (the petition is for the whole University, but apparently only the Math Dept has been formally notified of the redundancies at this stage).

You can also write directly to:

Professor Paul Boyle
President and Vice-Chancellor
University of Leicester
University Road,
Leicester, LE1 7RH,
United Kingdom

On pre-Courant algebroids

Janusz Grabowski and I have placed a prepint on the arXiv with the title Pre-Courant Algebroids.

In the `classical language’, a Courant algebroid is a vector bundle, whose sections come equipped with a bracket – bilinear map – together with an anchor map and a nondegenerate symmetric bilinear form that satisfy some compatibility conditions. The bracket on the space of sections is not a Lie bracket, but rather a non-skewsymmetric bracket that satisfies the Jacobi identity in Loday-Leibniz form. This bracket is usually called the Courant–Dorfman bracket.

A pre-Courant algebroid can be thought of as a Courant algebroid but without the Jacobi identity on the Courant–Dorfman pre-bracket.

It has long be known, due to Roytenberg [1], that Courant algebroids are `really’ symplectic Lie 2-algebroids. That is, we have an N-manifold of degree 2 (a supermanifold with a particular additional grading), equipped with a nondegenerate Poisson bracket of degree -2 and a homological vector field of degree 1 that is Hamiltonian. The brackets of Courant algebroid can then be recovered using the derived bracket formalism and the bilinear form is encoded in the symplectic structure.

Pre-Courant algebroids in the superlanguage
So, do we have a similar understanding of pre-Courant algebroids? The answer is yes…

First back to Courant algebroids. As stated above, they can be encoded in a Hamiltonian vector field – and so they can be encoded in a Grassmann odd Hamiltonian of degree/weight 3, which we denote as \( \Theta\). The fact that the Hamiltonian vector field is homological (Grassmann odd and squares to zero) is equivalent to

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

This condition encodes all the compatibility conditions between the bracket and the anchor map (a particular vector bundle map to the tangent bundle). More than that, this condition also encodes the Jacobi identity for the bracket. Thus, we need a weaker condition that is not too weak – we only want to lose the Jacobi identity and keep the other conditions. It turns out that we require

\( \{\{ \Theta, \Theta \}, f\} =0 \),

for all weight zero functions f, if we want to encode a pre-Courant algebroid in exactly the same way as we do a Courant algebroid. In the preprint we define what we call symplectic almost Lie 2-algebroids in this way and show how they correspond to pre-Courant algebroids.

Does this help any?
This change in starting position simplifies many basic facts about pre-Courant algebroids – just as it does with Courant algebroids. In particular, the notion a Dirac structures as a particular Lagrangian submanifolds is quite clear.

In the preprint was also show that including a compatible N-grading is quite simple when one uses the language of homogeneity structures [2]. One should also consult [3,4] where the notion of weighted Lie groupoids and weighted Lie algebroids are explored. As an example VB-Courant algebroids – Courant algebroids with a compatible vector bundle structure – are natural examples of weighted (pre-)Courant algebroids. This change of postion to `graded super bundles’ with some additional structures allows for a very neat understanding of weighted Dirac structure and in particular VB-Dirac structures. This framework simplifes the understanding of many thing.

Conclusion
The bottom line seems to be that Courant algebroids are `really’ sympelectic Lie 2-algebroids and pre-Courant algebroids are really symplectic almost Lie 2-algebroids.

References
[1] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in: Quantization, Poisson brackets and beyond (Manchester, 2001), 169–185, Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002.

[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21–36.

[3] A.J. Bruce, K. Grabowska & J. Grabowski, Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090.

[4] A.J. Bruce, K. Grabowska & J. Grabowski, Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101
(2016), 71–99.

Kirillov structures up to homotopy

My paper with Alfonso Tortorell on higher versions of Kirillov’s local Lie algebras has now been published in Diffrential Geometry and Applications [1]. If you have access to this journal you can follow this link.

