Death of Koszul


Jean-Louis Koszul died on Friday 12th January 2018 at the age of 97. I never met Koszul but I know his name from various sources, principally from the “ Koszul sign rule” in graded commutative algebra, for example the algebra of differential forms on a smooth manifold. He made many contributions to differential geometry and homological algebra.

My thoughts are with his family.

An IFS fractal


Another IFS pseudo-fractal image. I am now experimenting with how to colour them. Here have an opacity that encodes the number of times a point is visited, but also as a dynamical system the points are ordered. So I have added a colour based on the order at which the points are visited.

Moon pictures

Just a few snaps of the moon with my new camera, Cannon Powershot 420 is. It has been a while since I last observed or photographed the moon.


27th December 2017

28th December 2017

1st January 2018

5th January 2018

6th January 2018

Filtered bundles

board The paper `On the concept of a filtered bundle ‘ with Katarzyna Grabowska and Janusz Grabowski to appear in International Journal of Geometric Methods in Modern Physics is now `online ready’ and available for free for the rest of this October!

In the paper we generalise the notion of a graded bundle – a particularly nice kind on non-negatively graded manifold – allow for coordinate changes that do not strictly preserve the grading `on the nose’, but instead include lower degree terms. The coordinate changes are thus filtered. It turns out that many nice things from the theory for graded bundles can naturally be generalised to this filtered setting. One of the nice results is that any filtered bundle is non-canonically isomorphic to a graded bundle, and so furthermore any filtered bundle is non-canonically isomorphic to a Whitney sum of vector bundles. We also show that the linearisation process as given in [1] also carries over to this filtered setting.

Many examples of the polynomial bundles found in geometric mechanics and geometric formulations of field theories are not graded bundles, but rather they have a filtered structure. In the paper we take an abstraction of some of the basic structure of jet bundles and similar with an eye for future applications. Hopefully some of these ideas will be useful.

[1] 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.

Paper on supermechanics

board The paper `On a geometric framework for Lagrangian supermechanics‘ with Katarzyna Grabowska and Giovanni Moreno now been published in The Journal of Geometric Mechanics (JGM).

In the paper we re-examine Lagrangian mechanics on supermanifolds (loosely `manifolds’ with both commuting and anticommuting coordinates) in the geometric framework of Tulczyjew [2]. The only real deviation from the classical setting is that one now needs to understand curves in a more categorical framework as maps \(S \times \mathbb R \rightarrow M\), where \(M\) is the supermanifold understudy and \(S\) is some arbitrary supermanifold [1]. Thus one needs to think of families of curves parameterised by arbitrary supermanifolds and thus we have families of Lagrangians similarly parameterised. In our opinion, although the super-mechanics is now an old subject, none of the existing literature really explains what a solution to the dynamics is. We manage to describe the phase dynamics in this super-setting and give real meaning to solutions thereof, albeit one needs to think more categorically than the classical case.

Our philosophy is that time is the only true meaningful parameter describing dynamics on supermanifolds. The auxiliary parameterisations by \(S\) are necessary in order to property `track out’ paths on the supermanifold, but they must play no fundamental role in the theory other than that. Mathematically, this is described in terms of category theory as all the constructions are natural in \(S\). This basically means that if we change \(S\) then the theory is well behaved and that nothing fundamentally depends on our choice of \(S\). The theory holds for all \(S\), and to fully determine the dynamics one needs to consider all supermanifolds as `probes’ – in more categorical language we are constructing certain functors. This seems to take us away from the more familiar setting of classical mechanics, but it seems rather unavoidable. Supermanifolds are examples of `mild’ noncommutative spaces and as such we cannot expect there to be a very simple and universal notion of a curve – this is tied to the localisation problem in noncommutative geometry. Specifically, supermanifolds are not just collections of points together with a topology (or something similar).

The bottom line is that the classical framework of geometric mechanics following Tulczyjew generalises to the super-case upon taking some care with this `extended’ notion of a curve.

