Within `quantum algebras’ my own area of `quantum groups and noncommutative geometry’ lies right on the frontier between the most pure mathematics and the deepest foundations of theoretical physics. My interests cover pure mathematics including algebra, representation theory, category theory and K-theory, and mathematical physics including quantum gravity and string theory. Nowdays, this is a field that brings together everything from Lie algebras over fields of positive characteristic to ideas for the structure of space and time actually testable (at least in principle) by data from astronomical measurements such as gamma-ray bursts of cosmological origin.

From a pure-mathematical point of view, noncommutative geometry is about extending the same idea as underlying algebraic geometry, namely replacing a space by a suitable algebra of `coordinate functions', but taking it further to the case where this algebra could be noncommutative. In practice, it means going back to our basic ideas of differential and integral calculus, group theory, geometry etc and reinventing them in a way that is consistent with this noncommutativity. For example, take a look at the usual Leibniz rule d(ab)=(da)b+a(db) where a,b are elements of our algebra and d is the exterior derivative. Then d([a,b])=[da,b]+[a,db] implies that for an algebra which is very noncommutative and for d to be mainly non-trivial, you will need that differential forms do not commute with `functions'; (da)b and b(da) should not be assumed to coincide. It turns out that one can do practically all of geometry without assuming this either, one just takes the space of differential forms to be a bimodule over our algebra.

As well as such formal definitions, which tie in well with operator K-theory and other deep results, there are also plenty of examples. For some decades a main example was the exponentiated Heisenberg or Weyl algebra with relations vu=exp(i theta)uv. When completed to a C* algebra, i.e. at the level of analysis, even this simple algebra has a rich theory depending on the irrationality of theta. It is not so interesting at an algebraic level, but in the mid 1980s entire classes of other noncommutative algebraic examples appeared, quantum groups. Just as Lie groups and their homogeneous spaces are basic examples of differentiable manifolds, so quantum groups and their associated algebras should be basic examples of a theory of noncommutative geometry, which has led to a kind of `quantum groups approach' to noncommutative geometry. There are still other approaches as well.

In general, while one of the original motivations of the subject may have been quantum theory, there turn out to be many other motivations from deep within pure mathematics itself, such as discrete geometry, knot theory, representation theory and resolution of singularities. For example, discrete geometry and finite groups fit very nicely into this setting. On a discrete space the notion of `differential' is some kind of finite difference, which is necessarily associated to pairs of points (a bilocal object). As a result, finite differential forms do not naturally commute with the functions on a discrete space, one needs the kind of generalisation explained above even though the functions themselves commute. In this noncommutative geometry setting we can endow discrete spaces and simplicial complexes with the `differential geometric picture' that they deserve. Similarly, we can endow finite groups with (braided) Lie algebra structures. This is one of many potential applications of the subject to other areas of pure mathematics. An introduction for mathematicians is in these LMS lecture notes.

From a physical point of view, noncommutative geometry of spacetime itself amounts among other things to the postulation and potential detection of a new force of nature which has been called `cogravity'. If you know a bit of Fourier theory then what it corresponds to is gravity or curvature in momentum space Fourier conjugate to space time. The problem of quantum gravity then amounts to developing a theory with gravity and cogravity consistent with one another.

If you have some exposure to quantum theory then you may know that the whole point of the correspondence principle in quantum mechanics is that certain macroscopic concepts like position and momentum coordinates have analogues as noncommuting operators. But just how much of the macroscopic or geometrical world has its analogue in the quantum domain? The question was posed with the birth of quantum mechanics in the first decades of the 20th century. It is remarkable that it has taken the rest of the century to find a reasonable answer to this question, but I believe that it is now emerging. I suspect that we will look back to the emergence of noncommutative or `quantum' geometry as one of the main mathematical achievements of the 20th century, no less significant than the jump from Euclidean to non-Euclidean geometry. And once you have accepted the need for this then you may as well postulate that the spacetime position coordinates themselves may be noncommutative, as discussed above.

Quantum groups are the simplest convincing examples of noncommutative geometry. There are currently two main sources of true quantum groups. One is the construction of q-deformed enveloping algebras and coordinate rings of simple Lie groups arising in the theory of quantum integrable systems (Drinfeld, Jimbo, Reshetikhin etc.). The other source is the bicrossproduct construction associated to Lie group factorisations coming out of an algebraic approach to quanutm mechanics combined with gravity (as well as arising elsewhere). In a nutshell, if one seriously wants to unify the geometrical ideas about gravity that we learned from Newton and Einstein with quantum theory then it stands to reason that one should first cast both of them into the same language, which is that of algebra, and then modify both until they are consistent.

A second kind of noncommutativity was also discovered by physicists in the early part of the century, namely the statistical anticommutativity of independent fermionc systems. When fermions pass each other either in physical space or lexicographically during a calculation, their exchange involves an additional -1 factor. In recent years this idea has been generalised to a new kind of mathematics involving braiding operators in the role of -1. Braided groups are the simplest examples of this braided geometry. They arose in a categorical approach to q-deformation.