In this paper we take the point of view that Jacobi geometry is best understood as homogeneous Poisson geometry – that is Poisson geometry on principle \(\mathbb{R}^{\times}\)-bundles. Every line bundle over a manifold can be understood in terms of such a principle bundle.

The same holds try when we pass to supermanifolds. With this in mind Alfonso and I more-or-less just replace Poisson with higher or homotopy Poisson. This allows us to neatly define an \(L_{\infty}\)-algebra on the space of sections of an even line bundle in the categeory of supermanifolds. This algebra is the higher/homotopy generalisation of Kirillov’s local Lie algebra on the space of sections of a line bundle.

We show that the basic theorems from Kirillov’s local Lie algebras or Jacobi bundles all passes to this higher case.

Refrences
[1] Andrew James Bruce & Alfonso Giuseppe Tortorella, Kirillov structures up to homotopy, Differential Geometry and its Applications Volume 48, October 2016, Pages 72–86.

A geometric framework for supermechanics

K. Grabowska, Moreno and myself have placed a preprint on the arXiv called ‘On a geometric framework for Lagrangian supermechanics‘.

In this work we take the notion of a curve on a supermanifold to be an S-curve, which is an ‘element’ of the mapping supermanifold Hom(R,M) [1]. This mapping supermanifold is a generalised supermanifold and so it is a functor from the (opposite) category of supermanifolds to sets. Each ‘element’ needs to be ‘probed’ by a supermanifold, and so S-curves are ‘curves’ that are parameterised by all supermanifolds. Or maybe better to say that an S-curve is a family of functors paramaterised by time. At any given time and a given supermanifold S, we have a morphism of supermanifolds S → M. That is, an S-curve tracks out the S-points of M.

With this robust notion of a curve, we go on to define what we mean by an autonomous ordinary differential equation on a supermaifold, and more importantly what we mean by a solution. This seems to have been a notion not at all clearly defined in the existing literature. For us, a differential equation is a sub-structure of the tangent bundle of the said supermanifold, and solutions are S-curves on the supermanifold for which their tangent prolongation sit inside the differential equation. This is very close to the classical notions, but now we use S-points and not just the topological points.

We then take these notion and apply them to supermechanical systems given in terms of a Lagrangian. We use Tulczyjew’s geometric approach to Lagrangian mechanics, and really we only modify the notion of a curve and not the underlying geometry of Tulczyjew’s approach [2]. In doing so, we have a well defined notion of the phase dynamics, the Euler-Lagrange equations and solutions thereof for mechanical systems on supermanifolds. We present a few nice example, includinh Witten’s N=2 supersymmetric model [3] and geodesics on a super-sphere.

The importance of this work is not so much in the equations we present, these can be derived using formal variations. The point is we give some proper mathematical understanding of solutions to the equations.

References
[1] Andrew James Bruce, On curves and jets of curves on supermanifolds, Archivum Mathematicum, vol. 50 (2014), issue 2, pp. 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincare Sect. A (N.S.), 27(1):101–114, 1977.
[3] Edward Witten, Dynamical Breaking of Supersymmetry, Nucl. Phys. , B188:513, 1981.

What I have mostly been doing…

J. Grabowski, K. Grabowska and I have placed a preprint on the arXiv called ‘Introduction to Graded Bundles‘ [1], which is based on a talk given by Prof Grabowski at the First International Conference of Differential Geometry, Fez (Morocco), April 11-15, 2016.

The preprint outlines much of our recent work on graded bundles (a nice kind of graded manifold) and their linearisation (as a functor to k-fold vector bundles), as well as the notions of weighted Lie groupoids and algebroids, including the Lie theory.

One key observation that must be made is that there are many examples of graded bundles that appear in the existing literature, it is just that they are not recognised as such and their graded structure is not really exploited. The canonical example here are the higher order tangent bundles which are well studied from the perspective of higher order mechanics.