[1] A. J. Bruce, On curves and jets of curves on supermanifolds, Arch. Math. (Brno), 50 (2014), 115-130.
[2] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.

Remarks on Contact and Jacobi Geometry

board The paper `Remarks on Contact and Jacobi Geometry‘ with Katarzyna Grabowska and Janusz Grabowski now been published in SIGMA [1].

In the paper we present a rather general formalism to define and study Jacobi and Kirllov structures using principle \(\mathbb{R}^\times\)-bundles equipped with homogeneous Poisson structures. This approach was first described by Grabowski [2]. This set-up allows for a rather economical description of contact/Jacobi groupoids and related structures. Importantly, by using homogeneous Poisson structures we simplify the overall picture of contact/Jacobi/Kirillov geometry and show that many technical proofs of various statements in the theory are drastically simplified. We think that this approach gives new insight into the existing theory and hopefully the ideas will be useful to others.

Contact Geometry
Contact geometry is motivated by the formalism of classical mechanics, and in particular looking at constant energy surfaces in phase space. Jacobi geometry is the `degenerate brother’ of contact geometry, and Kirillov geometry is the `twisted sister’ of Jacobi geometry – for those that know think of the relation between symplectic and Poisson geometry, and then trivial and non-trivial line bundles. Contact geometry clearly from its conception has broad applications in physics, ranging from classical mechanics, geometric optics and thermodynamics. There are also some mathematical applications such as knot invariants and invariants in low dimensional topology.

Lie groupoids
Another facet of this paper are Lie groupoids, which should be through of as a wider setting to discuss symmetries than groups. Very loosely, a groupoid is a `many object’ group, and a Lie groupoid is a `geometric’ version of a groupoid. Associated with any Lie group is a Lie algebra, which describes infinitesimal (so `very small’) symmetries of geometric entities. Likewise, associated with any Lie groupoid is a Lie algebroid. Without any details, a Lie algebroid should be considered as describing `very small’ symmetries associated with a Lie groupoid. However, unlike Lie groups and algebras, not every Lie algebroid comes from a Lie groupoid!

Why study contact/Jacobi/Kirillov Groupoids?
Bringing contact and groupoids together is, in the standard setting, not so easy. Our formalism makes this much clearer and allows for direct generalisations to Jacobi geometry. But why bring them together in the first place?

Alan Weinstein [3] introduced the notion of a symplectic groupoid with the intention of extending methods from geometric quantisation to Poisson manifolds. Very loosely, the geometry of Lie groupoids is needed in geometric approaches that allow a passage from classical mechanics to quantum mechanics. In a sense, one can think of symplectic and Poisson groupoids as the Lie groupoid versions of the phase spaces found in classical mechanics, i.e., the spaces formed by position and momentum.

Since the initial work of Weinstein the topic of symplectic and Poisson groupoids has exploded, largely motivated by the geometry of classical mechanics – not that all practitioners see this!

Similarly, given the role of contact geometry in physics, it is natural to think about groupoid versions of contact and Jacobi geometry. More than this, it turns out that the integrating objects of Jacobi/Kirillov structures are precisely contact groupoids (as we define them). That is, as soon as one thinks about the `degenerate brother’ and `twisted sister’ of contact geometry one encounters contact groupoids as the `finite versions’.

For me, all this is strongly motivated by the basic questions of the geometry of classical mechanics. It is rather amazing that we are pushed rather quickly into more and more difficult ideas in geometry. And this is before we get into the quantum world!

I personally thank the anonymous referees for their effort in reading the paper and providing many helpful comments and suggestions. For sure the paper would not be what it is today without them.

[1] Andrew James Bruce, Katarzyna Grabowska and Janusz Grabowski, Remarks on Contact and Jacobi Geometry, SIGMA 13 (2017), 059, 22 pages.

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

[3] Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101-104.

