What is geometry?

This is a question I am not really sure how to answer. So I put it to Sir Michael Atiyah after his Frontiers talk in Cardiff. In essence he told me that geometry is any mathematics that you can imagine as pictures in your head.

To me this is in fact a very satisfactory answer. Geometry a word that literally means “Earth Measurement” has developed far beyond its roots of measuring distances, examining solid shapes and the axioms of Euclid.

Another definition of geometry would be the study of spaces. Then we are left with the question of what is a space?

Classically, one thinks of spaces, say topological or vector spaces as sets of points with some other properties put upon them. The notion of a point seems deeply tied into the definition on a space.

This is actually not the case. For example all the information of a topological space is contained in the continuous functions on that space. Similar statements hold differentiable manifolds for example. Everything here is encoded in the smooth functions on a manifold.

This all started with the Gelfand representation theorem of C*-algebras, which states that “commutative C*-algebras are dual to locally compact Hausdorff spaces”. I won’t say anything about C*-algebras right now.

In short one can instead of studying the space itself one can study the functions on that space. More than this, one can take the attitude that the functions define the space. In this way you can think of the points as being a derived notion and not a fundamental one.

This then opens up the possibility of non-commutative geometries by thinking of non-commutative algebras as “if they were” the algebra of functions on some non-commutative space.

Also, there are other constructions found in algebraic geometry that are not set-theoretical. Ringed spaces and schemes for example.

So, back to the opening question. Geometry seems more like a way of thinking about problems and constructions in mathematics rather than a “stand-alone” topic. Though the way I would rather put it that all mathematics is really geometry!

Should you beleive everything on the arXiv?

For those of you who do not know, the arXiv is an online repository of reprints in physics, mathematics, nonlinear science, computer science, qualitative biology, qualitative finance and statistics. In essence it is a place that scientists can share their work and work in progress, but note that it is not peer reviewed. The arXiv is owned and operated by Cornell University and all submissions should be in line with their academic standards.

So, can you believe everything on the arXiv?

In my opinion overall the arXiv is contains good material and is a vital resource for scientists to call upon. Many new works can be made public this way, before being published in a scientific journal. Indeed, most of the published papers I have had call to use have versions on the arXiv. Moreover, the service is free and requires no subscription.

However, there can be errors and mistakes in the preprints, both “editorial” but more importantly scientifically. Interestingly, overall the arXiv is not full of crackpot ideas despite it being quite open. There is a system of endorsement in place meaning that an established scientist should say that the first preprint you place on the arXiv is of general interest to the community. This stops the very eccentric quacks in their tracks.

There has been some widely publicised examples of preprints on the arXiv that have cursed a stir within the scientific community. Two well-known examples include

A. Garrett Lisi, An Exceptionally Simple Theory of Everything arXiv:0711.0770v1 [hep-th],

and more recently

V.G.Gurzadyan and R.Penrose, Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity arXiv:1011.3706v1 [astro-ph.CO],

both of which have received a lot of negative criticism. Neither has to date been published in a scientific journal.

Minor errors and editing artefacts can be corrected in updated versions of the preprints. Should preprints on the arXiv be found to be in grave error, the author can withdraw the preprint.

With that in mind, the arXiv can be a great place to generate feedback on your work. I have done this quite successfully in the past. This allowed me to get some useful comments and suggestion on work, errors and all.

My advice is to view all papers and preprints with some scepticism, even full peer review can not rule out errors. Though, always be more confident with published papers and arXiv preprints that have gone under some revision. Note that generally people who place preprints on the arXiv are not trying to con or trick anyone, all errors will be genuine mistakes.

Odd Jacobi structures and BV-gauge systems

In this paper we define Grassmann odd analogues of Jacobi structures on supermanifolds. We then examine their potential use in the Batalin-Vilkovisky formalism of classical gauge theories.

arXiv:1101.1844v1 [math-ph]

In my latest preprint I construct a Grassmann odd analogue of Jacobi structures on supermanifolds.

Without any details (being slack with signs) an odd Jacobi structure on a supermanifold is an ” almost Schouten structure”, \(S\) that is an odd function on the total space of the cotangent bundle of the supermanifold quadratic in fibre coordinates and a homological vector field \(Q\) on the supermanifold together with the compatibility conditions

\(L_{Q}S = \{\mathcal{Q}, S \} = 0\),
\( \{S, S \} = 2 S \mathcal{Q}\),

where \(\mathcal{Q} \in C^{\infty}(T^{*}M)\) is the “Hamiltonian” or principle symbol of the Homological vector field. The brackets here are the canonical Poisson brackets on the cotangent bundle.

An odd Lie bracket can then be constructed on \(C^{\infty}(M)\)

\([f,g] = \pm \{ \{ S,f \},g\} \pm \{ \mathcal{Q},fg \} \).

So, this odd bracket satisfies all the properties of a Schouten bracket i.e. symmetry and the Jacobi identity, but the Leibniz rule is not identically satisfied. There is an “anomaly” to the Leibniz rule of the form

\( [f,gh] = \pm [f,g]h \pm g [f,h] \pm [f,1] gh\)

In the preprint I examine the basic properties of odd Jacobi manifolds. The definition and study of odd Jacobi manifolds appears to be missing from the previous literature despite the wide interest in Schouten manifolds and Q-manifolds in mathematical physics.

One should note that for classical or even Jacobi structures (if you know what these are) the Reeb vector field has no constrain on it like being homological. For odd structures the homological condition is essential.

I also consider if the classical BV-antifield formalism can be generalised to odd Jacobi manifolds. In short, does one require the antibracket to be a Schouten bracket or can one weaken the Leibniz rule? I show that it looks possible to extend the BV formalism, classically anyway to odd Jacobi manifolds with the extra condition that the extended classical action not just be a Maurer-Cartan element,

\([s,s] = 0\),

but in addition should be Q-closed,

\(Qs =0\).

Much work needs to be done to generalise the BV formalism to odd Jacobi manifolds including adding the required gradings of ghost number, antifield number etc as well as understanding the quantum aspects.

UPDATE: 22 March 2011. I have found a mistake in one of the examples I suggest. This is corrected and an updated version of the preprint will appear in due course. The mistake does not really effect the rest of the preprint.