Anyway, if anyone want to get a quick overview of some of the ideas behind my work, then I direct them to this preprint. If you are interested in the applications to mechanics, then I suggest [2] as well as references therein.

References
[1] Introduction to graded bundles, Andrew J. Bruce, K. Grabowska, J. Grabowski, arXiv:1605.03296 [math.DG]

[2] New developments in geometric mechanics, A. J. Bruce, K. Grabowska, J. Grabowski, P. Urbanski, arXiv:1510.00296 [math-ph].

Can one disprove special relativity with high school mathematics?

Is it possible using mathematics that is not much beyond high school mathematics to prove that special relativity is wrong? And what does that even mean?

The mathematics of special relativity
It is more-or-less true that Einstein’s original works on special relativity do not really use any highbrow mathematics. In a standard undergraduate introduction to the subject no more than linear algebra is really used: vector spaces, matrices and quadratic forms.

So, as linear algebra is well-founded, one is not going to find some internal inconsistencies in special relativity.

Moreover, today we understand special relativity to be based on the geometry of Minkowski space-time. Basically, this is Euclidean with an awkward minus sign in the metric. Thus, special relativity, from a geometric perspective, is as well-founded as any thing in differential geometry.

So one is not going to mathematically prove that special relativity is wrong in any mathematical sense.

On to physics…
However, the theory of special relativity is falsifiable in the sense of Popper. That is, taking into account the domain of validity (ie., just the situations you expect the theory to work), experimental accuracy, statistical errors etc. one can compare the theoretical predictions with what is measured in experiments. If the predictions match the theory well, up to some pre-described level, then the theory is said to be ‘good’. Otherwise the theory is ‘bad’ and not considered to be a viable description of nature.

In this sense, using not much more that linear algebra one could in principle calculate something within special relativity that does not agree well with nature (being careful with the domain of validity etc). Thus, one can in principle show that special relativity is not a ‘good’ theory by finding some mismatch between the theory and observations. This must be the case if we want to consider special relativity as a scientific theory.

Is special relativity ‘good’ or ‘bad’?
Today we have no evidence, direct or indirect, to suggest that special relativity is not a viable description of nature (as ever taking into account the domain of validity). For example, the standard model of particle physics has at its heart special relativity. So far we have had great agreement with theory and experiment, the electromagnetic sector is extremely well tested. This tells us that special relativity is ‘good’.

Even the more strange predictions like time dilation are realised. For example the difference in the life-time of muons as measured at rest and at high speed via cosmic rays agrees very well with the predictions of special relativity.

Including gravity into the mix produces general relativity. However, we know that on small enough scales general relativity reduces to special relativity. Any evidence that general relativity is a ‘good’ theory also indirectly tells us that special relativity is ‘good’. Apart from all the other tests, I offer the discovery of gravitational waves as evidence that general relativity is ‘good’ and thus special relativity is also ‘good’.

The clause
The important thing to remember is that the domain of validity is vital in deciding if a theory is ‘good’ or ‘bad’. We know that physics depends on the scales at which you observe, so we in no way would expect special relativity be a viable description across all scales. For example, when gravity comes into play we have to consider general relativity.

On the very smallest length scales, outside of what we can probe, we expect the nature of space-time to be modified to take into account quantum mechanics. Thus, at these smallest length scales we would not expect the description of space-time using special relativity to be a very accurate one. So, no one is claiming that special relativity, nor general relativity is the final say on the structure of space and time. All we are claiming is that we do have ‘good’ theories by the widely accepted definition.

Are all claims that relativity is wrong bogus?
Well, one would have to examine all claims carefully to answer that…

However, in my experience most objections to special relativity are based on either philosophical grounds or misinterpreting the calculations. Neither of these are enough to claim that Einstein was completely wrong in regards to relativity.

The Polish and Welsh contributions to the discovery of gravitational waves

I just want to acknowledge the contributions of two teams to the discovery of gravitational waves. These groups are only part of the wider community and I highlight them for purely personal reasons.