Modular classes of Q-manifolds

board Q-manifolds are supermanifolds equipped with a Grassmann odd vector field that `squares to zero’, which is known as a homological vector field. Such things can be found behind the AKSZ-BV formalism in mathematical physics and in differential geometry they encode Lie algebroids and Courant algebroids amongst other things. The notion of the modular class of a Q-manifold is known to experts but there is not much in the literature to date.

In the preprint entitled “Modular classes of Q-manifolds: a review and some applications”, I review the notion of the modular class of a Q-manifold – which is understood as the obstruction to the existence of a Berezin volume that is invariant under the action of the homological vector field. The modular class is naturally defined in terms of the divergence of a chosen Berezin volume, but is independent of this choice. The notion directly generalises the notion of the modular class of a Poisson manifold (Koszul [1] and Weinstein [2]) and that of a Lie algebroid (Evans & Weinstein [3]).

I discuss the basic constructs and immediate consequences, all of which are probably known to the handful of experts. Maybe more interesting is that fact that I then apply this to double Lie algebroids ([4,5,6] ) and higher Poisson manifolds [7]. Along the way I make several observations which I believe maybe genuinely new. Either way, having these ideas written clearly in one place is beneficial to the community.

The basic idea
A Q-manifold is a pair \((M,Q)\), where \(M\) is a supermanifold and \(Q \in Vect(M)\) is an odd vector field that ‘self commutes’

\(Q^2 = \frac{1}{2} [Q,Q] =\frac{1}{2} \left( Q \circ Q – (-1)^{1} Q \circ Q \right)\),

note the extra minus sign as compared with the classical case of vector fields on a manifold. This means that `squaring to zero’ is a non-trivial condition. Moreover, as we have an odd vector field that squares to zero we have a differential and so a cohomology theory. In particular, \((C^{\infty}(M), Q )\) is a cochain complex and the related cohomology we refer to as the standard cohomology.

Given any Berezin volume \(\mathbf{\rho} = D[x] \rho(x)\), we can define the divergence of \(Q\) with respect to this volume:

\(L_{Q} \mathbf{\rho} = \mathbf{\rho} {Div}_{\rho}(Q). \)

Note that \({Div}_{\rho}(Q)\) is then a Grassmann odd function on \(M\) and it is \(Q\)-closed. Moreover, it turns out that under change of the Berezin volume the divergence of \(Q\) changes by a \(Q\)-exact term. Thus, we can define the modular class as the standard cohomology class of the divergence of the homological vector field and this does not depend on any chosen Berezin volume

\(Mod(Q) = [Div_{\mathbf{\rho}}(Q)]_{St}. \)

In local coordinates \(Q = Q^{a}(x)\frac{\partial}{\partial x^a}\) and so the modular class has a local characteristic representative

\(\phi_{Q}(x) = \frac{\partial Q^{a}}{\partial{x^a}}(x),\)

which corresponds to picking the standard coordinate volume (we simply drop the \(Q\)-exact term in the definition of the divergence). Moreover, we do not have a Poincare lemma here and so thinking of local representatives of cohomology classes makes sense in general.

In this way we associate to any Q-manifold a characteristic class in its standard cohomology. The modular class is one of the simplest such classes one can imagine on a Q-manifold. There are more complicated things, see [8].

I thank prof. Janauzs Grabowski for giving me the opportunity to present some of the ideas in this preprint at a Geometric Methods in Physics seminar in Warsaw on April 26th 2017. I also thank Florian Schatz for reading an earlier draft of this preprint.

[1] Koszul, J., Crochet de Schouten-Nijenhuis et cohomologie, The mathematical heritage of Elie Cartan (Lyon, 1984), Asterisque 1985, Numero Hors Serie, 257–271.

[2] Weinstein A., The modular automorphism group of a Poisson manifold, J. Geom. Phys. 23 (1997), 379–394.

[3] Evens, S., Lu, J.H., Weinstein, A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Ser. 2 50 (1999), 417–436.

