What are the branches of modern geometry

Non-commutative geometry and number theory

Research report 2003 - Max Planck Institute for Mathematics

Non-commutative geometry is a modern branch of mathematics that was started by Alain Connes in the early 1980s. It provides powerful tools that make it possible to examine `` quantized '' spaces. Unlike in the case of ordinary spaces, their coordinate algebras are non-commutative and can therefore model phenomena such as Heisenberg's uncertainty relation in quantum mechanics.
We describe how noncommutative geometry, a mathematical formulation of geometry adapted to quantum phenomena, interacts with number theory through quantum statistical mechanical systems with phase transitions and spontaneous symmetry breaking. This provides a unified setting for many important arithmetic results including the spectral realization of zeros of the Riemann zeta function and the Galois theory of modular functions.

What makes the theory particularly interesting is the fact that such quantized spaces appear in many different contexts. One directly encounters examples of such spaces if one considers equivalence relations that are so rough that they combine most of the points of the space into one class. Nevertheless, with this classification, one would like to retain enough information about the space in order to operate interesting geometry on it. In such cases, the non-commutative geometry provides a kind of "quantum cloud" around classical space, which contains the essential geometric information, even if the underlying classical space has degenerated extremely. All sophisticated tools of geometry and Analysis can be used - but only after appropriate adjustment. It has recently become increasingly clear that the tools of non-commutative geometry can provide new and important applications, particularly in the field of number theory. The latter is a completely different branch of pure mathematics, with a very long and glorious history. The connection between the two areas arose primarily from Connes' new approach to the Riemann Hypothesis (which is currently perhaps the most famous unsolved problem in mathematics). The first evidence of such a connection between non-commutative geometry and number theory had surfaced earlier when Bost and Connes discovered a very interesting non-commutative space with remarkable arithmetic properties. The physical system described by this space consists of quantized optical phases that are discretized with the help of different scalings. These scalings essentially correspond to the so-called "phasors" (pointers), which are also used in the modeling of quantum computers. A mechanism that controls the compatibility of scale changes divides the phasors into different classes by means of a kind of renormalization procedure (Fig. 1). In this way, the compatibility condition provides an equivalence relation which makes the set of equivalence classes a non-commutative space.

The system obtained in this way has an intrinsic dynamic, which describes how it changes over time, and one can consider corresponding thermodynamic equilibrium states at different temperatures. Above a certain critical temperature the phase distribution is essentially chaotic and there is a single state of equilibrium. At the so-called critical temperature the system experiences a phase shift with spontaneous symmetry breaking, and below the critical temperature the system exhibits many different states of equilibrium, which can be parameterized by arithmetic conditions. It is particularly interesting to investigate what happens at "zero temperature" (absolute zero). The arithmetic structure that describes the operation of the symmetry group of the extreme ground states was already well known to Gauss: he used (at the age of 17 ) exactly the same structure to solve the following famous geometrical problem: Which regular polygons (Fig. 2) can be constructed with a compass and ruler alone?

The decisive structural feature that allows a solution to this geometrical problem is the fact that, in addition to the obvious rotational symmetries of the regular polygon, there is another hidden and much more subtle symmetry that comes from the Galois group, a beautiful and mysterious object. This Galois group shows itself not only through the multiplicative operation of the roots of unity (namely as rotations of the polygon around its center), but also through the fact that these roots are raised to powers. From the example of the non-commutative space of Bost and Connes, a "dictionary" emerges that relates the phenomena of spontaneous symmetry breaking in statistical quantum mechanics to the mathematics of Galois extensions. In addition, the partition function of this statistical quantum mechanical system is an object of central interest in the Number theory: it is nothing other than the Riemann zeta function (Fig. 3).

More recently, further results have been obtained that indicate a deep-seated relationship between non-commutative geometry and number theory, in particular Connes and Moscovici found in work on the "modular Heck algebras" that the Rankin-Cohen brackets - an important algebraic structure on modular forms, which was studied in detail by Zagier (at the MPI) several years ago - have a natural interpretation in the language of non-commutative geometry. Module forms represent a very important class of functions that play a fundamental role in many areas of mathematics, especially in number theory and arithmetic geometry. They reveal, for example, very complex symmetry patterns that can be made evident in certain tilings of the hyperbolic planeFig. 4).

If one looks at the algebraic structures examined by Zagier from the point of view of the non-commutative geometer, they appear as an "incarnation" of a symmetry type of non-commutative spaces, which are linked to the transverse geometry of leaves in codimension ~ 1 (Fig. 5). The latter have been extensively studied in the work of Connes and Moscovici.