Last night (18th April) I attended a talk given by Prof. Alain Connes as part of the The Learned Society of Wales Frontiers lectrures. The talk was entitled “The spectral point of view on geometry and physics”.
The talk was very interesting and not too technical. Prof. Connes outlined his philosophy that the physical word should be described by spectral data. This idea really leads to the notion of noncommutative geometry, something Prof. Connes is well-know for.
Connes philosophy comes from many facts of physics. For example, our knowledge of the shape of the Universe comes from spectra data, the red shift and the CMBR. The meter is defined in a natural way in terms of the wavelength of the krypton-86 emission.
The key idea
The question that Connes really tackled was can we understand geometry spectrally? Connes was motivated by the Atiyah-Singer index theorem, which gives topological data about a space from analytical data about operators on that space.
The fundamental idea is that one can restate Riemannian geometry in terms of the spectra of the Dirac operator on that geometry. The topological (smooth) structure of the manifold is recovered from the algebra of (smooth) functions and the metric structure from the spectra of the Dirac operator.
One can then understand a smooth manifold with a Riemannian metric in terms of an algebra of functions and an operator acting on them.
Note that we do not need the notion of points in this spectal description. In fact, this reformulation of Rienamnnian geometry allows one to define metrics on non-commutative spaces, which are really just algebras.
Applications
One of the main hopes of Connes reformulation of Riemannian geometry is that, via non-commutative geometry, one can classically unite the standard model of particle physics with general relativity in a geometric way. In doing so, it may be possible to construct a unified theory, but Connes is not at that stage.
As it stands, Connes theory does not quite match the standard model and there is also the problem of Lorentzian signature metrics. Not having positve definite metrics almost always makes details of the mathematics tricky.
This should not distract from the fact that Connes is a pioneer of non-commutative geometry and mathematically his work is very important.
The talk itself
Connes is a good and entertaining speaker. If you get chance to listen to him, you should take it.
Bill Lumsden
Royal Fort House, University of Bristol. Picture courtesy of the YRM 2012 committee.