[4] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, I., Adv. Math. 94 (1992), no. 2, 180–239.

[5] Mackenzie, K.C.H., Double Lie algebroids and second-order geometry, II., Adv. Math. 154 (2000), no. 1, 46–75.

[6] Voronov, Th., Q-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), no. 2, 279–310.

[7] Voronov, Th., Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133–153.

[8] Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A., Characteristic classes of Q-manifolds: classification and applications, J. Geom. Phys. 60 (2010), no. 5, 729–759.

Representations theory of Lie algebroids and weighted Lie algebroids

board Weighted Lie algebroids are Lie algebroids in the category of graded bundles, or vice versa. It is well known that VB- algebroids (vector bundles in the category of Lie algebroids, or vice versa) are related to 2-term representations up to homotopy of Lie algebroids. Thus, it is natural to wonder if a similar relation holds for weighted Lie algebroids as these are a wide generalization fo VB-algebroids.

In a preprint entitled “Graded differential geometry and the representation theory of Lie algebroids” with Janusz Grabowski and Luca Vitagliano, we look at the relation between weighted Lie algebroids [1], Lie algebroid modules [2] and representations up to homotopy of Lie algebroids [3]. We show that associated with any weighted Lie algebroid is a series of canonical Lie algebroid modules over the underlying weight zero Lie algebroid. Moreover, we know, due to Mehta [4], that a Lie algebroid module is (up to isomorphisms classes) equivalent to a representation up to homotopy of the Lie algebroid.

Weighted Lie groupoids were first defined and studied in [5] and offer a wide generalisation of the notion of a VB-groupoid. We show that a refined version of the Van Est theorem [6] holds for weighted Lie groupoids, and in fact follows from minor adjustments to the ideas and proofs presented by Cabrera & Drummond [7].

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

[2] Vaintrob A.Yu., Lie algebroids and homological vector fields, Russ. Math. Surv. 52 (1997), 428–429.

[3] Abad C.A., Crainic M., Representations up to homotopy of Lie algebroids, J. Reine Angew.Math, 663 (2012), 91–126.

[4] Mehta R.A., Lie algebroid modules and representations up to homotopy. Indag. Math. (N.S.) 25 (2014), no. 5, 1122–1134.

[5] Bruce A.J., Grabowska K., Grabowski J., Graded Bundles in the Category of Lie Groupoids, SIGMA 11 (2015), 090, 25 pages.

[6] Crainic M., Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv, 78 (2003), 681–72.

[7] Cabrera A., Drummond T., Van Est isomorphism for homogeneous cochains, Pacific J. Math. 287 (2017), 297–336

Geometry and physics: Though lovers be lost love shall not

The title of this post comes from Dylan Thomas, And Death Shall Have No Dominion (1933). Here I give a non-technical essay on the interplay between geometry and physics, which I hope with give some of the readers a better idea of why I do what I do. Please enjoy and leave feedback if you like.

Geometry and physics: Though lovers be lost love shall not

To paraphrase a certain Polish mathematician: “the most important ideas in mathematics come from physics”. While there is no reason why mathematics — as mathematics — should come from physics, there is some deep connection between mathematics and our understanding of the Universe. Wigner in 1960 in his famous “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” article, noticed how the mathematical structure of physical theories can lead to new physical insight. And of course, physical insight can lead to new mathematics. By physics, I will mean the construction of mathematical models of natural phenomena and the comparing of the predictions of these models against nature.

From the very nature of physics it is clear that there is at least some superficial relation with mathematics. After all, physics uses mathematics. However, physics is not mathematics in the sense that mathematical constructions in physics should have (maybe not directly) some meaning. There must be some relation of the mathematics to a physical law. In mathematics, there is no such constraint that any of it have any meaning beyond what it mathematically means. It is a complete mystery as to why nature seems generally amenable to being understood in terms of abstract mathematics.

