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!

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 In essence it is a place that scientists can share their work and work in progress, but note that it is not peer reviewed.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.