 

The Polish group

The Virgo-POLGRAW group,  lead by   Prof. Andrzej Królak at IMPAN.

 

The Welsh group

The Cardiff Gravitational Physics Group,  and within that the Data Innovation Institute lead by Prof Bernard F Schutz.

 

 

 

All mathematics is already known?

An old friend of mine, who went into teaching chemistry at high school, was surprised that I am involved in mathematics research, or more correctly that anyone is.

Paraphrasing what he said:

Surely all mathematics was worked out and finalised years ago?

I think he was willing to accept that there are still some classical open problems, but essentially he thought that mathematics was now ‘done and dusted’.

Of course this cannot be the case, as evidence I offer all the preprints that appear on the arXiv everyday. Mathematics departments are not full of people who just teach linear algebra and calculus to engineering students! I also submit that my boss Prof. Grabowski would be wondering what I am doing day in day out!

But why would he think mathematics research is over?

High School Mathematics
I think this belief stems from mathematics teaching in schools. Let me explain…

Let us start with physics and science in general. Students and the public at large know that scientists are working on open problems and discovering new things. For example we hear about new materials (eg. graphene); we know that the likes of Hawking are wrestling with the theory of black holes; we see images of all kinds of things in observational cosmology; we hear about medical scientists working on cancer cures; biologist discovering new species can make the news; CERN discovered the higgs boson…

High school students are aware that science is far from over and the syllabus for A-level physics is periodically updated to reflect some of these new discoveries.

But what about mathematics?

Linear algebra first emerged in 1693 with the work of Leibniz. By about 1900 all the main ingredients were know, so vectors have a modern treatment by 1900. This is all quite dated, but some open questions remain (for example in relation to quantum information theory).

Quadratic functions were solved by Euclid (circa 300 BC) and ‘the formula’ was known to Brahmagupta by 628 AD.

Calculus the foundations are from the 17th century in the works of Newton and Leibniz.

Plane geometry goes back to 300 BC and Euclid. Coordinate geometry is due to Descartes in the 1600’s.

Probability theory has it origins in Cardano’s work in the 16th century. Fermat and Pascal in the 17th century also made fundamental advances here.

Logarithms and exponentials in their modern form is due to Euler in the 18th century.

Trigonometry has roots going back to the Greek mathematicians from the 3rd century BC. Islamic mathematicians by the 10th century were using all six trigonometric functions.

So in sort, much of the typical pure mathematics syllabus at advanced level in high school is quite old. This I think, together with the ‘unchanging’ nature of mathematics (once proven a statement is always true) leads to the idea that it is all done already and nothing new can be discovered.

It also take from my friends question that he understood that the applications of mathematics are important and that plenty of work in applied mathematics is going on, for example in computational approaches to chemical dynamics. However, the ideas that mathematics as mathematics is finished remained.

For me personally, these applications of mathematics can lead to new structures in mathematics and this is worth studying. Indeed much of my professional work is in studying geometries inspired by applications in physics, particularly mechanics and field theory.

What can be done?
The ‘unchanging’ nature of mathematics is hard to get around. In science some new evidence could come to light and change our views. Indeed the scientific method is an integral part of teaching physics at advanced level in the UK. This ‘flexibility’ of science to adapt is important in student understanding of the philosophy of science.

So, we could try to promote new discoveries in mathematics to the general public, including high school students. The problem is that the background needed to understand the questions, let alone have any idea about the solutions prevents wide public engagement. Astronomers are lucky, we have all seen stars in the sky and can admire nice pictures!

Trying to start at a much higher level of mathematics would be futile, given the prerequisites that are needed. Moreover, most students will not become researchers in mathematics and will only need to be comfortable applying basic mathematics to their later field of study and work.

In short I have no idea how to promote the idea that mathematics research is not over, but please take my word it is not over!

Random thoughts on mathematics, physics and more…