The deep interconnection between mathematics and physics seems especially true when focusing in on geometry: literally geometry means `Earth measurement’. At the most basic level, geometry is the study of spaces, which are understood as collections of points, together with a notion of points being `close to each other or not’, and usually with some further mathematical structures on them, such as a notion of the distance between to near by points. But this is definition in terms of points is not enough to cover the modern usage of `geometry’. So, what is geometry and where does it come from? Moreover, what has the study of spaces got to do with physics?

The first work on synthetic geometry is the book Elements written Euclid of Alexandria (c.325–265 BC). In this book an axiomatic approach to plane geometry, so parallel lines on flat surfaces etc., is established. For example, the internal angles of a triangle on the plane always add up to 180 degrees. However, curves, circles and spheres had been known about since antiquity. Solid geometry — the study of three dimensional objects — was needed as soon as humans started to imagine buildings such as domes and pyramids. In addition to this, the heavenly sky can be imagined as the inner surface of a dome speckled with stars — at least as we see it, and ancient astronomers saw it!

Methods of calculating the volume of simple regular three dimensional objects were developed. For example, the ancient Egyptians knew how to calculate the volume of pyramids and chambers therein: they were the mummy of all modern mathematicians! Archimedes (287–212 BC) in his `eureka’ moment realised that one could deduce the volume of three dimensional irregular objects based on the amount of water they displaced. However, Archimedes was unable to actually calculate volumes in any generality.

In another direction, Apollonius of Perga (c.262–190 BC) showed that the regular curves — circles; ellipses; parabola; and hyperbola — can be formed by cutting the cone, hence conic sections. Amazingly, in Newtonian gravity (circa 1686) the orbits of the two massive bodies are described by conic sections. This is part of the unifying power of mathematics: the mathematics involved in cutting cones is exactly the mathematics needed to describe orbits, for example the path of the Moon around the Earth! These mathematical coincidences are abound.

The most important mathematical works on conic sections — as far as our story goes — are that of Descartes (1596–1650) and Fermat (1601–1665), who in the 17th century brought algebra in to the game. Conic sections can be described by algebraic equations via coordinates — analytic and algebraic geometry were born! \par

The use coordinates (eg. x and x on the plane) opens up the use of calculus in geometry. Newton’s differential and integral calculus allows for methods of calculating gradients of curves, areas under curves, the volumes of objects etc. — calculus today is a common method of torturing undergraduate students! Differential geometry was born … or at least the seeds of the theory were planted by Newton (1642–1727) and Leibniz (1646–1716). One should not forget that much of Newton’s inspiration in developing calculus comes from his work on classical mechanics: so the mathematical description of the motion of massive bodies.

Curved surfaces – such as the sphere – represent non-Euclidean geometries. Lines drawn on them violate the axioms of Euclid’s plane geometry: this was seen as a real problem by mathematicians. It was Eugenio Beltrami (1835–1899) who showed that hyperbolic geometry is consistent: this is the geometry of surfaces of constant curvature for which the internal angles of a triangle add up to less that 180 degrees. Similar results were obtained for spherical geometries, so geometries of constant curvature for which the internal angles of triangles add up to more that 180 degrees.

Bernhard Riemann (1826–1866) in his PhD thesis extended the work of Beltrami to surfaces that have non-uniform curvature, and to higher dimensions. The work of Riemann allowed algebra and calculus to be applied to spaces known as smooth manifolds, i.e., spaces such that every `small piece’ of them looks like a `small piece’ of the n-dimensional plane for some integer n. One should keep in mind the relation between a globe and a map: any small piece of the globe can be represented on a sheet of paper as a map, and points on the globe are then represented by two numbers, the coordinates with respect to the given map. The notion of a smooth manifold underpins Einstein’s special and general relativity, as well as Maxwell’s theory of electromagnetism, Yang–Mills theories and classical mechanics: even thermodynamics has a geometric formulation!

