# Analysis and Geometry

**[Alain Connes]**

My research focuses on noncommutative geometry, the origins of which date back to Heisenberg’s discovery of the noncommutative nature of observable quantities for a physical system in quantum mechanics. The theory is interesting to mathematics in that the existence of many natural spaces, such as leaf space in a foliation, where a global set-theory description as a quotient does not allow the use of conventional tools in measure theory, topology and differential geometry. By replacing functions in such a space with convolution algebra of functions on the equivalence relation, you obtain a bridge between singular quotient spaces and noncommutative algebra.

Each of the well-known theories mentioned above extends to the noncommutative case: the main interest of this generalization arises from new phenomena, such as the emergence of dynamics (change over time) in a noncommutative space at the measure theory level. This has no equivalent in classical theory and interconnects aspects as different as characteristic classes (Godbillon-Vey invariant) and type of associated factors.

The extension of Riemann’s theory of spaces to the noncommutative case has demonstrated the crucial role of quantum formalism in the concept of the real variable as a self-adjoint operator in Hilbert spaces. The length element in Riemann’s theory becomes an operator: length measurement replaces the path length inf with the sup of real variable evaluations, thus verifying a commutation relation with the length element, which notably allows handling of discontinuous spaces. The geometric theory thus obtained is spectral – for instance, it results in an invariant which, together with the spectrum of the Dirac operator, makes it possible to reconstruct a Riemannian geometry based on its invariants. Naturally, the theory as a whole applies to spaces associated with noncommutative algebra. The group of diffeomorphisms for a smooth variety is the group of automorphisms for the algebra of smooth functions in this space.

For such groups, the connected component of the identity is usually a simple group, which is not the case for the Lagrangian invariance group in physics.

However, the group of automorphisms in noncommutative algebra – such as matrices – on the algebra of smooth functions on a variety has, thanks to the presence of the normal inner automorphisms subgroup, exactly the same structure as the Lagrangian invariance group in physics. Based on the fact that noncommutative geometry allows for the expression gravity coupled with the standard model of particle physics as a pure gravity on space-time whose fine structure is discrete.

In addition, the Lagrangian is given by a spectral invariant of the length element. We recently obtained (in collaboration with A. Chamseddine and S. Mukhanov) a Heisenberg-type equation, the solutions of which explain the particle content of the standard model coupled with gravity: this is potentially a major step towards a unified theory.

My current research also focuses on number theory and the discovery (in collaboration with C. Consani) of the arithmetic site giving a geometric framework in which, hopefully, we can transpose Weil’s proof of the Riemann hypothesis.