It is worth saying a little more about Einstein’s general relativity (1916). This theory is a theory of gravity, and to date it is the most accurate theory of gravity we have. Moreover four dimensional smooth manifolds are central to the theory. Einstein took the earlier idea that space and time should be unified into space-time seriously, we have one time coordinate and three space coordinates. Einstein then told us that gravity is not your typical force, but rather it really is due to the local shape of space-time! The mathematical theory of curved smooth manifolds is vital to our understanding of gravity and the Universe as a whole, and vice versa, physics has been the impetus for many mathematical works on curved smooth manifolds.

There is a duality between a space and the algebra of functions on that space ( i.e., maps from that space to the real or complex numbers). Loosely, if you know the algebra of functions on a space, then you know the space. The algebra of functions on a classical space is commutative: the order of pointwise multiplication does not matter. We can imagine a more general notion of a `space’ by considering any algebra — not necessarily commutative — as the algebra of functions on some `space’. The phase space of quantum mechanics, that is the `space’ of positions and momenta of a quantum particle, is a noncommutative geometry.

A quantum theory of gravity could be some kind of noncommutative geometry: both string theory and loop quantum gravity suggest noncommutativity of space-time at some level — both the loopers and p-braners agree on this! Trying to make sense of physics at the smallest scales pushes what we mean by geometry well beyond our original understanding. Noncommutative geometries are in general not set theoretical objects, i.e., they do not consist of a collection of points — it is all rather pointless!

There is a kind of `halfway house’ between classical and quantum geometry: here I refer to supermanifolds as defined by Berezin and Leites in 1976. Without details and being very loose, a supermanifold is a `manifold-like object’ which comes with some coordinates that commute with all the coordinates, and some coordinates that anticommute amongst themselves. By anticommute we mean that they pick up a minus sign when we exchange the order they appear in expressions. In particular we have some coordinates that square to zero!

Supermanifolds play the role of manifolds when, for example, fermions such as the electron are present in the theory. If we want to develop a `classical’ theory of fermions then we must employ objects that anticommute: one can justify this using the Pauli exclusion principle — no more than one fermion can be in a given quantum state, while for bosons there is no such restriction. Heuristically, one can say that bosons like to be together, while fermions are rather more like hermits.

Supermanifolds offer a conceptual and geometric way to treat bosons and fermions on equal footing: supermanifolds are the geometry of supersymmetry. In short, supersymmetry is an operation that allows us to `rotate’ a boson into a fermion and vice versa. It turns out that this is not just a neat way of unifying bosons and fermions, but theories that posses supersymmetry can have remarkable mathematical and phenomenological properties — we await CERN’s confirmation that nature uses supersymmetry!

Another amazing link between geometry and physics can be found in mirror symmetry which relates pairs of particular manifolds called Calabi-Yau manifolds. Superstring theory is 10 dimensional, yet our physical world appears 4 dimensional — one time and three space dimensions. To overcome this discrepancy one can postulate that 6 of these dimensions is `scrunched up tightly’, and all we see is four dimensions on all but the very smallest scales. These compactifications as they are known, are Calabi-Yau manifolds, and different compactifications in general lead to different physics. However, it was noticed in the late 1980s by Dixon, Lerche, Vafa, and Warner that two different versions of superstring theory (type IIA and IIB) can be compactified on two different Calabi-Yau manifolds, yet lead to the same physics. In this case the two Calabi-Yau manifolds are said to be mirror duals, and the symmetry between the physics is known as mirror symmetry. This pairing of Calabi-Yau manifolds is now an active area of mathematical research with much effort devoted to carefully understanding the intuitive physics based picture.

In conclusion, not only has geometry been essential in developing physical theories, but these theories then push our understanding of geometry and lead to new mathematics. I have only touched upon a tiny part of this interrelation. There are a great number of other things I could have described and new links are being uncovered all the time. What will future mathematicians understand by the term `geometry’ is anyone’s guess. However, I am sure it will be closely related to our understanding of the physical Universe.

Random thoughts on mathematics, physics and more…